Recent zbMATH articles in MSC 35https://zbmath.org/atom/cc/352021-01-08T12:24:00+00:00WerkzeugA new mixed finite volume element method for solving one-dimensional porous medium problems.https://zbmath.org/1449.652182021-01-08T12:24:00+00:00"Chen, Guofang"https://zbmath.org/authors/?q=ai:chen.guofang"Hei, Yuanyuan"https://zbmath.org/authors/?q=ai:hei.yuanyuan"Lv, Junliang"https://zbmath.org/authors/?q=ai:lv.junliangSummary: For the one-dimensional porous medium problem, the wave front of the numerical solution could not propagate forward when the standard mixed finite volume element method was used to solve them, we propose a new mixed finite volume element method for solving the degradation problem, in which the flux variable only includes the derivative of the original variable to the spacial variable. The results show that the method can avoid the phenomenon that the wave front of the numerical solution can not propagate forward, and can capture the interface of numerical solution well. The validity of the method is verified by numerical experiments.Soliton solutions of a class of differential-difference equations.https://zbmath.org/1449.351512021-01-08T12:24:00+00:00"Fan, Fangcheng"https://zbmath.org/authors/?q=ai:fan.fangcheng"Zhou, Ran"https://zbmath.org/authors/?q=ai:zhou.ranSummary: Firstly, based on Lax pairs of a class of differential-difference equations, the \(N\)-fold Darboux transformation was constructed. Then we obtained the exact solutions of the equation by using Darboux transformation. Through software drawing, we gave the one-, two-, three- and four-soliton solutions of the equation, and discussed the elastic effect among the three solitons and four solitons: soliton shapes and amplitudes do not change after interaction.Existence and uniqueness of solutions for a class of steady-state incompressible non-Newtonian Boussinesq equations.https://zbmath.org/1449.760052021-01-08T12:24:00+00:00"Yang, Hui"https://zbmath.org/authors/?q=ai:yang.hui.1"Wang, Changjia"https://zbmath.org/authors/?q=ai:wang.changjiaSummary: We considered the first boundary value problems for a class of steady-state incompressible non-Newtonian Boussinesq equations in a three-dimensional smooth bounded domain \(\Omega\). Under the conditions that the external force term was appropriately small, we proved the existence and uniqueness of regular solutions for the problem when the exponent \(p \in (1, 2)\) by using the iterative method.Finite time blow-up of solutions to parabolic type Kirchhoff equation with general nonlinearity.https://zbmath.org/1449.351172021-01-08T12:24:00+00:00"Li, Haixia"https://zbmath.org/authors/?q=ai:li.haixia"Cao, Chunling"https://zbmath.org/authors/?q=ai:cao.chunlingSummary: We considered the finite time blow-up of solutions to a class of parabolic type Kirchhoff equations with general nonlinearity. By using the first order differential inequality and convexity arguments, we gave some sufficient conditions for the solutions to blow up in finite time, and obtained the upper bound estimate of the blow-up time.Existence of ground state solution for a class of quasilinear biharmonic equations.https://zbmath.org/1449.352312021-01-08T12:24:00+00:00"Zhou, Sangsang"https://zbmath.org/authors/?q=ai:zhou.sangsang"Jia, Gao"https://zbmath.org/authors/?q=ai:jia.gaoSummary: Using the variational method, variable substitution and the Nehari manifold method, we obtained the existence of ground state solution for a class of quasilinear biharmonic equations by constructing the Nehari manifold and proving the properties of manifold when the nonlinearity term satisfied certain growth conditions.Existence of radial solutions for elliptic boundary value problems with gradient terms in annular domains.https://zbmath.org/1449.352012021-01-08T12:24:00+00:00"Li, Qixiang"https://zbmath.org/authors/?q=ai:li.qixiang"Li, Yongxiang"https://zbmath.org/authors/?q=ai:li.yongxiangSummary: Using the Leray-Schauder fixed point theorem, we consider the existence of radial solutions for elliptic boundary value problems with gradient term
\[\begin{cases}-\Delta u &= f (|x|, u, |\nabla u|), \; x \in \Omega, \\ u|_{\partial\Omega} &= 0,\end{cases}\]
where \(\Omega = \{x \in\mathbb{R}^N \mid r_1 < |x| < r_2\}\), \(N \ge 3\), \(f: [r_1, r_2] \times \mathbb{R}\times {\mathbb{R}^+} \to {\mathbb{R}}\) is continuous. The existence of radial solutions is obtained under the condition that allows \(f (r, u, \eta)\) superlinear growth on \(u\) and \(\eta\).An evolution \(p\)-Kirchhoff equation with power exponential nonlinearity and its steady state form.https://zbmath.org/1449.352582021-01-08T12:24:00+00:00"Chang, Xiuling"https://zbmath.org/authors/?q=ai:chang.xiuling"Gao, Wenjie"https://zbmath.org/authors/?q=ai:gao.wenjieSummary: We considered an initial boundary value problem of a class of \(p\)-Kirchhoff equation with exponential nonlinearity and its steady state form. For the evolution equation, we adopted the potential well method. We characterized the asymptotic behaviors of the general global solutions to the problem and the depth of the potential well could be attained by using the properties of potential well and estimates. For the steady state problem, we gave the existence of a ground state solution by using Lagrange multiplier method.Lower bound estimate of blow-up time for solutions to nonlinear hyperbolic equations with supercritical sources.https://zbmath.org/1449.351292021-01-08T12:24:00+00:00"Wang, Xue"https://zbmath.org/authors/?q=ai:wang.xue"Guo, Yue"https://zbmath.org/authors/?q=ai:guo.yue"Zu, Ge"https://zbmath.org/authors/?q=ai:zu.geSummary: By constructing a new control function with small dissipative term, using energy estimate inequalities and inverse Hölder inequality, the first order nonlinear differential inequality was established about the \({L^p}\) norm for the solutions of a class of nonlinear hyperbolic equations with supercritical source terms, and the accurate lower bound estimate of blow-up time for the solutions was obtained by discussing the properties of differential inequalities.New exact solutions of time-fractional Sharma-Tasso-Olver equation and Zakharov equations.https://zbmath.org/1449.354482021-01-08T12:24:00+00:00"Ren, Xiaojing"https://zbmath.org/authors/?q=ai:ren.xiaojing"Ge, Nannan"https://zbmath.org/authors/?q=ai:ge.nannanSummary: With the help of fractional traveling wave transformation and conformable fractional derivatives, we gave exact solutions of several hyperbolic functions for the nonlinear time-fractional generalized Sharma-Tasso-Olver equation and Zakharov equations by using the generalized Kudryashov method.Two-grid finite element discretization methods for a class of Poisson-Nernst-Planck equations.https://zbmath.org/1449.652592021-01-08T12:24:00+00:00"Tang, Ming"https://zbmath.org/authors/?q=ai:tang.ming"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.ying"Li, Xuefang"https://zbmath.org/authors/?q=ai:li.xuefangSummary: A two-grid finite element method was used to solve a class of Poisson-Nernst-Planck (PNP) equations discretely. By the two-grid discretization, the coupled PNP system was decoupled into a small scale linear symmetric system, which could effectively reduce the computational complexity. The theoretical results show that the linear symmetry two-grid algorithm has the same error order as the traditional finite element method. The numerical results show that the method has higher computational efficiency than the traditional finite element method.Inverse problem of heat source identification based on Bayesian differential evolution algorithm.https://zbmath.org/1449.652312021-01-08T12:24:00+00:00"Yin, Weishi"https://zbmath.org/authors/?q=ai:yin.weishi"Li, Jiaqi"https://zbmath.org/authors/?q=ai:li.jiaqiSummary: Using the Bayesian differential evolution algorithm, we discussed the two-dimensional heat conduction equation. The inversion estimation of the heat source position was given through the observation temperature at different time of an observation point. The numerical experiment results show that, with the increase of the number of iterations, the error of the position parameter of the heat source decreases. When the number of iterations reaches 120, the relative error of the parameter inversion is controlled within 2\%. When 5\% and 10\% white noise are added to the observed data, the relative error changes little, which indicates that the algorithm has good stability.Existence and local uniqueness of bubbling solutions for the Grushin critical problem.https://zbmath.org/1449.350692021-01-08T12:24:00+00:00"Gheraibia, Billel"https://zbmath.org/authors/?q=ai:gheraibia.billel"Wang, Chunhua"https://zbmath.org/authors/?q=ai:wang.chunhua"Yang, Jing"https://zbmath.org/authors/?q=ai:yang.jingThe authors consider the Grushin critical problem
\[-\Delta u(y,z)=\Phi(y,z)\frac{u^{\frac{N}{N-2}}(y,z)}{|y|},\quad u>0,\quad(y,z)\in\mathbb{R}^k\times\mathbb{R}^h\]
where \(k-1\geq h\geq 1\), \(k+h=N\geq 5\), and \(\Phi\in C^1(\mathbb{R}^N)\) is a nonnegative bounded nonconstant function, which is \(1\)-periodic in its the \(\bar{k}\) variables \(z_1,\dots,z_{\bar{k}}\), where \(1\leq\bar{k}<\frac{N-2}{N}\), and satisfies the following conditions: \(0\) is a critical point of \(\Phi\), with \(\Phi(0)>0\), and there exist \(\beta\in(N-2,N-1)\), \(\theta>0\), \(a_1,\dots,a_N\in\mathbb{R}\setminus\{0\}\), with \(\sum_{i=1}^Na_i<0\), and \(R\in C^{1,[\beta]}(\mathbb{R}^N)\) satisfying \(\sum_{s=0}^{[\beta]}|\nabla^sR(x)||x|^{-\beta+s}=O(|x|^{\theta})\) as \(x\rightarrow 0\), such that
\[\Phi(x)=\Phi(0)+\sum_{i=1}^Na_i|x_i|^{\beta}+R(x),\quad\text{for all }|x|\text{ small}.\]
With these assumptions, the authors show that, if \(\{P_i\}_{i\in\mathbb{N}}\) is a sequence of points in \(\mathbb{R}^{\bar{k}}\times\{0_{\mathbb{R}^{N-k-\bar{k}}}\}\) satisfying certain properties, then the problem admits a solution \(u\) with infinitely many bubbles concentrating at \(P_1,P_2,P_3,\dots\). To prove the existence of such a solution \(u\), the authors first consider the case of a finite sequence of points \(P_1,\dots,P_n\) and find a solution \(u_n\) with bubbles concentrating at \(P_1,P_2,\dots,P_n\). Then, using elliptic estimates, they get the solution \(u\) as a limit in \(C^2_{loc}(\mathbb{R}^N)\) of the sequence \(\{u_n\}\). Next, the authors prove a local uniqueness property which, in turn, they use to derive the periodicity with respect to the variables \(z_1,\dots,z_{\bar{k}}\) of the bubbling solution \(u\).
Reviewer: Giovanni Anello (Messina)Optimal control of resource in two competing species model with advection terms.https://zbmath.org/1449.490022021-01-08T12:24:00+00:00"Liang, Haojian"https://zbmath.org/authors/?q=ai:liang.haojian"Li, Huilai"https://zbmath.org/authors/?q=ai:li.huilaiSummary: We describe two competing species model by a reaction-diffusion equation with advection term. We hope that when resources were allocated, the weighted density of two species in the ecosystem reached the maximum during the whole time period and at the final time. By proving the existence of the solution of the equation, the existence of an optimal control and the necessary condition of the optimal control problem, the results show that the population density reaches the maximum when the resource allocation satisfies a bang-bang control.Shock layer solution of nonlinear wave system.https://zbmath.org/1449.353032021-01-08T12:24:00+00:00"Chen, Huaijun"https://zbmath.org/authors/?q=ai:chen.huaijun"Xu, Jianzhong"https://zbmath.org/authors/?q=ai:xu.jianzhong"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqiSummary: We considered the third boundary value problem of nonlinear singularly perturbed wave equation. Firstly, the outer solution was constructed by using singular perturbation method. Secondly, correction terms of the shock layer, initial layer and boundary layer were obtained by using the stretched variables. Finally, the asymptotic expansion of solution to the system was given, and the uniform validity of its asymptotic solution was proved.Bang-bang property of time optimal controls for the heat equation in the presence of a scale parameter.https://zbmath.org/1449.490222021-01-08T12:24:00+00:00"Benalia, Karim"https://zbmath.org/authors/?q=ai:benalia.karim"Ouakacha, Brahim"https://zbmath.org/authors/?q=ai:ouakacha.brahimSummary: In this paper we apply the classical control theory for the heat equation depending of a scale parameter. The main results establish a Pontryagyn type maximum principle and give sufficient conditions for the bang-bang property of optimal controls. In this fact, in first time we build exact solution. The dependence of this solution compared to the scale parameter thus lead to study the existence and uniqueness of the time optimal control for the heat equation. More precisely, supposing the \(L^\infty\)-controllability to zero, we can establish a bang-bang type property in the presence of a scale parameter. Numerical example is given in the last section to illustrate our main result.A class of constrained minimal elements for Schrödinger-Poisson equations.https://zbmath.org/1449.351902021-01-08T12:24:00+00:00"Lei, Yan"https://zbmath.org/authors/?q=ai:lei.yan"Guo, Zuji"https://zbmath.org/authors/?q=ai:guo.zuji"Wang, Shuli"https://zbmath.org/authors/?q=ai:wang.shuliSummary: The existence and the nonexistence of minimal elements with prescribed \({L^2}\)-norm for a class of Schrödinger-Poisson equations are considered by using variational methods. Firstly, by using Gagliardo-Nirenberg and Hardy-Littewood-Sobolev inequalities and selecting testing functions, some estimates are obtained. Secondly, in the discussion on the classification of exponent \(p\) of nonlinearity, by using the method of minimizing sequence, compact embedding lemma, Ekeland's variational principle, vanishing lemma, and Pohozaev's identity, the existence and the nonexistence of constrained minimal elements are proved.Lattice Boltzmann simulations for Turing pattern on sphere.https://zbmath.org/1449.652862021-01-08T12:24:00+00:00"Zhang, Jianying"https://zbmath.org/authors/?q=ai:zhang.jianying"Yan, Guangwu"https://zbmath.org/authors/?q=ai:yan.guangwuSummary: Using a lattice Boltzmann model in spherical coordinate system, we solved the Gierer-Meinhardt equation on spherical surface, and obtained the Turing patterns on spherical surface. Compared with the classical difference method, the model could be used to simulate the Turing pattern on the spherical surface.Dyons of unit topological charges in gauged Skyrme model.https://zbmath.org/1449.353702021-01-08T12:24:00+00:00"Wu, Zhonglin"https://zbmath.org/authors/?q=ai:wu.zhonglin"Li, Dongya"https://zbmath.org/authors/?q=ai:li.dongyaSummary: Dyons are an important family of topological solitons carrying both electric and magnetic charges and the presence of a nontrivial temporal component of the gauge field essential for the existence of electricity often makes the analysis of the underlying nonlinear equations rather challenging since the governing action functional assumes an indefinite form. In this work, by developing a constrained variational technique, we establish an existence theorem for the dyon solitons in a Skyrme model coupled with \(SO (3)\)-gauge fields. These solutions carry unit monopole and Skyrme charges.Quasimonotonicity and functional inequalities.https://zbmath.org/1449.354642021-01-08T12:24:00+00:00"Herzog, Gerd"https://zbmath.org/authors/?q=ai:herzog.gerd"Volkmann, Peter"https://zbmath.org/authors/?q=ai:volkmann.peterSummary: A comparison theorem for functional equations in ordered topological vector spaces will be given, which generalizes the results from \textit{P. Volkmann} [ISNM, Int. Ser. Numer. Math. 161, 269--273 (2012; Zbl 1253.26043); Ein Vergleichssatz für Integralgleichungen, KITopen, 3 p. (2016; \url{doi:10.5445/IR/1000061837})]. Quasimonotonicity is fundamental for these investigations.New periodic wave solutions of a \( (3+1)\)-dimensional generalized shallow water equation.https://zbmath.org/1449.350312021-01-08T12:24:00+00:00"Zhang, Shulin"https://zbmath.org/authors/?q=ai:zhang.shulin"Liu, Jian'gen"https://zbmath.org/authors/?q=ai:liu.jiangen"Liu, Wanli"https://zbmath.org/authors/?q=ai:liu.wanliSummary: A new three-wave method and the Hirota bilinear form are applied to investigate a \( (3+1)\)-dimensional generalized shallow water equation. As a result, new periodic wave solutions are obtained by two different cases involved the parameters. Furthermore, figures of some special periodic wave solutions are presented to illustrate the mechanical features of these solutions.New mixed solutions to the four-potential isospectral Ablowitz-Ladik equation.https://zbmath.org/1449.354032021-01-08T12:24:00+00:00"Zhang, Yiyan"https://zbmath.org/authors/?q=ai:zhang.yiyan"Chen, Shouting"https://zbmath.org/authors/?q=ai:chen.shoutingSummary: In this paper, the four-potential isospectral Ablowitz-Ladik equation is mainly discussed. By virtue of the double Casoratian technique and taking special cases in a general double Casoratian solution, the mixed solutions between the rational-like and complexiton cases in double Casoratian form are constructed. Moreover, the mixed solutions between the Matveev and complexiton cases are derived.Effective modules of relaxation of multi-component isotropic visco-elastic composite materials.https://zbmath.org/1449.740702021-01-08T12:24:00+00:00"Glushchenkov, V. S."https://zbmath.org/authors/?q=ai:glushchenkov.v-s"Lyulin, A. S."https://zbmath.org/authors/?q=ai:lyulin.a-s"Mantulenko, A. V."https://zbmath.org/authors/?q=ai:mantulenko.a-v"Saraev, A. L."https://zbmath.org/authors/?q=ai:saraev.a-lSummary: The macroscopic (effective) modules of relaxation of multicomponent isotropic viscoelastic composite materials of type ``matrix-sphere inclusions'' the components of which have the inherited properties are modeled.Tricomi problem analogue for loaded equation of hyperbolic-parabolic type with variable coefficients.https://zbmath.org/1449.353242021-01-08T12:24:00+00:00"Khubiev, K. U."https://zbmath.org/authors/?q=ai:khubiev.kazbek-uzeirovichSummary: An analogue of Tricomi problem for loaded equation of hyperbolic-parabolic type with variable coefficients is studied. The theorem on existence and uniqueness of solution is proved for special conditions on the coefficients.Solving a class of Hamilton-Jacobi-Bellman equations using pseudospectral methods.https://zbmath.org/1449.490072021-01-08T12:24:00+00:00"Mehrali-Varjani, Mohsen"https://zbmath.org/authors/?q=ai:mehrali-varjani.mohsen"Shamsi, Mostafa"https://zbmath.org/authors/?q=ai:shamsi.mostafa"Malek, Alaeddin"https://zbmath.org/authors/?q=ai:malek.alaeddinSummary: This paper presents a numerical approach to solve the Hamilton-Jacobi-Bellman (HJB) problem which appears in feedback solution of the optimal control problems. In this method, first, by using Chebyshev pseudospectral spatial discretization, the HJB problem is converted to a system of ordinary differential equations with terminal conditions. Second, the time-marching Runge-Kutta method is used to solve the corresponding system of differential equations. Then, an approximate solution for the HJB problem is computed. In addition, to get more efficient and accurate method, the domain decomposition strategy is proposed with the pseudospectral spatial discretization. Five numerical examples are presented to demonstrate the efficiency and accuracy of the proposed hybrid method.Global existence, asymptotic behavior and uniform attractors for a type III non-autonomous thermoelastic Timoshenko system.https://zbmath.org/1449.350772021-01-08T12:24:00+00:00"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yuming"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jieSummary: In this paper, we investigate a Timoshenko system of thermoelastic of type III. We prove the global existence and asymptotic behavior of solutions by using the semigroup method and multiplicative technique. Then we prove the existence of uniform attractor for a non-autonomous thermoelastic system by using the method of uniform contractive function.Non-local problem with fractional derivatives for one hyperbolic equation.https://zbmath.org/1449.354242021-01-08T12:24:00+00:00"Arlanova, E. Yu."https://zbmath.org/authors/?q=ai:arlanova.ekaterina-yurevnaSummary: We formulate and study the non-local boundary value problem with fractional integro-differentiation operators for one partial case of the moisture transfer equation. The uniqur solvability of this problem is proved.On Riemann method for solving a mixed problem.https://zbmath.org/1449.351762021-01-08T12:24:00+00:00"Mironov, A. N."https://zbmath.org/authors/?q=ai:mironov.a-nTranslation of the Russian annotation: For an equation with the highest partial derivative of a general form, a formula for solving the problem is constructed by the Riemann method, in which the solution is found in a characteristic parallelepiped with an angle cut off by a non-characteristic surface, and Cauchy conditions are set on the non-characteristic part of the boundary, and Goursat conditions are set on the characteristics adjacent to this part of the boundary.Exact solutions and the conservation law of fifth-order variable-coefficient equation.https://zbmath.org/1449.353042021-01-08T12:24:00+00:00"Dong, Mei"https://zbmath.org/authors/?q=ai:dong.mei"Li, Zhengyong"https://zbmath.org/authors/?q=ai:li.zhengyong"Liu, Hanze"https://zbmath.org/authors/?q=ai:liu.hanzeSummary: By using the improved CK method, the relation between fifth-order variable-coefficient equation and the corresponding constant-coefficient equation is obtained. Some new exact solutions of the corresponding constant-coefficient equation are given by applying the Lie group method. Then the explicit solutions of the fifth-order variable-coefficient equation are presented. Finally, we give the conservation laws of the fifth-order variable-coefficient equation in terms of the Lagrangian and adjoint equation method.Structural representation of inverse operator in Banach space.https://zbmath.org/1449.354622021-01-08T12:24:00+00:00"Tyan, V. K."https://zbmath.org/authors/?q=ai:tyan.v-kSummary: The problem of construction of an inverse operator for the equation \(Az=u\) (\(u\in U\), \(z\in F\); \(U\), \(F\) are metric spaces) is considered. The algorithm for solving the inverse problems based on structural representation of inverse operators as periodic structure is suggested. The necessary and sufficient conditions of fundamentality of periodic structure are formulated. It is shown that if these conditions are fulfilled the periodic structure operator converges to the inverse operator.The Lagrange stability of a class of impulsive differential equation.https://zbmath.org/1449.350472021-01-08T12:24:00+00:00"Dong, Hejin"https://zbmath.org/authors/?q=ai:dong.hejin"Shen, Jianhua"https://zbmath.org/authors/?q=ai:shen.jianhua.1|shen.jianhuaSummary: By using Moser's twist theorem, the Lagrange stability of a class of Duffing equations is proved to be preserved under suitable impulsive forcing.Stabilized IMLS based element free Galerkin method for stochastic elliptic partial differential equations.https://zbmath.org/1449.653162021-01-08T12:24:00+00:00"Izadpanah, Komeil"https://zbmath.org/authors/?q=ai:izadpanah.komeil"Mesforush, Ali"https://zbmath.org/authors/?q=ai:mesforush.ali"Nazemi, Ali"https://zbmath.org/authors/?q=ai:nazemi.ali-rezaSummary: In this paper, we propose a numerical method to solve the elliptic stochastic partial differential equations (SPDEs) obtained by Gaussian noises using an element free Galerkin method based on stabilized interpolating moving least square shape functions. The error estimates of the method is presented. The method is tested via several problems. The numerical results show the usefulness and accuracy of the new method.Numerical solution of singularly perturbed parabolic problems by a local kernel-based method with an adaptive algorithm.https://zbmath.org/1449.350342021-01-08T12:24:00+00:00"Rafieayanzadeh, Hossein"https://zbmath.org/authors/?q=ai:rafieayanzadeh.hossein"Mohammadi, Maryam"https://zbmath.org/authors/?q=ai:mohammadi.maryam-beyg"Babolian, Esmail"https://zbmath.org/authors/?q=ai:babolian.esmailSummary: Global approaches make troubles and deficiencies for solving singularly perturbed problems. In this work, a local kernel-based method is applied for solving singularly perturbed parabolic problems. The kernels are constructed by the Newton basis functions (NBFs) on stencils selected as thin regions of the domain of problem that leads to increasing accuracy with less computational costs. In addition, position of nodes may affect significantly on accuracy of the method, therefore, the adaptive residual subsampling algorithm is used to locate optimal position of nodes. Finally, some problems are solved by the proposed method and the accuracy and efficiency of the method is compared with results of some other methods.A high-order compact difference method for fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions.https://zbmath.org/1449.652052021-01-08T12:24:00+00:00"Wang, Yuan-Ming"https://zbmath.org/authors/?q=ai:wang.yuanmingSummary: In a recent paper, \textit{L. Ren} and \textit{L. Liu} [Comput. Appl. Math. 37, No. 5, 6252--6269 (2018; Zbl 1413.65329)] proposed and analyzed a high-order compact finite difference method for a class of fractional sub-diffusion equations with variable coefficients and nonhomogeneous Neumann boundary conditions. In this paper, we point out some deficiencies and errors found in that paper and make the corresponding revisions.Weak Galerkin finite element method for an inhomogeneous Brusselator model with cross-diffusion.https://zbmath.org/1449.652492021-01-08T12:24:00+00:00"Khaled-Abad, Leila Jafarian"https://zbmath.org/authors/?q=ai:khaled-abad.leila-jafarian"Salehi, Rezvan"https://zbmath.org/authors/?q=ai:salehi.rezvanSummary: A new weak Galerkin finite element method is applied for time dependent Brusselator reaction-diffusion systems by using discrete weak gradient operators over discontinuous weak functions. In this work, we consider the lowest order weak Galerkin finite element space \((P_0,P_0,RT_0)\). Discrete weak gradients are defined in Raviart-Thomas space. Thus, we employ this approximate space on triangular mesh for solving unknown concentrations \((u,v)\) in Brusselator reaction-diffusion systems. Based on a weak variational form, semi-discrete and fully-discrete weak Galerkin finite element scheme are obtained. In addition, the paper presents some numerical results to illustrate the power of the proposed method.Numerical solutions of time-fractional coupled viscous Burgers' equations using meshfree spectral method.https://zbmath.org/1449.652732021-01-08T12:24:00+00:00"Hussain, Manzoor"https://zbmath.org/authors/?q=ai:hussain.manzoor"Haq, Sirajul"https://zbmath.org/authors/?q=ai:haq.sirajul"Ghafoor, Abdul"https://zbmath.org/authors/?q=ai:ghafoor.abdul"Ali, Ihteram"https://zbmath.org/authors/?q=ai:ali.ihteramSummary: In this article, we compute numerical solutions of time-fractional coupled viscous Burgers' equations using meshfree spectral method. Radial basis functions (RBFs) and spectral collocation approach are used for approximation of the spatial part. Temporal fractional part is approximated via finite differences and quadrature rule. Approximation quality and efficiency of the method are assessed using discrete \(E_2, E_{\infty }\) and \(E_{\text{rms}}\) error norms. Varying the number of nodal points \(M\) and time step-size \(\Delta t\), convergence in space and time is numerically studied. The stability of the current method is also discussed, which is an important part of this paper.Discrete energy behavior of a damped Timoshenko system.https://zbmath.org/1449.652572021-01-08T12:24:00+00:00"Sabrine, Chebbi"https://zbmath.org/authors/?q=ai:sabrine.chebbi"Makram, Hamouda"https://zbmath.org/authors/?q=ai:makram.hamoudaSummary: In this article, we consider a one-dimensional Timoshenko system subject to different types of dissipation (linear and nonlinear damping). Based on a combination between the finite element and the finite difference methods, we design a discretization scheme for the different Timoshenko systems under consideration. We first come up with a numerical scheme to the free-undamped Timoshenko system. Then we adapt this numerical scheme to the corresponding linear and nonlinear damped systems. Interestingly, this scheme reaches to reproduce the most important properties of the discrete energy, namely we show for the discrete energy the positivity, the energy conservation property and the different decay rate profiles. We numerically reproduce the known analytical results established on the decay rate of the energy associated with each type of dissipation.The application of differential characteristic set method to pseudo differential operator and Lax representation.https://zbmath.org/1449.354692021-01-08T12:24:00+00:00"Jia, Yifeng"https://zbmath.org/authors/?q=ai:jia.yifeng"Xiao, Dongliang"https://zbmath.org/authors/?q=ai:xiao.dongliangSummary: Differential characteristic set method is applied to the calculation of pseudo differential operators and Lax representation of nonlinear evolution equations. Firstly, differential characteristic set method and differential division with remainder are used for the calculation of inverse and extraction root of pseudo differential operator, such that the process is simplified since it is unnecessary to solve ordinary differential equation systems and substitute the solutions. Secondly, using differential characteristic set method, the nonlinear partial differential equation systems derived from the generalized Lax equation and Zakharov-Shabat equation, are reduced, and the corresponding nonlinear evolution equation is obtained. The related programs are compiled in Mathematica. A computer-based computer algebra system, and Lax representation of some nonlinear evolution equations can be calculated with the aid of the computer.A cardinal method to solve coupled nonlinear variable-order time fractional sine-Gordon equations.https://zbmath.org/1449.354372021-01-08T12:24:00+00:00"Heydari, Mohammad Hossein"https://zbmath.org/authors/?q=ai:heydari.mohammadhossein"Avazzadeh, Zakieh"https://zbmath.org/authors/?q=ai:avazzadeh.zakieh"Yang, Yin"https://zbmath.org/authors/?q=ai:yang.yin"Cattani, Carlo"https://zbmath.org/authors/?q=ai:cattani.carloSummary: In this study, a computational approach based on the shifted second-kind Chebyshev cardinal functions (CCFs) is proposed for obtaining an approximate solution of coupled variable-order time-fractional sine-Gordon equations where the variable-order fractional operators are defined in the Caputo sense. The main ideas of this approach are to expand the unknown functions in tems of the shifted second-kind CCFs and apply the collocation method such that it reduces the problem into a system of algebraic equations. To algorithmize the method, the operational matrix of variable-order fractional derivative for the shifted second-kind CCFs is derived. Meanwhile, an effective technique for simplification of nonlinear terms is offered which exploits the cardinal property of the shifted second-kind CCFs. Several numerical examples are examined to verify the practical efficiency of the proposed method. The method is privileged with the exponential rate of convergence and provides continuous solutions with respect to time and space. Moreover, it can be adapted for other types of variable-order fractional problems straightforwardly.New numerical studies for Darcy's problem coupled with the heat equation.https://zbmath.org/1449.353492021-01-08T12:24:00+00:00"Dib, Dayana"https://zbmath.org/authors/?q=ai:dib.dayana"Dib, Séréna"https://zbmath.org/authors/?q=ai:dib.serena"Sayah, Toni"https://zbmath.org/authors/?q=ai:sayah.toniSummary: In this article, we consider the heat equation coupled with Darcy's law by a nonlinear viscosity depending on the temperature. We recall two numerical schemes and introduce a new non-stabilized one, we show the existence and uniqueness of the solutions and we establish an a priori error estimates using the Brezzi-Rappaz-Raviart theorem. Numerical investigations are preformed and showed.Solution of the non-local problem for the hyperbolic equation in the closed form.https://zbmath.org/1449.353142021-01-08T12:24:00+00:00"Salikhov, R. N."https://zbmath.org/authors/?q=ai:salikhov.r-nSummary: The non-local boundary value problem for the degenerate hyperbolic equation in the area \(D\), which is the union of two areas in the upper and lower half-planes, is examined. The proof of the existence and uniqueness of this problem solution is reduced to the question of the solvability of a singular integral equation.On one nonlocal problem for the heat equation with an integral condition.https://zbmath.org/1449.352432021-01-08T12:24:00+00:00"Danilkina, O. Yu."https://zbmath.org/authors/?q=ai:danilkina.o-yuSummary: The solvability of the non-local problem with the integral condition for the heat-transfer equation is studied. The existence and uniqueness theorem for the generalized solution is proved.Existence of entropy solutions for an elliptic equation with degenerate coercivity.https://zbmath.org/1449.352342021-01-08T12:24:00+00:00"Dai, Lili"https://zbmath.org/authors/?q=ai:dai.liliSummary: In this paper, we use the truncation method to investigate the existence of solutions for degenerate elliptic problems with variable exponent in weighted Sobolev spaces. With the help of the Marcinkiewicz estimate and using some a priori estimates for the sequence of solutions of the approximate problem, we choose suitable test functions for the approximate equation and obtain the needed estimates. Then, we obtain the entropy solutions for the elliptic equation in weighted Sobolev spaces with a variable exponent.Two-mode coupled KdV equation: multiple-soliton solutions and other exact solutions.https://zbmath.org/1449.351582021-01-08T12:24:00+00:00"Zhao, Qian"https://zbmath.org/authors/?q=ai:zhao.qian"Bai, Xirui"https://zbmath.org/authors/?q=ai:bai.xiruiSummary: In this paper, multiple-soliton solutions for a new two-mode coupled KdV (nTMcKdV) equation are obtained by using the simplified Hirota's method and the Cole-Hopf transformation. It is shown that these types of multiple solutions exist only for models in which specific values for the nonlinearity and dispersion parameters are included in the models. Furthermore, other exact solutions for an nTMcKdV equation using general values of these parameters are derived by using several different expansion methods such as the tanh/coth method and the Jacobi elliptic function method.An initial-boundary value problem for the generalized Sasa-Satsuma equation on the half-line.https://zbmath.org/1449.351852021-01-08T12:24:00+00:00"Dong, Fengjiao"https://zbmath.org/authors/?q=ai:dong.fengjiao"Hu, Beibei"https://zbmath.org/authors/?q=ai:hu.beibeiSummary: In this paper, we implement the Fokas unified transform method to study initial-boundary value problems of the generalized Sasa-Satsuma equation on the half-line. Assuming that the solution \(u (x, t)\) of the generalized Sasa-Satsuma equation exists, we will prove that it can be expressed in terms of the unique solution of a \(3\times 3\) matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter \(\lambda\).Comparison principle and stability analysis of a class of stochastic parabolic equations with delay and Markovian switching.https://zbmath.org/1449.350552021-01-08T12:24:00+00:00"Li, Zhao"https://zbmath.org/authors/?q=ai:li.zhao"Li, Shuyong"https://zbmath.org/authors/?q=ai:li.shuyongSummary: This paper investigates the mean square stability for stochastic parabolic equations with delay and Markovian switching. By establishing the comparison principle, using delay differential inequality and stochastic analysis techniques, the mean square stability, mean square uniform stability, mean square asymptotic stability and mean square exponential stability for the system are obtained. Finally, an example is given to illustrate the main theoretical result.Efficient and accurate numerical methods for long-wave short-wave interaction equations in the semiclassical limit regime.https://zbmath.org/1449.652012021-01-08T12:24:00+00:00"Wang, Tingchun"https://zbmath.org/authors/?q=ai:wang.tingchun"Zhao, Xiaofei"https://zbmath.org/authors/?q=ai:zhao.xiaofei"Peng, Mao"https://zbmath.org/authors/?q=ai:peng.mao"Wang, Peng"https://zbmath.org/authors/?q=ai:wang.peng|wang.peng.1|wang.peng.2Summary: This paper focuses on performance of several efficient and accurate numerical methods for the long-wave short-wave interaction equations in the semiclassical limit regime. The key features of the proposed methods are based on: (i) the utilization of the first-order or second-order time-splitting method to the nonlinear wave interaction equations; (ii) the application of Fourier pseudo-spectral method or compact finite difference approximation to the linear subproblem and the spatial derivatives; (iii) the adoption of the exact integration of the nonlinear subproblems and the ordinary differential equations in the phase space. The numerical methods under study are efficient, unconditionally stable and higher-order accurate, they are proved to preserve two invariants including the position density in \({L^1}\). Numerical results are reported for case studies with different types of initial data. These results verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter \(\varepsilon\), and demonstrate the behavior of nonlinear dispersive waves in the semiclassical limit regime.A decoupling two-grid method for the steady-state Poisson-Nernst-Planck equations.https://zbmath.org/1449.653292021-01-08T12:24:00+00:00"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.ying"Lu, Benzhuo"https://zbmath.org/authors/?q=ai:lu.benzhuo"Xie, Yan"https://zbmath.org/authors/?q=ai:xie.yanSummary: Poisson-Nernst-Planck equations are widely used to describe the electrodiffusion of ions in a solvated biomolecular system. Two kinds of two-grid finite element algorithms are proposed to decouple the steady-state Poisson-Nernst-Planck equations by coarse grid finite element approximations. Both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two-grid algorithms for solving Poisson-Nernst-Planck equations.A fourth-order compact and conservative difference scheme for the generalized Rosenau-Korteweg de Vries equation in two dimensions.https://zbmath.org/1449.652002021-01-08T12:24:00+00:00"Wang, Jue"https://zbmath.org/authors/?q=ai:wang.jue"Zeng, Qingnan"https://zbmath.org/authors/?q=ai:zeng.qingnanSummary: In this paper, a conservative difference scheme for the Rosenau-Korteweg de Vries (RKdV) equation in 2D is proposed. The system satisfies the conservative laws in energy and mass. Existence and uniqueness of its difference solution are shown. The order of \(O ({\tau^2} + {h^4})\) in the discrete \({L^\infty}\)-norm with time step \(\tau\) and mesh size \(h\) is obtained. Some important lemmas are proposed to prove the high order convergence. We prove that the present scheme is unconditionally stable. Numerical results are also given in order to check the properties of analytical solution.Superconvergence analysis for time-fractional diffusion equations with nonconforming mixed finite element method.https://zbmath.org/1449.652662021-01-08T12:24:00+00:00"Zhang, Houchao"https://zbmath.org/authors/?q=ai:zhang.houchao"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, a fully discrete scheme based on the \(L1\) approximation in temporal direction for the fractional derivative of order \(\alpha\) in \( (0,1)\) and nonconforming mixed finite element method (MFEM) in spatial direction is established. First, we prove a novel result of the consistency error estimate with order \(O (h^2)\) of \(EQ_1^{rot}\) element (see Lemma 2.3). Then, by using the proved character of \(EQ_1^{rot}\) element, we present the superconvergent estimates for the original variable \(u\) in the broken \({H^1}\)-norm and the flux \(\vec p = \nabla u\) in the \( (L^2)^2\)-norm under a weaker regularity of the exact solution. Finally, numerical results are provided to confirm the theoretical analysis.Interior estimates of semidiscrete finite element methods for parabolic problems with distributional data.https://zbmath.org/1449.652442021-01-08T12:24:00+00:00"Guo, Li"https://zbmath.org/authors/?q=ai:guo.li.1|guo.li.2"Li, Hengguang"https://zbmath.org/authors/?q=ai:li.hengguang"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.2|yang.yang.5|yang.yang.4|yang.yang.1|yang.yang.3Summary: Let \(\Omega \subset {\mathbb{R}^d}\), \(1 \le d \le 3\), be a bounded \(d\)-polytope. We consider the parabolic equation on \(\Omega\) with the Dirac delta function on the right hand side. We study the well-posedness, regularity, and the interior error estimate of semidiscrete finite element approximations of the equation. In particular, we derive that the interior error is bounded by the best local approximation error, the negative norms of the error, and the negative norms of the time derivative of the error. This result implies different convergence rates for the numerical solution in different interior regions, especially when the region is close to the singular point. Numerical test results are reported to support the theoretical prediction.Unconditional superconvergence analysis of an \({H^1}\)-Galerkin mixed finite element method for two-dimensional Ginzburg-Landau equation.https://zbmath.org/1449.652582021-01-08T12:24:00+00:00"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyang"Wang, Junjun"https://zbmath.org/authors/?q=ai:wang.junjunSummary: An \({H^1}\)-Galerkin mixed finite element method (MFEM) is discussed for the two-dimensional Ginzburg-Landau equation with the bilinear element and zero order Raviart-Thomas element \( (Q_{11} + Q_{10} \times Q_{01})\). A linearized Crank-Nicolson fully-discrete scheme is developed and a time-discrete system is introduced to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, the regularity of the time-discrete system is deduced through the temporal error estimation. On the other hand, the superconvergent estimates of \(u\) in \({H^1}\)-norm and \(\vec q\) in \(H ({\mathrm{div}}; \Omega)\)-norm with order \(O ({h^2} + {\tau^2})\) are obtained unconditionally based on the achievement of the spatial result. At last, a numerical experiment is included to illustrate the feasibility of the proposed method. Here, \(h\) is the subdivision parameter and \(\tau\) is the time step.An unfitted \(hp\)-interface penalty finite element method for elliptic interface problems.https://zbmath.org/1449.653262021-01-08T12:24:00+00:00"Wu, Haijun"https://zbmath.org/authors/?q=ai:wu.haijun"Xiao, Yuanming"https://zbmath.org/authors/?q=ai:xiao.yuanmingSummary: An \(hp\) version of interface penalty finite element method (\(hp\)-IPFEM) is proposed to solve the elliptic interface problems in two and three dimensions on unfitted meshes. Error estimates in broken \({H^1}\) norm, which are optimal with respect to \(h\) and suboptimal with respect to \(p\) by half an order of \(p\), are derived. Both symmetric and non-symmetric IPFEM are considered. Error estimates in \({L_2}\) norm are proved by the duality argument. All the estimates are independent of the location of the interface relative to the meshes. Numerical examples are provided to illustrate the performance of the method.Alternating direction implicit schemes for the two-dimensional time fractional nonlinear super-diffusion equations.https://zbmath.org/1449.651812021-01-08T12:24:00+00:00"Huang, Jianfei"https://zbmath.org/authors/?q=ai:huang.jianfei"Zhao, Yue"https://zbmath.org/authors/?q=ai:zhao.yue"Arshad, Sadia"https://zbmath.org/authors/?q=ai:arshad.sadia"Li, Kuangying"https://zbmath.org/authors/?q=ai:li.kuangying"Tang, Yifa"https://zbmath.org/authors/?q=ai:tang.yifaSummary: As is known, there exist numerous alternating direction implicit (ADI) schemes for the two-dimensional linear time fractional partial differential equations. However, if the ADI schemes for linear problems combined with local linearization techniques are applied to solve nonlinear problems, the stability and convergence of the methods are often not clear. In this paper, two ADI schemes are developed for solving the two-dimensional time fractional nonlinear super-diffusion equations based on their equivalent partial integro-differential equations. In these two schemes, the standard second-order central difference approximation is used for the spatial discretization, and the classical first-order approximation is applied to discretize the Riemann-Liouville fractional integral in time. The solvability, unconditional stability and \({L_2}\) norm convergence of the proposed ADI schemes are proved rigorously. The convergence order of the schemes is \(O\left ({\tau + h_x^2 + h_y^2} \right)\), where \(\tau\) is the temporal mesh size, \({h_x}\) and \({h_y}\) are spatial mesh sizes in the \(x\) and \(y\) directions, respectively. Finally, numerical experiments are carried out to support the theoretical results and demonstrate the performances of two ADI schemes.Analysis of the improved element-free Galerkin method for nonlinear Poisson-Boltzmann equation.https://zbmath.org/1449.653322021-01-08T12:24:00+00:00"Zhong, Siyao"https://zbmath.org/authors/?q=ai:zhong.siyao"Li, Xiaolin"https://zbmath.org/authors/?q=ai:li.xiaolinSummary: The purpose of this paper is to solve the nonlinear Poisson-Boltzmann equation by the improved element-free Galerkin method. Combining the improved moving least square approximation with Galerkin weak form, the improved element-free Galerkin method is establish for the nonlinear Poisson-Boltzmann equation. Based on the error results of the improved moving least square approximation, the error of the improved element-free Galerkin method for nonlinear Poisson-Boltzmann equation is derived theoretically. Error estimation is obtained in the Sobolev space. Numerical examples verify the theoretical analysis. The proposed method has higher calculation accuracy and better stability. The errors decrease as the nodal space reduces.On the quasilinear Poisson equations in the complex plane.https://zbmath.org/1449.352232021-01-08T12:24:00+00:00"Gutlyanskii, V.Ya."https://zbmath.org/authors/?q=ai:gutlyanskii.vladimir-ya"Nesmelova, O. V."https://zbmath.org/authors/?q=ai:nesmelova.o-v"Ryazanov, V. I."https://zbmath.org/authors/?q=ai:ryazanov.vladimir-iSummary: First, we study the existence and regularity of solutions for the linear Poisson equations \(\Delta U(z) = g(z)\) in bounded domains \(D\) of the complex plane \(\mathbb{C}\) with charges g in the classes \(L^1(D)\cap L_{loc}^p(D), p > 1\). Then, applying the Leray-Schauder approach, we prove the existence of Hölder continuous solutions \(U\) in the class \(W_{loc}^{2,p}(D)\) for the quasilinear Poisson equations of the form \(\Delta U(z) = h(z) f (U(z))\) with \(h\) in the same classes as \(g\) and continuous functions \(f:\mathbb{R} \to\mathbb{R}\) such that \(f (t)/t\to 0\) as \( t\to\infty\). These results can be applied to various problems of mathematical physics.The existence of the global solution for a nonlinear wave equation arising in elastic waveguide model.https://zbmath.org/1449.353122021-01-08T12:24:00+00:00"Zhang, Yuanyuan"https://zbmath.org/authors/?q=ai:zhang.yuanyuanSummary: By the method of Galerkin's approximation, the paper studies the existence and uniqueness of the global solution to the initial boundary value problem for a nonlinear wave equation arising in elastic waveguide model. It is proved that when the space dimension \(N = 1\), under rather mild conditions the above-mentioned problem possesses a global solution.Existence and uniqueness of ground state solutions for a class of \(p\)-Kirchhoff equations.https://zbmath.org/1449.351972021-01-08T12:24:00+00:00"Wang, Zhuangzhuang"https://zbmath.org/authors/?q=ai:wang.zhuangzhuang"Zeng, Xiaoyu"https://zbmath.org/authors/?q=ai:zeng.xiaoyuSummary: For the following \(p\)-Kirchhoff type functional
\[ E_p(u) = \frac{a}{p}\int_{\mathbb{R}^n}|\nabla u|^p \,dx + \frac{b}{2p}\left (\int_{\mathbb{R}^n}|\nabla u|^p \,dx \right)^2 -\frac{1}{s}\int_{\mathbb{R}^n}|u|^s \,dx,\]
we prove the existence and uniqueness of global minimum or mountain pass type critical points on the \({L^p}\)-normalized manifold \({S_c}: = \{u\in W^{1,p} (\mathbb{R}^n):\int_{\mathbb{R}^n} |u|^p \,dx = {c^p}\}\). We show that these critical points indeed are optimizers of a certain Gagliardo-Nirenberg inequality. Especially, when \(p\in (1, 2]\), they are unique up to translations. We extend some known results for the case of \(p = 2\) in previous papers.Time-dependent pullback attractors for nonclassical diffusion equations with time delays.https://zbmath.org/1449.351002021-01-08T12:24:00+00:00"Wang, Fangping"https://zbmath.org/authors/?q=ai:wang.fangping"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: We first proved the well-posedness of weak solutions for the nonclassical diffusion equations with time delays by using the Faedo-Galerkin method, and then we gave the pullback \(\mathcal{D}\)-asymptotic compactness by using the contraction function method, which proved the existence of time-dependent pullback attractor.Nonlinear Schrödinger equation for envelope Rossby waves with complete Coriolis force and its solution.https://zbmath.org/1449.353992021-01-08T12:24:00+00:00"Yin, Xiaojun"https://zbmath.org/authors/?q=ai:yin.xiaojun"Yang, Liangui"https://zbmath.org/authors/?q=ai:yang.liangui"Yang, Hongli"https://zbmath.org/authors/?q=ai:yang.hongli"Zhang, Ruigang"https://zbmath.org/authors/?q=ai:zhang.ruigang"Su, Jinmei"https://zbmath.org/authors/?q=ai:su.jinmeiSummary: The physical features of the equatorial envelope Rossby waves including with complete Coriolis force and dissipation are investigated analytically. Staring with a potential vorticity equation, the wave amplitude evolution of equatorial envelope Rossby waves is described as a nonlinear Schrödinger equation by employing multiple scale analysis and perturbation expansions. The equation is more suitable for describing envelope Rossby solitary waves when the horizontal component of Coriolis force is stronger near the equator. Then, based on the Jacobi elliptic function expansion method and trial function method, the classical Rossby solitary wave solution and the corresponding stream function of the envelope Rossby solitary waves are obtained, respectively. With the help of these solutions, the effect of dissipation and the horizontal component of Coriolis parameter are discussed in detail by graphical presentations. The results reveal the effect of the horizontal component of Coriolis force and dissipation on the classical Rossby solitary waves.A reaction-diffusion model for a class of nonlinear parabolic equations with moving boundaries: existence, uniqueness, exponential decay and simulation.https://zbmath.org/1449.352632021-01-08T12:24:00+00:00"Robalo, Rui J."https://zbmath.org/authors/?q=ai:robalo.rui-j"Almeida, Rui M. P."https://zbmath.org/authors/?q=ai:almeida.rui-m-p"Coimbra, Maria do Carmo"https://zbmath.org/authors/?q=ai:do-carmo-coimbra.maria"Ferreira, Jorge"https://zbmath.org/authors/?q=ai:ferreira.jorge-n-m|ferreira.jorge-carregal|ferreira.jorge-a-f|ferreira.jorge-s|marques-ferreira.jorge-nelioSummary: The aim of this paper is to establish the existence, uniqueness and asymptotic behaviour of a strong regular solution for a class of nonlinear equations of reaction-diffusion nonlocal type with moving boundaries:
\[
\begin{cases} u_t - a \left(\int_{{\Omega}_t} u(x, t){dx}\right) u_{{xx}} = f(x, t),\quad (x, t) \in Q_t, \\ u(\alpha(t), t) = u(\beta(t), t) = 0,\quad t > 0, \\ u(x, 0) = u_0(x),\quad x \in{\Omega}_0 =] \alpha(0), \beta(0) [, \end{cases}
\]
where \(Q_t\) is a bounded non-cylindrical domain defined by
\[
Q_t = \left\{(x, t) \in \mathbb{R}^2 : \alpha(t) < x < \beta(t), \quad \text{ for all } 0 < t < T\right\}.
\]
Moreover, we study the properties of the solution and implement a numerical algorithm based on the Moving Finite Element Method (MFEM) with polynomial approximations of any degree, to solve this class of problems. Some numerical tests are investigated to evaluate the performance of our Matlab code based on the MFEM and illustrate the exponential decay of the solution.Longitudinal impact on rigid barrier homogeneous and stepped bar is lowered longitudinal stiffness.https://zbmath.org/1449.652252021-01-08T12:24:00+00:00"Bityurin, Anatoliĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:bityurin.anatolii-aleksandrovich"Manzhosov, Vladimir Kuz'mich"https://zbmath.org/authors/?q=ai:manzhosov.vladimir-kuzmichSummary: Mathematical modelling of longitudinal elastic central blow of non-uniform rod system about a rigid barrier is carried out, at not keeping communications by the analytical decision of the wave differential equation by a method of d'Alambert. The rod system consists of a homogeneous core of constant cross-section section and the step non-uniform core having two homogeneous sites of various length and the area of cross-section sections. Ties with a rigid barrier and between cores are not witholding.On the decay of solutions to a class of Hartree equations.https://zbmath.org/1449.353982021-01-08T12:24:00+00:00"Tarulli, Mirko"https://zbmath.org/authors/?q=ai:tarulli.mirko"Venkov, George"https://zbmath.org/authors/?q=ai:venkov.georgeIn the paper under consideration nonlinear defocusing Schrödinger equations with Hartree-type nonlinearity is studied. The authors prove that global solution of the Cauchy problem for such equation has peculiar decay property. To do this a combination of a localization trick, the nonlinear interaction Morawetz estimate and interpolation is applied. In this way the long-time behavior of the solutions of the problem under consideration in the space \( L^{q}(\mathbb{R}^{d})\) is obtained which leads to the scattering in the energy space.
Reviewer: Angela Slavova (Sofia)Estimations of a differential operator in spectral parameter problems for elliptic equations with discontinuous nonlinearities.https://zbmath.org/1449.352022021-01-08T12:24:00+00:00"Potapov, Dmitriĭ Konstantinovich"https://zbmath.org/authors/?q=ai:potapov.dmitrii-konstantinovichSummary: The basic boundary value problems for semilinear equations of elliptic type with a spectral parameter and discontinuous nonlinearity are considered in a bounded domain with a sufficiently smooth boundary. The parameter values for which the corresponding problem has the nonzero solution are called eigenvalues. The existence of eigenvalue problem solutions for equations of elliptic type with discontinuous nonlinearities is considered in this paper. Estimations of the differential operator are obtained for these problems.The boundary value problem for the loaded equation of mixed parabolic-hyperbolic type in rectangular area.https://zbmath.org/1449.353302021-01-08T12:24:00+00:00"Tarasenko, Anna Valer'evna"https://zbmath.org/authors/?q=ai:tarasenko.a-vSummary: In work necessary and sufficient conditions of uniqueness of the decision of a regional problem for the loaded equation mixed parabolic hyperbolic type in rectangular area are established. The problem decision is constructed in the form of the number sum on own functions of a corresponding one-dimensional problem on own values.A mixed problem for one 3D space analogue of hyperbolic type equation.https://zbmath.org/1449.352922021-01-08T12:24:00+00:00"Dolgopolov, Mikhaĭl Vyacheslavovich"https://zbmath.org/authors/?q=ai:dolgopolov.mikhail-vyacheslavovich"Rodionova, Irina Nikolaevna"https://zbmath.org/authors/?q=ai:rodionova.irina-nikolaevnaSummary: It is well known that differential equations with an operator are used for study of the processes connected with appearances of vibration and other mechanics problems, and also play an essential role in the theory of approximation and mapping. In the present work a unique solution for the mixed problem of the full hyperbolic equation of the third order with constant factors, in a three-dimensional Euclidean space, was obtain with the Riemann method, which then becomes considerably simpler at the expense of integral representation of one of boundary conditions. Owing to this it can be used for statement and a solution of new boundary value problems.Estimates in Morrey-Campanato spaces of a suitable weak solution of the Navier-Stokes equations, satifying an extra-condition.https://zbmath.org/1449.760162021-01-08T12:24:00+00:00"Mauro, Jmmy Alfonso"https://zbmath.org/authors/?q=ai:mauro.jmmy-alfonsoIn this paper the non-stationary Navier-Stokes equations with unit viscosity and zero body force are studied. The regularity of suitable weak solutions of the Cauchy problem satisfying a suitable extra-condition is investigated. To do this the author applies the theory of Morrey-Campanato spaces and obtains suitable estimates.
Reviewer: Angela Slavova (Sofia)Orbital stability of solitary waves to fourth order dispersive equations with quadratic nonlinearity.https://zbmath.org/1449.351162021-01-08T12:24:00+00:00"Kolkovska, Natalia"https://zbmath.org/authors/?q=ai:kolkovska.natalia-t"Dimova, Milena"https://zbmath.org/authors/?q=ai:dimova.milena"Kutev, Nikolaj"https://zbmath.org/authors/?q=ai:kutev.nikolai-dIn this paper fourth order dispersive equation is studied. The authors prove orbital stability of solitary waves of such equation with constant velocity. Investigation of the region of stability is presented in details. The proof is direct without results of Grillakis, Shatach and Strauss's. Moreover, dependence of the stability on the parameters is obtained very precisely.
Reviewer: Angela Slavova (Sofia)On an example of derivative nonlinear Schrödinger equation with \( \mathbb{D}_{2} \) reduction.https://zbmath.org/1449.353942021-01-08T12:24:00+00:00"Gerdjikov, Vladimir"https://zbmath.org/authors/?q=ai:gerdzhikov.vladimir-stefanov"Stefanov, Alexander"https://zbmath.org/authors/?q=ai:stefanov.aleksander-aIn the paper under consideration integrable derivative nonlinear Schrödinger equation (DNLS) are discussed. New integrable DNLS equation is derived by additional \( \mathbb{Z}_{2} \) invariance imposed on the Lax pair. Spectral properties of the \( \mathbb{D}_{2} \)-invariant Lax operator \( \mathbb{L} \) are analyzed. Fundamental analytical solutions are constructed. By means of the dressing Zakharov-Shabat method soliton solutions are obtained for the new system.
Reviewer: Angela Slavova (Sofia)The boson star equation with Hartree type non-linearity: global existence in \(H^{\frac{1}{2}}(\mathbb{R}^2)\).https://zbmath.org/1449.353932021-01-08T12:24:00+00:00"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-s"Shakarov, Boris"https://zbmath.org/authors/?q=ai:shakarov.borisIn this paper the nonlinear boson star equation with Hartree type nonlinearity is studied. The authors prove local and global existence of the Cauchy problem with initial data in the critical space \(H^{\frac12}(\mathbb{R}^2) \). To prove local existence Strichartz estimates and a contraction method are applied. Global existence is obtained by means of harmonic analysis and conservations laws. Moreover, bilinear estimate is proved from which global existence result follows.
Reviewer: Angela Slavova (Sofia)Remark on the global non-existence of semirelativistic equations with non-gauge invariant power nonlinearity with mass.https://zbmath.org/1449.353612021-01-08T12:24:00+00:00"Fujiwara, Kazumasa"https://zbmath.org/authors/?q=ai:fujiwara.kazumasaThis paper deals with Cauchy problem for the semirelativistic equation with gauge invariant power type nonlinearity and mass \( m \geq 0 \). At first the author defines weak solution to such equation and its life span. In a previous paper of the same author global non-existence result in appropriate functional space \( X(T) \) for sufficiently large \( T > 0 \) was proved in the massless case \( m = 0 \). A non-existence of global solution to the equation under consideration and in the case with mass \( m > 0 \) is shown in Proposition 7 - main result of this paper. As a corollary of Proposition 7 small data blow up with small mass \( m > 0 \) is obtained in Corollary 3.
Reviewer: Petar Popivanov (Sofia)Modified FVK model.https://zbmath.org/1449.354112021-01-08T12:24:00+00:00"Del Corso, Giulio"https://zbmath.org/authors/?q=ai:del-corso.giulio"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-sThe paper under consideration deals with a generalization of Föppl-von Kármán (FVK) model for elastic plate depending on the real parameter \( \sigma \). \( \sigma \) participates in the second equation of the system as \( (- \Delta)^{\sigma} \). The energy of this system is conserved quantity. In the first part of the work the case \( \sigma =1 \) is studied and global wellposedness of the Cauchy problem for small initial data is proved. To do this Strichartz estimates for vibrating plate are applied and several properties of Riesz transform are used. This result is generalized in the second part of the work to a perturbed version of the original case \( \sigma = 2 \) by using the Yukawa (or Bessel) potential \( Y(x) \).
Reviewer: Petar Popivanov (Sofia)Existence of classical solutions of linear non-cooperative elliptic systems.https://zbmath.org/1449.352102021-01-08T12:24:00+00:00"Boyadzhiev, Georgi"https://zbmath.org/authors/?q=ai:bojadziev.georgi"Kutev, Nikolaj"https://zbmath.org/authors/?q=ai:kutev.nikolai-dLet \( \Omega \subset \mathbb{R}^n \) be a bounded domain with smooth boundary \( \partial \Omega \). This paper deals with the classical solvability in \( C^{2}(\Omega) \cap C(\bar{\Omega}) \) of the weakly coupled linear elliptic system. To do this the authors prove at first the validity of the comparison principle of such system and then apply the method of super and subsolutions. The existence of classical solution is shown in Theorem 1 for competitive elliptic systems. The existence result for general non-cooperative systems is proved in Theorem 2 and is based on Theorem 1 and the comparison principle.
Reviewer: Petar Popivanov (Sofia)Recent developments of the methodology of the modified method of simplest equation with application.https://zbmath.org/1449.351712021-01-08T12:24:00+00:00"Vitanov, Nikolay"https://zbmath.org/authors/?q=ai:vitanov.nikolay-v|vitanov.nikolay-kIn this paper modified method of simplest equations is applied for obtaining exact solutions of nonlinear partial differential equations. The algorithm of this extension is give schematically in seven steps and is compared with the previous version of the method. The new presented technique is based on the possibility of use of more than one simplest equation which allows obtaining of multisolitons. Two examples are provided in order to illustrate the obtained results - bisoliton solution of the Kortweg-de-Vries equation and generalized Maxwell-Cataneo equation.
Reviewer: Angela Slavova (Sofia)Note for global existence of semilinear heat equation in weighted \(L^\infty\) space.https://zbmath.org/1449.352602021-01-08T12:24:00+00:00"Fujiwara, Kazumasa"https://zbmath.org/authors/?q=ai:fujiwara.kazumasa"Georgiev, Vladimir"https://zbmath.org/authors/?q=ai:georgiev.vladimir-s"Ozawa, Tohru"https://zbmath.org/authors/?q=ai:ozawa.tohruThis paper deals with the local and global existence of the Cauchy problem for semilinear heat equation with small data \(u_0(x)\) and nonlinearity \(F(u) =|u|^p\) or \(|u|^{p-1}\) with \(p > p_F = 1 + \frac{2}{n}\), \(0 \leq t < T\). \( u_0\) has a singularity localized near \(0\) and such that \(u_0 \in |x|^{-k}L^{\infty} \), \( 0 \leq k \leq \frac{2}{p-1} \). The main result here claims that if \(|u_0|\) is sufficiently small and \( k = \frac{2}{p-1} \) then there exists a unique global solution \( u \) of the Cauchy problem and \( |u(t,x)| \leq C(t+|x|^{2})^{- \frac{1}{p-1}}, 0 < C \) being sufficiently small. if \( k < \frac{2}{p-1} \) we have a local solution for \( (t,x) \in [0,T) \times \mathbb{R}^n\) with some \(T > 0\).
Reviewer: Petar Popivanov (Sofia)Tear film dynamics with evaporation, osmolarity and surfactant transport.https://zbmath.org/1449.920092021-01-08T12:24:00+00:00"Siddique, J. I."https://zbmath.org/authors/?q=ai:siddique.javed-i"Braun, R. J."https://zbmath.org/authors/?q=ai:braun.richard-jSummary: In this article we develop a model for the evaporation and rupture of the tear film. The tear film is generally considered a multi-layer structure which we simplify to a single layer in our modeling. We examine how well the floating lipid layer can be approximated by a mobile insoluble surfactant monolayer in the context of lubrication theory with film rupture ``breakup'' in the tear film literature. This model includes the effects of surface tension, insoluble surfactant monolayer transport, solutal Marangoni effects, evaporation, osmolarity transport, osmosis and wettability of corneal surface. Evaporation is hypothesized to be dependent on pressure, temperature and surface concentration at the surface of the film. A focus of this paper is to study the competition between the effect of increasing surfactant concentration to (1) slowing down evaporation and (2) lowering surface tension. The solutal Marangoni effect, for local increases in surfactant concentration, can induce local thinning and this effect always seems to dominate the reduction in thinning rate due to evaporation in our model. It also seems to eliminate any localized area of increased evaporation due to reduced surfactant concentration. Osmolarity in the tear film increases because water lost to the average evaporation rate and to a lesser extent by flow inside the film. The presence of van der Waals conjoining pressure is only significant when osmosis is very small or absent. The model predicts that the Marangoni effect coupled with evaporation can determine the location of first breakup; it also agrees with another model of breakup that predicts elevated osmolarity when breakup occurs.On a homogenous thermoconvection model of the non-compressible viscoelastic Kelvin-Voight fluid of the non-zero order.https://zbmath.org/1449.354572021-01-08T12:24:00+00:00"Sukacheva, Tamara Gennad'evich"https://zbmath.org/authors/?q=ai:sukacheva.tamara-gennadevich"Matveeva, Ol'ga Pavlovna"https://zbmath.org/authors/?q=ai:matveeva.olga-pavlovnaSummary: The homogeneous thermoconvection problem of the non-compressible viscoelastic Kelvin-Voight fluid of the non-zero order is considered. The conducted research is based on the results of the semilinear Sobolev type equations theory, because the first initial value problem for the corresponding system of the differential equations in private derivatives is reduced to the abstract Cauchy problem for the specified equation. The concepts of the \(p\)-sectorial operator and the resolving semigroup of operators of the Cauchy problem for the corresponding linear homogeneous Sobolev type equation are used. The existence and uniqueness theorem of the solution which is a quasi-stationary semi-trajectory is proved. The complete description of the phase space is obtained.An efficient numerical approach for solving nonlinear coupled hyperbolic partial differential equations with nonlocal conditions.https://zbmath.org/1449.652412021-01-08T12:24:00+00:00"Bhrawy, A. H."https://zbmath.org/authors/?q=ai:bhrawy.ali-h"Alghamdi, M. A."https://zbmath.org/authors/?q=ai:alghamdi.mohammed-ali"Alaidarous, Eman S."https://zbmath.org/authors/?q=ai:alaidarous.eman-sSummary: One of the most important advantages of collocation method is the possibility of dealing with nonlinear partial differential equations (PDEs) as well as PDEs with variable coefficients. A numerical solution based on a Jacobi collocation method is extended to solve nonlinear coupled hyperbolic PDEs with variable coefficients subject to initial-boundary nonlocal conservation conditions. This approach, based on Jacobi polynomials and Gauss-Lobatto quadrature integration, reduces solving the nonlinear coupled hyperbolic PDEs with variable coefficients to a system of nonlinear ordinary differential equation which is far easier to solve. In fact, we deal with initial-boundary coupled hyperbolic PDEs with variable coefficients as well as initial-nonlocal conditions. Using triangular, soliton, and exponential-triangular solutions as exact solutions, the obtained results show that the proposed numerical algorithm is efficient and very accurate.Simulation of heat waves in an nonlinear anisotropic space.https://zbmath.org/1449.800032021-01-08T12:24:00+00:00"Formalëv, Vladimir Fedorovich"https://zbmath.org/authors/?q=ai:formalev.vladimir-fedorovich"Selin, Il'ya Aleksandrovich"https://zbmath.org/authors/?q=ai:selin.ilya-aleksandrovich"Kuznetsova, Ekaterina L'vovna"https://zbmath.org/authors/?q=ai:kuznetsova.ekaterina-lvovnaSummary: The authors obtain for the first time an analytical solution of the problem with boundary conditions in the non-linear anisotropic space for the quasilinear parabolic heat equation, where heat conductivity tensor's components are temperature functions. This solution shows a wave type of the heat diffusion. Wave's fronts are elliptical in the anisotropic space at different time moments. The existence of the solution in wave's front and the time derivative and space derivatives different orders is analysed depending on the power of temperature which influences the heat conductivity tensor's components.Two dimension spatial pattern formation in a coupled autocatalysis system.https://zbmath.org/1449.920552021-01-08T12:24:00+00:00"Li, Zhang"https://zbmath.org/authors/?q=ai:li.zhang"Chen, Weisheng"https://zbmath.org/authors/?q=ai:chen.weishengSummary: This paper addresses the cubic autocatalator kinetics modeling of coupling via diffusion interchange of autocatalyst. By incorporating the effect of two identical cells, each governed by cubic autocatalator kinetics, considering the possibility of the spatiotemporal structures of two dimensional Turing patterns, a new model is proposed. Unlike previous models, the proposed model has two dimensional spatial variation. First, the equations and the local stability are obtained by linearizing about the spatially uniform solutions. It is shown that the necessary condition for the model undergoes bifurcation by using the singular perturbation theory. Next Landau constant and amplitude functions of two dimensional Turing patterns consisting of rhombic arrays of rectangles and hexagonal is obtain by singular perturbations theory. Finally, by the method of computer simulation of the model, we describe two different patterns.Cauchy problem for fractional nonlinear equation in defined class with local-nonlocal setting.https://zbmath.org/1449.354312021-01-08T12:24:00+00:00"Chadaev, V. A."https://zbmath.org/authors/?q=ai:chadaev.v-aSummary: Theorem of existence and uniqueness of Cauchy problem solution for fractional nonlinear equation with local-nonlocal setting is proved.Non-classic 3D Goursat problem for one hyperbolic equation with discontinuous coefficients.https://zbmath.org/1449.354722021-01-08T12:24:00+00:00"Mamedov, Il'gar Gurdam"https://zbmath.org/authors/?q=ai:mamedov.ilgar-gurdamSummary: For a differential equation of hyperbolic type with discontinuous coefficients a 3D Goursat problem with nonclassical boundary conditions is considered, which requires no matching conditions. Equivalence of these conditions boundary condition is substantiated classical, in the case if the solution of the problem in the anisotropic S. L. Sobolev's space is found.On solvability of a certain nonlocal problem for a Laplace equation.https://zbmath.org/1449.352322021-01-08T12:24:00+00:00"Beĭlina, Natal'ya Viktorovna"https://zbmath.org/authors/?q=ai:beilina.natalya-viktorovnaSummary: In this article, we prove the solvability of a nonlocal problem with integral conditions for the Laplace equation in a rectangular domain.Mathematical model of copper corrosion.https://zbmath.org/1449.741652021-01-08T12:24:00+00:00"Clarelli, F."https://zbmath.org/authors/?q=ai:clarelli.fabrizio"De Filippo, B."https://zbmath.org/authors/?q=ai:de-filippo.b"Natalini, R."https://zbmath.org/authors/?q=ai:natalini.robertoSummary: A new partial differential model for monitoring and detecting copper corrosion products (mainly brochantite and cuprite) is proposed to provide predictive tools suitable for describing the evolution of damage induced on bronze specimens by sulfur dioxide (\(\mathrm{SO}_{2}\)) pollution. This model is characterized by the movement of a double free boundary. Numerical simulations show a nice agreement with experimental result.Pattern formation in two-component reaction-diffusion systems in fluctuate environment.https://zbmath.org/1449.350672021-01-08T12:24:00+00:00"Kurushina, Svetlana Evgen'evna"https://zbmath.org/authors/?q=ai:kurushina.svetlana-evgenevna"Zhelnov, Yuriĭ Valer'evich"https://zbmath.org/authors/?q=ai:zhelnov.yurii-valerevich"Zavershinskiĭ, Igor' Petrovich"https://zbmath.org/authors/?q=ai:zavershinskii.igor-petrovich"Maksimov, Valeriĭ Vladimirovich"https://zbmath.org/authors/?q=ai:maksimov.valerii-vladimirovichSummary: The influence of multiplicative fluctuations of parameters on pattern formation was researched. It was received the system which described interaction of undamped modes when soft mode instability were developed. It was shown that the fluctuations of parameters lead to changing of eigenvalues of unstable modes. The computational modeling of spatial structures evolution was conducted. Changing of fluctuating level of dynamic variables in process of dissipative pattern formation which conditioned by changing of external random field parameters were investigated.Cauchy problem for the nonlocal equation diffusion-advection radon in fractal media.https://zbmath.org/1449.653482021-01-08T12:24:00+00:00"Parovik, Roman Ivanovich"https://zbmath.org/authors/?q=ai:parovik.roman-ivanovichSummary: In this article, using the Green's function method solved the Cauchy problem for the equation of anomalous diffusion-advection of radon in a fractal medium, which is represented by a fractional derivative of the Caputo time fractional derivative and Riesz-Weil on the spatial coordinate.Global existence and blow-up for a two-dimensional attraction-repulsion chemotaxis system.https://zbmath.org/1449.350012021-01-08T12:24:00+00:00"Xiao, Min"https://zbmath.org/authors/?q=ai:xiao.min"Li, Zhongping"https://zbmath.org/authors/?q=ai:li.zhongpingSummary: This paper is devoted to dealing with the parabolic-elliptic-elliptic attraction-repulsion chemotaxis system. We aim to understand the competition among the repulsion, the attraction, the nonlinear productions and give conditions of global existence and blow-up for the two-dimensional attraction-repulsion chemotaxis system.Boundary integral equation method for holomorphic vector for problems monitoring elasticity field.https://zbmath.org/1449.354172021-01-08T12:24:00+00:00"Schwab, Al'bert Aleksandrovich"https://zbmath.org/authors/?q=ai:schwab.albert-aleksandrovichSummary: Consideration is given to a class of non-classical problems in elasticity theory which concerns the restoration of a full tensor of stresses in a body in the case where for one part of the body surface the loading vector and the displacement vector are prescribed, for other parts vector of displacements or vector of loads is known and for some parts of the body surface (its measure is not equal zero) the conditions are unknown. For solving of this problem boundary integral equations method for the holomorphic vector is proposed. It is shown that this method is efficient and can be applied to the non-classical problems of mechanics.Approximate solution of time-fractional fuzzy partial differential equations.https://zbmath.org/1449.652942021-01-08T12:24:00+00:00"Senol, Mehmet"https://zbmath.org/authors/?q=ai:senol.mehmet"Atpinar, Sevda"https://zbmath.org/authors/?q=ai:atpinar.sevda"Zararsiz, Zarife"https://zbmath.org/authors/?q=ai:zararsiz.zarife"Salahshour, Soheil"https://zbmath.org/authors/?q=ai:salahshour.soheil"Ahmadian, Ali"https://zbmath.org/authors/?q=ai:ahmadian.aliSummary: In this study, we develop perturbation-iteration algorithm (PIA) for numerical solutions of some types of fuzzy fractional partial differential equations (FFPDEs) with generalized Hukuhara derivative. We also present the convergence analysis of the method. The proposed approach reveals fast convergence rate and accuracy of the present method when compared with exact solutions of crisp problems. The main efficiency of this method is that while scaling support zone of uncertainty for the fractional partial differential equations, it eliminates over calculation and produces highly approximate and accurate results. Error analysis of the PIA for the FFPDEs is also illustrated within examples.Problem with conjugation on the characteristic plane for one 3D space analogue of hyperbolic type equation.https://zbmath.org/1449.351822021-01-08T12:24:00+00:00"Dolgopolov, Vyacheslav Mikhaĭlovich"https://zbmath.org/authors/?q=ai:dolgopolov.vyacheslav-mikhailovich"Rodionova, Irina Nikolaevna"https://zbmath.org/authors/?q=ai:rodionova.irina-nikolaevnaSummary: The value boundary problem with conjugation together on the characteristic plane in the special class, entered by authors, is solved for the one 3D space analogue of the hyperbolic type equation in the area, which parts of border are the planes of singularity of the given equation factors.Hoff equation stability on a graph.https://zbmath.org/1449.350592021-01-08T12:24:00+00:00"Sviridyuk, Georgiĭ Anatol'evich"https://zbmath.org/authors/?q=ai:sviridyuk.georgii-anatolevich"Zagrebina, Sof'ya Aleksandrovna"https://zbmath.org/authors/?q=ai:zagrebina.sofya-aleksandrovna"Pivovarova, Polina Olegovna"https://zbmath.org/authors/?q=ai:pivovarova.polina-olegovnaSummary: We consider the stability of stationary solutions of the Hoff equation on a graph, which is a model design of I-beams. The basic approach second Lyapunov method, modified according to our situation. In the end explains the technical meaning of the parameter \(\lambda_0\).A blow-up result for a class of doubly nonlinear parabolic equations with variable-exponent nonlinearities.https://zbmath.org/1449.351152021-01-08T12:24:00+00:00"Hu, Qingying"https://zbmath.org/authors/?q=ai:hu.qingying"Li, Donghao"https://zbmath.org/authors/?q=ai:li.donghao"Zhang, Hongwei"https://zbmath.org/authors/?q=ai:zhang.hongweiSummary: This paper deals with the following doubly nonlinear parabolic equations \( (u+{|u|^{r (x)-2}u})_t - {\mathrm{div}} (|\nabla u|^{m (x)-2}\nabla u) = |u|^{p(x)-2}u\), where the exponents of nonlinearity \(r (x)\), \(m (x)\) and \(p(x)\) are given functions. Under some appropriate assumptions on the exponents of nonlinearity, and with certain initial data, a blow-up result is established with positive initial energy.A general filter regularization method to solve the three dimensional Cauchy problem for inhomogeneous Helmholtz-type equations: theory and numerical simulation.https://zbmath.org/1449.653062021-01-08T12:24:00+00:00"Quoc Viet Tran"https://zbmath.org/authors/?q=ai:quoc-viet-tran."Huy Tuan Nguyen"https://zbmath.org/authors/?q=ai:huy-tuan-nguyen."Van Thinh Nguyen"https://zbmath.org/authors/?q=ai:van-thinh-nguyen."Duc Trong Dang"https://zbmath.org/authors/?q=ai:duc-trong-dang.Summary: In this paper, we solve the Cauchy problem for an inhomogeneous Helmholtz-type equation with homogeneous Dirichlet and Neumann boundary condition. The proposed problem is ill-posed. Up to now, most investigations on this topic focus on very specific cases, and with Dirichlet boundary condition. Recently, the second author et al. solve this problem in 2D for an inhomogeneous modified Helmholtz equation [ibid. 37, No. 3, 793--814 (2013; Zbl 1352.65442)]. This work is a continuous expansion of our previous results. Herein we introduce a general filter regularization (GFR) method, and then from the GFR we deduce two concrete filters, which are a foundation to implement a numerical procedure. In addition, we develop a numerical model for solving this problem in three dimensional region. The proposed filter method has been verified by numerical experiments.PDF control of nonlinear stochastic systems based on MGC method.https://zbmath.org/1449.932442021-01-08T12:24:00+00:00"Yang, Hengzhan"https://zbmath.org/authors/?q=ai:yang.hengzhan"Fu, Yueyuan"https://zbmath.org/authors/?q=ai:fu.yueyuan"Gao, Song"https://zbmath.org/authors/?q=ai:gao.song"Qian, Fucai"https://zbmath.org/authors/?q=ai:qian.fucaiSummary: For nonlinear stochastic systems, it is difficult to meet the actual control requirements by taking the low-order statistical characteristics such as mean and variance as the research objects, and the higher order statistical characteristics need to be considered. The probability density function (PDF) contains complete statistical characteristics, therefore, PDF control can achieve effective control of all moments. In this paper, aiming at the nonlinear stochastic system excited by Gaussian white noise, the Fokker-Planck-Kolmogrov (FPK) equation is taken as the research tool, and a PDF control method based on the multi-Gaussian closure (MGC) method is proposed. Firstly, according to the shape of the target PDF, a PDF superimposed by multiple Gaussian PDFs is constructed. Then, an optimization problem is built to make the PDF approximate the target PDF. Furthermore, the state equation of the controlled system is obtained by solving the FPK equation. Finally, the control function is calculated according to the original state equation, and the tracking control of the target PDF is implemented. Simulation results of different target PDFs show the feasibility and the effectiveness of the proposed method.Conditional Lie-Bäcklund symmetries to inhomogeneous nonlinear diffusion equations.https://zbmath.org/1449.352592021-01-08T12:24:00+00:00"Di, Yanmei"https://zbmath.org/authors/?q=ai:di.yanmei"Zhang, Danda"https://zbmath.org/authors/?q=ai:zhang.danda"Shen, Shoufeng"https://zbmath.org/authors/?q=ai:shen.shoufeng"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.3Summary: In this paper, conditional Lie-Bäcklund symmetry method is used to classify a class of inhomogeneous nonlinear diffusion equations \(u_t = e^{-{qx}} \left(e^{{px}} w(u) u_x\right)_x\). Equations admitted conditional Lie-Bäcklund symmetries can be either solved exactly or reduced to finite-dimensional dynamical systems. A number of concrete examples defined on the exponential and trigonometric invariant subspaces are considered to illustrate this method.Asymptotic stability of self-similar solutions for dissipative systems modeling electrohydrodynamics.https://zbmath.org/1449.353572021-01-08T12:24:00+00:00"Zhao, Jihong"https://zbmath.org/authors/?q=ai:zhao.jihong"Li, Xiurong"https://zbmath.org/authors/?q=ai:li.xiurongSummary: The authors consider a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics in the whole space \({\mathbb{R}^n}\), \(n \ge 3\). By making use of the generalized \({L^p}\)-\({L^q}\) heat semigroup estimates in the Lorentz spaces and the generalized Hardy-Littlewood-Sobolev inequality, the authors first prove the global existence and uniqueness of the self-similar solutions in the Lorentz spaces, then establish the asymptotic stability of the self-similar solutions as time goes to infinity. Since the authors cope with the initial data in the Lorentz spaces, the global existence and asymptotic stability of the self-similar solutions corresponding to the initial data are small homogeneous functions.An optimal sixth-order finite difference scheme for the Helmholtz equation in one-dimension.https://zbmath.org/1449.652992021-01-08T12:24:00+00:00"Liu, Xu"https://zbmath.org/authors/?q=ai:liu.xu"Wang, Haina"https://zbmath.org/authors/?q=ai:wang.haina"Hu, Jing"https://zbmath.org/authors/?q=ai:hu.jingSummary: In this paper, we present an optimal 3-point finite difference scheme for solving the 1D Helmholtz equation. We provide a convergence analysis to show that the scheme is sixth-order in accuracy. Based on minimizing the numerical dispersion, we propose a refined optimization rule for choosing the scheme's weight parameters. Numerical results are presented to demonstrate the efficiency and accuracy of the optimal finite difference scheme.On a novel modification of the Legendre collocation method for solving fractional diffusion equation.https://zbmath.org/1449.354422021-01-08T12:24:00+00:00"Jaleb, Hosein"https://zbmath.org/authors/?q=ai:jaleb.hosein"Adibi, Hojatollah"https://zbmath.org/authors/?q=ai:adibi.hojatollahSummary: In this paper, a modification of the Legendre collocation method is used for solving the space fractional differential equations. The fractional derivative is considered in the Caputo sense along with the finite difference and Legendre collocation schemes. The numerical results obtained by this method have been compared with other methods. The results show the capability and efficiency of the proposed method.A nearly analytic discrete method for one-dimensional unsteady convection-dominated diffusion equations.https://zbmath.org/1449.652902021-01-08T12:24:00+00:00"Kim, Yonchol"https://zbmath.org/authors/?q=ai:kim.yonchol"Yun, Nam"https://zbmath.org/authors/?q=ai:yun.nam-yeol"Chai, Dongho"https://zbmath.org/authors/?q=ai:chai.donghoSummary: In this paper, a nearly analytic discretization method for one-dimensional linear unsteady convection-dominated diffusion equations and viscous Burgers' equation as one of the nonlinear equation is considered. In the case of linear equations, we find that the local truncation error of the scheme is \(O ({\tau^2} + {h^4})\) and consider the stability analysis of the method on the basis of the classical von Neumann's theory. In addition, the nearly analytic discretization method for the one-dimensional viscous Burgers' equation is also constructed. The numerical experiments are performed for several benchmark problems presented in some literatures to illustrate the theoretical results. Theoretical and numerical results show that our method is to be higher accurate and nonoscillatory and might be helpful particularly in computations for the unsteady convection-dominated diffusion problems.The Hamiltonian structures and algebro-geometric solution of the generalized Kaup-Newell soliton equations.https://zbmath.org/1449.353692021-01-08T12:24:00+00:00"Wei, Hanyu"https://zbmath.org/authors/?q=ai:wei.hanyu"Pi, Guomei"https://zbmath.org/authors/?q=ai:pi.guomeiSummary: Starting from a new spectral problem, a hierarchy of the generalized Kaup-Newell soliton equations is derived. By employing the trace identity, their Hamiltonian structures are also generated. Then, the generalized Kaup-Newell soliton equations are decomposed into two systems of ordinary differential equations. The Abel-Jacobi coordinates are introduced to straighten the flows, from which the algebro-geometric solutions of the generalized Kaup-Newell soliton equations are obtained in terms of the Riemann theta functions.Application of the invariant subspace method in conjunction with the fractional Sumudu's transform to a nonlinear conformable time-fractional dispersive equation of the fifth order.https://zbmath.org/1449.354382021-01-08T12:24:00+00:00"Hosseini, Kamyar"https://zbmath.org/authors/?q=ai:hosseini.kamyar"Ayati, Zainab"https://zbmath.org/authors/?q=ai:ayati.zainab"Ansari, Reza"https://zbmath.org/authors/?q=ai:ansari.rezaSummary: During the past years, a wide range of distinct approaches has been developed to solve nonlinear fractional differential equations (NLFDEs). In this paper, the invariant subspace method (ISM) in conjunction with the fractional Sumudu's transform (FST) in the conformable context is formally adopted to deal with a nonlinear conformable time-fractional dispersive equation of the fifth order. As an outcome, a new exact solution of the model is determined.Exponential decay in a Timoshenko-type system of thermoelasticity of type III.https://zbmath.org/1449.350782021-01-08T12:24:00+00:00"Qin, Yuming"https://zbmath.org/authors/?q=ai:qin.yuming"Liu, Zili"https://zbmath.org/authors/?q=ai:liu.ziliSummary: In this work, a Timoshenko system of type III of thermoelasticity with frictional versus viscoelastic under Dirichlet-Dirichlet-Neumann boundary conditions was considered. By exploiting energy method to produce a suitable Lyapunov functional, we establish the global existence and exponential decay of type-III case.Qualitative analysis of a class of predator-prey model with fear effect.https://zbmath.org/1449.352712021-01-08T12:24:00+00:00"Wang, Rong"https://zbmath.org/authors/?q=ai:wang.rong"Yang, Wenbin"https://zbmath.org/authors/?q=ai:yang.wenbin"Li, Yanling"https://zbmath.org/authors/?q=ai:li.yanlingSummary: The nature of the reaction-diffusion system solution contains abundant information, which is of great significance to the study of population ecological phenomena. In this paper, the existence of positive solutions to the steady-state system of a predator-prey model with fear effect is studied. Firstly, some priori estimates of positive constant steady-state solution are obtained using the maximum principle, which lay the foundation for the subsequent research. Secondly, the sufficient condition of the unique existence for the equilibrium solution is given, and the stability of positive constant steady-state solution is discussed using the stability theory of linear operators. Finally, on the basis of the degree theory, the existence of non-constant steady-state solution is given.A new moving mesh method for solving the two-dimensional Navier-Stokes equation.https://zbmath.org/1449.653412021-01-08T12:24:00+00:00"Duan, Xianbao"https://zbmath.org/authors/?q=ai:duan.xianbao"Cao, Qinqin"https://zbmath.org/authors/?q=ai:cao.qinqin"Tan, Hongxia"https://zbmath.org/authors/?q=ai:tan.hongxiaSummary: In order to reduce the computational cost of solving partial differential equation (PDE), whose solution has strong singularity or drastic change in a small local area, a moving mesh method based on equation solution is proposed and applied to solve the two-dimensional incompressible Navier-Stokes equations. Different from the most existing moving mesh methods, the moving distance of the nodes is obtained by solving a variable-coefficient diffusion equation, which avoids regional mapping and does not need to smooth the monitoring function, so the algorithm is easier to program and implement. Numerical examples show that the proposed algorithm can refine the mesh in the position where the gradient of the solution changes drastically, which can save a lot of computation time on the premise of improving the resolution of the numerical solution. Due to the typicality of the Navier-Stokes equations, the proposed algorithm can be generalized to solve many similar partial differential equations numerically.A finite element variational multiscale method based on Crank-Nicolson scheme for the unsteady Navier-Stokes equations.https://zbmath.org/1449.652642021-01-08T12:24:00+00:00"Xue, Jufeng"https://zbmath.org/authors/?q=ai:xue.jufeng"Shang, Yueqiang"https://zbmath.org/authors/?q=ai:shang.yueqiangSummary: The incompressible viscous flows are fluid movements that do not change in density. They are used to describe many important physical phenomena such as weather, ocean currents, flow around airfoil, and blood flow within the arteries. The Navier-Stokes equations are the basic equations for incompressible viscous flows. Therefore, the numerical method for solving Navier-Stokes equations has been paid more and more attention in recent decades. In this paper, we mainly study a two-level fully discrete finite element variational multiscale method based on Crank-Nicolson scheme for the unsteady Navier-Stokes equations. The method is carried out in two steps. A stabilized nonlinear Navier-Stokes system is solved on a coarse grid at the first step, and the second step is that a stabilized linear problem is solved on a fine grid to correct the coarse grid solution. Error estimate of the velocity which is derived via the two-level finite element variational multiscale method is of second-order in time. Numerical experiments show that the method of this paper can save a lot of computation time compared with the finite element variational method which uses a one-level grid directly on the fine grid in the case of coarse grid matching.Wind stability finite point method for soil solute transport equations.https://zbmath.org/1449.652552021-01-08T12:24:00+00:00"Qin, Xinqiang"https://zbmath.org/authors/?q=ai:qin.xinqiang"Su, Lijun"https://zbmath.org/authors/?q=ai:su.lijun"Wang, Xing"https://zbmath.org/authors/?q=ai:wang.xing"Li, Yongzhen"https://zbmath.org/authors/?q=ai:li.yongzhen"Wang, Yueling"https://zbmath.org/authors/?q=ai:wang.yuelingSummary: A finite point method for windward stability of soil solute transport equation is proposed. This method adopts an adaptive windward format to make its support field lean to the windward side, so it can obtain upstream information and avoid numerical oscillation when convection is dominant. The convergence, the order of convergence and the stability of the numerical solution of the new algorithm are analyzed in detail under the action of different distribution points, different time steps and different influence factors through the numerical calculation of the one-dimensional and two-dimensional soil solute transport equation. Numerical results show that the proposed method can effectively improve the computational accuracy and eliminate the numerical oscillation in the boundary region and gradient variations large region.Three positive solutions of \(N\)-dimensional \(p\)-Laplacian with indefinite weight.https://zbmath.org/1449.352382021-01-08T12:24:00+00:00"Chen, Tianlan"https://zbmath.org/authors/?q=ai:chen.tianlan"Ma, Ruyun"https://zbmath.org/authors/?q=ai:ma.ruyunSummary: This paper is concerned with the global behavior of components of positive radial solutions for the quasilinear elliptic problem with indefinite weight \[\text{div}(\varphi_p(\nabla u))+\lambda h(x)f(u)=0, \; \text{in}\ \; B,\] \[u=0, \; \text{on}\; \partial B,\] where \(\varphi_p(s)=\vert s\vert ^{p-2}s\), \(B\) is the unit open ball of \(\mathbb{R}^N\) with \(N\geq1\), \(1<p<\infty\), \(\lambda>0\) is a parameter, \(f\in C([0, \infty), [0, \infty))\) and \(h\in C(\bar{B})\) is a sign-changing function. We manage to determine the intervals of \(\lambda\) in which the above problem has one, two or three positive radial solutions by using the directions of a bifurcation.The generalized solution for transitional shock layer to singularly perturbed nonlinear hyperbolic type differential system with two parameters.https://zbmath.org/1449.352912021-01-08T12:24:00+00:00"Han, Xianglin"https://zbmath.org/authors/?q=ai:han.xianglin"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqiSummary: A class of nonlinear hyperbolic type differential system to singularly perturbed initial-boundary problem with two parameters is studied. Firstly, using singular perturbation theory and method, the outer solution for the problem is structured related to two small parameters. Secondly, using the multi-scale and stretched variables, the transitional shock layer, boundary layer and initial layer corrective terms are obtained for the original problem respectively. Finally, the asymptotic expansion of solution for the original problem is given. The uniform validity of its asymptotic solution is proved by using the theory of fixed point of functional analysis. The method which is used to obtain the asymptotic solution of the original problem, can also carry on analytical operation for the differential and integral and so on. More behaviors of the transitional shock layer of the solution are known. It is proved that this method has a good application prospect.The solution of soliton to generalized higher dimensions Klein-Gordon forced disturbed equation.https://zbmath.org/1449.351532021-01-08T12:24:00+00:00"Han, Xianglin"https://zbmath.org/authors/?q=ai:han.xianglin"Wang, Weigang"https://zbmath.org/authors/?q=ai:wang.weigang"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqiSummary: A class of nonlinear generalized forced disturbed Klein-Gordon equation is considered by using the homotopic mapping method. Firstly, an approximate solution of soliton to typical nonlinear equation is solved using the method of undetermined coefficients for the hyperbolic tangent functions. Then, the approximate solution of soliton to nonlinear forced disturbed equation is obtained using the homotopic mapping principle. Finally, it is pointed out that the approximate solution of soliton is an analytic expression, so we can carry on analytic operation to it. However, one can not obtain the approximate solution by using the simple simulated method.The non-Fourier effect on the fin performance under periodic thermal conditions.https://zbmath.org/1449.800142021-01-08T12:24:00+00:00"Lin, Jae-Yuh"https://zbmath.org/authors/?q=ai:lin.jaeyuhSummary: This paper investigates the effect of thermal relaxation time (the non-Fourier effect) on the thermal performance of a convective fin under periodic thermal conditions. The environment heat transfer coefficient is assumed to be spatially varying. The periodic oscillation of the base temperature is considered. An efficient numerical scheme involving the hybrid application of the Laplace transform and control volume methods in conjunction with hyperbolic shape functions is used to solve the linear hyperbolic heat conduction equation. The thermal performance of the fin predicted by using the non-Fourier heat conduction model is compared with that predicted by using the Fourier model. Results show that the effects of the thermal relaxation time on the fin performance are significant for a short time after the initial transient. For the steady state, the non-Fourier effect is still great if the frequency of the temperature oscillation is high.q-homotopy analysis method for solving the seventh-order time-fractional Lax's Korteweg-de Vries and Sawada-Kotera equations.https://zbmath.org/1449.354272021-01-08T12:24:00+00:00"Akinyemi, Lanre"https://zbmath.org/authors/?q=ai:akinyemi.lanreSummary: This article presents exact and approximate solutions of the seventh order time-fractional Lax's Korteweg-de Vries (7TfLKdV) and Sawada-Kotera (7TfSK) equations using the modification of the homotopy analysis method called the q-homotopy analysis method. Using this method, we construct the solutions to these problems in the form of recurrence relations and present the graphical representation to verify all obtained results in each case for different values of fractional order. Error analysis is also illustrated in the present investigation.Liouville correspondence between the short-wave model of Novikov hierarchy and the Sawada-Kotera hierarchy.https://zbmath.org/1449.353832021-01-08T12:24:00+00:00"Kang, Ting"https://zbmath.org/authors/?q=ai:kang.ting"Guo, Xu"https://zbmath.org/authors/?q=ai:guo.xu"Guo, Mingyue"https://zbmath.org/authors/?q=ai:guo.mingyue"Shi, Zhenhua"https://zbmath.org/authors/?q=ai:shi.zhen-huaSummary: In this paper, we study an explicit correspondence between the integrable short-wave model of Novikov hierarchy and the Sawada-Kotera hierarchy. A Liouville transformation between the isospectral problems of short-wave model of Novikov and Sawada-Kotera equation is used to link their respective recursion operators, and thus the one-to-one correspondence between each pair of integrable equations and each pair of Hamiltonian conservation laws is established.Highly oscillatory behavior of solutions of space fractional semi-classical Schrödinger equation.https://zbmath.org/1449.350152021-01-08T12:24:00+00:00"Sun, Suzhen"https://zbmath.org/authors/?q=ai:sun.suzhen"Wang, Dongling"https://zbmath.org/authors/?q=ai:wang.donglingSummary: In this paper, we mainly use strong splitting Fourier spectral method to study highly oscillatory behavior of solutions of space fractional semi-classical Schrödinger equation. Compared with the highly oscillatory behavior of integer order, it turns out that this highly oscillatory behavior also exists in fractional Schrödinger equations as well, and the order of space fractional Laplacian operator has a direct influence on the highly oscillatory behavior of the solution in some cases.Application of the SBA method to solve the nonlinear biological population models.https://zbmath.org/1449.652912021-01-08T12:24:00+00:00"Mayembo, Stevy Mikamona"https://zbmath.org/authors/?q=ai:mayembo.stevy-mikamona"Bonazebi-Yindoula, Joseph"https://zbmath.org/authors/?q=ai:bonazebi-yindoula.joseph"Paré, Youssouf"https://zbmath.org/authors/?q=ai:pare.youssouf"Bissanga, Gabriel"https://zbmath.org/authors/?q=ai:bissanga.gabrielSummary: This paper presents numerical solutions of some nonlinear degenerate parabolic equations arising in the spatial diffusion of biological populations. The SBA method based on a combination of the Adomian decomposition method, the principle of Picard and successive approximations is used for solving these equations. The obtained analytical solutions show that the SBA method leads to more accurate results.A spectral method for fourth-order boundary value problems.https://zbmath.org/1449.352082021-01-08T12:24:00+00:00"Bouarroudj, Nadra"https://zbmath.org/authors/?q=ai:bouarroudj.nadra"Belaib, Lekhmissi"https://zbmath.org/authors/?q=ai:belaib.lekhmissi"Messirdi, Bekkai"https://zbmath.org/authors/?q=ai:messirdi.bekkaiSummary: Boundary-value problems for fourth-order partial differential equations are studied in this paper; more precisely, vibrational phenomena of plates in an incompressible non-viscous fluid along the edge are mathematically analyzed. The spectral method via the variational formulation is used to prove existence, uniqueness and regularity theorems for the strong solution. We discuss also a discrete variational formulation for the considered problem.Some problems in radiation transport fluid mechanics and quantum fluid mechanics.https://zbmath.org/1449.353512021-01-08T12:24:00+00:00"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.boling"Wu, Jun"https://zbmath.org/authors/?q=ai:wu.jun.1|wu.jun.2|wu.junSummary: We introduce the radiation transport equations, the radiation fluid mechanics equations and the fluid mechanics equations with quantum effects. We obtain the unique global weak solution for the radiation transport fluid mechanics equations under certain initial and boundary values. In addition, we also obtain that the periodic region problem of the compressible N-S equation with quantum effect has weak solutions under some conditions.Global existence and stability to a prey-taxis model with porous medium diffusion and indirect signal production.https://zbmath.org/1449.350642021-01-08T12:24:00+00:00"Zhang, Limin"https://zbmath.org/authors/?q=ai:zhang.limin"Xu, Haiyan"https://zbmath.org/authors/?q=ai:xu.haiyan"Jin, Chunhua"https://zbmath.org/authors/?q=ai:jin.chunhuaSummary: In this paper, we consider the following prey-taxis model with nonlinear diffusion and indirect signal production \[\begin{cases}{u_t} = \Delta {u^{m_1}} -\chi\nabla \cdot (u\nabla w),\\{w_t} = \Delta w - \mu w + \alpha v{F_0} (u), \\{v_t} = \Delta {v^{m_2}} + \lambda v\left (1-\frac{v}{k}\right)-v{F_0} (u),\end{cases}\] in a bounded domain of \({\mathbb{R}^3}\) with zero-flux boundary condition. It is shown that for any \({m_1} > 1\), \({m_2} > 1\), there exists a global bounded weak solution for any large initial datum. Based on the uniform boundedness property, we also studied the large time behavior of solutions. The global asymptotic stability of the constant steady states was established. More precisely, we showed that when \(\lambda = 0\), \(\alpha \ge 0\), the global weak solution converges to \( ({\bar u}_0, 0, 0)\) in the large time limit. When \(\lambda > 0\), \(\alpha = 0\), the global weak solution converges to \( ({\bar u}_0, 0, 0)\) if \(\lambda < {F_0} ({\bar u})\), and the global weak solution converges to \(\left ({\bar u}_0, 0, k\left (1 - \frac{{F_0} ({\bar u})}{\lambda}\right)\right)\) if \(\lambda > {F_0} ({\bar u})\).Singularity and decay of solutions for a degenerate semilinear elliptic equation.https://zbmath.org/1449.352252021-01-08T12:24:00+00:00"Li, Dongyan"https://zbmath.org/authors/?q=ai:li.dongyan"Dong, Yan"https://zbmath.org/authors/?q=ai:dong.yanSummary: In this paper, we establish a singularity and decay of solutions for a degenerate semilinear elliptic equation based on re-scaling arguments combined with a doubling property. As an application, we derive a priori bounds of solutions of a boundary value problem.Existence and uniqueness of global solution for the treatment model of cancer vaccine and checkpoint inhibitor.https://zbmath.org/1449.352692021-01-08T12:24:00+00:00"Liu, Chunyan"https://zbmath.org/authors/?q=ai:liu.chunyan"Wei, Xuemei"https://zbmath.org/authors/?q=ai:wei.xuemei"Feng, Zhaoyong"https://zbmath.org/authors/?q=ai:feng.zhaoyong"Liu, Chengxia"https://zbmath.org/authors/?q=ai:liu.chengxiaSummary: A combined treatment model of cancer vaccine and checkpoint inhibitor, which contains nine coupled reaction diffusion equations, is studied. By using the Banach fixed point theorem and \({L^p}\)-theory of parabolic equation, the existence and uniqueness of the model in the local solution are proved. And then the existence and uniqueness of the global solution are obtained by using the extension method.BKM's blow-up criterion in homogeneous Triebel-Lizorkin spaces for the 3D dissipative system in electro-hydrodynamics.https://zbmath.org/1449.351212021-01-08T12:24:00+00:00"Li, Xiurong"https://zbmath.org/authors/?q=ai:li.xiurong"Liang, Hong"https://zbmath.org/authors/?q=ai:liang.hongSummary: In this paper, we study the breakdown of local smooth solutions for a class of nonlinear dissipative electro-hydrodynamics system. This system is a strongly coupled system by the well-known incompressible Navier-Stokes equations in hydromechanics and Poisson-Nernst-Planck equations in electrodynamics, modeling the drift, diffusion and convection phenomena of charged particle in an isothermal, incompressible and viscous fluids. Based on the Littlewood-Paley decomposition theory, the BKM's blow-up criterion in terms of horizontal component of velocity field in homogeneous Triebel-Lizorkin spaces is established for local smooth solutions, and generalizes the previous results. In particular, this blow-up criterion reveals that the horizontal component of velocity field is more important than the density functions of charged particles in the blow-up theory of the system.The linear instability of the gravity driven viscous Navier-Stokes flow in three-dimension.https://zbmath.org/1449.350612021-01-08T12:24:00+00:00"Yang, Jing"https://zbmath.org/authors/?q=ai:yang.jing"Ge, Qing"https://zbmath.org/authors/?q=ai:ge.qing"Li, Ji'na"https://zbmath.org/authors/?q=ai:li.jinaSummary: We investigate the instability of some steady-states of a three-dimensional nonhomogeneous incompressible viscous flow driven by gravity. When the steady density is heavier with increasing height (i.e., the Rayleigh-Taylor steady-state), we show that the steady-state is linear unstable by constructing a (standard) energy functional and exploiting the modified variational method.The new method of solving solutions of the \( (3+1)\)-dimension Klein-Gordon equation.https://zbmath.org/1449.353852021-01-08T12:24:00+00:00"Taogetusang"https://zbmath.org/authors/?q=ai:taogetusang."Yi, Lina"https://zbmath.org/authors/?q=ai:yi.linaSummary: A method combing the first kind of elliptic equation with the function transformation is presented. By some steps, many kinds of new solutions of the \( (3+1)\)-dimension Klein-Gordon equation are constructed. Step1, according to the nature of the Jacobi elliptic function, some kinds of new solutions of the first kind of elliptic equation are obtained. Step2, by the method combing the first kind of elliptic equation with the function transformation, the problem of solving solutions of the \( (3+1)\)-dimension Klein-Gordon equation is changed to the problem of solving solutions of the nonlinear algebraic equation. Step3, with the help of the symbol calculation system Mathematica, the solutions of the equation set are solved. Some new compound solutions, such as double periodic solutions and double soliton solutions, are constructed by the combination of Riemann \(\theta\) function, Jacobi elliptic function, hyperbolic function and trigonometric function.The small initial value problem for the fourth order parabolic equation under the weighted space condition.https://zbmath.org/1449.352532021-01-08T12:24:00+00:00"Zhao, Jianfeng"https://zbmath.org/authors/?q=ai:zhao.jianfeng"Wang, Shuying"https://zbmath.org/authors/?q=ai:wang.shuying"Hao, Fang"https://zbmath.org/authors/?q=ai:hao.fang"Su, Xiao"https://zbmath.org/authors/?q=ai:su.xiaoSummary: In this paper, we study a small initial value problem for a class of fourth order parabolic equations in weighted space of \({L^P}\). By means of Sobolev embedding theorem, Fourier transform and symbolic operator and compression mapping, the existence and uniqueness of the small initial value solution and the optimal decay estimate for the four order parabolic equations in the critical state of \(\sigma = \frac{4}{n}\) are proved.Long time behaviors of the solutions to stochastic wave equations with damping.https://zbmath.org/1449.350602021-01-08T12:24:00+00:00"Wang, Suxin"https://zbmath.org/authors/?q=ai:wang.suxinSummary: The long time behavior of the solution to a stochastic wave equation with damping is considered. Under some appropriate conditions, the exponential stability of the solutions holds almost surely. Finally two examples to illustrate the results are given.Traveling wave solutions of the generalized hyperelastic-rod wave equation.https://zbmath.org/1449.351442021-01-08T12:24:00+00:00"Gu, Yongyi"https://zbmath.org/authors/?q=ai:gu.yongyi"Yuan, Wenjun"https://zbmath.org/authors/?q=ai:yuan.wenjun"Wu, Yonghong"https://zbmath.org/authors/?q=ai:wu.yonghong.1Summary: In this paper, we study the generalized hyperelastic-rod wave equation. We changed the generalized hyperelastic-rod wave equation into a complex differential equation by using traveling wave transform and show that meromorphic solutions of the complex differential equation belong to the class \(W\) by the weak \(\langle {h, k} \rangle\) condition and the Fuchs index. Furthermore, we find out all meromorphic solutions of the complex differential equation. Then we obtain the traveling wave solutions of the generalized hyperelastic-rod wave equation. We can apply the idea of this study to some related mathematical physics equations.Exponential time differencing methods for the time-space-fractional Schrödinger equation.https://zbmath.org/1449.651892021-01-08T12:24:00+00:00"Liang, Xiao"https://zbmath.org/authors/?q=ai:liang.xiao"Bhatt, Harish"https://zbmath.org/authors/?q=ai:bhatt.harish-pSummary: In this paper, exponential time differencing schemes with Padé approximation to the Mittag-Leffler function are proposed for the time-space-fractional nonlinear Schrödinger equations. Ways of increasing the efficiency of the proposed schemes are discussed. Numerical experiments are performed on the time-space-fractional nonlinear Schrödinger equations with various parameters. The accuracy, efficiency, and reliability of the proposed method are illustrated by numerical results.Gradient estimates for a weighted nonlinear equation on complete noncompact manifolds.https://zbmath.org/1449.580042021-01-08T12:24:00+00:00"Li, Jing"https://zbmath.org/authors/?q=ai:li.jing.11"He, Guoqing"https://zbmath.org/authors/?q=ai:he.guoqing"Zhao, Peibiao"https://zbmath.org/authors/?q=ai:zhao.peibiaoSummary: \textit{B. Ma} et al. [Proc. Am. Math. Soc. 146, No. 11, 4993--5002 (2018; Zbl 1398.58007)] considered \(\Delta u+cu^{\alpha}=0 \; (\alpha < 0)\) with Ric\(_{ij}\) \(\ge K g_{ij}\), and obtained some gradient estimates. In the present paper, we investigate the weighted nonlinear equation \(\Delta_f u+cu^{\alpha}=0\) with Ric\(_{f}^N\) \(\ge-K\), where \(f\) is a smooth real-valued function on a complete noncompact Riemannian manifold \((M^n,g), \; \alpha> 0\) and \(c\) are two real constants, and we achieve some gradient estimates for positive solutions of this weighted nonlinear equation. The results posed in this paper can be regarded as a natural generalization of the above mentioned paper.A constrained variational problem of Kirchhoff type equation with ellipsoid-shaped potential.https://zbmath.org/1449.351962021-01-08T12:24:00+00:00"Li, Rongxing"https://zbmath.org/authors/?q=ai:li.rongxing"Wang, Wenqing"https://zbmath.org/authors/?q=ai:wang.wenqing"Zeng, Xiaoyu"https://zbmath.org/authors/?q=ai:zeng.xiaoyuSummary: In this paper, we are concerned with a constrained variational problem for certain type of Kirchhoff equation with trapping potential and the bottom of the potential is an ellipsoid. We are interested in the asymptotic behavior of solutions of variational problem and we prove that the minimizers of the minimization problem blows up at one of the endpoints of the major axis of the ellipsoid as the related parameter approaches a critical value.Global existence and exponential decay of solutions for a class of high-order nonlinear wave equations.https://zbmath.org/1449.353112021-01-08T12:24:00+00:00"Wang, Jianping"https://zbmath.org/authors/?q=ai:wang.jianping.1"Ye, Yaojun"https://zbmath.org/authors/?q=ai:ye.yaojun"Ye, Ziqing"https://zbmath.org/authors/?q=ai:ye.ziqingSummary: A initial-boundary value problem for some nonlinear wave equations with damping and source terms
\[ u_{tt} + Au + u_t + aAu_t = b|u|^{q-1}u \]
in a bounded domain is studied, where \(A = (-\Delta)^m,\ m \ge 1\) is a natural number, \(a \ge 0,\ b > 0\) and \(q > 1\) are real numbers. The existence of global solutions for this problem is proved by constructing the stable sets. The exponential decay estimates of the global solutions are established as time goes to infinity by applying the multiplier method. Meanwhile, under the conditions of the nonnegative initial energy and \(a = 0\), it is shown that the solution blows up in finite time.Traveling wave fronts for the nonlocal dispersal equation with state-dependent delay.https://zbmath.org/1449.351472021-01-08T12:24:00+00:00"Wan, Yuji"https://zbmath.org/authors/?q=ai:wan.yuji"Yu, Zhixian"https://zbmath.org/authors/?q=ai:yu.zhixian"Meng, Yanling"https://zbmath.org/authors/?q=ai:meng.yanlingSummary: This paper is concerned with traveling waves for the nonlocal dispersal equation with the state-dependent delay. If the birth function is monotone, then the existence and nonexistence of monotone traveling waves are established. By a prior estimate and Ikehara's theorem, we obtain the asymptotic behavior of critical traveling wave fronts. Finally, by introducing two auxiliary quasi-monotone equations, we improve our results of existence to the non-quasi-monotone equation.Spectral problem of bacterial population model with generalized boundary conditions.https://zbmath.org/1449.353322021-01-08T12:24:00+00:00"Wang, Shenghua"https://zbmath.org/authors/?q=ai:wang.shenghua"Wu, Hongxing"https://zbmath.org/authors/?q=ai:wu.hongxingSummary: The objective of this paper is to research the model of structured bacterial population with generalized boundary conditions in \({L_1}\) space. We discuss the spectral analysis of corresponding transport operators for this model. We obtain that the spectra of the transport operators only consist of finitely isolate eigenvalues with finite algebraic multiplicities in the right half plane and so on.Explicit solutions of the coupled Kaup-Kupershmidt equations.https://zbmath.org/1449.351462021-01-08T12:24:00+00:00"Wang, Hui"https://zbmath.org/authors/?q=ai:wang.hui.5|wang.hui.4|wang.hui|wang.hui.1|wang.hui.2Summary: In this paper, coupled Kaup-Kupershmidt equations are studied by using the tanh function method. By traveling wave reduction, two fifth-order nonlinear evolution equations are transformed into ODEs. Combining with the properties of Riccati equation, an algebraic system of some parameters is obtained. With the help of Mathematica symbolic computing function, the explicit traveling wave solutions of the above equation, including soliton-like solutions, triangular periodic solutions and rational solutions, are finally obtained.The constrained minimizers of Schrödinger-Poisson equations.https://zbmath.org/1449.351932021-01-08T12:24:00+00:00"Zhou, Xiaomin"https://zbmath.org/authors/?q=ai:zhou.xiaomin"Wang, Shuli"https://zbmath.org/authors/?q=ai:wang.shuli"Guo, Zuji"https://zbmath.org/authors/?q=ai:guo.zujiSummary: In this paper, we concerned with the existence and the nonexistence of constrained minimizers for a class of Schrödinger-Poisson equations. By means of the categorized discussion for parameter \(p\) in the nonlinearity of the equation, the corresponding conclusion was proved by using minimizing sequence method, Ekeland variational principle, vanishing lemma, the Gagliardo-Nirenberg identity, the Hardy-Littewood-Sobolev inequality and the Pohozaev identity in the variational methods.On \({W^{1,p}}\) regularity of a system arising from electromagnetic fields.https://zbmath.org/1449.351342021-01-08T12:24:00+00:00"Chen, Zhihong"https://zbmath.org/authors/?q=ai:chen.zhihong"Li, Dongsheng"https://zbmath.org/authors/?q=ai:li.dongshengSummary: We establish the fundamental \(W^{1,p}\) estimate for the weak solution of a system in a bounded domain \(\Omega\) in \(\mathbb{R}^3\). The system is related to the steady-state of Maxwell's equations for the magnetic field. The inverse of the principle coefficient matrix is assumed to be in the VMO space. We transform the system to scalar elliptic equations by using the properties of curl and divergence of vector fields in \(\mathbb{R}^3\). By the regularity theory of elliptic equations, we get the \(W^{1,p}\) estimate for \(1 < p < \infty\).Numerical solution of reaction-diffusion model in the biological pattern and its parameters inversion.https://zbmath.org/1449.652682021-01-08T12:24:00+00:00"Zhang, Shimei"https://zbmath.org/authors/?q=ai:zhang.shimei"Sun, Yao"https://zbmath.org/authors/?q=ai:sun.yao"Min, Tao"https://zbmath.org/authors/?q=ai:min.taoSummary: The study of numerical solution of reaction-diffusion model in the biological pattern and its parameters inversion is very interesting and important. This paper mainly investigates positive and inverse problem of Gray-Scott model in biological pattern and proposes a new algorithm, DE-ME algorithm. Besides, the feasibility and effectiveness of the algorithm are verified by a numerical example. The results show that the hybrid algorithm can effectively solve this kind of problem of parameters inversion about reaction diffusion model.Biorthogonal systems of solutions of the Helmholtz equation in a cylindrical coordinate system.https://zbmath.org/1449.351922021-01-08T12:24:00+00:00"Sukhorol's'kyĭ, M. A."https://zbmath.org/authors/?q=ai:sukhorolskyi.m-a"Dostoĭna, V. V."https://zbmath.org/authors/?q=ai:dostoina.veronika-v"Veselovs'ka, O. V."https://zbmath.org/authors/?q=ai:veselovska.olga-vSummary: We constructed a system of solutions of the Helmholtz equation in cylindrical coordinates in the form of homogeneous polynomials by two biorthogonal systems of functions.Global existence of self-similar Leray weak solution for three dimensional incompressible magnetohydrodynamics equations.https://zbmath.org/1449.351612021-01-08T12:24:00+00:00"Guo, Hua"https://zbmath.org/authors/?q=ai:guo.hua"Yuan, Rong"https://zbmath.org/authors/?q=ai:yuan.rongSummary: This paper deals with the Cauchy problem of the three dimensional incompressible magnetohydrodynamics equations. Using the regularity estimation of the local space near the initial time and the Leray-Schauder fixed point theorem, the global existence of a smooth self-similar Leray weak solution to the Cauchy problem with the smooth and scale-invariant initial data is achieved.The global attractor for the beam with nonlinear damp term.https://zbmath.org/1449.351102021-01-08T12:24:00+00:00"Yao, Huazhen"https://zbmath.org/authors/?q=ai:yao.huazhen"Zhang, Jianwen"https://zbmath.org/authors/?q=ai:zhang.jianwenSummary: Considering the initial boundary value problem for a class of beam equation with nonlinear damping and external force and using the classical theory of operator semigroup, we prove the existence and uniqueness of the solution for this system. Then we introduce a operator semigroup. The existence of global attractors is proved by the classical operator semigroup decomposition method.Superconvergence analysis of an \({H^1}\)-Galerkin mixed finite element method for nonlinear parabolic equation.https://zbmath.org/1449.652612021-01-08T12:24:00+00:00"Wang, Junjun"https://zbmath.org/authors/?q=ai:wang.junjun"Yang, Xiaoxia"https://zbmath.org/authors/?q=ai:yang.xiaoxiaSummary: Nonlinear parabolic equation is studied by \({H^1}\)-Galerkin mixed finite element method. The bilinear element and the zero-order Raviart-Thomas elements are utilized to discuss superclose properties of the original variable \(u\) in \({H^1} (\Omega)\) and the flux \(\vec p = \nabla u\) in \(H ({\mathrm{div}}; \Omega)\) under the semi-discrete scheme and Euler fully-discrete scheme. During the process, the splitting technique is used and the regularity of \(u\) and \({\vec p}\) are not improved. The numerical example confirms the theory.New numerical process solving nonlinear infinite-dimensional equations.https://zbmath.org/1449.651122021-01-08T12:24:00+00:00"Khellaf, Ammar"https://zbmath.org/authors/?q=ai:khellaf.ammar"Merchela, Wassim"https://zbmath.org/authors/?q=ai:merchela.wassim"Benarab, Sarra"https://zbmath.org/authors/?q=ai:benarab.sarraSummary: Solving a nonlinear equation in a functional space requires two processes: discretization and linearization. In recent paper [\textit{L. Grammont} et al., J. Integral Equations Appl. 26, No. 3, 413--436 (2014; Zbl 1307.65077)], the authors study the difference between applying them in one and in the other order. Linearizing first the nonlinear problem and discretizing the linear problem will be called option (B). Discretizing first the nonlinear problem and linearizing the discrete nonlinear problem will be called option (C). In this paper, we propose a new numerical process equivalent to the option (B): we linearize first the original nonlinear problem with an alternative linearization scheme than that used in the option (B), then we discretize the resulting iterative linear equations using a projection method to implement the corresponding finite-dimensional problem. The aims of this new process are intended to get weaker the theoretical assumptions and to give a powerful numerical performance. We give sufficient conditions to deal with the convergence results. Finally, as a numerical application, we solve a system of Fredholm equations of the second kind. The accuracy and efficiency of this process are illustrated in some numerical examples.On the fractional Newton and wave equation in one space dimension.https://zbmath.org/1449.354512021-01-08T12:24:00+00:00"Velasco, M. P."https://zbmath.org/authors/?q=ai:velasco.maria-pilar"Vázquez, L."https://zbmath.org/authors/?q=ai:vazquez.luis|vazquez.luis-a|vazquez.l-v|vazquez.leonorSummary: We consider the new mathematical scenarios in the framework of the Fractional Calculus. In this context, we study the generalized fractional virial theorem as well as the fractional plane wave solutions and the fractional dispersion relations for the fractional wave equation.On the Hartman effect and velocity of propagating the electromagnetic wave in the tunneling process.https://zbmath.org/1449.810352021-01-08T12:24:00+00:00"Chuprikov, Nikolaĭ Leonidovich"https://zbmath.org/authors/?q=ai:chuprikov.nikolai-lSummary: A new approach to the problem of scattering the plane electromagnetic TE wave on a homogeneous dielectric layer is presented, which does not predict, unlike the standard model, the Hartman effect for the tunneling time in the case of scattering the TE wave in the regime of a frustrated total internal refection (FTIR). The basic idea of this approach is that a correct definition of the tunneling velocity and time is possible if only the dynamics of both its subprocesses -- transmission and reflection -- is known at all stages of the scattering process investigated. It is shown that the Wigner (group) tunneling time was introduced without taking into account of this requirement, and, as a consequence, both this characteristic itself and the associated with it Hartman effect have no relation to transferring the light energy through the layer. The dwell transmission time, which is directly related to it, does not lead to the Hartman effect.High accuracy analysis of linear triangular element for time fractional diffusion equations.https://zbmath.org/1449.653232021-01-08T12:24:00+00:00"Shi, Yanhua"https://zbmath.org/authors/?q=ai:shi.yanhua"Zhang, Yadong"https://zbmath.org/authors/?q=ai:zhang.yadong"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenling"Zhao, Yanmin"https://zbmath.org/authors/?q=ai:zhao.yanmin"Wang, Pingli"https://zbmath.org/authors/?q=ai:wang.pingliSummary: In this paper, based on linear triangular element and improved \(L1\) approximation, a fully-discrete scheme is proposed for time fractional diffusion equations with \(\alpha\) order Caputo fractional derivative. Firstly, the unconditional stability is proved. Secondly, by employing the properties of the element and Ritz projection operator, superclose analysis for the projection operator is deduced with order \(O ({h^2} + {\tau^{2 - \alpha}})\). Furthermore, combining with relationship between the interpolation operator and Ritz projection, superclose analysis for the interpolation operator is also investigated with order \(O ({h^2} + {\tau^{2 - \alpha}})\). And then, the superconvergence result is obtained through the interpolated postprocessing technique. Finally, numerical results are provided to show the validity of our theoretical analysis.Analysis of a fully implicit SUPG scheme for a filtration and separation model.https://zbmath.org/1449.652632021-01-08T12:24:00+00:00"Wilson, A. B."https://zbmath.org/authors/?q=ai:wilson.a-b"Jenkins, E. W."https://zbmath.org/authors/?q=ai:jenkins.eleanor-wSummary: We describe and analyze a streamline-upwinded, Petrov-Galerkin (SUPG) method for numerical simulation of transport models with adsorption kinetics. Specifically, we rewrite the adsorption term to pose the problem using a single mass accumulation term for the liquid phase concentration. The rate of adsorption is used in the nonlinear coefficient for the mass accumulation. We use streamline upwinding to stabilize the advection-dominated applications we consider. We provide a priori error analysis verified by numerical results, and we provide additional numerical results demonstrating the usefulness of our scheme. We consider a variety of adsorption model parameters and describe the performance of our algorithm in these cases.A novel hierarchy of differential equations, conservation laws and Darboux transformation.https://zbmath.org/1449.353052021-01-08T12:24:00+00:00"He, Guoliang"https://zbmath.org/authors/?q=ai:he.guoliang"Zheng, Zhenzhen"https://zbmath.org/authors/?q=ai:zheng.zhenzhenSummary: With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a \(3 \times 3\) matrix spectral problem was proposed. Based on two linear spectral problems, the infinite many conservation laws of the first two members and explicit solutions constructed from the Darboux transformation of the first member in the hierarchy were obtained.An effective approximation to the dispersive soliton solutions of the coupled KdV equation via combination of two efficient methods.https://zbmath.org/1449.652962021-01-08T12:24:00+00:00"Başhan, Ali"https://zbmath.org/authors/?q=ai:bashan.aliSummary: The numerical solutions of the coupled Korteweg-de Vries equation is investigated via combination of two efficient methods that includes Crank-Nicolson scheme for time integration and quintic B-spline based differential quadrature method for space integration. The differential quadrature method has the advantage of using less number of grid points. And the advantage of the Crank-Nicolson scheme is the prevention of long and tedious algebraic computations for time integration. Those advantages come together and produce better results. To display the accuracy and efficiency of the present hybrid method three well-known test problems, namely single soliton, interaction of two soliton and birth of solitons are solved and the error norms \(L_2\) and \(L_\infty\) are computed and compared with earlier works. Present hybrid method obtained superior results than earlier works by using the same parameters and less number of grid points. This situation is shown by comparison of the earlier works. At the same time, two lowest invariants and numerical and analytical values of the amplitudes of the solitons during the simulations are computed and tabulated. Besides those, relative changes of invariants are computed. Properties of solitons observed clearly at the all of the test problems and figures of the all of the simulations are given.The eigenfunctions of curl, gradient of divergence and Stokes operators. Applications.https://zbmath.org/1449.353312021-01-08T12:24:00+00:00"Saks, Romèn Semenovich"https://zbmath.org/authors/?q=ai:saks.romen-semenovichSummary: We consider the spectral problems for curl, gradient of divergence and Stokes operators. The eigenvalues are defined by zeroes of half-integer order Bessel functions and derivatives thereof. The eigenfunctions are given in an explicit form by half-integer order Bessel functions and spherical harmonics. Their applications are described. The completeness of eigenfunctions of curl operator in \(\mathbf{L}_2(B)\) is proved.The Cauchy problems for dissipative hyperbolic mean curvature flow.https://zbmath.org/1449.580062021-01-08T12:24:00+00:00"Lv, Shixia"https://zbmath.org/authors/?q=ai:lv.shixia"Wang, Zenggui"https://zbmath.org/authors/?q=ai:wang.zengguiSummary: In this paper, we investigate initial value problems for hyperbolic mean curvature flow with a dissipative term. By means of support functions of a convex curve, a hyperbolic Monge-Ampère equation is derived, and this equation could be reduced to the first order quasilinear systems in Riemann invariants. Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems, we discuss lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow.Stability of solution of one nonlinear initial-boundary problem of aeroelasticity.https://zbmath.org/1449.740852021-01-08T12:24:00+00:00"Ankilov, Andreĭ Vladimirovich"https://zbmath.org/authors/?q=ai:ankilov.andrey-v"Vel'misov, Pëtr Aleksandrovich"https://zbmath.org/authors/?q=ai:velmisov.petr-aleksandrovich"Kazakova, Yuliya Aleksandrovna"https://zbmath.org/authors/?q=ai:kazakova.yuliya-aleksandrovnaSummary: The dynamic stability of an elastic element of the channel wall under the subsonic stream of an ideal compressible fluid (gas) is studied. Determination of the stability of an elastic body corresponds to the concept of stability of dynamical systems by Lyapunov. The sufficient conditions for stability are obtained. Conditions impose limitations on the speed of the uniform stream of gas, compressed (tensile) element of efforts, the elastic element stiffness and other parameters of the mechanical system.Decay rate of wave equations with boundary memory damping.https://zbmath.org/1449.353082021-01-08T12:24:00+00:00"Li, Donglin"https://zbmath.org/authors/?q=ai:li.donglin"Liu, Yao"https://zbmath.org/authors/?q=ai:liu.yaoSummary: In this paper, a class of semilinear wave equations with acoustic boundary conditions and boundary memory damping is studied to obtain the energy decay rate of the system. Firstly, the energy function of the system is constructed. Then, the Lyapunov function of the system is established. Finally, the energy decay results are obtained by means of the Lyapunov function of the system.Identifying inverse source for fractional diffusion equation with Riemann-Liouville derivative.https://zbmath.org/1449.354502021-01-08T12:24:00+00:00"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yong.1"Long, Le Dinh"https://zbmath.org/authors/?q=ai:long.le-dinh"Can, Nguyen Huu"https://zbmath.org/authors/?q=ai:can.nguyen-huuSummary: In this work, we study an inverse problem to determine an unknown source term for fractional diffusion equation with Riemann-Liouville derivative. In general, the problem is severely ill posed in the sense of Hadamard. To regularize the unstable solution of the problem, we have applied the quasi-boundary value method. In the theoretical result, we show the error estimate between the exact solution and regularized solution with a priori parameter choice rules and analyze it. Eventually, a numerical example has been carried out, the result shows that our regularization method is converged.Existence, uniqueness and blow-up of solutions for the 3D Navier-Stokes equations in homogeneous Sobolev-Gevrey spaces.https://zbmath.org/1449.351132021-01-08T12:24:00+00:00"Braz e Silva, P."https://zbmath.org/authors/?q=ai:braz-e-silva.pablo"Melo, W. G."https://zbmath.org/authors/?q=ai:melo.wilberclay-g"Rocha, N. F."https://zbmath.org/authors/?q=ai:rocha.nata-firminoSummary: We show existence and uniqueness of solutions for the classical Navier-Stokes equations in Sobolev-Gevrey spaces \(\dot{H}_{a,\sigma}^s(\mathbb{R}^3)\), where \(s\in (1/2,3/2)\), \(a>0\) and \(\sigma \geq 1\); furthermore, we present some blow-up criteria considering these same spaces with \(\sigma >1\).Highly accurate technique for solving distributed-order time-fractional-sub-diffusion equations of fourth order.https://zbmath.org/1449.652702021-01-08T12:24:00+00:00"Abdelkawy, M. A."https://zbmath.org/authors/?q=ai:abdelkawy.mohamed-a"Babatin, Mohammed M."https://zbmath.org/authors/?q=ai:babatin.mohammed-m"Lopes, António M."https://zbmath.org/authors/?q=ai:lopes.antonio-mSummary: This paper presents a new method for calculating the numerical solution of distributed-order time-fractional-sub-diffusion equations (DO-TFSDE) of fourth order. The method extends the shifted fractional Jacobi (SFJ) collocation scheme for discretizing both the time and space variables. The approximate solution is expressed as a finite expansion of SFJ polynomials whose derivatives are evaluated at the SFJ quadrature points. The process yields a system of algebraic equations that are solved analytically. The new method is compared with alternative numerical algorithms when solving different types of DO-TFSDE. The results show that the proposed method exhibits superior accuracy with an exponential convergence rate.Positive solutions of a predator-prey model with cross diffusion.https://zbmath.org/1449.350272021-01-08T12:24:00+00:00"Yuan, Hailong"https://zbmath.org/authors/?q=ai:yuan.hailong"Wang, Yuping"https://zbmath.org/authors/?q=ai:wang.yuping.1"Li, Yanling"https://zbmath.org/authors/?q=ai:li.yanlingSummary: A predator-prey model with cross diffusion under homogeneous Dirichlet boundary conditions is investigated. Firstly, the existence of positive solutions can be established by the Leray-Schauder degree theory. Secondly, we consider the existence of positive solutions of the regular perturbation system and the singular perturbation system when \(m = \beta \) is sufficiently large, respectively. Moreover, we show that the positive solutions of the singular perturbation system will blow up along the continuum at \({a^*}\) by the bifurcation theory. Finally, the multiplicity results of positive solutions of the system are also considered.Sufficient and necessary condition for the existence of positive entire solutions of a nonlinear biharmonic equations on \({\mathbb{R}^N}\).https://zbmath.org/1449.352172021-01-08T12:24:00+00:00"Ou, Xiaohang"https://zbmath.org/authors/?q=ai:ou.xiaohangSummary: The aim of this paper is to study the nonlinear biharmonic equations of the following form \({\Delta ^2}u = f\left ({\left| x \right|, u, \left| {\nabla u} \right|} \right)\left ({x \in {\mathbb{R}^N}, N > 2} \right)\). The sufficient and necessary condition for the existence of positive entire solutions is proved, and some properties of the solutions are obtained.Bounded weak solutions to an elliptic equation with lower order terms and degenerate coercivity.https://zbmath.org/1449.351682021-01-08T12:24:00+00:00"Li, Zhongqing"https://zbmath.org/authors/?q=ai:li.zhongqing"Gao, Wenjie"https://zbmath.org/authors/?q=ai:gao.wenjieSummary: A boundary value problem to a class of elliptic equations with lower order terms and degenerate coercivity is studied in this paper. With the help of De Giorgi iterative technique and Boccardo-Brezis's test function, the \({L^\infty}\) estimate to weak solutions of the problem is obtained. Based upon the uniform \({L^\infty}\) bound, the existence of bounded solution is proved.Study on weak solution and strong solution of incompressible MHD equations with damping in three-dimensional systems.https://zbmath.org/1449.351652021-01-08T12:24:00+00:00"Li, Kai"https://zbmath.org/authors/?q=ai:li.kai"Yang, Han"https://zbmath.org/authors/?q=ai:yang.han"Wang, Fan"https://zbmath.org/authors/?q=ai:wang.fanSummary: In this paper, the Cauchy problem of the MHD equations with damping is studied. When \(\beta \ge 1\) and initial data satisfy \({u_0}, {b_0} \in {L^2}\left(\mathbb{R}^3\right)\), the Galerkin method is used to prove the global weak solution of the equations. When the initial data satisfy \({u_0} \in H_0^1 \cap {L^{\beta + 1}}\left(\mathbb{R}^3\right), {b_0} \in H_0^1\left(\mathbb{R}^3\right)\), it is possible to obtain a unique local strong solution for the equation group.Blow-up phenomena of solutions for a class of viscous Cahn-Hilliard equation with gradient dependent potentials and sources.https://zbmath.org/1449.351222021-01-08T12:24:00+00:00"Long, Qunfei"https://zbmath.org/authors/?q=ai:long.qunfei"Chen, Jianqing"https://zbmath.org/authors/?q=ai:chen.jianqingSummary: In this manuscript, we study the blow-up phenomena of solutions for a class of viscous Cahn-Hilliard equation with gradient dependent potentials and source. We establish a criterion for blow-up and determine the upper bound for blow-up time by the energy method, the differential inequality and the derivative formula of the product. We determine the lower bounds for blow-up time by the differential inequality and the derivative formula of the product.Dynamical analysis and traveling wave solutions for generalized \( (3+1)\)-dimensional Kadomtsev-Petviashvili equation.https://zbmath.org/1449.351502021-01-08T12:24:00+00:00"Zhang, Xue"https://zbmath.org/authors/?q=ai:zhang.xue"Sun, Yuhuai"https://zbmath.org/authors/?q=ai:sun.yuhuaiSummary: Dynamical analysis and explicit solutions for generalized \( (3+1)\)-dimension Kadomtsev-Petviashvili equation have been carried out. The singular solution is obtained by the ansatz method. The bifurcation phase portraits and corresponding explicit solution are also constructed by the approach of dynamical analysis.Periodic solutions of a semi-linear Klein-Gordon equations with high frequencies.https://zbmath.org/1449.350302021-01-08T12:24:00+00:00"Tong, Changqing"https://zbmath.org/authors/?q=ai:tong.changqing"Zheng, Jing"https://zbmath.org/authors/?q=ai:zheng.jingSummary: In this paper, we prove the existence of periodic solutions with high frequencies of some semi-linear Klein-Gordon equations. We only assume that the nonlinearities are \({C^k}\) regular and without smallness. Using Nash-Moser iteration, we obtain some periodic solutions in Sobolev space.Fractional-order mathematical model for calcium distribution in nerve cells.https://zbmath.org/1449.354432021-01-08T12:24:00+00:00"Joshi, Hardik"https://zbmath.org/authors/?q=ai:joshi.hardik"Jha, Brajesh Kumar"https://zbmath.org/authors/?q=ai:jha.brajesh-kumarSummary: Calcium (Ca\(^{2+})\) ions are known as a second messenger, involved in a variety of signalling process. It regulates various physiological processes like signal transduction, proliferation, etc. Calbindin-D\(_{28k}\) reacts with free Ca\(^{2+}\) ions and significantly lowers down the cytosolic free calcium concentration ([Ca\(^{2+}])\) in nerve cell. Voltage-gated calcium channel (VGCC) works as an outward source of Ca\(^{2+}\) which initiates and sustains Ca\(^{2+}\) signalling process for the smooth functioning of the cells. An alteration in Ca\(^{2+}\) signalling process leads to the early symptoms of Parkinson's disease (PD). In this piece of work two major analyses has done. First, we have developed a mathematical model in the form of a fractional reaction-diffusion equation. Second, the obtained results are interpreted with a neuronal disorder, i.e., PD. Hence the physiological role of calbindin-D\(_{28k}\) and VGCC in PD is analyze using mathematical model by incorporating all important parameters like diffusion coefficient, fluxes, etc. Appropriate initial and boundary conditions are used according to the biophysical conditions of the cell.Global existence and self-similar blow-up of Landau-Lifshitz-Gilbert equation on hyperbolic space.https://zbmath.org/1449.350032021-01-08T12:24:00+00:00"Zhong, Penghong"https://zbmath.org/authors/?q=ai:zhong.penghong"Yang, Ganshan"https://zbmath.org/authors/?q=ai:yang.ganshan"Ma, Xuan"https://zbmath.org/authors/?q=ai:ma.xuanSummary: By the generalized Hasimoto transformation, we deduce an equivalent system of the Landau-Lifshitz-Gilbert equation on hyperbolic space \(\mathbb{H}^2\). Based on this equivalent model, we prove the global existence of the Landau-Lifshitz-Gilbert equation with the small initial condition. Until now, we have not seen a paper discussing the explicit dynamic solution of the complete equation with a damping term on this target. We construct an explicit small data global solution by the equivalent system obtained in this paper. A self-similar finite blow-up solution is also presented for the equivalent system. In the previous paper, we constructed a finite time blow-up solution without Gilbert damping on \(\mathbb{H}^2\). The question of whether a solution of the complete equation with a Gilbert term can develop a finite time blow-up from \(\mathbb{H}^2\) and smooth initial data is not clear. The self-similar finite time blow-up solution we presented here is a finite energy solution on the entire spacial domain. Our result gives a positive answer to this question.The integrability of the KdV-shallow water waves equation.https://zbmath.org/1449.353802021-01-08T12:24:00+00:00"Hao, Xiaohong"https://zbmath.org/authors/?q=ai:hao.xiaohong"Cheng, Zhilong"https://zbmath.org/authors/?q=ai:cheng.zhilongSummary: In this paper, the binary Bell polynomials are used to construct bilinear forma, bilinear Bäcklund transformation, Lax pair of the KdV-shallow water waves equation. Through bilinear Bäcklund transformation, some soliton solutions are presented. Moreover, the infinite conservation laws are also derived by Bell polynomials. All conserved densities and fluxes are given with explicit recursion formulas.The asymptotic behaviors and phase separation for a class of subcritical Bose-Einstein condensation system.https://zbmath.org/1449.350882021-01-08T12:24:00+00:00"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.4|zhang.jing.10|zhang.jing.2|zhang.jing.6|zhang.jing.12|zhang.jing.5|zhang.jing.1|zhang.jing.3|zhang.jing.9|zhang.jing.11|zhang.jing.7|zhang.jing.8Summary: In this paper, we study the phase separation phenomena of the limit profile as the coupling constant tending to minus infinity for some Bose-Einstein condensation system with subcritical exponent in a general smooth bounded domain via variational methods and elliptic equations theories.Implicit Runge-Kutta and spectral Galerkin methods for Riesz space fractional/distributed-order diffusion equation.https://zbmath.org/1449.652832021-01-08T12:24:00+00:00"Zhao, Jingjun"https://zbmath.org/authors/?q=ai:zhao.jingjun"Zhang, Yanming"https://zbmath.org/authors/?q=ai:zhang.yanming"Xu, Yang"https://zbmath.org/authors/?q=ai:xu.yang.1Summary: A numerical method with high accuracy both in time and in space is constructed for the Riesz space fractional diffusion equation, in which the temporal component is discretized by an \(s\)-stage implicit Runge-Kutta method and the spatial component is approximated by a spectral Galerkin method. For an algebraically stable Runge-Kutta method of order \(p\) \((p\ge s+1)\), the unconditional stability of the full discretization is proven and the convergence order of \(s+1\) in time is obtained. The optimal error estimate in space, with convergence order only depending on the regularity of initial value and \(f\), is also derived. Meanwhile, this kind of method is applied to the Riesz space distributed-order diffusion equation, and similar stability and convergence results are obtained. Finally, numerical experiments are provided to illustrate the theoretical results.A novel bat algorithm based on cross boundary learning and uniform explosion strategy.https://zbmath.org/1449.350622021-01-08T12:24:00+00:00"Yong, Jia-Shi"https://zbmath.org/authors/?q=ai:yong.jia-shi"He, Fa-Zhi"https://zbmath.org/authors/?q=ai:he.fazhi"Li, Hao-Ran"https://zbmath.org/authors/?q=ai:li.haoran"Zhou, Wei-Qing"https://zbmath.org/authors/?q=ai:zhou.wei-qingSummary: Population-based algorithms have been used in many real-world problems. Bat algorithm (BA) is one of the states of the art of these approaches. Because of the super bat, on the one hand, BA can converge quickly; on the other hand, it is easy to fall into local optimum. Therefore, for typical BA algorithms, the ability of exploration and exploitation is not strong enough and it is hard to find a precise result. In this paper, we propose a novel bat algorithm based on cross boundary learning (CBL) and uniform explosion strategy (UES), namely BABLUE in short, to avoid the above contradiction and achieve both fast convergence and high quality. Different from previous opposition-based learning, the proposed CBL can expand the search area of population and then maintain the ability of global exploration in the process of fast convergence. In order to enhance the ability of local exploitation of the proposed algorithm, we propose UES, which can achieve almost the same search precise as that of firework explosion algorithm but consume less computation resource. BABLUE is tested with numerous experiments on unimodal, multimodal, one-dimensional, high-dimensional and discrete problems, and then compared with other typical intelligent optimization algorithms. The results show that the proposed algorithm outperforms other algorithms.Analysis and simulation of a PDE model for surface relaxation.https://zbmath.org/1449.352642021-01-08T12:24:00+00:00"Versieux, H. M."https://zbmath.org/authors/?q=ai:versieux.henrique-mSummary: We study a fourth-order, singular, nonlinear PDE model for surface relaxation. A weak solution for the model is defined using an inequality formulation. A numerical scheme based on semi-implicit time stepping, mixed finite elements and regularization is proposed to approximate the PDE model. We investigate the convergence of the scheme with respect to the discretization and the regularization parameters. Finally, formal arguments show that the model can be viewed as a gradient flow with respect to an appropriate Riemannian metric.Pullback attractors for strongly damped wave equation with delays.https://zbmath.org/1449.351072021-01-08T12:24:00+00:00"Xu, Guigui"https://zbmath.org/authors/?q=ai:xu.guigui"Wang, Libo"https://zbmath.org/authors/?q=ai:wang.libo"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguangSummary: In this paper, we study the existence of the pullback attractors for strongly damped wave equation with delay. By means of constructing the energy functional and combining with the idea of contractive function method, we verify the compactness for the process \(\{U (t,\tau)\}_{t \ge \tau}\) in \(C_{V,H}\) generated by the strongly damped wave equation with delay, and then we obtain the existence of the pullback attractors in \(C_{V,H}\) for the process \(\{U (t,\tau)\}_{t \ge \tau}\).Blow-up of the solution to the Cauchy problem for a class of fourth-order nonlinear wave equations.https://zbmath.org/1449.351192021-01-08T12:24:00+00:00"Li, Ning"https://zbmath.org/authors/?q=ai:li.ning"Li, Tianrui"https://zbmath.org/authors/?q=ai:li.tianrui"Chen, Qiaoling"https://zbmath.org/authors/?q=ai:chen.qiaolingSummary: In this paper, the fourth order wave equation with nonlinear damping and source term is considered. We show that the solution blows up in finite time if \(m < p\), the initial energy is positive and the initial value satisfies a suitable condition.The wave length of periodic waves of a short pulse equation.https://zbmath.org/1449.354562021-01-08T12:24:00+00:00"Sun, Min"https://zbmath.org/authors/?q=ai:sun.min"Zhang, Kelei"https://zbmath.org/authors/?q=ai:zhang.keleiSummary: The wave length of periodic waves of a short pulse equation with parameters is studied. The short pulse equation is transformed into a plane polynomial differential system through variable transformation. When the parameter \(\alpha < 0\), the short pulse equation has a smooth periodic wave. The qualitative theory and analysis on dynamical systems are used to study this polynomial differential system. The main result is the monotonicity of \(T (h)\), which is affected by the parameter \(\beta\).Dynamic behavior for a non-autonomous wave equation with white noise on unbounded domains.https://zbmath.org/1449.351022021-01-08T12:24:00+00:00"Wang, Miaomiao"https://zbmath.org/authors/?q=ai:wang.miaomiao"Jiang, Yong"https://zbmath.org/authors/?q=ai:jiang.yong.1|jiang.yong"Yang, Xiao"https://zbmath.org/authors/?q=ai:yang.xiaoSummary: In this paper, we study the existence of random attractor for stochastic non-autonomous wave equation with additive noise on unbounded domains, where the nonlinear term satisfies the critical exponent. By estimate of solutions of transformed system, we get the existence of asymptotically compact D-pullback absorbing set, and then get the existence of random attractor of the original system.Discrete cosine pseudo spectral method for solving reaction diffusion equation.https://zbmath.org/1449.652812021-01-08T12:24:00+00:00"Zhang, Rongpei"https://zbmath.org/authors/?q=ai:zhang.rongpei"Yang, Chengcheng"https://zbmath.org/authors/?q=ai:yang.chengcheng"Liu, Jia"https://zbmath.org/authors/?q=ai:liu.jiaSummary: Discrete cosine transform has been widely used in signal processing, image compression and partial differential equation. Firstly, the properties of discrete cosine transform are discussed. Based on the orthogonality of the family of cosine functions on the set of discrete points, the least square cosine interpolation polynomial of the approximate function is obtained. A mapping between triangular interpolation coefficients and function values is established, and the discrete cosine transform is obtained. Secondly, the discrete cosine transform method is used to solve the reaction diffusion equation. The cosine pseudo-spectral method is obtained. The reaction diffusion equation is discretized by backward Euler time method, and then the cosine transformation of the discrete diffusion equation is carried out, the diffusion equation is transformed into a nonlinear algebra equation. Picard iteration method is used to solve the problem, and then the solution of the original equation is obtained by inverse cosine transformation. In the end, a numerical example is given to solve Gray-Scott equation which is widely used in chemical reactions. Some numerical results are obtained to verify the effectiveness of the method.Multi-domain decomposition pseudospectral method for nonlinear Fokker-Planck equations.https://zbmath.org/1449.652792021-01-08T12:24:00+00:00"Sun, Tao"https://zbmath.org/authors/?q=ai:sun.tao"Wang, Tian-Jun"https://zbmath.org/authors/?q=ai:wang.tianjunSummary: Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application, a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.A reduced-order extrapolating Crank-Nicolson finite difference scheme for the Riesz space fractional order equations with a nonlinear source function and delay.https://zbmath.org/1449.651732021-01-08T12:24:00+00:00"Cao, Yanhua"https://zbmath.org/authors/?q=ai:cao.yanhua"Luo, Zhendong"https://zbmath.org/authors/?q=ai:luo.zhendongSummary: This article mainly studies the order-reduction of the classical Crank-Nicolson finite difference (CNFD) scheme for the Riesz space fractional order differential equations (FODEs) with a nonlinear source function and delay on a bounded domain. For this reason, the classical CNFD scheme for the Riesz space FODE and the existence, stability, and convergence of the classical CNFD solutions are first recalled. And then, a reduced-order extrapolating CNFD (ROECNFD) scheme containing very few degrees of freedom but holding the fully second-order accuracy for the Riesz space FODEs is established by means of proper orthogonal decomposition and the existence, stability, and convergence of the ROECNFD solutions are discussed. Finally, some numerical experiments are presented to illustrate that the ROECNFD scheme is far superior to the classical CNFD one and to verify the correctness of theoretical results. This indicates that the ROECNFD scheme is very effective for solving the Riesz space FODEs with a
nonlinear source function and delay.Solving parabolic integro-differential equations with purely nonlocal conditions by using the operational matrices of Bernstein polynomials.https://zbmath.org/1449.354252021-01-08T12:24:00+00:00"Bencheikh, Abdelkrim"https://zbmath.org/authors/?q=ai:bencheikh.abdelkrim"Chiter, Lakhdar"https://zbmath.org/authors/?q=ai:chiter.lakhdar"Li, Tongxing"https://zbmath.org/authors/?q=ai:li.tongxingSummary: Some problems from modern physics and science can be described in terms of partial differential equations with nonlocal conditions. In this paper, a numerical method which employs the orthonormal Bernstein polynomials basis is implemented to give the approximate solution of integro-differential parabolic equation with purely nonlocal integral conditions. The properties of orthonormal Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given integro-differential parabolic equation to the solution of algebraic equations. An illustrative example is given to demonstrate the validity and applicability of the new technique.Existence of solutions for Schrödinger-Poisson system with asymptotically periodic terms.https://zbmath.org/1449.352112021-01-08T12:24:00+00:00"Wang, Da-Bin"https://zbmath.org/authors/?q=ai:wang.dabin"Ma, Lu-Ping"https://zbmath.org/authors/?q=ai:ma.luping"Guan, Wen"https://zbmath.org/authors/?q=ai:guan.wen"Wu, Hong-Mei"https://zbmath.org/authors/?q=ai:wu.hongmeiSummary: In this paper, we consider the following nonlinear Schrödinger-Poisson system \[ \begin{cases} -\Delta u + V(x)u+K(x)\phi u= f(x,u), \quad & x\in \mathbb{R}^3,\\ -\Delta \phi=K(x)u^{2}, & x\in \mathbb{R}^3, \end{cases} \] where \(V, K\in L^{\infty}(\mathbb{R}^3)\) and \(f:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous. We prove that the problem has a nontrivial solution under asymptotically periodic case of \(V\), \(K\), and \(f\) at infinity. Moreover, the nonlinear term \(f\) does not satisfy any monotone condition.Solving explicit new traveling wave solutions of KdV-Burgers-Kuramoto equation by Riccati equation.https://zbmath.org/1449.351452021-01-08T12:24:00+00:00"Lin, Fubiao"https://zbmath.org/authors/?q=ai:lin.fubiao"Zhang, Qianhong"https://zbmath.org/authors/?q=ai:zhang.qianhongSummary: Firstly, 8 types explicit new analytical solutions of the Riccati equation are presented by the trial function method combing with the related properties of solutions for the Riccati equation. Secondly, the reduced equations and invariant solutions of KdV-Burgers-Kuramoto (KBK) equation are given by the Lie group analysis method. Finally, the extended tanh-function method and 8 types explicit new analytical solutions of the Riccati equation are used to solve the reduced equation of KBK equation. Moreover, many types explicit new traveling wave solutions of KBK equation are found. In addition, periodic types, rational types of exponential function and trigonometric function of explicit new traveling wave solutions of other similar nonlinear partial differential equations can be obtained by use of the extended tanh-function method, 8 types explicit new analytical solutions of the Riccati equation and the Lie group analysis method.Fractional Tikhonov method of a non-characteristic Cauchy problem for a parabolic equation.https://zbmath.org/1449.652272021-01-08T12:24:00+00:00"Chen, Yawen"https://zbmath.org/authors/?q=ai:chen.yawen"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: The ill-posed non-characteristic Cauchy problem for a parabolic equation is considered. A fractional Tikhonov regularization method is applied to solve the the problem. Some stability error estimates under a-priori and a-posteriori choice rules are given.Continuous dependence on viscosity coefficient for primitive equations.https://zbmath.org/1449.350372021-01-08T12:24:00+00:00"Li, Yuanfei"https://zbmath.org/authors/?q=ai:li.yuanfeiSummary: The continuous dependence of the solutions of the 3D primitive equations of large scale ocean under oscillating random force in a cylindrical region is considered. Using the technique of differential inequality, the a prior bounds of the solutions of the equations are derived. By the energy analysis methods, the continuous dependence on the viscosity coefficient of the solution of the three-dimensional viscous primitive equation of large-scale ocean is obtained.Attractor family and dimension for a class of high-order nonlinear Kirchhoff equations.https://zbmath.org/1449.350952021-01-08T12:24:00+00:00"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguang"Li, Zhuoxi"https://zbmath.org/authors/?q=ai:li.zhuoxiSummary: The initial boundary value problem for a class of high-order Kirchhoff equations with nonlinear nonlocal source terms and strong damping terms is studied. For the nonlinear nonlocal source term and the Kirchhoff stress term, the existence and uniqueness of the global solution of the equation are firstly proved by Galerkin finite element method and a prior estimate. Then the bounded absorption set is obtained by a prior estimate, so the global attractor family of high-order nonlinear Kirchhoff equation is obtained. By linearizing the equation and proving the Frechet differentiability of the solution semigroup, we further prove the decay of the volume element of the linearization problem. Finally, the Hausdorff dimension and fractal dimension of the global attractor family are proved to be finite.General energy decay of solutions for a wave equation with nonlocal damping and nonlinear boundary damping.https://zbmath.org/1449.350712021-01-08T12:24:00+00:00"Li, Donghao"https://zbmath.org/authors/?q=ai:li.donghao"Zhang, Hongwei"https://zbmath.org/authors/?q=ai:zhang.hongwei"Hu, Qingying"https://zbmath.org/authors/?q=ai:hu.qingyingSummary: In this paper, we consider a nonlinear wave equation with nonlocal damping and nonlinear boundary damping. We prove a general energy decay property for solutions in terms of coefficient of the frictional boundary damping by using the multiplier technique. Our result extends and improves the result in the literature in which only exponential energy decay was considered. Furthermore, we get also the energy decay for the equation with nonlocal damping only but without nonlinear boundary damping.Nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data.https://zbmath.org/1449.352652021-01-08T12:24:00+00:00"Zaki, Khaled"https://zbmath.org/authors/?q=ai:zaki.khaled"Redwane, Hicham"https://zbmath.org/authors/?q=ai:redwane.hichamSummary: An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established. A comparison result is also proved for such solutions.Dynamics of a nonlinear stochastic viscoelastic equation with multiplicative noise.https://zbmath.org/1449.350462021-01-08T12:24:00+00:00"Caraballo, Tomas"https://zbmath.org/authors/?q=ai:caraballo.tomas"Pina, Nicolas"https://zbmath.org/authors/?q=ai:pina.nicolas"Munoz, Jaime"https://zbmath.org/authors/?q=ai:munoz.jaimeSummary: The well-posedness and stability properties of a stochastic viscoelastic equation with multiplicative noise, Lipschitz and locally Lipschitz nonlinear terms are investigated. The method of Lyapunov functions is used to investigate the asymptotic dynamics when zero is not a solution of the equation by using an appropriate cocycle and random dynamical system. The stability of mild solutions is proved in both cases of Lipschitz and locally Lipschitz nonlinear terms. Furthermore, we investigate the existence of a non-trivial stationary solution which is exponentially stable, by using a general random fixed point theorem for general cocycles. In this case, the stationary solution is generated by the composition of random variable and Wiener shift. In addition, the theory of random dynamical system is used to construct another cocycle and prove the existence of a random fixed point exponentially attracting every path.Solitary waves of 1-nonlinear Schrödinger equation in the composite right- and left-handed metamaterial.https://zbmath.org/1449.351542021-01-08T12:24:00+00:00"Houwe, A."https://zbmath.org/authors/?q=ai:houwe.alphonse"Klofai, Yerima"https://zbmath.org/authors/?q=ai:klofai.yerima"Justin, Mibaile"https://zbmath.org/authors/?q=ai:justin.mibaile"Gambo, Betchewe"https://zbmath.org/authors/?q=ai:gambo.betchewe"Doka, Serge Y."https://zbmath.org/authors/?q=ai:doka.serge-yamignoSummary: In this article, we analyze solitary waves in nonlinear left-handed transmission line with nonlinear diodes (Schottkys) which is an important issue, especially for soliton devices. By applying the Kirchhoffs laws and reductive direct method, the voltage in the spectral domain was obtained. Considering the Taylor series around a certain modulation frequency, we obtained one dimensional nonlinear Schrödinger equation (NSE), which support envelops soliton, and bright soliton solutions. Using sine-cosine mathematical method, soliton solutions of the standard nonlinear Schrödinger equation are obtained. The method used is straightforward and concise and can be further applied to solve the nonlinear PDEs in mathematical physics.Performance output tracking based on error feedback for one-dimension heat equation.https://zbmath.org/1449.931222021-01-08T12:24:00+00:00"Bai, Lihua"https://zbmath.org/authors/?q=ai:bai.lihuaSummary: In this paper, the output tracking problem for a one-dimensional heat equation is discussed, where the performance output is non-collocated to the control actuation. Firstly, the non-collocated problem is converted into a collocated one by constructing an auxiliary system. Secondly, both the observer and the controller are designed by using the error feedback between the performance output and the reference signal only, which makes the performance output converge to the reference signal, and the closed-loop system is uniformly bounded.An accurate numerical method for solving the generalized time-fractional diffusion equation.https://zbmath.org/1449.354492021-01-08T12:24:00+00:00"Syam, Muhammed"https://zbmath.org/authors/?q=ai:syam.muhammed-i"Al-Subaihi, Ibrahim"https://zbmath.org/authors/?q=ai:al-subaihi.ibrahim-aSummary: In this paper, a formulation for the fractional Legendre functions is constructed to solve a class of time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre. Analysis for the presented method is given and numerical results are presented.Application of CRE method in Boussinesq-Burgers equations.https://zbmath.org/1449.353652021-01-08T12:24:00+00:00"Ge, Nannan"https://zbmath.org/authors/?q=ai:ge.nannan"Ren, Xiaojing"https://zbmath.org/authors/?q=ai:ren.xiaojingSummary: The definition of CRE solvability is given. The CRE solvable concept is used to prove the CRE solvability of Boussinesq-Burgers equations. According to this property, the soliton-conoidal wave solution of the Boussinesq-Burgers equation is constructed under the assumption of negative exponents. In order to study the nature of the solution, the corresponding solution is given by selecting the appropriate parameters.On Hausdorff dimension of random attractors for a stochastic wave equation.https://zbmath.org/1449.350902021-01-08T12:24:00+00:00"Ban, Ailing"https://zbmath.org/authors/?q=ai:ban.ailingSummary: This paper mainly examines the upper bound estimation of the Hausdorff dimension of random attractors for strongly damped stochastic wave equations with critical growth exponents. We prove that the obtained upper bound of the Hausdorff dimension of random attractor deceases as the strongly damped coefficient grows and is uniformly bounded when the strongly damped coefficient is sufficiently large.A series expansion method for solving the boundary value problem connected with the Helmholtz equation.https://zbmath.org/1449.653052021-01-08T12:24:00+00:00"Du, Xinwei"https://zbmath.org/authors/?q=ai:du.xinweiSummary: A boundary value problem connected with the Helmholtz equation is studied in a smooth bounded domain. A series expansion method is proposed for obtaining an approximate solution to the problem. Tikhonov regularization is applied to the problem with noisy data. Numerical experiments are presented to show the effectiveness of the proposed method.On the global asymptotic stability of solutions to a predator-prey system.https://zbmath.org/1449.350662021-01-08T12:24:00+00:00"Zhu, Zirui"https://zbmath.org/authors/?q=ai:zhu.zirui"Tian, Meimei"https://zbmath.org/authors/?q=ai:tian.meimei"Xu, Yancong"https://zbmath.org/authors/?q=ai:xu.yancongSummary: The global dynamic behavior of the solution of a predator-prey model is studied in this paper, especially by using analysis and the Lyapunov function, asymptotic stability of the solution of constraints and the sufficient condition of global asymptotic stability of steady-state solutions are obtained. At the end of the paper, two examples are also given, through the numerical simulation to better explain the theorem and the corresponding constraints.Dynamic bifurcation of Cahn-Hilliard equation.https://zbmath.org/1449.350412021-01-08T12:24:00+00:00"Wu, Ruili"https://zbmath.org/authors/?q=ai:wu.ruili"Chai, Rongqian"https://zbmath.org/authors/?q=ai:chai.rongqian"Qian, Xiaorui"https://zbmath.org/authors/?q=ai:qian.xiaoruiSummary: With the guidance of spectrum theory of the linear completely continuous fields, center manifolds reduction method and transition theory of nonlinear dissipative system, this paper investigates dynamic bifurcation of Cahn-Hilliard equation. The conditions of the divergence, its critical point and the expression of the stable singularity attractor and saddle points of the equation with Neumann boundary condition are given in this paper.Finite difference method for Riesz space fractional diffusion equations with delay and a nonlinear source term.https://zbmath.org/1449.652072021-01-08T12:24:00+00:00"Yang, Shuiping"https://zbmath.org/authors/?q=ai:yang.shuipingSummary: In this paper, we propose a finite difference method for the Riesz space fractional diffusion equations with delay and a nonlinear source term on a finite domain. The proposed method combines a time scheme based on the predictor-corrector method and the Crank-Nicolson scheme for the spatial discretization. The corresponding theoretical results including stability and convergence are provided. Some numerical examples are presented to validate the proposed method.Mixed virtual element methods for parabolic equations.https://zbmath.org/1449.652432021-01-08T12:24:00+00:00"Guo, Hairong"https://zbmath.org/authors/?q=ai:guo.hairong"Wang, Feng"https://zbmath.org/authors/?q=ai:wang.feng|wang.feng.4|wang.feng.3|wang.feng.2|wang.feng.1Summary: Virtual elements are defined on arbitrary polygonal or polyhedral grids. In this paper, mixed virtual elements are proposed for parabolic equations. We present a priori error estimates, which are verified by some numerical experiments.Dielectric waveguides of arbitrary cross sectional shape.https://zbmath.org/1449.780112021-01-08T12:24:00+00:00"Horikis, Theodoros P."https://zbmath.org/authors/?q=ai:horikis.theodoros-pSummary: Numerical guided mode solutions of arbitrary cross sectional shaped waveguides are obtained using a finite difference (FD) technique. The standard FD scheme is appropriately modified to capture all discontinuities, due to the change of the refractive index, across the waveguides' interfaces taking into account the shape of each interface at the same time. The method is applied to the vector Helmholtz equation formulated to describe the electric or magnetic fields in the waveguide (one field is retrieved from the other through Maxwell's equations). The computational cost is kept to a minimum by exploiting sparse matrix algebra. The waveguides under study have arbitrary cross sectional shape and arbitrary refractive index profile.A robust \({C^0}{P_1}\)-\({P_3}\) space-time finite element method for elastic vibration problems.https://zbmath.org/1449.652452021-01-08T12:24:00+00:00"Guo, Yuling"https://zbmath.org/authors/?q=ai:guo.yuling"Huang, Jianguo"https://zbmath.org/authors/?q=ai:huang.jianguoSummary: This paper devises a robust \({C^0}{P_1}\)-\({P_3}\) space-time finite element method for elastic vibration equations. The temporal discretization is obtained by the \({C^0}{P_1}\) DG method, and the spatial discretization is given by the \({P_3}\)-nonconforming element method, leading to a \({C^0}{P_1}\)-\({P_3}\) space-time fully discrete scheme for the problem. Numerical results demonstrate the robustness of the proposed method.Chebyshev polynomial regularization method for solving the unknown source on Poisson equation.https://zbmath.org/1449.653042021-01-08T12:24:00+00:00"Dai, Pei"https://zbmath.org/authors/?q=ai:dai.pei"Xiong, Xiangtuan"https://zbmath.org/authors/?q=ai:xiong.xiangtuanSummary: The problem of determining unknown source on Poisson equation is a severely ill-posed problem. Because the standard Tikhonov regularization method has saturation restriction, in this paper, we use Chebyshev polynomial regularization method to obtain the regularization solution for the unknown source. The error estimates under an a priori parameter choice rule and an a posteriori parameter choice rule are given, respectively.New exact solutions of \( (2+1)\)-dimensional Boiti-Leon-Manna-Pempinelli equation.https://zbmath.org/1449.353642021-01-08T12:24:00+00:00"Deng, Changrui"https://zbmath.org/authors/?q=ai:deng.changrui"Zhou, Xiaohong"https://zbmath.org/authors/?q=ai:zhou.xiaohongSummary: Nonlinear evolution equations are widely used in real physical model, such as high polymer physics, fluid dynamics, solid state physics, plasma physics and so on. This paper mainly studies the \( (2+1)\)-dimensional Boti-Leon-Manna-Pempinelli equation. Firstly, the Hirota bilinear form of the equation is obtained by Hirota method, and then the extended three-wave test method is used to obtain the periodic soliton solution, periodic double soliton solution and double periodic double soliton solution of the \( (2+1)\)-dimensional Boti-Leon-Manna-Pempinelli equation.Rational solutions to a generalized \( (2 + 1)\)-dimensional shallow-water-wave-like equation.https://zbmath.org/1449.351592021-01-08T12:24:00+00:00"Du, Yahong"https://zbmath.org/authors/?q=ai:du.yahong"Yin, Shan"https://zbmath.org/authors/?q=ai:yin.shanSummary: In this paper, we study the rational solutions of \( (2 + 1)\)-dimensional shallow-water-wave-like equation. By using the generalized bilinear operators, the polynomial solutions of the generalized bilinear equation with the prime number of \(p = 3\) are solved, and four classes of rational solutions to the equation are obtained.New exact solutions for Burgers-KPP equations.https://zbmath.org/1449.353882021-01-08T12:24:00+00:00"Wang, Xin"https://zbmath.org/authors/?q=ai:wang.xin.9|wang.xin.1|wang.xin|wang.xin.7|wang.xin.2|wang.xin.10|wang.xin.6|wang.xin.5|wang.xin.12|wang.xin.13|wang.xin.4|wang.xin.8|wang.xin.3|wang.xin.11Summary: In this paper, we study the explicit and exact solutions of the Burgers-KPP equation. Based on the \( (G'/G)\) expansion method, we construct a kind of \(G\) expansion method for satisfying a class of variable coefficient equations. The Burgers-KPP equation is solved by this kind of expansion method. Several new explicit traveling wave solutions of trigonometric functions and hyperbolic functions are obtained, which enrich the scope of the solutions of the equation.Variational reduction for semi-stiff Ginzburg-Landau vortices.https://zbmath.org/1449.352132021-01-08T12:24:00+00:00"Rodiac, Rémy"https://zbmath.org/authors/?q=ai:rodiac.remyIn this article, the existence of solutions to the Ginzburg-Landau problem \[-\varepsilon^2\Delta u=(1-|u|^2)u\quad\text{ in }\Omega,\] is investigated. Here \(\Omega\subset\mathbb{R}^2\) is a smooth and bounded domain. The above equation is complemented with the boundary conditions \[|u|=1\quad\text{ and }\quad\mathrm{Im}(\overline{u}\partial_\nu u)=0\qquad\text{ on }\partial\Omega.\] The approach relies on a variational reduction method in the spirit of [\textit{M. del Pino} et al., J. Funct. Anal. 239, No. 2, 497--541 (2006; Zbl 1387.35561)]. In particular, it is shown that: \begin{enumerate}\item If \(\Omega\) is simply connected, then a solution with degree one on the boundary always exists;\item If \(\Omega\) is not simply connected then for any \(k\geq 1\), a solution with \(k\) vortices of degree one exists.\end{enumerate}
Reviewer: Marius Ghergu (Dublin)Local regularity for strongly degenerate elliptic equations and weighted sum operators.https://zbmath.org/1449.351312021-01-08T12:24:00+00:00"Di Fazio, G."https://zbmath.org/authors/?q=ai:di-fazio.giuseppe"Fanciullo, M. S."https://zbmath.org/authors/?q=ai:fanciullo.maria-stella"Zamboni, P."https://zbmath.org/authors/?q=ai:zamboni.pietroThe authors give a contribution to the project of showing regularity properties for solutions of degenerate elliptic equations. In order to gain the objective, Di Fazio, Fanciullo and Zamboni use the explicit representation formulas and Poincaré inequality adapted to the geometry induced by the system \(X\) a system of first order locally Lipschitz vector fields in \(\mathbb{R}^n\) of the given vector fields. As a direct consequence, they get continuity and Hölder continuity of the weak solutions of the equations. Let us point out that the assumptions on the lower order terms are very sharp in the sense that they are also necessary in some particular cases.
Reviewer: Maria Alessandra Ragusa (Catania)The ground state of the Chern-Simons-Schrödinger system.https://zbmath.org/1449.351952021-01-08T12:24:00+00:00"Yu, Chun"https://zbmath.org/authors/?q=ai:yu.chun"Wan, Youyan"https://zbmath.org/authors/?q=ai:wan.youyanSummary: We study the nonlinear Chern-Simons-Schrödinger system with superlinear nonlinearities. By the concentration compactness principle combined with the Nehari manifold, we prove the existence of positive ground state to this problem. Moreover, we obtain that the ground state has exponential decay at infinity.The anisotropic \(\infty\)-Laplacian eigenvalue problem with Neumann boundary conditions.https://zbmath.org/1449.353352021-01-08T12:24:00+00:00"Piscitelli, Gianpaolo"https://zbmath.org/authors/?q=ai:piscitelli.gianpaoloThe author studies the limiting process as \(p\to+\infty\) for an anisotropic \(p\)-Laplacian eigenvalue problem with a Neumann boundary condition. The solutions are considered in a viscosity sense. The comparison with the corresponding Dirichlet problem is discussed. Results dealing with existence and geometric properties of the first \(\infty\)-eigenfunctions are presented.
Reviewer: Dumitru Motreanu (Perpignan)The concavity of \(p\)-Renyi entropy power for the weighted doubly nonlinear diffusion equations on weighted Riemannian manifolds.https://zbmath.org/1449.580052021-01-08T12:24:00+00:00"Wang, Yuzhao"https://zbmath.org/authors/?q=ai:wang.yuzhao"Zhang, Huiting"https://zbmath.org/authors/?q=ai:zhang.huitingSummary: In this paper, we study the concavity of the entropy power on Riemannian manifolds. By using the nonlinear Bochner formula and Bakry-Emery method, we prove that \(p\)-Renyi entropy power is concave for positive solutions to the weighted doubly nonlinear diffusion equations on the weighted closed Riemannian manifolds with \(CD (-K, m)\) condition for some \(K \ge 0\) and \(m \ge n\), which generalizes the cases of porous medium equation and nonnegative Ricci curvature.An evolutionary perspective on cancer, with applications to anticancer drug resistance modelling and perspectives in therapeutic control.https://zbmath.org/1449.354202021-01-08T12:24:00+00:00"Clairambault, Jean"https://zbmath.org/authors/?q=ai:clairambault.jeanSummary: The question of a mathematical representation and theoretical overcoming by optimized therapeutic strategies of drug-induced drug resistance in cancer cell populations is tackled here from the point of view of adaptive dynamics and optimal population growth control, using integro-differential equations. Combined impacts of external continuous-time functions, standing for drug actions, on targets in a plastic (i.e., able to quickly change its phenotype in deadly environmental conditions) cell population model, represent a therapeutical control to be optimized. A justification for the introduction of the adaptive dynamics setting, retaining such plasticity for cancer cell populations, is firstly presented in light of the evolution of multicellular species and disruptions in multicellularity coherence that are characteristics of cancer and of its progression. Finally, open general questions on cancer and evolution in the Darwinian sense are listed, that may open innovative tracks in modelling and treating cancer by circumventing drug resistance. This study sums up results that were presented at the international NUMACH workshop, Mulhouse, France, in July 2018.A nonstandard higher-order variational model for speckle noise removal and thin-structure detection.https://zbmath.org/1449.652292021-01-08T12:24:00+00:00"Hamdi, Houichet"https://zbmath.org/authors/?q=ai:hamdi.houichet"Anis, Theljani"https://zbmath.org/authors/?q=ai:anis.theljani"Badreddine, Rjaibi"https://zbmath.org/authors/?q=ai:badreddine.rjaibi"Maher, Moakher"https://zbmath.org/authors/?q=ai:maher.moakherSummary: We propose a multiscale approach for a nonstandard higher-order PDE based on the \(p (\cdot)\)-Kirchhoff energy. We first use the topological gradient approach for a semi-linear case in order to detect important objects of the image. We consider a fully nonlinear \(p (\cdot)\)-Kirchhoff equation with variable-exponent functions that are chosen adaptively based on the map provided by the topological gradient in order to preserve important features of the image. Then, we consider the split Bregman method for the numerical implementation of the proposed model. We compare our model with other classical variational approaches such as the TVL and bi-harmonic restoration models. Finally, we present some numerical results to illustrate the effectiveness of our approach.On self-adjoint realizations of sign-indefinite Laplacians.https://zbmath.org/1449.351912021-01-08T12:24:00+00:00"Pankrashkin, Konstantin"https://zbmath.org/authors/?q=ai:pankrashkin.konstantinLet \(\Omega \subset \mathbb{R}^d\) be a domain divided onto two parts \(\Omega^{\pm}\) having a joint smooth surface. The author considers the spectral problem for the operator \(-\nabla \cdot h \nabla\) in \(L_2(\Omega)\) with the Dirichlet problem on \(\partial \Omega\). The function \(h\) is equal to unity in \(\Omega^{+}\) and \(-\mu <0\) in \(\Omega^{-}\). This case of different signs of \(h\) corresponds to ``metamaterials''. The model case when \(\Omega^{\pm}\) are two rectangles having the joint side is investigated. The obtained results are extended to general case.
Reviewer: Vladimir Mityushev (Kraków)Essential spectrum, quasi-orbits and compactifications: application to the Heisenberg group.https://zbmath.org/1449.470282021-01-08T12:24:00+00:00"Mougel, Jérémy"https://zbmath.org/authors/?q=ai:mougel.jeremyLet \(H\) be the Heisenberg group. Using the natural bijection between \(H\) and \(\mathbb{R}\sp{3}\), there is introduced a compactification \(\bar{H}\) of \(H\) induced by the spherical compactification of \(\mathbb{R}\sp{3}\). Let \(T=-\Delta + V\) be the Schrödinger-type operator on \(L\sp{2}(H)\), where \(V\) is a continuous function on \(\bar{H}\). The main result of the paper gives a representation of the essential spectrum of \(T\) as the union of spectra of some simpler operators. There are also obtained some similar results.
Reviewer: Vladimir S. Pilidi (Rostov-na-Donu)Positive solutions for \((p,2)\)-equations with superlinear reaction and a concave boundary term.https://zbmath.org/1449.352392021-01-08T12:24:00+00:00"Papageorgiou, Nikolaos"https://zbmath.org/authors/?q=ai:papageorgiou.nikolaos-s"Scapellato, Andrea"https://zbmath.org/authors/?q=ai:scapellato.andreaSummary: We consider a nonlinear boundary value problem driven by the \((p,2)\)-Laplacian, with a \((p-1)\)-superlinear reaction and a parametric concave boundary term (a ``concave-convex'' problem). Using variational tools (critical point theory) together with truncation and comparison techniques, we prove a bifurcation type theorem describing the changes in the set of positive solutions as the parameter \(\lambda>0\) varies. We also show that for every admissible parameter \(\lambda>0\), the problem has a minimal positive solution \(\overline{u}_\lambda\) and determine the monotonicity and continuity properties of the map \(\lambda \mapsto \overline{u}_\lambda\).A blow-up estimate of the solution for nonlinear fractional Schrödinger equation.https://zbmath.org/1449.351242021-01-08T12:24:00+00:00"Peng, Congming"https://zbmath.org/authors/?q=ai:peng.congming"Zhao, Dun"https://zbmath.org/authors/?q=ai:zhao.dunSummary: A blow-up estimate was investigated for nonlinear fractional Schrödinger equation, and the lower bounds estimate was obtained for the blow-up solutions obtained by means of the fractional Leibniz formula and Gagliardo-Nirenberg inequality. The result gave a complement for the blow-up theory of a reference.Gluing action groupoids: Fredholm conditions and layer potentials.https://zbmath.org/1449.354672021-01-08T12:24:00+00:00"Carvalho, Catarina"https://zbmath.org/authors/?q=ai:carvalho.catarina-a|carvalho.catarina-c"Côme, Rémi"https://zbmath.org/authors/?q=ai:come.remi"Qiao, Yu"https://zbmath.org/authors/?q=ai:qiao.yu.1|qiao.yu|qiao.yu.2Relevant subject of research, in the modern theory of the pseudodifferential operators, is the study of elliptic equations on non-compact manifolds, or manifolds with boundary and singularities, in particular conical domains. Pseudodifferential operators on Lie groupoids are intended to give a unified presentation of the problems, by recapturing as examples several results of the preceding literature, see \textit{V. Nistor} et al. [Pac. J. Math. 189, No. 1, 117--152 (1999; Zbl 0940.58014)]. In this line of ideas the authors of the present paper introduce a new class of groupoids, called boundary action groupoids, which are obtained by gluing reductions of action groupoids. The main result is a conditions for the Fredholm property (operators with parametrix and finite index). The condition involves standard ellipticity at the interior points, and invertibility of vector valued symbols at the singular points. Several applications are detailed.
Reviewer: Luigi Rodino (Torino)Finite volume method of option pricing model under uncertain volatility.https://zbmath.org/1449.652192021-01-08T12:24:00+00:00"Gan, Xiaoting"https://zbmath.org/authors/?q=ai:gan.xiaoting"Xu, Dengguo"https://zbmath.org/authors/?q=ai:xu.dengguo"Zhao, Renqing"https://zbmath.org/authors/?q=ai:zhao.renqingSummary: In view of the numerical solution of option pricing model under uncertain volatility, we constructed a fully implicit finite volume scheme of the nonlinear HJB (Hamilton-Jacobi-Bellman) equation, and proved the stability, existence and uniqueness of the scheme. The robustness and effectiveness of the proposed method were verified by numerical experiments.Periodic wave shock solutions of Burgers equations. A new approach.https://zbmath.org/1449.353752021-01-08T12:24:00+00:00"Bendaas, Saida"https://zbmath.org/authors/?q=ai:bendaas.saida"Alaa, Noureddine"https://zbmath.org/authors/?q=ai:alaa.noureddineSummary: In this paper we investigate the exact periodic wave shock solutions of the Burgers equations. Our purpose is to describe the asymptotic behavior of the solution in the Cauchy problem for viscid equation with small parameter \(\varepsilon\) and to discuss in particular the case of periodic wave shock. We show that the solution of this problem approaches the shock type solution for the Cauchy problem of the inviscid Burgers equation. The results are formulated in classical mathematics and proved with infinitesimal techniques of non standard analysis.Population model with age structure solved by finite element method.https://zbmath.org/1449.652462021-01-08T12:24:00+00:00"Hao, Yongle"https://zbmath.org/authors/?q=ai:hao.yongle"Zuo, Ping"https://zbmath.org/authors/?q=ai:zuo.ping"Zhu, Qing"https://zbmath.org/authors/?q=ai:zhu.qingSummary: We considered the numerical solution of population model with age structure and random diffusion term. Firstly, we integrated the model, obtained the simplified form of partial differential equation, and gave its variational treatment. Secondly, we used finite element method with equal partition as finite interval to solve the model numerically, and gave the full discrete form and error analysis. Finally, we gave a numerical solution and error analysis of the population model in numerical simulation.Solving a nonlinear inverse system of Burgers equations.https://zbmath.org/1449.652322021-01-08T12:24:00+00:00"Zeidabadi, Hamed"https://zbmath.org/authors/?q=ai:zeidabadi.hamed"Pourgholi, Reza"https://zbmath.org/authors/?q=ai:pourgholi.reza"Tabasi, Seyyed Hashem"https://zbmath.org/authors/?q=ai:tabasi.seyyed-hashemSummary: By applying finite difference formula to time discretization and the cubic B-splines for spatial variable, a numerical method for solving the inverse system of Burgers equations is presented. Also, the convergence analysis and stability for this problem are investigated and the order of convergence is obtained. By using two test problems, the accuracy of presented method is verified. Additionally, obtained numerical results of the cubic B-spline method are compared to trigonometric cubic B-spline method, exponential cubic B-spline method and radial basis function method. Implementation simplicity and less computational cost are the main advantages of proposed scheme compared to previous proposals.Estimation of lower bounds of blow-up time for variable sign solutions of semilinear parabolic equations with nonlocal sources.https://zbmath.org/1449.351262021-01-08T12:24:00+00:00"Sun, Aihui"https://zbmath.org/authors/?q=ai:sun.aihui"Chen, Peng"https://zbmath.org/authors/?q=ai:chen.peng"Li, Yan"https://zbmath.org/authors/?q=ai:li.yan.8|li.yan.5|li.yan.3|li.yan.2|li.yan|li.yan.9|li.yan.6|li.yan.7|li.yan.4|li.yan.1|li.yan.10"Bao, Kaihua"https://zbmath.org/authors/?q=ai:bao.kaihuaSummary: We considered the blow-up properties of solutions of Neumann boundary value problems for a class of semilinear parabolic equations with nonlocal sources. By constructing auxiliary functions and using the first order differential inequality, we gave the estimation of lower bound of blow-up time for solutions of the equations.Asymptotic behavior of solutions for a class of non-autonomous predator-prey systems with diffusion and stage structure.https://zbmath.org/1449.350702021-01-08T12:24:00+00:00"Hu, Huashu"https://zbmath.org/authors/?q=ai:hu.huashu"Pu, Zhilin"https://zbmath.org/authors/?q=ai:pu.zhilin"Shen, Yixin"https://zbmath.org/authors/?q=ai:shen.yixinSummary: In this paper, we study the asymptotic behavior of solutions for a class of non-autonomous predator-prey systems with diffusion and stage structure, including the forward and pullback behavior and the existence of pullback attractors. By using the sub-super solution method, the spectral theory for linear elliptic equations and the theory of attractors for non-autonomous differential equations, we obtain the solution estimates and the existence of pullback attractors.Analysis solution of lattice Boltzmann model for complex Ginzburg-Landau equation.https://zbmath.org/1449.354052021-01-08T12:24:00+00:00"Zhang, Jianying"https://zbmath.org/authors/?q=ai:zhang.jianying"Yan, Guangwu"https://zbmath.org/authors/?q=ai:yan.guangwu"Li, Ting"https://zbmath.org/authors/?q=ai:li.tingSummary: By using the method of Chapman analysis in the lattice Boltzmann model, we gave the general forms of a series of partial differential equations and Chapman polynomials. By solving the equilibrium distribution functions for complex Ginzburg-Landau equation, we gave the expressions of distribution functions on different time scales, and then obtained analysis solutions of the lattice Boltzmann for complex Ginzburg-Landau equation.Existence of exponential attractors for Boussinesq equations with memory.https://zbmath.org/1449.351012021-01-08T12:24:00+00:00"Wang, Meixia"https://zbmath.org/authors/?q=ai:wang.meixia"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: We considered the long-time dynamical behavior of solutions of Boussinesq equations with memory. Firstly, the original equation was transformed into a dynamical system by introducing a new variable. Secondly, the compactness of corresponding solutions semigroups associated with the problem we studied was proved by using the technique of operator decomposition and the compactness theorem on historical spaces. Finally, the exponential attractor of the memory-type Boussinesq equation was obtained by combining the existence of the exponential attractor, and the finite fractal dimension of the global attractor of the problem was also obtained.Random attractors for Berger equation with white noise.https://zbmath.org/1449.351032021-01-08T12:24:00+00:00"Wang, Xuan"https://zbmath.org/authors/?q=ai:wang.xuan"Song, An"https://zbmath.org/authors/?q=ai:song.anSummary: We considered the random asymptotic behaviors of solutions for the Berger equation with white noise. By introducing the equivalent process of isomorphic mapping, and using asymptotic a priori estimation technique and operator decomposition method, we proved the existence of random attractor in \( ({H^2} (U) \bigcap H_0^1 (U)) \times {L^2} (U)\).Strong relaxation limits of higher multidimensional Euler equations.https://zbmath.org/1449.353452021-01-08T12:24:00+00:00"Xu, Zhilin"https://zbmath.org/authors/?q=ai:xu.zhilin"Lin, Chunjin"https://zbmath.org/authors/?q=ai:lin.chunjinSummary: Using the method of energy integral, we discussed the global existence and strong relaxation limit of smooth solutions for higher dimensional Euler equations with damping when the initial values were sufficiently small, and obtained the uniform a priori estimates of the solutions. We proved that the asymptotic behavior of the global solution was controlled by the porous media equation when the relaxation time tended to zero.The decay property of solutions near the traveling waves for the second-order Camassa-Holm equation.https://zbmath.org/1449.351432021-01-08T12:24:00+00:00"Ding, Danping"https://zbmath.org/authors/?q=ai:ding.danping"Wang, Kai"https://zbmath.org/authors/?q=ai:wang.kai.2|wang.kai|wang.kai.3|wang.kai.1|wang.kai.4Summary: The decay properties of solutions around the traveling waves for Cauchy problem of the second-order Camassa-Holm equation is studied. Applying the extended pseudo-conformal transformation methods that appeared in the relevant works on the generalized Korteweg-de Vries equation, the solution with initial exponential decay value is obtained. It is proved that the solution can be controlled by an exponential function with decay.Exact solutions of generalized Burgers equation and \( (2+1)\)-dimensional Burgers equation.https://zbmath.org/1449.353822021-01-08T12:24:00+00:00"Jiang, Guifeng"https://zbmath.org/authors/?q=ai:jiang.guifengSummary: Since Burgers equation is a kind of important equation in nonlinear partial differential equation, a suitable function is constructed to transform the nonlinear partial differential equation into ordinary differential equation and then it can be solved. Some exact solutions of generalized Burgers equation and \( (2+1)\)-dimensional Burgers equation are obtained.Spatio-temporal deformation of kink-breather to the \( (2+1)\)-dimensional Nizhnik-Novikov-Veselov equation.https://zbmath.org/1449.353772021-01-08T12:24:00+00:00"Fu, Haiming"https://zbmath.org/authors/?q=ai:fu.haiming"Dai, Zhengde"https://zbmath.org/authors/?q=ai:dai.zhengdeSummary: By replacing the test function in Hirota method with a cyclic three-wave function, a new periodic soliton solution and a periodic dual soliton wave solution of the Nizhnik-Novikov-Veselov equation are obtained. Then the space-time bifurcation problem of the dynamical system described by the solution is discussed.A numerical solution of the pricing model of Asian options under sub-fractional jump-diffusion process.https://zbmath.org/1449.651832021-01-08T12:24:00+00:00"Hu, Pan"https://zbmath.org/authors/?q=ai:hu.panSummary: Under the assumption of the sub-fractional Ho-Lee stochastic interest rate model, this research firstly uses the delta hedging principle and establishes the partial differential equation of geometric average Asian options under the sub-fractional jump-diffusion process with transaction costs and dividends. Secondly, the pricing model is simplified to the Cauchy problem by using the variable substitution. Finally, a numerical solution of the pricing model is given by using the finite difference method and the composite trapezoid method. An example is also given to verify the effectiveness of the algorithm design.Reduced-order modelling for the Allen-Cahn equation based on scalar auxiliary variable approaches.https://zbmath.org/1449.652842021-01-08T12:24:00+00:00"Zhou, Xiaolan"https://zbmath.org/authors/?q=ai:zhou.xiaolan"Azaiez, Mejdi"https://zbmath.org/authors/?q=ai:azaiez.mejdi"Xu, Chuanju"https://zbmath.org/authors/?q=ai:xu.chuanjuSummary: In this article, we study the reduced-order modelling for Allen-Cahn equation. First, a collection of phase field data, i.e., an ensemble of snapshots of at some time instances is obtained from numerical simulation using a time-space discretization. The full discretization makes use of a temporal scheme based on the scalar auxiliary variable approach and a spatial spectral Galerkin method. It is shown that the time stepping scheme is unconditionally stable. Then a reduced order method is developed by using proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM). It is well-known that the Allen-Cahn equations have a nonlinear stability property, i.e., the free-energy functional decreases with respect to time. Our numerical experiments show that the discretized Allen-Cahn system resulting from the POD-DEIM method inherits this favorable property by using the scalar auxiliary variable approach. A few numerical results are presented to illustrate the performance of the proposed reduced-order method. In particular, the numerical results show that the computational efficiency is significantly enhanced as compared to directly solving the full order system.Existence of time-scale class of three dimensional fractional differential equations.https://zbmath.org/1449.354402021-01-08T12:24:00+00:00"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waell"Darus, Maslina"https://zbmath.org/authors/?q=ai:darus.maslinaSummary: The holomorphic results for fractional differential operator formals have been established. The analytic continuation of these outcomes has been studied for the fractional differential formal \[\begin{cases}
\frac{\partial^\alpha v(\wp,z)}{\partial\wp^\alpha}=\mathfrak{h}(\wr,z,v,\frac{\partial v}{\partial z},\frac{\partial^2v}{\partial z^2}),\quad \alpha\in[0,1),\\ v(a,z)=\psi(z),\quad \text{in a proximity to } z\in U,
\end{cases}\] where \(U\) is the open unit disk. The benefit of such a problem is that a generalization of two significant problems: the Cauchy problem and the diffusion problem. Moreover, the analytic solution is given inside the open unit disk, this leads to discuss the solution geometrically. The upper bound of outcomes is determined by suggesting a majorant analytic function in \(U\) (for two functions characterized by a power series, a majorant is the summation of a power series with positive coefficients which are not less than the absolute values of the conforming coefficients of the assumed series). This technique is very useful in approximation theory.Adaptive mesh method for solving a second-order hyperbolic equation.https://zbmath.org/1449.652152021-01-08T12:24:00+00:00"Zhou, Qin"https://zbmath.org/authors/?q=ai:zhou.qin"Yang, Yin"https://zbmath.org/authors/?q=ai:yang.yinSummary: In this paper, we study a class of second-order hyperbolic equations with small parameters. An adaptive moving mesh method for solving the equation with finite difference scheme is proposed, and the moving mesh algorithm is given. The superiority of the method is verified by numerical experiments, and the result on uniform mesh is improved.An approximate solution of the inverse spectral problem for the Laplace operator.https://zbmath.org/1449.354632021-01-08T12:24:00+00:00"Zakirova, G. A."https://zbmath.org/authors/?q=ai:zakirova.galiya-amrullovnaSummary: The paper is devoted to an approximate solution of the inverse spectral problem for the Laplace operator with the multiple spectrum.Express-control of atmospheric pollution source on the basis of serial functional specification.https://zbmath.org/1449.354602021-01-08T12:24:00+00:00"Chubatov, A. A."https://zbmath.org/authors/?q=ai:chubatov.andrey-alexeevich"Karmazin, V. N."https://zbmath.org/authors/?q=ai:karmazin.v-nSummary: The approach described allows to estimate the intensity of atmospheric pollution source step-by step on the basis of impurities concentration measurements in several stationary control points. The inverse problem was solved with the help of sequential functional specification method. The solution is presented in the form of a digital filter.The explicit splitting scheme for Maxwell's equations.https://zbmath.org/1449.780122021-01-08T12:24:00+00:00"Mingalev, I. V."https://zbmath.org/authors/?q=ai:mingalev.i-v"Mingalev, O. V."https://zbmath.org/authors/?q=ai:mingalev.oleg-v"Ahmetov, O. I."https://zbmath.org/authors/?q=ai:ahmetov.o-i"Suvorova, Z. V."https://zbmath.org/authors/?q=ai:suvorova.z-vSummary: This paper presents a new explicit scheme for the numerical integration of Maxwell's equations in isotropic and anisotropic dielectrics and conductors, as well as in the plasma in the case of the Vlasov-Maxwell system. In this scheme, the electric and magnetic fields are calculated in the same time points in the same spatial grid nodes, and a splitting in spatial directions and physical processes has been used. The scheme is monotonic and has 2nd order accuracy in time and 3rd order accuracy in the spatial variables. The presented scheme allows us to use a much larger step of time integration in modeling the propagation of low-frequency signals in the ionosphere than the widely used method of finite differences in the time domain with the same accuracy.Spatial convergence for semi-linear backward stochastic differential equations in Hilbert space: a mild approach.https://zbmath.org/1449.601052021-01-08T12:24:00+00:00"Abidi, Hani"https://zbmath.org/authors/?q=ai:abidi.hani"Pettersson, Roger"https://zbmath.org/authors/?q=ai:pettersson.rogerSummary: In this paper, we present convergence results of a spatial semi-discrete approximation of a Hilbert space-valued backward stochastic differential equations with noise driven by a cylindrical \(Q\)-Wiener process. Both the solution and its space discretization are formulated in mild forms. Under suitable assumptions of the final value and the drift, a convergence rate is established.Riemann function formulation for two equations with leading partial derivatives.https://zbmath.org/1449.351772021-01-08T12:24:00+00:00"Mironov, A. N."https://zbmath.org/authors/?q=ai:mironov.a-nSummary: Riemann functions were constructed for two linear equations with leading partial derivatives in two- and three-dimentional spaces.Boundary value problem solution of a second type confluent \(B\)-elliptic equation.https://zbmath.org/1449.352332021-01-08T12:24:00+00:00"Chebatoreva, E. V."https://zbmath.org/authors/?q=ai:chebatoreva.e-vSummary: Fundamental solution and potential of simple fiber and double layer types for confluent \(B\)-elliptic equation of the second kind are built in the given research work. with the help of above mentioned potentials boundary problems are converted into Fredholm integral equations of the second kind.The solution of the Tricomi problem for a mixed type equation with Bessel operator by the method of the theory of integral equations.https://zbmath.org/1449.353192021-01-08T12:24:00+00:00"Safina, R. M."https://zbmath.org/authors/?q=ai:safina.rimma-mSummary: In this article the existence and uniqueness of Tricomi problem solution is proven for Lavrentiev-Bitsadze equation with the Bessel operator
\[ x^{-k}\frac\partial{\partial x}\left(x^k\frac{\partial u}{\partial x}\right)+\operatorname{sign}y\frac{\partial^2u}{\partial y^2}=0 \]
in \(D\) area bounded by the rectifiable \(\Gamma\) curve, \(Oy\) axis and \(OC: x+y=0\) and \(BC: x-y=1\) characteristics, by the method of integral equations.Local and nonlocal value boundary problems for a third-order mixed-type equation equipped with Tricomi operator in its hyperbolic part.https://zbmath.org/1449.353232021-01-08T12:24:00+00:00"Balkizov, Zh. A."https://zbmath.org/authors/?q=ai:balkizov.zhuraslan-anatolevich|balkizov.zhiraslan-anatolevichSummary: The existence and uniqueness of local and nonlocal value boundary problems for third-order mixed-type equations with multiple characteristics is proved. Uniqueness of the problem solution is proved with energy-integral method. The existence of the solution is proved with equivalent reduction method to Fredholm integral equations of the second kind with the help of Green's function.Classic theorem by Lyapunov for differential equations in Hilbert spaces.https://zbmath.org/1449.350402021-01-08T12:24:00+00:00"Vavilov, S. A."https://zbmath.org/authors/?q=ai:vavilov.sergey-a"Fedotova, V. S."https://zbmath.org/authors/?q=ai:fedotova.v-sSummary: A theorem analogical to Lyapunov Classic Theorem is formulated for differential equations in Hilbert spaces. Example from the theory of partial differential equations is presented. The result automatically demonstrates the well-know conditions of continuum existence for periodic solutions of ordinary differential equations systems. Moreover, by applying the topological degree theory, these conditions can be set as less rigid than those formulated in Hopf Bifurcation Theory.Exact-analytic solutions and their dynamical properties of the fractional-order telegraph equation.https://zbmath.org/1449.354552021-01-08T12:24:00+00:00"Zhang, Qin"https://zbmath.org/authors/?q=ai:zhang.qin"Wu, Chun"https://zbmath.org/authors/?q=ai:wu.chunSummary: In order to study the exact solutions of three classes of classical fractional-order telegraph equations, a method combining the separation variable method and the homogeneous balance principle is used. The exact solutions of the space-fractional telegraph equation, the time-fractional telegraph equation and the time-space fractional telegraph equation are obtained by special transformation. The mechanical properties and the evolution of these solutions are further analyzed. The graphs of partial exact solutions with time and evolution of space development are given. Compared with the results in the existing literature, the exact solutions obtained here are new results, and the solving method and technique are much simpler than those in the previous literature.Qualitative analysis of a predator-prey system with Smith growth for prey.https://zbmath.org/1449.352662021-01-08T12:24:00+00:00"Lian, Tong"https://zbmath.org/authors/?q=ai:lian.tong"Yang, Wenbin"https://zbmath.org/authors/?q=ai:yang.wenbin"Li, Yanling"https://zbmath.org/authors/?q=ai:li.yanlingSummary: A Holling type-III predator-prey model with Smith growth subject to the homogeneous Neumann boundary condition is investigated in this paper. Firstly, the local stability of positive constant steady state solution is discussed. Secondly, we give a priori estimate of positive solutions. Finally, some sufficient conditions for the nonexistence and existence of nonconstant positive steady state solutions are derived.Influence of temporal form of laser pulse on surface temperature change by heating.https://zbmath.org/1449.780092021-01-08T12:24:00+00:00"Gureev, G. D."https://zbmath.org/authors/?q=ai:gureev.g-d"Gureev, D. M."https://zbmath.org/authors/?q=ai:gureev.dmitrii-mikhailovichSummary: The modeling of a laser influence on a half-endless medium surface by pulses of a different temporal forms in the one-measured linear approximation is performed. A comparative analysis of the change of heating temperature on the surface considering the time of laser influence is conducted. Optimum temporal parameters of the laser technological processes of a surface modification are determined.Study of the cosmic rays transport problems using second order parabolic type partial differential equation.https://zbmath.org/1449.850132021-01-08T12:24:00+00:00"Gil, Agnieszka"https://zbmath.org/authors/?q=ai:gil.agnieszka"Alania, Michael V."https://zbmath.org/authors/?q=ai:alania.michael-vSummary: It has been exactly 100 years since Hess's historical discovery: an extraterrestrial origin of cosmic rays [\textit{E. N. Parker}, ``Dynamics of the interplanetary gas and magnetic fields'', Astrophys. J. 128, 664--676 (1958; \url{doi:10.1086/146579})]. Galactic cosmic rays (GCR) being charged particles, penetrate the heliosphere and are modulated by the solar magnetic field. The propagation of cosmic rays is described by Parker's transport equation [\textit{V. F. Hess}, ``Über Beobachtungen der durchdringenden Strahlung bei sieben Freiballonfahrten'', Phys. Z. 13, 1084--1091 (1912)], which is a second order parabolic type partial differential equation. It is time dependent 4-variables (with \(r\), \(\theta\), \(\varphi\), \(R\), meaning: distance from the Sun, heliolatitudes, heliolongitudes and particles' rigidity, respectively) equation which is a well known tool for studying problems connected with solar modulation of cosmic rays. Transport equation contains all fundamental processes taking place in the heliosphere: convection, diffusion, energy changes of the GCR particles owing to the interaction with solar wind's inhomogeneities, drift due to the gradient and curvature of the regular interplanetary magnetic field and on the heliospheric current sheet.
In our paper we investigate a topic of the 27-day variation of the galactic cosmic rays intensity, which is connected with solar rotation. We numerically solve the Parker's transport equation involving in situ measurements of solar wind and magnetic field.Approach to solution of hyperbolic type equations by method of boundary elements.https://zbmath.org/1449.653392021-01-08T12:24:00+00:00"Fedotov, V. P."https://zbmath.org/authors/?q=ai:fedotov.vladimir-petrovich"Konteev, A. A."https://zbmath.org/authors/?q=ai:konteev.a-aSummary: The paper presents a method of solution of hyperbolic type problems by the boundary elements method. Using example of an one-dimensional case a resolving boundary equation was obtained and applicability of the method was shown. In particular, the paper shows a possibility of precise calculation of partial derivative, that is of high practical importance, as it allows to discover stresses correctly even for problems with singularities.Existence and uniqueness of mild solutions for Boltzmann equation under different potentials.https://zbmath.org/1449.353392021-01-08T12:24:00+00:00"Meng, Fei"https://zbmath.org/authors/?q=ai:meng.feiSummary: In this paper, the global existence and uniqueness of mild solutions for the Boltzmann equation are discussed. The method developed in a literature is used to extend some previous results from the hard sphere model to other potentials. Specifically, the existence and uniqueness of mild solutions in the case of non-hard sphere model are proved when the initial data decay exponentially with respect to variables \(x\). Since the non-hard sphere model is considered, the estimate of the collision operator in the case of soft potentials is introduced. Compared to the existing results, the space we build in this paper depends on the parameter in the collision kernel.The discrete Poisson equation and the heat equation with the exponential nonlinear term.https://zbmath.org/1449.352442021-01-08T12:24:00+00:00"Li, Yafeng"https://zbmath.org/authors/?q=ai:li.yafeng"Xin, Qiao"https://zbmath.org/authors/?q=ai:xin.qiao"Mu, Chunlai"https://zbmath.org/authors/?q=ai:mu.chunlaiSummary: This paper mainly studies the relations between the solution of the discrete Poisson equation and the solution of the discrete heat equation with exponential nonlinear term by monotone iterative method and comparison principle. When the solutions of the discrete Poisson equation exist, we discuss the asymptotic stability of the solutions to the discrete heat equation with exponential nonlinear term.Growth of smooth solutions of the incompressible Euler equations with external force on the disk.https://zbmath.org/1449.353442021-01-08T12:24:00+00:00"Gan, Lei"https://zbmath.org/authors/?q=ai:gan.lei"Deng, Dawen"https://zbmath.org/authors/?q=ai:deng.dawenSummary: We proved that for the incompressible Euler equations with external force on the disk, there was a smooth solution with vorticity gradient growing double exponentially. The result had been proved for the case without external force. We need to estimate the velocity field more carefully to get the same result when external force is present. The incompressible Euler equations with external force are similar to the inviscid Boussinesq system without heat conduction in both systems, the vorticity equation has a force term. Investigating the former may throw light on how to investigate the latter.Existence of entropy solutions to a doubly nonlinear integro-differential equation.https://zbmath.org/1449.450212021-01-08T12:24:00+00:00"Scholtes, Martin"https://zbmath.org/authors/?q=ai:scholtes.martin"Wittbold, Petra"https://zbmath.org/authors/?q=ai:wittbold.petraThe authors consider a class of doubly nonlinear problems with memory. They consider kernels of the type \(k(t)=t^{-\alpha}/\Gamma(1-\alpha)\). Doing so, the time-derivatives side becomes the fractional derivative of order \(\alpha\in(0,1)\) in the sense of Riemann-Liouville. The uniqueness of entropy solutions has been shown in a previous work. In this paper, the authors prove the existence of entropy solutions for general \(L^1\)-data and Dirichlet boundary conditions. The main idea of the existence proof is a modification of the regularization method by \textit{R. Landes} [J. Reine Angew. Math. 393, 21--38 (1989; Zbl 0664.35027)].
Reviewer: Vincenzo Vespri (Firenze)Existence and multiplicity of solutions for equations of \(p(x)\)-Laplace type in \(\mathbb{R}^N\) without AR-condition.https://zbmath.org/1449.351642021-01-08T12:24:00+00:00"Kim, Jae-Myoung"https://zbmath.org/authors/?q=ai:kim.jaemyoung"Kim, Yun-Ho"https://zbmath.org/authors/?q=ai:kim.yunho"Lee, Jongrak"https://zbmath.org/authors/?q=ai:lee.jun-ikSummary: We are concerned with the following elliptic equations with variable exponents \[ -\text{div}(\varphi(x,\nabla u))+V(x)|u|^{p(x)-2}u=\lambda f(x,u)\quad\text{in}\quad\mathbb{R}^N, \] where the function \(\varphi(x,v)\) is of type \(|v|^{p(x)-2}v\) with continuous function \(p:\mathbb{R}^N\to(1,\infty)\), \(V:\mathbb{R}^N\to(0,\infty)\) is a continuous potential function, and \(f:\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}\) satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass theorem and fountain theorem. Second, we determine the precise positive interval of \(\lambda\)'s for which our problem admits a nontrivial solution with simple assumptions in some sense.Logarithmic decay of wave equations with Cauchy-Ventcel boundary conditions.https://zbmath.org/1449.352822021-01-08T12:24:00+00:00"Fu, Xiaoyu"https://zbmath.org/authors/?q=ai:fu.xiaoyu"Liu, Xu"https://zbmath.org/authors/?q=ai:liu.xu"Zhu, Xianzheng"https://zbmath.org/authors/?q=ai:zhu.xianzhengSummary: This paper is devoted to a study of decay properties for a class of wave equations with Cauchy-Ventcel boundary conditions and a local internal damping. Based on an estimate on the resolvent operator, solutions of the wave equations under consideration are proved to decay logarithmically without any geometric control condition. The proof of the decay result relies on the interpolation inequalities for an elliptic equation with Cauchy-Ventcel boundary conditions and the estimate of the resolvent operator for that equation. The main tool to derive the desired interpolation inequality is global Carleman estimate.On the global well-posedness of 3-D Navier-Stokes equations with vanishing horizontal viscosity.https://zbmath.org/1449.353472021-01-08T12:24:00+00:00"Abidi, Hammadi"https://zbmath.org/authors/?q=ai:abidi.hammadi"Paicu, Marius"https://zbmath.org/authors/?q=ai:paicu.mariusSummary: We study, in this paper, the axisymmetric 3-D Navier-Stokes system where the horizontal viscosity is zero. We prove the existence of a unique global solution to the system with initial data in Lebesgue spaces.Asymptotic behavior for a class of non-autonomous nonclassical parabolic equations with delay on unbounded domain.https://zbmath.org/1449.350872021-01-08T12:24:00+00:00"Zhang, Fanghong"https://zbmath.org/authors/?q=ai:zhang.fanghong"Bai, Lihong"https://zbmath.org/authors/?q=ai:bai.lihongSummary: In this article, we investigate the longtime behavior for the following non-autonomous nonclassical parabolic equations on unbounded domain \[{u_t}-\Delta {u_t}-\Delta u+\lambda u = f (x,u (x,t-\rho (t))) + g (x,t).\] Under some suitable conditions on the delay term \(f\) and the non-autonomous forcing term \(g\), we prove the existence of uniform attractors in Banach space \(C_{H^1 (\mathbb{R}^\mathbb{N})}\) for the multi-valued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.Elementary energy estimates of three-dimensional axially symmetric nonhomogeneous incompressible MHD equations under the rectangular coordinate system.https://zbmath.org/1449.353542021-01-08T12:24:00+00:00"Liu, Fangjun"https://zbmath.org/authors/?q=ai:liu.fangjun"Ma, Qian"https://zbmath.org/authors/?q=ai:ma.qianSummary: It is known that elementary energy estimates of three-dimensional axially symmetric nonhomogeneous incompressible MHD equations is crucial for the global existence of the subsequent solution to MHD equations. In order to make it applied widely, by relying on elementary energy estimates of three-dimensional axially symmetric MHD equations under the cylindrical coordinate system and removing the constraint condition of the cylindrical coordinate, the problem-solving process of each of the items in elementary energy estimates of MHD equations is explicitly given under the rectangular coordinate system. Finally, by making full use of mass conservation formula and the incompressible constraint condition of MHD equations under the rectangular coordinate system, the elementary energy estimates of MHD equations under the rectangular coordinate system are successfully deduced.Global existence of solutions to the Keller-Segel system with initial data of large mass.https://zbmath.org/1449.350042021-01-08T12:24:00+00:00"Shi, Renkun"https://zbmath.org/authors/?q=ai:shi.renkunSummary: In this paper, the Cauchy problem of the classical Keller-Segel chemotaxis model with initial data of large mass is considered. By the Green's function method and scaling technique, we prove that if \(||u_0||_{L^1}^{\frac{1}{n}}||u_0||_{L^\infty}^{1-\frac{1}{n}}\) and \(\kappa ||\nabla v_0||_{L^p}\) are less than some constant, then the problem always admits a unique global classical solution, even when the initial mass \(||u_0||_{L^1}\) is large. Moreover, the decay rates of the classical solution are also obtained.Well-posedness for a plate equation with nonlocal source term.https://zbmath.org/1449.351862021-01-08T12:24:00+00:00"Liu, Gongwei"https://zbmath.org/authors/?q=ai:liu.gongwei"Zhao, Ruimin"https://zbmath.org/authors/?q=ai:zhao.ruimin"Zhang, Hongwei"https://zbmath.org/authors/?q=ai:zhang.hongweiSummary: In this paper, we investigate the initial boundary value problem for a plate equation with nonlocal source term. The local, global existence and exponential decay result are established under certain conditions. Moreover, we also prove the blow-up in finite time and the lifespan of solution under certain conditions.A regularity criterion via the pressure on the three-dimensional Boussinesq fluid equations.https://zbmath.org/1449.351362021-01-08T12:24:00+00:00"Li, Xiao"https://zbmath.org/authors/?q=ai:li.xiao"Li, Yingchao"https://zbmath.org/authors/?q=ai:li.yingchaoSummary: In this paper, we investigate the regularity criterion via the pressure of weak solutions to the Boussinesq fluid equations in three dimensions. We obtain that for \(\frac{2}{q} + \frac{3}{\gamma} \le 2\), \(\frac{3}{2} \le \gamma \le 3\) if \({\partial_3}\pi \in {L^\gamma} (0,T;{L^q})\), then the weak solution (\(u,\theta\)) is regular on \( (0,T]\).On convergence of the iterative process for the third order pseudo-parabolic equation with nonlocal boundary value conditions in a multidimensional domain.https://zbmath.org/1449.354712021-01-08T12:24:00+00:00"Beshtokov, Murat Khamidbievich"https://zbmath.org/authors/?q=ai:beshtokov.murat-khamidbievichSummary: In this paper the nonlocal boundary value problem for the pseudo-parabolic equation of the third-order in a multidimensional domain is considered. Using an iterative method, the solving process of the nonlocal boundary value problem is reduced to solving the series of some local problems. An a priori estimate for the convergence of the iterative method in the norm \(W^1_2(G)\) is obtained.Globally bounded solutions in a chemotaxis model of quasilinear parabolic type.https://zbmath.org/1449.352772021-01-08T12:24:00+00:00"Liu, Bingchen"https://zbmath.org/authors/?q=ai:liu.bingchen"Dong, Mengzhen"https://zbmath.org/authors/?q=ai:dong.mengzhenSummary: In this paper, we consider a quasilinear parabolic-parabolic chemotaxis model with nonlinear diffusivity, aggregation and logistic damping source: \[\begin{cases}{u_t} =\nabla\cdot (D (u)\nabla u)-\nabla\cdot (S (u)\nabla v)+g (u), \\{v_t} = \Delta v - v + u,\end{cases}\] where \({k_1}{e^{pu}}\le D (u)\) or \({k_1}{u^p}\le D (u)\); \({k_2}{e^{qu}}\le S (u)\le {k_3}{e^{qu}}\); \(g (u)\le a-b{e^{ku}}\). It is proved that, if \(q< k-1\) or \(q = k-1\) and \(b> {b_0}\) for some constant \({b_0}>0\), then there exists a unique classical solution which is globally bounded. The results show the effectiveness of the aggregation and the logistic damping source on the existence of globally bounded solutions.Finite difference methods for the time fractional advection-diffusion equation.https://zbmath.org/1449.651942021-01-08T12:24:00+00:00"Ma, Yan"https://zbmath.org/authors/?q=ai:ma.yan"Musbah, F. S."https://zbmath.org/authors/?q=ai:musbah.f-sSummary: In this paper, three implicit finite difference methods are developed to solve a one dimensional time fractional advection-diffusion equation. The fractional derivative is treated by applying the right shifted Grunwald-Letnikov formula of order \(\alpha \in (0, 1)\). We investigate the stability analysis by using the von Neumann method with mathematical induction and prove that these three proposed methods are unconditionally stable. Numerical results are presented to demonstrate the effectiveness of the schemes mentioned in this paper.Classification of blow-up limits for the sinh-Gordon equation.https://zbmath.org/1449.352242021-01-08T12:24:00+00:00"Jevnikar, Aleks"https://zbmath.org/authors/?q=ai:jevnikar.aleks"Wei, Juncheng"https://zbmath.org/authors/?q=ai:wei.juncheng"Yang, Wen"https://zbmath.org/authors/?q=ai:yang.wenSummary: The aim of this paper is to use a selection process and a careful study of the interaction of bubbling solutions to show a classification result for the blow-up values of the elliptic sinh-Gordon equation \[\Delta u+h_1e^u-h_2e^{-u}=0\qquad\text{in }B_1\subset\mathbb{R}^2.\] In particular, we get that the blow-up values are multiple of \(8\pi\). It generalizes the result of \textit{J. Jost} et al. [Calc. Var. Partial Differ. Equ. 31, No. 2, 263--276 (2008; Zbl 1137.35061)] where the extra assumption \(h_1=h_2\) is crucially used.The sinc-Galerkin methods of the Burgers' equation based on the Hopf-Cole transformation.https://zbmath.org/1449.652652021-01-08T12:24:00+00:00"Yang, Mei"https://zbmath.org/authors/?q=ai:yang.mei"Zhao, Fengqun"https://zbmath.org/authors/?q=ai:zhao.fengqun"Guo, Chong"https://zbmath.org/authors/?q=ai:guo.chongSummary: In this paper, the Sinc-Galerkin method is used to solve the initial boundary value problem of the Burgers equation. Firstly, the Hopf-Cole transform is used to transform the second-order nonlinear Burgers equation into a second-order linear equation, while the first type of boundary condition is changed into the second type of boundary condition. Then the time derivative is discretized in \(\theta\)-weighted scheme, and the spatial derivative is discretized by the Sinc-Galerkin method. For the second type of boundary condition, the basis functions are introduced at the ends based on Hermite interpolation method. Finally, the validity and exponential convergence of the Sinc-Galerkin method is verified by numerical examples. A comparison between the numerical solution and the exact solution shows that the numerical scheme constructed in this paper has high accuracy, and can effectively capture physical phenomena such as shock waves.Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture.https://zbmath.org/1449.900932021-01-08T12:24:00+00:00"Yang, Qi"https://zbmath.org/authors/?q=ai:yang.qi"Wang, Lei"https://zbmath.org/authors/?q=ai:wang.lei.14"Feng, Enmin"https://zbmath.org/authors/?q=ai:feng.enmin"Yin, Hongchao"https://zbmath.org/authors/?q=ai:yin.hongchao"Xiu, Zhilong"https://zbmath.org/authors/?q=ai:xiu.zhilongSummary: In this paper, we present a framework to infer the possible transmembrane transport of intracellular substances. Considering four key enzymes, a modified fourteen-dimensional nonlinear hybrid dynamic system is established to describe the microbial continuous culture with enzyme-catalytic and genetic regulation. A novel quantitative definition of biological robustness is proposed to characterize the system's resilience when system parameters were perturbed. It not only considers the expectation of system output data after parameter disturbance but also considers the influence of the variance of these data. In this way, the definition can be used as an objective function of the system identification model due to the lack of data on the concentration of intracellular substances. Then, we design a parallel computing method to solve the system identification model. Numerical results indicate that the most likely transmembrane mode of transport is active transport coupling with passive diffusion for glycerol and 1,3-propanediol.Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations.https://zbmath.org/1449.652932021-01-08T12:24:00+00:00"Saratha, S. R."https://zbmath.org/authors/?q=ai:saratha.s-r"Bagyalakshmi, M."https://zbmath.org/authors/?q=ai:bagyalakshmi.morachan"Sai Sundara Krishnan, G."https://zbmath.org/authors/?q=ai:saisundarakrishnan.gSummary: This paper presents a novel hybrid technique which is developed by incorporating the fractional derivatives in the generalised integral transform method. Homotopy analysis method is combined with fractional generalised integral transform method to solve the fractional order nonlinear differential equations. The performance of the proposed method is analysed by solving various categories of nonlinear fractional differential equations like Navier Stokes's model and Riccatti equations, etc. Unlike the other analytical methods, the hybrid method provides a better way to control the convergence region of the obtained series solution through an auxiliary parameter \(h\). Furthermore, as proposed in this paper, the `Fractional Generalised Homotopy Analysis Method' along with the several examples reveal that this method can be effectively used as a tool for solving various kinds of nonlinear fractional differential equations.A geometric proof of \({W^{1,p}}\) interior estimate for the fractional Laplace equations.https://zbmath.org/1449.354342021-01-08T12:24:00+00:00"Chen, Runhua"https://zbmath.org/authors/?q=ai:chen.runhua"Cao, Yi"https://zbmath.org/authors/?q=ai:cao.yiSummary: In order to study the solution of fractional Laplace equation on \({B_2}\), a geometric method is considered, in which \({B_2}\) is a sphere with the origin as the center and 2 as the radius. The \({W^{1,2}}\) estimate of the solution of the fractional Laplace equations is established by fundamental solution of \( (-\Delta)^{\sigma/2} (0 <\sigma < 2)\). By using geometric method, the \({W^{1,p}}\) interior estimate of the solution of this equation is given based on the maximal function and the covering lemma.Critical curves and non-extinction condition for non-Newtonian filtration equations coupled via boundary sources.https://zbmath.org/1449.352622021-01-08T12:24:00+00:00"Ling, Zhengqiu"https://zbmath.org/authors/?q=ai:ling.zhengqiuSummary: This paper is concerned with the critical curves and non-extinction condition of the solutions for a non-Newtonian polytropic filtration equation coupled via nonlinear boundary sources in \(\mathbb{R}^N\). The critical global existence curve and the critical Fujita curve are given by means of various self-similar supersolutions and subsolutions. In particular, it is shown that the above two critical curves depend not only on the parameters in the problem, but also the dimension \(N\) of space. This differs greatly from the known results for dimension \(N = 1\). In addition, the non-extinction conditions of solutions for this problem are given.Traveling wave solutions of a epidemic model with nonlocal diffusion, immigration and spatio-temporal delays.https://zbmath.org/1449.351492021-01-08T12:24:00+00:00"Zhang, Lijuan"https://zbmath.org/authors/?q=ai:zhang.lijuan"Wang, Fuchang"https://zbmath.org/authors/?q=ai:wang.fuchangSummary: An epidemic model with time-delay and spatial diffusion is established, and the existence of traveling wave solutions is discussed. Firstly, in view of the phenomenon that the infected individuals in the population have free movement and transmission of diseases during the incubation period, the existence of traveling wave solutions is determined by using the basic reproduction number and the minimum speed. Then, Schauder fixed point theorem is used to determine the existence of traveling wave solutions. The existence of fixed point is proved by fixed point theorem, and the existence of traveling wave solution is obtained.Multiple solutions for a class of elliptic systems with variable exponent.https://zbmath.org/1449.352202021-01-08T12:24:00+00:00"Zhang, Shengui"https://zbmath.org/authors/?q=ai:zhang.shenguiSummary: By using the theory of variable exponent Sobolev spaces and the Clark's theorem in critical point theory, a class of boundary value problems of elliptic systems wiDE071770077
th variable exponent is studied. When the nonlinearity has a \(p^-\)-sublinear growth near zero, the existence of infinitely many solutions for this system is obtained.Properties of solutions of \(n\)-dimensional incompressible Navier-Stokes equations.https://zbmath.org/1449.353432021-01-08T12:24:00+00:00"Zhang, Linghai"https://zbmath.org/authors/?q=ai:zhang.linghaiSummary: This paper considers the $n$-dimensional incompressible Navier-Stokes equations $$\begin{array}{*{20}{c}}\frac{\partial}{\partial t}u - \alpha\Delta u + (u\cdot \nabla)u + \nabla p = f (x,t),\;\; \nabla\cdot u = 0,\;\; \nabla\cdot f = 0,\\ u (x,0) = {u_0} (x),\;\; \nabla \cdot {u_0} = 0.\end{array}$$ There exists a global weak solution under some assumptions on the initial function and the external force. It is well known that the global weak solutions become sufficiently small and smooth after a long time. Here are several very interesting questions about the global weak solutions of the Cauchy problems for the $n$-dimensional incompressible Navier-Stokes equations.\\$\bullet$ Can we establish better decay estimates with sharp rates not only for the global weak solutions but also for all order derivatives of the global weak solutions?\\$\bullet$ Can we accomplish the exact limits of all order derivatives of the global weak solutions in terms of the given information?\\$\bullet$ Can we use the global smooth solution of the linear heat equation, with the same initial function and the external force, to approximate the global weak solutions of the Navier-Stokes equations?\\$\bullet$ If we drop the nonlinear terms in the Navier-Stokes equations, will the exact limits reduce to the exact limits of the solutions of the linear heat equation?\\$\bullet$ Will the exact limits of the derivatives of the global weak solutions of the Navier-Stokes equations and the exact limits of the derivatives of the global smooth solution of the heat equation increase at the same rate as the order $m$ of the derivative increases? In another word, will the ratio of the exact limits for the derivatives of the global weak solutions of the Navier-Stokes equations be the same as the ratio of the exact limits for the derivatives of the global smooth solutions for the linear heat equation? \\The positive solutions to these questions obtained in this paper will definitely help us to better understand the properties of the global weak solutions of the incompressible Navier-Stokes equations and hopefully to discover new special structures of the Navier-Stokes equations.A fluid-particle model with electric fields near a local Maxwellian with rarefaction wave.https://zbmath.org/1449.353362021-01-08T12:24:00+00:00"Wang, Teng"https://zbmath.org/authors/?q=ai:wang.teng"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.2Summary: The paper is concerned with time-asymptotic behavior of solution near a local Maxwellian with rarefaction wave to a fluid-particle model described by the Vlasov-Fokker-Planck equation coupled with the compressible and inviscid fluid by Euler-Poisson equations through the relaxation drag frictions, Vlasov forces between the macroscopic and microscopic momentums and the electrostatic potential forces. Precisely, based on a new micro-macro decomposition around the local Maxwellian to the kinetic part of the fluid-particle coupled system, we show the time-asymptotically nonlinear stability of rarefaction wave to the one-dimensional compressible inviscid Euler equations coupled with both the Vlasov-Fokker-Planck equation and Poisson equation.On the well-posedness of weak solutions.https://zbmath.org/1449.351352021-01-08T12:24:00+00:00"Liu, Taiping"https://zbmath.org/authors/?q=ai:liu.tai-pingSummary: This is to comment on the well-posedness of weak solutions for the initial value problem for partial differential equations. In recent decades, and particularly in recent years, there have been substantial progresses on construction by convex integration for the study of non-uniqueness of solutions for incompressible Euler equations, and even for compressible Euler equations. This prompts the question of whether it is possible to give a sense of well-posedness, which is narrower than the canonical Hadamard sense, so that the evolutionary equations are well-posed. We give a brief and partial review of the related results and offer some thoughts on this fundamental topic.Multiple vortices for the shallow water equation in two dimensions.https://zbmath.org/1449.352222021-01-08T12:24:00+00:00"Cao, Daomin"https://zbmath.org/authors/?q=ai:cao.daomin"Liu, Zhongyuan"https://zbmath.org/authors/?q=ai:liu.zhongyuanSummary: In this paper, we construct stationary classical solutions of the shallow water equation with vanishing Froude number \(Fr\) in the so-called lake model. To this end we need to study solutions to the following semilinear elliptic problem \[\begin{cases} -{\varepsilon^2}{\mathrm{div}}\left (\frac{\nabla u}{b}\right) = b\left (u-q\log\frac{1}{\varepsilon}\right)_+^p, & {\mathrm{in}}\;\Omega,\\ u=0, & {\mathrm{on}}\; \partial\Omega,\end{cases}\] for small \(\varepsilon > 0\), where \(p > 1\), \({\mathrm{div}}\left (\frac{\nabla q}{b}\right) = 0\) and \(\Omega \subset \mathbb{R}^2\) is a smooth bounded domain. We show that if \(\frac{q^2}{b}\) has \(m\) strictly local minimum (maximum) points \({{\bar z}_i}, i = 1,\cdots, m\), then there is a stationary classical solution approximating stationary \(m\) points vortex solution of shallow water equations with vorticity \(\sum\limits_{i=1}^m \frac{2\pi q ({\bar z}_i)}{b ({\bar z}_i)}\). Moreover, strictly local minimum points of \(\frac{q^2}{b}\) on the boundary can also give vortex solutions for the shallow water equation.Local times of the solution to stochastic heat equation with fractional noise.https://zbmath.org/1449.601232021-01-08T12:24:00+00:00"Wang, Zhi"https://zbmath.org/authors/?q=ai:wang.zhi"Yan, Litan"https://zbmath.org/authors/?q=ai:yan.litan"Yu, Xianye"https://zbmath.org/authors/?q=ai:yu.xianyeSummary: In this paper, we study the collision and intersection local times of the solution to stochastic heat equation with additive fractional noise. We mainly prove its existence and smoothness properties through local nondeterminism and chaos expansion.Stability of traveling waves in a population dynamic model with delay and quiescent stage.https://zbmath.org/1449.350652021-01-08T12:24:00+00:00"Zhou, Yonghui"https://zbmath.org/authors/?q=ai:zhou.yonghui"Yang, Yunrui"https://zbmath.org/authors/?q=ai:yang.yunrui"Liu, Kepan"https://zbmath.org/authors/?q=ai:liu.kepanSummary: This article is concerned with a population dynamic model with delay and quiescent stage. By using the weighted-energy method combining continuation method, the exponential stability of traveling waves of the model under non-quasi-monotonicity conditions is established. Particularly, the requirement for initial perturbation is weaker and it is uniformly bounded only at \(x = +\infty\) but may not be vanishing.Asymptotic behavior of solutions of the bipolar quantum drift-diffusion model in the quarter plane.https://zbmath.org/1449.350732021-01-08T12:24:00+00:00"Liu, Fang"https://zbmath.org/authors/?q=ai:liu.fang|liu.fang.1"Li, Yeping"https://zbmath.org/authors/?q=ai:li.yepingSummary: In this study, we consider the one-dimensional bipolar quantum drift-diffusion model, which consists of the coupled nonlinear fourth-order parabolic equation and the electric field equation. We first show the global existence of the strong solution of the initial boundary value problem in the quarter plane. Moreover, we show the self-similarity property of the strong solution of the bipolar quantum drift-diffusion model in the large time. Namely, we show the unique global strong solution with strictly positive density to the initial boundary value problem of the quantum drift-diffusion model, which in large time, tends to have a self-similar wave at an algebraic time-decay rate. We prove them in an energy method.Global existence and \({L_p}\) decay estimate of solutions for viscous Cahn-Hilliard equation with inertial term.https://zbmath.org/1449.350022021-01-08T12:24:00+00:00"Xu, Hongmei"https://zbmath.org/authors/?q=ai:xu.hongmei"Shi, Yu"https://zbmath.org/authors/?q=ai:shi.yuSummary: In this paper, Cauchy problem of viscous Cahn-Hilliard equation with inertial term in multi-space dimension is considered. Based on the detailed analysis of Green function, using fixed point theorem, we get the global in-time existence of classical solution. Furthermore, we get \({L_p}\) decay rate of the solution.On a proplem for generalized Boussinesq-Love equation.https://zbmath.org/1449.352952021-01-08T12:24:00+00:00"Zhegalov, Valentin Ivanovich"https://zbmath.org/authors/?q=ai:zhegalov.valentin-ivanovichSummary: For a fourth-order equation with two independent variables a variant of the Goursat problem with data on two intersecting characteristics is considered. It includes not only the construction of the desired function, but also the coefficients of the equation. Thus, we are talking about the inverse problem of determining the coefficients of the equation. The method of construction of conditions providing allocation of infinite number of sets of this type equations is offered, for which the problem under consideration is solvable in quadratures. Instead of introducing additional boundary conditions, restrictions on the structure of the equation are proposed, related to the possibilities of its factorization.Dirichlet problem for mixed type equation with characteristic degeneration.https://zbmath.org/1449.353282021-01-08T12:24:00+00:00"Sabitova, Yuliya Kamilevna"https://zbmath.org/authors/?q=ai:sabitova.yuliya-kamilevnaSummary: For a mixed elliptic-hyperbolic type equation with characteristic degeneration, the first boundary value problem in a rectangular region is investigated. The criterion for the uniqueness of the solution of the problem is established. Earlier, in proving the uniqueness of solutions of boundary value problems for equations of mixed type, the extremum principle or the method of integral identities was used. The uniqueness of the solution to this problem is established on the basis of the completeness of the system of eigenfunctions of the corresponding one-dimensional spectral problem. The solution of the problem is constructed as a sum of a series in the system of eigenfunctions. When we proved the convergence of the obtained series, the problem of small denominators of a more complicated structure than in other known works arose. These denominators contain a parameter depending on the lengths of the sides of the rectangle in the hyperbolic part of the domain and the exponent of the degree of degeneration. In this connection, estimates are established about separation from zero with the corresponding asymptotics, in cases where this parameter is a natural, rational and algebraic irrational number of degree two. If this parameter is not an algebraic irrational number of degree two, then the solution of the problem as a sum of a series does not exist. Using the obtained estimates, the uniform convergence of the constructed series in the class of regular solutions is justified under certain sufficient conditions with respect to the boundary functions. The stability of the solution of the problem with respect to the boundary functions in the norms of the space of summable functions and in the space of continuous functions is also proved.Second boundary-value problem for the generalized Aller-Lykov equation.https://zbmath.org/1449.354442021-01-08T12:24:00+00:00"Kerefov, Marat Aslanbievich"https://zbmath.org/authors/?q=ai:kerefov.marat-aslanbievich"Gekkieva, Sakinat Khadanovna"https://zbmath.org/authors/?q=ai:gekkieva.sakinat-khadanovnaSummary: The equations that describe a new type of wave motion arise in the course of mathematical modeling for continuous media with memory. This refers to differential equations of fractional order, which form the basis for most mathematical models describing a wide class of physical and chemical processes in media with fractal geometry. The paper presents a qualitatively new equation of moisture transfer, which is a generalization of the Aller-Lykov equation, by introducing the concept of the fractal rate of change in humidity clarifying the presence of flows affecting the potential of humidity. We have studied the second boundary value problem for the Aller-Lykov equation with the fractional Riemann-Liouville derivative. The existence of a solution to the problem has been proved by the Fourier method. To prove the uniqueness of the solution we have obtained an a priori estimate, in terms of a fractional Riemann-Liouville using the energy inequality method.On some classes of nonlocal problems in Musielak-Sobolev spaces.https://zbmath.org/1449.352062021-01-08T12:24:00+00:00"Avci, Mustafa"https://zbmath.org/authors/?q=ai:avci.mustafaSummary: In the present paper, we study the existence of solutions to some classes of nonlocal elliptic equations under different type of boundary conditions. Problems are settled in Musielak-Sobolev spaces. The main tool is variational approach, however, various auxiliary tools from the theory of nonlinear functional analysis, convex analysis and critical point theory are also applied.Stability and convergence of difference schemes for the multi-term time-fractional diffusion equation with generalized memory kernels.https://zbmath.org/1449.651882021-01-08T12:24:00+00:00"Khibiev, Aslanbek Khizirovich"https://zbmath.org/authors/?q=ai:khibiev.aslanbek-khizirovichSummary: In this paper, a priori estimate for the corresponding differential problem is obtained by using the method of the energy inequalities. We construct a difference analog of the multi-term Caputo fractional derivative with generalized memory kernels (analog of L1 formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the \((2-\alpha_0)\)-th order in time for the generalized multi-term time-fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid \(L_2 \)-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by numerical calculations carried out for some test problems.Simple proof of the adiabatic theorem.https://zbmath.org/1449.353622021-01-08T12:24:00+00:00"Katanaev, Mikhaĭl Orinovich"https://zbmath.org/authors/?q=ai:katanaev.mikhail-orinovichSummary: Simple proof of the adiabatic theorem is given in a finite dimensional case for nondegenerate as well as degenerate states. The estimate is obtained for the deviation of the norm of the solution of the Schrödinger equation which is uniform on the parameter in the Hamiltonian.On the correctness of boundary value problems for the mixed type equation of the second kind.https://zbmath.org/1449.353292021-01-08T12:24:00+00:00"Sabitov, Kamil' Basirovich"https://zbmath.org/authors/?q=ai:sabitov.kamil-basirovich"Egorova, Irina Petrovna"https://zbmath.org/authors/?q=ai:egorova.irina-petrovnaSummary: In this paper, the intervals of change in the exponent of the degree of degeneration of a mixed-type equation with characteristic degeneration are established. The first boundary problem and the modified boundary problem (analogue of the Keldysh problem) with the conditions of periodicity are correctly set. In the case of the first problem, a criterion for the uniqueness of its solution is manifested. It is shown that the solution of the analogue of the Keldysh problem is unique up to a term of a linear function. Solutions are constructed as the sum of series of eigenfunctions of the corresponding one-dimensional spectral problem. In justifying the convergence of a series representing the solution of the first boundary-value problem, the problem of small denominators of a more complex structure arises in the class of regular solutions of this equation than in previously known works. The estimate on separation from zero is established with the corresponding asymptotic. Based on this estimate, sufficient conditions are found for the boundary functions to substantiate the uniform convergence of the series and their derivatives up to the second order inclusive.On a mathematical model of non-isothermal creeping flows of a fluid through a given domain.https://zbmath.org/1449.353502021-01-08T12:24:00+00:00"Domnich, Anastasiya Aleksandrovna"https://zbmath.org/authors/?q=ai:domnich.anastasiya-aleksandrovna"Baranovskiĭ, Evgeniĭ Sergeevich"https://zbmath.org/authors/?q=ai:baranovskii.evgenii-sergeevich"Artemov, Mikhaĭl Anatol'evich"https://zbmath.org/authors/?q=ai:artemov.mikhail-anatolevichSummary: We study a mathematical model describing steady creeping flows of a non-uniformly heated incompressible fluid through a bounded 3D domain with locally Lipschitz boundary. The model under consideration is a system of second-order nonlinear partial differential equations with mixed boundary conditions. On in-flow and out-flow parts of the boundary the pressure, the temperature and the tangential component of the velocity field are prescribed, while on impermeable solid walls the no-slip condition and a Robin-type condition for the temperature are used. For this boundary-value problem, we introduce the concept of a weak solution (a pair ``velocity-temperature''), which is defined as a solution to some system of integral equations. The main result of the work is a theorem on the existence of weak solutions in a subspace of the Cartesian product of two Sobolev's spaces. To prove this theorem, we give an operator interpretation of the boundary value problem, derive a priori estimates of solutions, and apply the Leray-Schauder fixed point theorem. Moreover, energy equalities are established for weak solutions.Flux approximation to the zero dissipation limit to rarefaction wave for 1-D compressible isentropic Navier-Stokes equations.https://zbmath.org/1449.353412021-01-08T12:24:00+00:00"Wang, Jinni"https://zbmath.org/authors/?q=ai:wang.jinni"Liu, Jinjing"https://zbmath.org/authors/?q=ai:liu.jinjingSummary: The paper studies the flux approximation to the zero dissipation limit to rarefaction wave for the one-dimensional compressible isentropic Navier-Stokes equations. Given that the solution of the corresponding Euler equations is rarefaction wave with one-side vacuum state, we employ the flux approximation method to control the degeneracies caused by the vacuum in the rarefaction wave, and we construct a sequence of solutions to the compressible isentropic Navier-Stokes equations. Then we adopt the elementary energy method to prove that the solutions converge to the rarefaction wave as the viscosity vanishes. In addition, the uniform convergence rate is obtained.General scheme of solution for a boundary value problem of non-steady thermal conductivity with inner thermal sources for multi-layer constructions.https://zbmath.org/1449.800102021-01-08T12:24:00+00:00"Averin, B. V."https://zbmath.org/authors/?q=ai:averin.b-vSummary: A general closed analytical solution for non-stationary thermal conductivity is obtained for non-steady thermal conductivity in multi-layer walls of a plane, cylindrical and spherical forms with inner thermal sources.A new variational approach for inverse source problems.https://zbmath.org/1449.350062021-01-08T12:24:00+00:00"Hu, Qiya"https://zbmath.org/authors/?q=ai:hu.qiya"Shu, Shi"https://zbmath.org/authors/?q=ai:shu.shi"Zou, Jun"https://zbmath.org/authors/?q=ai:zou.junSummary: We propose a new variational approach for recovering a general source profile in an elliptic system, using measurement data from the interior of the physical domain. The solution of the ill-posed inverse source problem is achieved by solving only one well-posed direct elliptic problem, resulting in the same computational cost as the one for the direct problem, and hence making the whole solution process of the inverse problem much less expensive than most existing methods. The resulting approximate solution is shown to be stable with respect to the change of the noise in the observation data, and a desired error estimate is also established in terms of the mesh size and the noise level in observation data. Numerical experiments are presented to confirm the theoretical predictions.Fractional evolution equations with nonlocal conditions in partially ordered Banach space.https://zbmath.org/1449.351742021-01-08T12:24:00+00:00"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Yang, He"https://zbmath.org/authors/?q=ai:yang.he"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-pSummary: In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form \[\begin{cases} ^CD^\sigma_tx(t)+Ax(t)=f(t,x(t)),\quad t\in J:= [0,b],\\ x(0)=x_0\in X,\end{cases}\tag{A}\] where \({}^CD^\sigma_t\) denotes the Caputo fractional derivative of order \(\sigma\in(0,1),-A:D(A)\subset X\to X\) generates a positive \(C_0\)-semigroup \(T(t)(t\ge 0)\) of uniformly bounded linear operator in \(X,b >0\) is a constant, \(f\) is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone \(K\) is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.Pullback \(D\)-attractor of coupled rod equations with nonlinear moving heat source.https://zbmath.org/1449.350992021-01-08T12:24:00+00:00"Wang, Danxia"https://zbmath.org/authors/?q=ai:wang.danxia"Zhang, Jianwen"https://zbmath.org/authors/?q=ai:zhang.jianwen"Wang, Yinzhu"https://zbmath.org/authors/?q=ai:wang.yinzhuSummary: We consider the pullback \(D\)-attractor for the nonautonomous nonlinear equations of thermoelastic coupled rod with a nonlinear moving heat source. By Galerkin method, the existence and uniqueness of global solutions are proved under homogeneous boundary conditions and initial conditions. By prior estimates combined with some inequality skills, the existence of the pullback \(D\)-absorbing set is obtained. By proving the properties of compactness about the nonlinear operator \(g_1(\cdot), g_2(\cdot)\), and then proving the pullback \(D\)-condition (C), the existence of the pullback \(D\)-attractor of the equations previously mentioned is given.Existence and stability for a generalized differential mixed quasi-variational inequality.https://zbmath.org/1449.490132021-01-08T12:24:00+00:00"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei-wayne|li.wei.7|li.wei.9|li.wei.5|li.wei|li.wei.10|li.wei.8"Xiao, Yi-bin"https://zbmath.org/authors/?q=ai:xiao.yibin"Wang, Xing"https://zbmath.org/authors/?q=ai:wang.xing"Feng, Jun"https://zbmath.org/authors/?q=ai:feng.junSummary: In the present paper, we investigate a generalized differential mixed quasi-variational inequality consisting of a system of an ordinary differential equation and a generalized mixed quasi-variational inequality. By using an important result concerning the measurable selection, we prove the existence of Carathéodory weak solution to the generalized differential mixed quasi-variational inequality. Then, with the existence result, we establish two stability results for the generalized differential mixed quasi-variational inequality under different conditions, i.e., upper semicontinuity and lower semicontinuity of the Carathéodory weak solution with respect to the parameter, which is a perturbation of some mappings in the generalized mixed quasi-variational inequality.On thermal stability of multi layer plane walls when heating by temperature-dependent internal heat sources.https://zbmath.org/1449.800092021-01-08T12:24:00+00:00"Averin, B. V."https://zbmath.org/authors/?q=ai:averin.b-vSummary: We obtain an analytical solution for a stationary non-linear problem of thermal conductivity for a multi-layer plane wall with temperature-dependent internal heat source is presented. Curves are plotted that allow to separate the region of stationary temperature distribution from the region of unrestricted growth of the temperature (thermal explosion region).The optimal error estimate of finite element method with Crank-Nicolson scheme for Poisson-Nernst-Planck equations.https://zbmath.org/1449.652692021-01-08T12:24:00+00:00"Zhu, Wanwan"https://zbmath.org/authors/?q=ai:zhu.wanwan"Shen, Ruigang"https://zbmath.org/authors/?q=ai:shen.ruigang"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.yingSummary: In this paper, the error estimate of finite element method is studied for time-dependent Poisson-Nernst-Planck equations. We use the Crank-Nicolson scheme for the time discretization. The theoretical results show that the error estimate of the finite element solution is optimal order in \({L^2}\) norm.Solution of the singular Cauchy problem for a general inhomogeneous Euler-Poisson-Darboux equation.https://zbmath.org/1449.353152021-01-08T12:24:00+00:00"Shishkina, Elina"https://zbmath.org/authors/?q=ai:shishkina.elina-leonidovnaSummary: In this paper, we solve Cauchy problem for a general form of an inhomogeneous Euler-Poisson-Darboux equation, where Bessel operator acts instead of the each second derivative. In the classical formulation, the Cauchy problem for this equation is not correct. However, for a specially selected form of the initial conditions, the equation has a solution. The general form of the Euler-Poisson-Darboux equation with such conditions we will call the singular Cauchy problem.Mixed problems for degenerate abstract parabolic equations and applications.https://zbmath.org/1449.352562021-01-08T12:24:00+00:00"Shakhmurov, Veli. B."https://zbmath.org/authors/?q=ai:shakhmurov.veli-b"Sahmurova, Aida"https://zbmath.org/authors/?q=ai:sahmurova.aidaSummary: Degenerate abstract parabolic equations with variable coefficients are studied. Here the boundary conditions are nonlocal. The maximal regularity properties of solutions for elliptic and parabolic problems and Strichartz type estimates in mixed Lebesgue spaces are obtained. Moreover, the existence and uniqueness of optimal regular solution of mixed problem for nonlinear parabolic equation is established. Note that these problems arise in fluid mechanics and environmental engineering.Mathematical modeling of longitudinal blow of the system of homogeneous rods about rigid barrier at not-holding connections.https://zbmath.org/1449.652242021-01-08T12:24:00+00:00"Bityurin, A. A."https://zbmath.org/authors/?q=ai:bityurin.anatolii-aleksandrovich"Manzhosov, V. K."https://zbmath.org/authors/?q=ai:manzhosov.vladimir-kuzmichSummary: Mathematical modeling of the longitudinal elastic central blow of the rod system, consisting of two homogeneous rods of various lengths and the area of cross section over a rigid barrier is implemented at not-holding connections.An implicit difference approximation for the variable coefficients space fractional diffusion equation.https://zbmath.org/1449.651922021-01-08T12:24:00+00:00"Liu, Dongbing"https://zbmath.org/authors/?q=ai:liu.dongbing"Tan, Qianrong"https://zbmath.org/authors/?q=ai:tan.qianrong"Liu, Tao"https://zbmath.org/authors/?q=ai:liu.tao.1Summary: The variable coefficient space fractional diffusion equation is considered. An implicit difference scheme is constructed, which is unconditionally stable and is super-linearly convergent about space step, and the convergence order of the method is \(O ({h^\alpha} + \tau)\) \((1 < \alpha \le 2)\). Finally, an example is presented to show that the numerical analysis is right and the method is feasible and efficient.Statistical estimates for the degrees of Green operator.https://zbmath.org/1449.650022021-01-08T12:24:00+00:00"Kuznetsov, A. N."https://zbmath.org/authors/?q=ai:kuznetsov.andrei-nikolavich"Sipin, A. S."https://zbmath.org/authors/?q=ai:sipin.aleksandr-stepanovichSummary: In this paper we examine known and some new algorithms for calculation of degrees of Green operator using Monte Carlo methods. Statistical estimations used for solving Dirichlet problem for Helmholtz equation with complex parameter. The efficiency of numerical realization of these algorithms is also considered.The study on the dark-soliton solutions to the fifth-order variable-coefficient nonlinear Schrödinger equation.https://zbmath.org/1449.351572021-01-08T12:24:00+00:00"Wu, Suqin"https://zbmath.org/authors/?q=ai:wu.suqin"Cheng, Yan"https://zbmath.org/authors/?q=ai:cheng.yan"Xu, Daojun"https://zbmath.org/authors/?q=ai:xu.daojun"Li, Guowang"https://zbmath.org/authors/?q=ai:li.guowangSummary: Soliton analysis of the high-order nonlinear Schrödinger equations has been an advancing research in soliton area, which has a great application in optical fiber communications. In this paper, a fifth-order variable-coefficient nonlinear Schrödinger equation is investigated, which describes the propagation of the attosecond pulses in an optical fiber. Via the Hirota method and auxiliary functions, bilinear forms and dark one-, two-, three-soliton solutions are obtained. Propagation and interaction of the solitons are discussed and the following conclusions are derived. Firstly, solitonic velocities are only related to the coefficients of the second-, third-, fourth- and fifth-order terms of the equation, while the solitonic amplitudes are related to these coefficients as well as the wave number. Secondly, when the above coefficients are constants, or the linear, quadratic and trigonometric functions of the space, we obtain the linear, parabolic, cubic and periodic dark solitons, respectively. Thirdly, interactions of the solitons can be regarded to be elastic because the solitonic amplitudes remain unchanged except for some phase shifts after each interaction.Global solvability of two unknown variables identification problem in one inverse problem for the integro-differential wave equation.https://zbmath.org/1449.352862021-01-08T12:24:00+00:00"Durdiev, D. K."https://zbmath.org/authors/?q=ai:durdiev.durdimurod-kalandarovichSummary: Identification problem of the two coefficients, one of which is located under the integral sign in a hyperbolic equation and represents memory of the medium is studied, the other one defines a regular part of an impulse source. As an additional information Fourier image of the solution's trace of a direct problem on the hyperplane \(y=0\) for two different values of transformation parameters is applied. The global solvability and the estimate of stability of the inverse problem are defined.Stability of solutions of Oskolkov linear equations on a geometrical graph.https://zbmath.org/1449.350582021-01-08T12:24:00+00:00"Sviridyuk, G. A."https://zbmath.org/authors/?q=ai:sviridyuk.georgii-anatolevich"Shipilov, A. S."https://zbmath.org/authors/?q=ai:shipilov.a-sSummary: Stability and unstability of steady-state solutions of Oskolkov linear equations \(\lambda u_{jt}-u_{jtxx}=\nu u_{jxx}\) on a geometrical graph depending on parameter \(\lambda\in\mathbb{R}\) are studied.Initial value problems for system of wave equations.https://zbmath.org/1449.352842021-01-08T12:24:00+00:00"Leksina, S. V."https://zbmath.org/authors/?q=ai:leksina.s-vSummary: We present solutions to the Cauchy problem and the Goursat problem for system of wave equations. Discharged analogue d'Alembert formula for system of wave equations.Construction of general solution of degenerating Pohlmeyer-Lund-Regge system.https://zbmath.org/1449.370472021-01-08T12:24:00+00:00"Gur'eva, A. M."https://zbmath.org/authors/?q=ai:gureva.a-mSummary: It is demonstrated that degenerating Pohlmeyer-Lund-Regge system is a Liouville-type system, formulas are obtained for \(x\)- and \(y\)-integrals at the first and the second orders. It is demonstrated how they can be used in order to construct a general solution on this equation system.Numerical method of solution of the problem on transposition of two-sided derivative of the fractional order.https://zbmath.org/1449.653032021-01-08T12:24:00+00:00"Beĭbalaev, V. D."https://zbmath.org/authors/?q=ai:beibalaev.vetlugin-dzhabrilovich|beibalaev.vertlugin-dzhabrailovichSummary: Numerical method of the solution of the problem of heat transposition with two-sided derivative of the fractional order along the space variable and with the fractional order derivative in time is studied. Finite-different scheme was constructed and the stability of this different scheme was proven.Global strong solution for incompressible MHD system with variable magnetic diffusion and magnetic dissipation coefficients.https://zbmath.org/1449.351692021-01-08T12:24:00+00:00"Chen, Hong"https://zbmath.org/authors/?q=ai:chen.hong|chen.hong.1"Yuan, Rong"https://zbmath.org/authors/?q=ai:yuan.rongSummary: In this paper, we study the initial boundary value problem of three dimensional incompressible MHD system with variable magnetic diffusion and magnetic dissipation coefficients in a bounded region \(\Omega \subset {R^3}\) with smooth boundary. The existence of the unique local strong solution to the MHD system is proved, and the local strong solution can be extended to the global strong solution of the MHD system.Boundedness of Schrödinger type operators on generalized Morrey spaces.https://zbmath.org/1449.351942021-01-08T12:24:00+00:00"Ren, Feng"https://zbmath.org/authors/?q=ai:ren.fengSummary: In this paper, we study Schrödinger type operators \({V^{\beta_1}}\nabla (-\Delta + V)^{-\beta_2}\). By the decomposition technique of function, the boundedness of these operators on generalized Morrey spaces is obtained when the potential \(V\) belongs to the reverse Hölder class \(RH_s\) with \({s > n/2}\). The results enrich and improve some existing ones.The local well-posedness of the strong solution of Boussinesq equations with Navier-slip boundary conditions.https://zbmath.org/1449.353522021-01-08T12:24:00+00:00"Guo, Lianhong"https://zbmath.org/authors/?q=ai:guo.lianhongSummary: In this paper, we considered the Boussinesq Equations in a boundary smooth domain \(\Omega\) of \({R^3}\) with Navier-slip boundary conditions, obtained the local well-posedness of the strong solution of Boussinesq equations by Galerkin approximation method.An \({H^1}\)-Galerkin mixed finite element method for semi-linear dual phase lagging heat conduction equations.https://zbmath.org/1449.652532021-01-08T12:24:00+00:00"Ma, Ge"https://zbmath.org/authors/?q=ai:ma.ge"Dong, Lijiao"https://zbmath.org/authors/?q=ai:dong.lijiao"Hu, Shuangnian"https://zbmath.org/authors/?q=ai:hu.shuangnianSummary: An \({H^1}\)-Galerkin mixed finite element method for semi-linear dual phase lagging heat conduction equations is constructed by incomplete biquadratic element \({Q_2^-}\) and first order BDFM element. Based on special characters of the interpolation operators of the elements instead of the projection operator ones, the superclose estimates of order \(O ({h^3})\) and \(O (h^3 + (\Delta t)^2)\) are derived for the primitive variable in \({H^1}\)-norm and the flux in \(H\) (div)-norm under semi-discrete scheme and a fully-discrete scheme, respectively.Modified Cauchy problem for one hyperbolic equation of the third order in three-dimensional space.https://zbmath.org/1449.351812021-01-08T12:24:00+00:00"Dolgopolov, V. M."https://zbmath.org/authors/?q=ai:dolgopolov.vyacheslav-mikhailovich"Rodionova, I. N."https://zbmath.org/authors/?q=ai:rodionova.irina-nikolaevnaSummary: Existence and uniqueness of the solution for the modified Cauchy problem with data located on the plane of singularity of a hyperbolic equation of the third order coefficients in three-dimensional space is proved by Riemann method.Boundary value problem for the equation of mixed type with singular coefficient in the domain where the elliptic part is a half-band.https://zbmath.org/1449.353272021-01-08T12:24:00+00:00"Ruziev, M. Kh."https://zbmath.org/authors/?q=ai:ruziev.menglibay-kholtojibaevichSummary: Boundary value problem is studied in the unbounded domain. Uniqueness of the problem is proven with the help of the extreme principle. Existence of the problem is established by methods of separation of variables and integral equations.Non-local problem with non-linear conditions for a hyperbolic equation.https://zbmath.org/1449.352892021-01-08T12:24:00+00:00"Dmitriev, V. B."https://zbmath.org/authors/?q=ai:dmitriev.v-bSummary: Non-local problem with non-linear integral condition for the hyperbolic equation with random size space variables is studied. Existence of the unique generalized solution is proved.On gas flow beyond strong shock wave front, form of which approaches a certain curve.https://zbmath.org/1449.760312021-01-08T12:24:00+00:00"Bogatko, V. I."https://zbmath.org/authors/?q=ai:bogatko.v-i"Kolton, G. A."https://zbmath.org/authors/?q=ai:kolton.g-a"Potekhina, E. A."https://zbmath.org/authors/?q=ai:potekhina.e-a|potekhina.elena-aSummary: The plane auto model problem of the in viscid gas motion beyond intensive shock wave is studied. It is supposed, that shock wave front approaches some curve, the form of which is known. Solution is constructed in the form of series on the small parameter degrees. This parameter characterizes the relation of gas densities at shock wave front. Certain cases are studied as examples: when intensive shock wave front form is closely approximated to the straight line or to the circle. Solution of the problem is reduced to the Euler-Darboux equation integration.On direct and inverse problems for the Hoff equations on a graph.https://zbmath.org/1449.354222021-01-08T12:24:00+00:00"Sviridyuk, G. A."https://zbmath.org/authors/?q=ai:sviridyuk.georgii-anatolevich"Bayazitova, A. A."https://zbmath.org/authors/?q=ai:bayazitova.alfiya-adygamovaSummary: The Hoff equations defined on a graph describe the dynamics of H-beams construction buckling. Generalization of the direct problem which is the Cauchy problem is obtained. For the first time the inverse coefficient problem is studied which is modeling the experiment that allows with additional measurements not only to study the construction buckling dynamics but also the characteristics of beam material. The unique solution of this problem is demonstrated.Exact solutions of a \( (3+1)\)-dimensional extended Jimbo-Miwa equation.https://zbmath.org/1449.353722021-01-08T12:24:00+00:00"Zhang, Shulin"https://zbmath.org/authors/?q=ai:zhang.shulin"Liu, Jiangen"https://zbmath.org/authors/?q=ai:liu.jiangen"Liu, Wanli"https://zbmath.org/authors/?q=ai:liu.wanliSummary: In this paper, a new three-wave test function is applied to investigate a \( (3+1)\)-dimensional extended Jimbo-Miwa equation with the Hirota bilinear method. As a result, twelve classes of its exact solutions are generated by Maple symbolic computations, whose analyticity can be easily achieved by taking special choices of the involved parameters. Furthermore, figures of some exact solutions are presented to illustrate the mechanical features of these solutions.A generalized multi-symplectic method for stochastic space-fractional nonlinear Schrödinger equation with multiplicative noise.https://zbmath.org/1449.653512021-01-08T12:24:00+00:00"Liu, Ziyuan"https://zbmath.org/authors/?q=ai:liu.ziyuan"Liang, Jiarui"https://zbmath.org/authors/?q=ai:liang.jiarui"Qian, Xu"https://zbmath.org/authors/?q=ai:qian.xu"Song, Songhe"https://zbmath.org/authors/?q=ai:song.songheSummary: The stochastic space-fractional nonlinear Schrödinger equation with multiplicative noise is an important equation which describes the evolution of an open nonlocal quantum system. In this paper we prove that this system is an infinite-dimensional stochastic fractional Hamiltonian system, and satisfies both the mass and the generalized multi-symplectic conservation law. After that, with the Fourier pseudo-spectral approximation to the spatial fractional Laplacian operator and the implicit mid-point method for time discretization, we propose a mass-conserving generalized stochastic multi-symplectic method. Numerical simulations are presented for soliton solution and plane wave solution. The results demonstrate the effectiveness and conservative property of the proposed methods. Furthermore, the results show that the mean square convergence order on time is approximately 0.5 to 1.The multiscale algorithms for the Maxwell-Dirac system in matrix form with quadratic correction.https://zbmath.org/1449.651782021-01-08T12:24:00+00:00"Fu, Yaoyao"https://zbmath.org/authors/?q=ai:fu.yaoyao"Cao, Liqun"https://zbmath.org/authors/?q=ai:cao.liqunSummary: The Maxwell-Dirac system with quadratic correction has a wide applications in materials science such as topological insulators, graphene, superconductors and so on. In this paper, we first present the Dirac equation in matrix form with quadratic correction. Combining the Maxwell's equations, we present the homogenization method and the multiscale asymptotic method for the modified Maxwell-Dirac system in matrix form with rapidly oscillating discontinuous coefficients in a bounded Lipschitz convex domain under the Weyl gauge. Based on the multiscale asymptotic expansions of the solution of the Maxwell-Dirac system, by using the Crank-Nicolson finite difference method and the adaptive edge element method, we developed the multiscale algorithms for solving the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients. Numerical examples are then carried out to validate the method presented in this paper.Homogenization of a degenerate PDE with a nonlinear Neumann boundary condition.https://zbmath.org/1449.601062021-01-08T12:24:00+00:00"Coulibaly, A."https://zbmath.org/authors/?q=ai:coulibaly.alioune"Diedhiou, A."https://zbmath.org/authors/?q=ai:diedhiou.alassane"Sane, I."https://zbmath.org/authors/?q=ai:sane.ibrahimaSummary: We establish homogenization results of a degenerate semilinear PDE with a nonlinear Neumann boundary condition. Our approach is entirely probabilistic, and extends the result of \textit{Y. Ouknine} and \textit{É. Pardoux} [Prog. Probab. 52, 229--242 (2002; Zbl 1038.60049)].Positive solutions for Kirchhoff-type equations with an asymptotically nonlinearity.https://zbmath.org/1449.350262021-01-08T12:24:00+00:00"Xu, Liping"https://zbmath.org/authors/?q=ai:xu.liping"Chen, Haibo"https://zbmath.org/authors/?q=ai:chen.haiboSummary: We focus on a class of nonlinear Kirchhoff-type equation. The nonlinear function \(f (x, u)\) is either asymptotically linear or asymptotically nonlinear with respect to \(u\) at infinity. Under certain conditions on the potential function \(V (x)\) and the nonlinear term \(f (x, u)\), the existence of positive solutions is obtained without using the compactness of embedding of the working space.Strongly quasilinear parabolic systems in divergence form with weak monotonicity.https://zbmath.org/1449.352572021-01-08T12:24:00+00:00"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farahSummary: The existence of solutions to the strongly quasilinear parabolic system \[\frac{\partial u}{\partial t}-\operatorname{div},\sigma(x,t,u,Du)+g(x,t,u,Du)=f,\]
is proved, where the source term \(f\) is assumed to belong to \(L^{p'}(0,T; W^{-1,p'}(\Omega;\mathbb{R}^m))\). Further, we prove the existence of a weak solution by means of the Young measures under mild monotonicity assumptions on \(\sigma\).On solutions for a class of \(p\)-Laplace equation with nonlocal boundary condition.https://zbmath.org/1449.352502021-01-08T12:24:00+00:00"Fu, Meimei"https://zbmath.org/authors/?q=ai:fu.meimei"Xie, Junhui"https://zbmath.org/authors/?q=ai:xie.junhuiSummary: In this article, we study a class of \(p\)-Laplace parabolic equations with nonlocal boundary condition. By the method of upper and lower solutions of parabolic equations and some basic theory of parabolic equations, we prove the global existence of solution, blow-up in finite time as well as the rate of blow-up.Asymptotic dynamics of non-autonomous modified Swift-Hohenberg equations with multiplicative noise on unbounded domains.https://zbmath.org/1449.350742021-01-08T12:24:00+00:00"Mohamed, Yagoub Abaker"https://zbmath.org/authors/?q=ai:mohamed.yagoub-abaker"Liu, Tingting"https://zbmath.org/authors/?q=ai:liu.tingting"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: We investigate the dynamical behavior of the stochastic non-autonomous modified Swift-Hohenberg equation with time-dependent forcing term and multiplicative noise on \(\mathbb{R}^2\). In order to overcome the difficulty that Sobolev embedding is not compact in the unbounded domain, we first define a continuous cocycle associated with the problem in \({L^2} (\mathbb{R}^2)\), and make some uniform estimates on the tails of solutions for large space variables. With the aid of uniform estimates of solution, we verify the pullback asymptotic compactness of the random dynamical system, and further obtain the existence of random attractors.Superconvergence analysis of a mixed finite element method for two-dimension Ginzburg-Landau equations.https://zbmath.org/1449.653202021-01-08T12:24:00+00:00"Li, Qingfu"https://zbmath.org/authors/?q=ai:li.qingfu"Wang, Junjun"https://zbmath.org/authors/?q=ai:wang.junjunSummary: \(EQ_1^{\mathrm{rot}}\) finite element and zero order Raviart-Thomas element are applied to discuss a kind of mixed finite element method (MFEM) for the two-dimension Ginzburg-Landau equations. The superclose results of original variant \(u\) in \({H^1}\)-norm and flux variant \(H ({\mathrm{div}}; \Omega)\) in \({L^2}\)-norm with \(O ({h^2} + {\tau^2})\) are derived technically under the semi-discrete scheme and the linearized Euler fully-discrete scheme. At last, a numerical experiment is included to illustrate the feasibility of the proposed method.A blow-up result for the dissipative Boussinesq equation.https://zbmath.org/1449.351282021-01-08T12:24:00+00:00"Su, Xiao"https://zbmath.org/authors/?q=ai:su.xiao"Wang, Shubin"https://zbmath.org/authors/?q=ai:wang.shubin"Song, Ruili"https://zbmath.org/authors/?q=ai:song.ruiliSummary: This paper is devoted to the finite time blow-up of the weak solutions of the initial-boundary value problem for the dissipative Boussinesq equations. We provide the sufficient and necessary conditions of finite time blow-up of weak solutions with the initial data from the potential well, and give the lower bounds of the lifespan time.Residual symmetry and interaction solution of the \( (2+1)\)-dimensional Kadomtsev-Petviashvili equation.https://zbmath.org/1449.353782021-01-08T12:24:00+00:00"Ge, Nannan"https://zbmath.org/authors/?q=ai:ge.nannan"Ren, Xiaojing"https://zbmath.org/authors/?q=ai:ren.xiaojingSummary: The truncated Painlevé expansion method is developed to obtain the nonlocal residual symmetry and Bäcklund transformation for the \( (2+1)\)-dimensional Kadomtsev-Petviashvili (KP) equation. The nonlocal symmetry cannot directly solve the \( (2+1)\)-dimensional KP equation. It is necessary to localize the nonlocal symmetry. Then, using the solvable concept of the consistent Riccati expansion method (CRE), the \( (2+1)\)-dimensional KP equation is proved to be consistent Riccati expansion solvable, and the new interaction solution of the \( (2+1)\)-dimensional KP equation is obtained.Numerical treatment for a class of partial integro-differential equations with a weakly singular kernel using Chebyshev wavelets.https://zbmath.org/1449.653382021-01-08T12:24:00+00:00"Xu, Xiaoyong"https://zbmath.org/authors/?q=ai:xu.xiaoyong"Zhou, Fengying"https://zbmath.org/authors/?q=ai:zhou.fengying"Xie, Yu"https://zbmath.org/authors/?q=ai:xie.yuSummary: In this paper, a numerical method based on fourth kind Chebyshev wavelet collocation method is applied for solving a class of partial integro-differential equations (PIDEs) with a weakly singular kernel under three types of boundary conditions. Fractional integral formula of a single Chebyshev wavelet in the Riemann-Liouville sense is derived by means of shifted Chebyshev polynomials of the fourth kind. By implementing fractional integral formula and two-dimensional fourth kind Chebyhev wavelets together with collocation method, PIDEs with a weakly singular kernel are converted into system of algebraic equation. The convergence analysis of two-dimensional fourth kind Chebyhev wavelets is investigated. Some numerical examples are included for demonstrating the efficiency of the proposed method.The boundary layer for MHD equations in a plane-parallel channel.https://zbmath.org/1449.353562021-01-08T12:24:00+00:00"Wang, Na"https://zbmath.org/authors/?q=ai:wang.na"Wang, Shu"https://zbmath.org/authors/?q=ai:wang.shuSummary: In this paper, we study the boundary layer problem for the incompressible MHD equations in a plane-parallel channel. Using the multiscale analysis and the careful energy method, we prove the convergence of the solution of viscous and diffuse MHD equations to that of the ideal MHD equations as the viscosity and magnetic diffusion coefficients tend to zero.Finite difference method for simulating phonon heat transport process.https://zbmath.org/1449.800262021-01-08T12:24:00+00:00"Chen, Jianling"https://zbmath.org/authors/?q=ai:chen.jianling"Feng, Yangde"https://zbmath.org/authors/?q=ai:feng.yangdeSummary: The equation of phonon radiative transfer is a differential-integral equation. The finite difference method is used to numerically solve the equation of phonon radiative transfer with boundary conditions. The Gauss-Seidel method and the finite difference discretization can be used to ensure stable numerical solutions. Through an one-dimensional numerical simulation of the phonon heat transport process of Ge/Si/Ge film at room temperature and using the diffuse mismatch interface model at the interface, we can evaluate the influence of the temperature along the normal direction of the film and the thickness ratio of the Ge/Si film and the temperature jump at the interface on the thermal conductivity of the overal-1 structure of the material, and estimate the change of the thermal conductivity of the film as the thickness of the film increases. Through a two-dimensional numerical simulation of the phonon heat transport process of the silicon film, we can obtain the temperature along the normal and orientation-oriented distribution of the film, and the change in temperature and the thermal conductivity when the film width and thickness ratio are different.A compact difference scheme for nonlinear parabolic equation.https://zbmath.org/1449.652142021-01-08T12:24:00+00:00"Zhao, Xinyi"https://zbmath.org/authors/?q=ai:zhao.xinyi"Dong, Mingzhe"https://zbmath.org/authors/?q=ai:dong.mingzheSummary: A compact difference scheme for nonlinear parabolic equation is proposed. Firstly, the nonlinear term in the parabolic equation is linearized, and a compact difference scheme for nonlinear parabolic equation is derived referred to the derivation of the compact difference scheme for linear parabolic equation. The truncation error of the scheme is analyzed. Secondly, a priori estimation of the solution of the compact difference scheme is given using the energy analysis method, which proves the solvability, stability and convergence of the scheme. The convergence order is \(O ({\tau^2} + {h^4})\). Thirdly, the Richardson's extrapolation method is applied to the compact difference scheme and the solution with accuracy \(O ({\tau^4} + {\tau^2}{h^4} + {h^6})\) is obtained through once extrapolation. Finally, the high accuracy of the compact difference scheme for nonlinear parabolic equation and its extrapolation scheme are proved through a concrete numerical example.Analytical modeling of non-Fickian wave-diffusion of gas in heterogeneous media.https://zbmath.org/1449.800152021-01-08T12:24:00+00:00"Rasmussen, Maurice L."https://zbmath.org/authors/?q=ai:rasmussen.maurice-l"Civan, Faruk"https://zbmath.org/authors/?q=ai:civan.farukSummary: An isothermal transient-state non-Fickian diffusion model is developed and analytically solved for the description of gas dissolution in locally heterogeneous media suddenly exposed to a high pressure gas. The full-, short-, and long-time analytical solutions are used to establish the significance of the non-Fickian gas dissolution in heterogeneous media compared to the Fickian diffusion assumption. Parametric studies are carried out by means of the special analytical-solutions obtained for gas transport in the semi-infinite and finite-thickness heterogeneous media involving a delay time. The profiles of concentration and diffusion flux obtained for the non-Fickian wave-diffusion case are compared with the Fickian pure-diffusion case. The initial propagation of a right-running wave and its reflection from the wall are illustrated for the concentrations and diffusion fluxes. The small-time behavior is shown to be inherently wave-like and the discontinuity wave front propagates into the medium with the speed decreasing with time. For small times, the differences between the wave- and pure-diffusion cases are found to be significant depending on the magnitude of the delay time. For sufficiently large times, the wave behavior dies out and the wave solutions approach the equilibrium pure-diffusion solutions, except very near the decaying wave front. The formulations presented in this paper are of practical importance because they can be instrumental in determination of the diffusivity, interface surface mass-transfer coefficient, and rate of dissolution of gases in heterogeneous medium. A parameter estimation method is also proposed and elaborated for the estimation of the diffusion and interface surface mass-transfer coefficients from measured pressure decay data.The Pogorelov interior \(C^2\) estimate of \(\sigma_2\) Hessian equations and its application.https://zbmath.org/1449.352162021-01-08T12:24:00+00:00"Miao, Zhengwu"https://zbmath.org/authors/?q=ai:miao.zhengwuSummary: The Hessian equation is an important class of completely nonlinear partial differential equations. In this paper, the author re-proves the concaveness by using the Lagrange multiplier method and defines a convex cone \({\widetilde{\Gamma}_3} = \{\lambda = ({\lambda_1}, {\lambda_2},\ldots, {\lambda_n}) \in\mathbb{R}^n:{\sigma_1} (\lambda) > 0, {\sigma_2} (\lambda|i) > 0, 1 \le i \le n\}\). He further uses the optimal concave of \({\sigma_2}\) operator to give the Pogorelov interior \({C^2}\) estimate of \({\sigma_2}\) Hessian equations. Then, he proves that the \({\widetilde{\Gamma}_3}\)-convex entire solution of \({\sigma_2} ({D^2}u (x)) = 1, x \in \mathbb{R}^n\) is a quadratic polynomial if \(u\) satisfies a quadratic growth condition.Hysteresis and temperature-induced transitions in ferromagnetic materials.https://zbmath.org/1449.780012021-01-08T12:24:00+00:00"Berti, Alessia"https://zbmath.org/authors/?q=ai:berti.alessia"Giorgi, Claudio"https://zbmath.org/authors/?q=ai:giorgi.claudio"Vuk, Elena"https://zbmath.org/authors/?q=ai:vuk.elenaSummary: In this paper we present two basic one-dimensional models for the temperature-induced phase-changes in a ferromagnetic material. In the framework of the Ginzburg-Landau theory, we construct suitable thermodynamic potentials from which thermodynamically-consistent evolution equations for the magnetization are derived. For both soft and hard materials these models account for saturation and provide an effective description of the transition from paramagnetic to ferromagnetic regimes by displaying the onset of hysteresis loops when the temperature decreases below the Curie critical value. The temperature enters the model as a parameter by way of the magnetic susceptibility. Such a dependence is discussed in order to comply with both Bloch's law (below the critical value) and Curie-Weiss law (far above the critical value). Focusing on uniform processes, numerical simulations of the magnetic responses at different temperatures are performed.Solitary wave solutions and traveling wave solutions for systems of time-fractional nonlinear wave equations via an analytical approach.https://zbmath.org/1449.931232021-01-08T12:24:00+00:00"Thabet, Hayman"https://zbmath.org/authors/?q=ai:thabet.hayman"Kendre, Subhash"https://zbmath.org/authors/?q=ai:kendre.s-d|kendre.subhash-dhondiba"Peters, James"https://zbmath.org/authors/?q=ai:peters.james-e|peters.james-v|peters.james-f-iii"Kaplan, Melike"https://zbmath.org/authors/?q=ai:kaplan.melikeSummary: This paper introduces a new approximate-analytical approach for solving systems of fractional nonlinear partial differential equations (FNPDEs). However, the main advantage of this new approximate-analytical approach is to obtain the analytical solution for general systems of FNPDEs in forms of convergent series with easily computable components using Caputo fractional partial derivative. Moreover, the convergence theorem and error analysis of the proposed method are also shown. Solitary wave solutions and travelling wave solutions for the system of fractional dispersive wave equations and the system of fractional long water wave equations are successfully obtained. The numerical solutions are also obtained in forms of tables and graphs to confirm the accuracy and efficiency of the suggested method.Nonlinear Galerkin finite element methods for fourth-order bi-flux diffusion model with nonlinear reaction term.https://zbmath.org/1449.652472021-01-08T12:24:00+00:00"Jiang, Maosheng"https://zbmath.org/authors/?q=ai:jiang.maosheng"Bevilacqua, Luiz"https://zbmath.org/authors/?q=ai:bevilacqua.luiz"Zhu, Jiang"https://zbmath.org/authors/?q=ai:zhu.jiang"Yu, Xijun"https://zbmath.org/authors/?q=ai:yu.xijunSummary: A fourth-order diffusion model is presented with a nonlinear reaction term to simulate some special chemical and biological phenomenon. To obtain the solutions to those problems, the nonlinear Galerkin finite element method under the framework of the Hermite polynomial function for the spatial domain is utilized. The Euler backward difference method is used to solve the equation in the temporal domain. Subject to the Dirichlet and Navier boundary conditions, the numerical experiments for Bi-flux Fisher-Kolmogorov model present excellent convergence, accuracy and acceleration behavior. Also, the numerical solutions to the Bi-flux Gray-Scott model, subject to no flux boundary conditions, show excellent convergence, accuracy and symmetry.Compact finite-difference method for 2D time-fractional convection-diffusion equation of groundwater pollution problems.https://zbmath.org/1449.651912021-01-08T12:24:00+00:00"Li, Lingyu"https://zbmath.org/authors/?q=ai:li.lingyu"Jiang, Ziwen"https://zbmath.org/authors/?q=ai:jiang.ziwen"Yin, Zhe"https://zbmath.org/authors/?q=ai:yin.zheSummary: In this work, we provide a compact finite-difference scheme (CFDS) of 2D time-fractional convection-diffusion equation (TF-CDE) for solving fluid dynamics problem, especially groundwater pollution. The successful predication of the pollutants concentration in groundwater will greatly benefit the protection of water resources for provide the fast and intuitive decision-makings in response to sudden water pollution events. Here, we creatively use the dimensionality reduction technology (DRT) to rewrite the original 2D problem as two equations, and we handle each one as a 1D problem. Particularly, the spatial derivative is approximated by fourth-order compact finite-difference method (CFDM) and time-fractional derivative is approximated by \(L_1\) interpolation of Caputo fractional derivative. Based on the approximations, we obtain the CFDS with fourth-order in spatial and \((2-\alpha )\)-order in temporal by adding two 1D results. In addition, the unique solvability, unconditional stability, and convergence order \(\mathcal{O}(\tau^{2-\alpha} +h_1^4+h_2^4)\) of the proposed scheme are studied. Finally, several numerical examples are carried out to support the theoretical results and demonstrate the effectiveness of the CFDS based DRT strategy. Obviously, the method developed in 2D TF-CDE of groundwater pollution problem can be easily extended for the other complex problems.A new stable finite difference scheme and its convergence for time-delayed singularly perturbed parabolic PDEs.https://zbmath.org/1449.651982021-01-08T12:24:00+00:00"Pramod Chakravarthy, Podila"https://zbmath.org/authors/?q=ai:pramod-chakravarthy.podila"Kumar, Kamalesh"https://zbmath.org/authors/?q=ai:kumar.kamaleshSummary: In this study, we consider the time-delayed singularly perturbed parabolic PDEs (SPPPDEs). We know that the classical finite difference scheme will not produce good results for singular perturbation problems on a uniform mesh. Here, we propose a new stable finite difference (NSFD) scheme, which produces good results on a uniform mesh and also on an adaptive mesh. The NSFD scheme is constructed based on the stability of the analytical solution. Results are compared with the results available in the literature and observed that the proposed method is efficient over the existing methods for solving SPPPDEs.Semi-uniform dynamics for non-autonomous Kuramoto-Sivashinsky equations.https://zbmath.org/1449.350972021-01-08T12:24:00+00:00"She, Lianbing"https://zbmath.org/authors/?q=ai:she.lianbing"Zhang, Wenlin"https://zbmath.org/authors/?q=ai:zhang.wenlin"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrongSummary: We introduce a concept of a semi-uniform attractor for an evolution process. A theorem for existence result of semi-uniform attractor is given in this paper. Although the invariance is not involved, but it can induce a semi-uniformly compact pullback attractor. Moreover, under some suitable assumptions, we show that the non-autonomous Kuramoto-Sivashinsky equation has a semi-uniform attractor and a semi-uniformly compact pullback attractor.Asymptotic behavior of solutions for a class of nonlinear higher-order Kirchhoff-type equations.https://zbmath.org/1449.350722021-01-08T12:24:00+00:00"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguang"Zhu, Changqing"https://zbmath.org/authors/?q=ai:zhu.changqingSummary: The initial-boundary value problems for a class of nonlinear nonlocal higher-order Kirchhoff partial differential equations is studied. Firstly, the existence and uniqueness of the global solution of the equation in space \(H_0^{m + k} (\Omega) \times H_0^k (\Omega)\) are proved by prior-estimation and Galerkin method. Then, the compact method is used to prove that the solution semigroup \(S (t)\) generated by the problem has a compact global attractor family \({A_k}\). Finally, the semigroup of operators is proved by linearization method. The Hausdorff dimension and Fractal dimension estimation of the global attractor family are obtained by using the Frechet differentiability and the attenuation of the volume element for the linearization problem.Convergence analysis and error estimates of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time.https://zbmath.org/1449.651402021-01-08T12:24:00+00:00"Shen, Quan"https://zbmath.org/authors/?q=ai:shen.quan"Ding, Rui"https://zbmath.org/authors/?q=ai:ding.rui"Zhu, Zhengcheng"https://zbmath.org/authors/?q=ai:zhu.zhengchengSummary: This paper is presented for the convergence analysis of the interpolating element-free Galerkin method for the evolutionary variational inequality of the second-order in time, which arises from the theory of viscoelastic materials with edge friction. First, the existence and uniqueness of the solutions for the evolutionary variational inequality of the second-order in time are proved, which are mainly based on the fixed point theorem. Second, the convergence analysis of the interpolating element-free Galerkin method is presented for them. The error estimates show that the convergence order depends not only on the number of basis functions in the interpolating moving least-squares approximation but also the relationship with the time step and the spatial step. Numerical examples verify the convergence analysis and the error estimates.Nonlocal flocking dynamics: learning the fractional order of PDEs from particle simulations.https://zbmath.org/1449.354472021-01-08T12:24:00+00:00"Mao, Zhiping"https://zbmath.org/authors/?q=ai:mao.zhiping"Li, Zhen"https://zbmath.org/authors/?q=ai:li.zhen"Karniadakis, George Em"https://zbmath.org/authors/?q=ai:karniadakis.george-emSummary: Flocking refers to collective behavior of a large number of interacting entities, where the interactions between discrete individuals produce collective motion on the large scale. We employ an agent-based model to describe the microscopic dynamics of each individual in a flock, and use a fractional partial differential equation (fPDE) to model the evolution of macroscopic quantities of interest. The macroscopic models with phenomenological interaction functions are derived by applying the continuum hypothesis to the microscopic model. Instead of specifying the fPDEs with an ad hoc fractional order for nonlocal flocking dynamics, we learn the effective nonlocal influence function in fPDEs directly from particle trajectories generated by the agent-based simulations. We demonstrate how the learning framework is used to connect the discrete agent-based model to the continuum fPDEs in one- and two-dimensional nonlocal flocking dynamics. In particular, a Cucker-Smale particle model is employed to describe the microscale dynamics of each individual, while Euler equations with nonlocal interaction terms are used to compute the evolution of macroscale quantities. The trajectories generated by the particle simulations mimic the field data of tracking logs that can be obtained experimentally. They can be used to learn the fractional order of the influence function using a Gaussian process regression model implemented with the Bayesian optimization. We show in one- and two-dimensional benchmarks that the numerical solution of the learned Euler equations solved by the finite volume scheme can yield correct density distributions consistent with the collective behavior of the agent-based system solved by the particle method. The proposed method offers new insights into how to scale the discrete agent-based models to the continuum-based PDE models, and could serve as a paradigm on extracting effective governing equations for nonlocal flocking dynamics directly from particle trajectories.A compact difference scheme for multi-point boundary value problems of heat equations.https://zbmath.org/1449.652032021-01-08T12:24:00+00:00"Wang, Xuping"https://zbmath.org/authors/?q=ai:wang.xuping"Sun, Zhizhong"https://zbmath.org/authors/?q=ai:sun.zhizhongSummary: In this paper, a compact difference scheme is established for the heat equations with multi-point boundary value conditions. The truncation error of the difference scheme is \(O(\tau^2+h^4),\) where \(\tau\) and \(h\) are the temporal step size and the spatial step size. A prior estimate of the difference solution in a weighted norm is obtained. The unique solvability, stability and convergence of the difference scheme are proved by the energy method. The theoretical statements for the solution of the difference scheme are supported by numerical examples.A split-step predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions.https://zbmath.org/1449.651862021-01-08T12:24:00+00:00"Kazmi, Kamran"https://zbmath.org/authors/?q=ai:kazmi.kamran"Khaliq, Abdul"https://zbmath.org/authors/?q=ai:khaliq.abdul-q-mSummary: A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence. The matrix transfer technique is used for spatial discretization of the problem. The method is shown to be unconditionally stable and second-order convergent. Numerical experiments are performed to confirm the stability and second-order convergence of the method. The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps. Our method is seen to produce reliable and oscillation-free results for any time step when implemented on numerical examples with nonsmooth initial data. We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.Numerical methods for solving space fractional partial differential equations using Hadamard finite-part integral approach.https://zbmath.org/1449.652042021-01-08T12:24:00+00:00"Wang, Yanyong"https://zbmath.org/authors/?q=ai:wang.yanyong"Yan, Yubin"https://zbmath.org/authors/?q=ai:yan.yubin"Hu, Ye"https://zbmath.org/authors/?q=ai:hu.yeSummary: We introduce a novel numerical method for solving two-sided space fractional partial differential equations in two-dimensional case. The approximation of the space fractional Riemann-Liouville derivative is based on the approximation of the Hadamard finite-part integral which has the convergence order \(O(h^{3-\alpha})\), where \(h\) is the space step size and \(\alpha \in (1, 2)\) is the order of Riemann-Liouville fractional derivative. Based on this scheme, we introduce a shifted finite difference method for solving space fractional partial differential equations. We obtained the error estimates with the convergence orders \(O(\tau +h^{3-\alpha}+ h^\beta)\), where \(\tau\) is the time step size and \(\beta >0\) is a parameter which measures the smoothness of the fractional derivatives of the solution of the equation. Unlike the numerical methods for solving space fractional partial differential equations constructed using the standard shifted Grünwald-Letnikov formula or higher order Lubich's methods which require the solution of the equation to satisfy the homogeneous Dirichlet boundary condition to get the first-order convergence, the numerical method for solving the space fractional partial differential equation constructed using the Hadamard finite-part integral approach does not require the solution of the equation to satisfy the Dirichlet homogeneous boundary condition. Numerical results show that the experimentally determined convergence order obtained using the Hadamard finite-part integral approach for solving the space fractional partial differential equation with non-homogeneous Dirichlet boundary conditions is indeed higher than the convergence order obtained using the numerical methods constructed with the standard shifted Grünwald-Letnikov formula or Lubich's higher order approximation schemes.Numerical simulation of two-dimensional tempered fractional diffusion equation.https://zbmath.org/1449.651742021-01-08T12:24:00+00:00"Chen, Jinghua"https://zbmath.org/authors/?q=ai:chen.jinghua"Chen, Xuejuan"https://zbmath.org/authors/?q=ai:chen.xuejuan"Zhang, Hongmei"https://zbmath.org/authors/?q=ai:zhang.hongmeiSummary: This paper discusses a two-dimensional tempered fractional diffusion equation, in which the tempered fractional derivative is the extension of fractional derivative. The scheme can model the transition from super-diffusion early time to diffusive late-time behavior. We apply the alternating directions implicit approach and the Crank-Nicolson (C-N) algorithm to establish our numerical discretization scheme. Furthermore, we obtain the second-order accurate difference method by a Richardson extrapolation. The stability and the convergence of the numerical scheme are proven via the technique of matrix analysis. A numerical example is given to demonstrate the efficiency of the designed schemes.A semi-Lagrangian spectral method for the Vlasov-Poisson system based on Fourier, Legendre and Hermite polynomials.https://zbmath.org/1449.652722021-01-08T12:24:00+00:00"Fatone, Lorella"https://zbmath.org/authors/?q=ai:fatone.lorella"Funaro, Daniele"https://zbmath.org/authors/?q=ai:funaro.daniele"Manzini, Gianmarco"https://zbmath.org/authors/?q=ai:manzini.gianmarcoSummary: In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on some standard benchmark problems including the two-stream instability and the Landau damping test cases. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.Cauchy problem for a class of pseudo-differential equations on \(\mathbb{Q}_p^n\).https://zbmath.org/1449.354702021-01-08T12:24:00+00:00"Wu, Bo"https://zbmath.org/authors/?q=ai:wu.bo"Qian, Dandan"https://zbmath.org/authors/?q=ai:qian.dandanSummary: This paper considers a class of pseudo-differential equations on \(\mathbb{Q}_p^n\),
\[ \frac{{\partial^2}u (t,x)}{\partial{t^2}} + 2{a^2}\Delta_p^{\alpha/2}\frac{\partial u (t,x)}{\partial t} + {b^2}\Delta_p^\alpha u (t,x) + {c^2}u (t,x) = q (t,x), \]
where \(t \in [0, T]\) and \({\Delta_p}\) is the Laplacian of \(\mathbb{Q}_p^n\). When nonlinear term and initial value satisfy some conditions, the solutions of pseudo-differential equations are obtained by applying fundamental solutions.Jacobi-Sobolev orthogonal polynomials and spectral methods for elliptic boundary value problems.https://zbmath.org/1449.330142021-01-08T12:24:00+00:00"Yu, Xuhong"https://zbmath.org/authors/?q=ai:yu.xuhong"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqing"Li, Huiyuan"https://zbmath.org/authors/?q=ai:li.huiyuanSummary: Generalized Jacobi polynomials with indexes \(\alpha\), \(\beta \in \mathbb{R}\) are introduced and some basic properties are established. As examples of applications, the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered, and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems, the Jacobi-Sobolev orthogonal basis functions are constructed, which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.On gradient calculation in flux correction method.https://zbmath.org/1449.652392021-01-08T12:24:00+00:00"Bakhvalov, P. A."https://zbmath.org/authors/?q=ai:bakhvalov.pavel-aSummary: Flux Correction method is a family of edge-based schemes for solving hyperbolic systems on unstructured meshes. The cruical operation there is a nodal gradient calculation of physical variables with at least second order of accuracy. There are two well-known procedures meeting this condition. One is based on Least Squares method and the other one is based on spectral elements. In this paper we compare resulting schemes and discuss their problems.Solving interface problems of the Helmholtz equation by immersed finite element methods.https://zbmath.org/1449.653192021-01-08T12:24:00+00:00"Lin, Tao"https://zbmath.org/authors/?q=ai:lin.tao"Lin, Yanping"https://zbmath.org/authors/?q=ai:lin.yanping"Zhuang, Qiao"https://zbmath.org/authors/?q=ai:zhuang.qiaoSummary: This article reports our explorations for solving interface problems of the Helmholtz equation by immersed finite elements (IFE) on interface independent meshes. Two IFE methods are investigated: the partially penalized IFE (PPIFE) and discontinuous Galerkin IFE (DGIFE) methods. Optimal convergence rates are observed for these IFE methods once the mesh size is smaller than the optimal mesh size which is mainly dictated by the wave number. Numerical experiments also suggest that higher degree IFE methods are advantageous because of their larger optimal mesh size and higher convergence rates.Domain decomposition preconditioners for mixed finite-element discretization of high-contrast elliptic problems.https://zbmath.org/1449.653272021-01-08T12:24:00+00:00"Xie, Hui"https://zbmath.org/authors/?q=ai:xie.hui"Xu, Xuejun"https://zbmath.org/authors/?q=ai:xu.xuejunSummary: In this paper, we design an efficient domain decomposition (DD) preconditioner for the saddle-point problem resulting from the mixed finite-element discretization of multiscale elliptic problems. By proper equivalent algebraic operations, the original saddle-point system can be transformed to another saddle-point system which can be preconditioned by a block-diagonal matrix efficiently. Actually, the first block of this block-diagonal matrix corresponds to a multiscale \(H(\operatorname{div})\) problem, and thus, the direct inverse of this block is unpractical and unstable for the large-scale problem. To remedy this issue, a two-level overlapping DD preconditioner is proposed for this \(H(\operatorname{div})\) problem. Our coarse space consists of some velocities obtained from mixed formulation of local eigenvalue problems on the coarse edge patches multiplied by the partition of unity functions and the trivial coarse basis (e.g., Raviart-Thomas element) on the coarse grid. The condition number of our preconditioned DD method for this multiscale \(H(\operatorname{div})\) system is bounded by \(C(1+\frac{H^2}{\hat{\delta}^2})(1+\log^4(\frac{H}{h}))\), where \(\hat{\delta}\) denotes the width of overlapping region, and \(H\), \(h\) are the typical sizes of the subdomain and fine mesh. Numerical examples are presented to confirm the validity and robustness of our DD preconditioner.Superconvergence of energy-conserving discontinuous Galerkin methods for linear hyperbolic equations.https://zbmath.org/1449.652502021-01-08T12:24:00+00:00"Liu, Yong"https://zbmath.org/authors/?q=ai:liu.yong.1|liu.yong.5|liu.yong.3|liu.yong.4"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wang"Zhang, Mengping"https://zbmath.org/authors/?q=ai:zhang.mengpingSummary: In this paper, we study the superconvergence properties of the energy-conserving discontinuous Galerkin (DG) method in [\textit{G. Fu} and \textit{C.-W. Shu}, J. Comput. Appl. Math. 349, 41--51 (2019; Zbl 1407.65187)] for one-dimensional linear hyperbolic equations. We prove the approximate solution superconverges to a particular projection of the exact solution. The order of this superconvergence is proved to be \(k+2\) when piecewise \(\mathbb{P}^k\) polynomials with \(k \geq 1\) are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise \(\mathbb{P}^k\) polynomials with arbitrary \(k \geq 1\). Furthermore, we find that the derivative and function value approximations of the DG solution are superconvergent at a class of special points, with an order of \(k+1\) and \(k+2\), respectively. We also prove, under suitable choice of initial discretization, a \((2k+1)\)-th order superconvergence rate of the DG solution for the numerical fluxes and the cell averages. Numerical experiments are given to demonstrate these theoretical results.Analysis and approximation of gradient flows associated with a fractional order Gross-Pitaevskii free energy.https://zbmath.org/1449.653022021-01-08T12:24:00+00:00"Ainsworth, Mark"https://zbmath.org/authors/?q=ai:ainsworth.mark"Mao, Zhiping"https://zbmath.org/authors/?q=ai:mao.zhipingSummary: We establish the well-posedness of the fractional PDE which arises by considering the gradient flow associated with a fractional Gross-Pitaevskii free energy functional and some basic properties of the solution. The equation reduces to the Allen-Cahn or Cahn-Hilliard equations in the case where the mass tends to zero and an integer order derivative is used in the energy. We study how the presence of a non-zero mass affects the nature of the solutions compared with the Cahn-Hilliard case. In particular, we show that, analogous to the Cahn-Hilliard case, the solutions consist of regions in which the solution is a piecewise constant (whose value depends on the mass and the fractional order) separated by an interface whose width is independent of the mass and the fractional derivative. However, if the average value of the initial data exceeds some threshold (which we determine explicitly), then the solution will tend to a single constant steady state.Optimal location of heat sources inside areas of complex geometric forms.https://zbmath.org/1449.800192021-01-08T12:24:00+00:00"Osipov, O. V."https://zbmath.org/authors/?q=ai:osipov.o-v"Brusentsev, A. G."https://zbmath.org/authors/?q=ai:brusentsev.a-gSummary: Algorithms for the optimal arrangement of heat sources with volumetric heat release within regions of a complex geometric shape are considered. The distribution we found has the minimum total power and provides the temperature in the given temperature corridor. Finite-dimensional approximations of the original problem are constructed in the form of a linear programming problem. A method is given for constructing a finite-difference scheme for solving the heat equation, a brief description of the developed software modules for constructing grids and solving equations. Several computer experiments are carried out using the developed programs.Stability of the backward problem of parabolic equation with time-dependent reaction coefficient.https://zbmath.org/1449.350572021-01-08T12:24:00+00:00"Ma, Zongli"https://zbmath.org/authors/?q=ai:ma.zongli"Yue, Sufang"https://zbmath.org/authors/?q=ai:yue.sufangSummary: The backward problem of two-dimensional parabolic equation with a time-dependent coefficient was considered. This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the given data. Using the method of regularization, an optimal stability estimation of the solution was derived. A numerical example shows the effectiveness of the presented method.A nondivergence parabolic problem with a fractional time derivative.https://zbmath.org/1449.354282021-01-08T12:24:00+00:00"Allen, Mark"https://zbmath.org/authors/?q=ai:allen.mark-gThe author considers a nonlocal nonlinear parabolic problem with a fractional time derivative. A Krylov-Safonov type result is proved. In particular, a Hölder regularity of solutions is shown. The estimates given in the article remain uniform as the order of the fractional time derivative \(\alpha\to 1\).
Reviewer: Abdallah Bradji (Annaba)The interaction solution to generalized \( (3+1)\)-dimensional shallow wave equation.https://zbmath.org/1449.353672021-01-08T12:24:00+00:00"Meng, Yong"https://zbmath.org/authors/?q=ai:meng.yongSummary: By using the Hirota bilinear derivative method and the symbolic computation software Maple, the lump solution and the respiratory wave solution to the generalized \( (3+1)\)-dimensional shallow water wave equation were obtained. In combination with images, the dynamic properties (position, height, depth, velocity, and trajectory) of the lump-type soliton were studied. Finally, the interaction between different types of solutions were discussed, and it was found that lump-type solitons were phagocytosed by kink waves.Lie group analysis on Brownian motion and thermophoresis effect on free convective boundary-layer flow on a vertical cylinder embedded in a nanofluid-saturated porous medium.https://zbmath.org/1449.760392021-01-08T12:24:00+00:00"Ferdows, Mohammad"https://zbmath.org/authors/?q=ai:ferdows.mohammad"Hamad, Mohammed Abdul Ali"https://zbmath.org/authors/?q=ai:hamad.mohammed-abdul-ali"Ali, Mohamed"https://zbmath.org/authors/?q=ai:ali.mohamed-m|ali.mohamed-e|ali.mohamed-afsar|ali.mohamed-s-s|ali.mohamed-rSummary: Natural convective boundary-layer flow of a nanofluid on a heated vertical cylinder embedded in a nanofluid-saturated porous medium is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. Lie groups analysis is used to get the similarity transformations, which transform the governing partial differential equations to a system of ordinary differential equations. Two groups of similarity transformations are obtained. Numerical solutions of the resulting ordinary differential systems are obtained and discussed for various values of the governing parameters.Dynamical behavior for stochastic wave equation with nonlinear damping and white noise on unbounded domains.https://zbmath.org/1449.350752021-01-08T12:24:00+00:00"Qiao, Dan"https://zbmath.org/authors/?q=ai:qiao.dan"Wang, Miaomiao"https://zbmath.org/authors/?q=ai:wang.miaomiao"Li, Xiaojun"https://zbmath.org/authors/?q=ai:li.xiaojunSummary: The existence of random attractor for stochastic non-autonomous wave equation with nonlinear damping and additive noise on unbounded domains is studied. By using the uniform estimates of solutions of transformed system and the decomposition technique on the space, the existence of asymptotically compact \(\mathcal{D}\)-pullback absorbing set is obtained, and then the existence of random attractor of original system is given.On a nonlocal damping model in ferromagnetism.https://zbmath.org/1449.354082021-01-08T12:24:00+00:00"Moumni, M."https://zbmath.org/authors/?q=ai:moumni.mohammed"Tilioua, M."https://zbmath.org/authors/?q=ai:tilioua.mouhcineSummary: We consider a mathematical model describing nonlocal damping in magnetization dynamics. The model consists of a modified form of the Landau-Lifshitz-Gilbert (LLG) equation for the evolution of the magnetization vector in a rigid ferromagnet. We give a global existence result and characterize the long time behaviour of the obtained solutions. The sensitivity of the model with respect to large and small nonlocal damping parameters is also discussed.Inverse spectral problem for a pair of self-adjoint Hankel operators.https://zbmath.org/1449.350322021-01-08T12:24:00+00:00"Gérard, Patrick"https://zbmath.org/authors/?q=ai:gerard.patrick"Grellier, Sandrine"https://zbmath.org/authors/?q=ai:grellier.sandrineSummary: We give a precise inverse spectral result for compact self-adjoint Hankel operators. From Megretskii-Peller-Treil, a necessary and sufficient condition on a sequence of non zero real numbers, finite or infinite but tending to zero, to be a sequence of eigenvalues of some self-adjoint and compact Hankel operator is that the multiplicity of an eigenvalue \(\lambda\) should differ from the multiplicity of \(-\lambda\) at most by one. Under this condition, we describe precisely the set of symbols for which the Hankel operator has a given sequence of eigenvalues. This theorem is a consequence of a general inverse spectral result that we proved for non-necessarily self-adjoint Hankel operators. As a by-product, we show how we recover the Megretskii-Peller-Treil condition.
For the entire collection see [Zbl 1404.42002].Solution of the Cauchy problem for system of the Euler-Poisson-Darboux equations.https://zbmath.org/1449.353022021-01-08T12:24:00+00:00"Maksimova, Ekaterina Alekseevna"https://zbmath.org/authors/?q=ai:maksimova.ekaterina-alekseevnaSummary: The system of Euler-Poisson-Darboux equations is considered, the Cauchy problem is solved for the case, when characteristic numbers of matrix-coefficient are complex conjugate and having real part in the interval \((-1/2, 0)\).On the problem for mixed type equation with M. Saigo operators.https://zbmath.org/1449.353222021-01-08T12:24:00+00:00"Arlanova, Ekaterina Yur'evna"https://zbmath.org/authors/?q=ai:arlanova.ekaterina-yurevnaSummary: In this paper boundary value problem with M. Saigo operators is considered for the equation of the mixed type. This equation is represented as an equation of fractional diffusion in upper half-plane and as an equation of moisture transfer in lower half-plane. One-valued solvability of the problem is proved.An influence of the false bottom on the nonlinear dynamics of the water freezing process.https://zbmath.org/1449.860022021-01-08T12:24:00+00:00"Nizovtseva, Irina Gennad'evna"https://zbmath.org/authors/?q=ai:nizovtseva.irina-gennadevna"Alexandrov, Dmitriĭ Valer'evich"https://zbmath.org/authors/?q=ai:aleksandrov.dmitrii-valerevichSummary: The current paper casts the light on the processes of structural-phase transitions during the freezing salt water, including the false bottom effects. A nonlinear mathematical model of heat and mass transfer was obtained. It takes into account the presence of three moving boundaries of phase transition and turbulent fluid flows from the ocean side by the surface of the false bottom. The exact analytical solutions of the nonlinear model were obtained -- in their turn, they takes into account the time dependence of temperature and salinity at the depth and fluctuations of friction velocity. The distribution of temperature and salinity, the concentration of solids, the laws of motion of the boundaries of the phase transition, ``salt water -- the two-phase zone'', ``two-phase zone -- melted water'' and ``melt-water -- ice'' were found. The heat flux at the lower boundary of a false bottom was specified. The latter can change its direction at the time oscillations of the sea water temperature and friction velocity. Also it was shown, that structural transitions in the ice thickness are strictly associated with the processes of evolution of a false bottom.On the essential spectrum of a model operator associated with the system of three particles on a lattice.https://zbmath.org/1449.810212021-01-08T12:24:00+00:00"Rasulov, Tulkin Khusenovich"https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovichSummary: A model operator \(H\) associated with the system of three-identical particles on a lattice \(\mathbb{Z}^3\) is considered. The location of the essential spectrum of \(H\) is described by the spectrum of the corresponding Friedrichs model, that is, the two-particle and three-particle branches of the essential spectrum of \(H\) are singled out. It is proved that the essential spectrum of \(H\) consists of no more than three bounded closed intervals. An appearance of two-particle branches on the both sides of the three-particle branch is shown. Moreover, we obtain an analogue of the Faddeev equation and its symmetric version, for the eigenfunctions of \(H\).The Goursat problem for one hyperbolic system of the third order differential equations with two independent variables.https://zbmath.org/1449.352962021-01-08T12:24:00+00:00"Andreev, Aleksandr Anatol'evich"https://zbmath.org/authors/?q=ai:andreev.aleksandr-anatolevich"Yakovleva, Yuliya Olegovna"https://zbmath.org/authors/?q=ai:yakovleva.yuliya-olegovnaSummary: The solution of the Goursat problem for the system of the differential equations of the third order is received on the basis of Riemann method. The Riemann matrix expressed in terms of hypergeometric functions with matrix argument is received, using it the solution of Goursat problem for the system of the linear hyperbolic equations of the third order is constructed.On one non-local problem for axisymmetric Helmholtz equation.https://zbmath.org/1449.351882021-01-08T12:24:00+00:00"Abashkin, Anton Aleksandrovich"https://zbmath.org/authors/?q=ai:abashkin.anton-aleksandrovichSummary: Non-local boundary problem for the axisymmetric Helmholtz equation is explored. The uniqueness of the solution is proved by the spectral method. The conditions of solvability are found. The solution of the problem is constructed in the form of the biorthogonal series.On some conjugation problems of parabolic and hyperbolic equations with integro-differential conditions on the separating boundary.https://zbmath.org/1449.353162021-01-08T12:24:00+00:00"Eleev, Valeriĭ Abdurakhmanovich"https://zbmath.org/authors/?q=ai:eleev.valerii-abdurakhmanovich"Balkizova, Alëna Khamudbievna"https://zbmath.org/authors/?q=ai:balkizova.alena-khamudbievnaSummary: The unique solvability of the problems of conjugation of hyperbolic and parabolic equations in finite domains is proved by the method of equivalent reduction to the Volterra integral equation of the second kind.On some classes of solutions of aerohydromechanic equations.https://zbmath.org/1449.353532021-01-08T12:24:00+00:00"Kazakova, Yuliya Aleksandrovna"https://zbmath.org/authors/?q=ai:kazakova.yuliya-aleksandrovnaSummary: The article gives a review of the method of constructing parametric solutions of polynomial form for systems of partial differential equations and its application to some aerohydromechanic equations. For example the parametric solutions are constructed and their classification is carried out for the system of equations, describing the plane flows of the ideal gas with the constant entropy. Particularly, the solutions of ``simple wave'' type are demonstrated. Also the solutions of transonic equations, describing flows of gas with local supersonic zones in Laval nozzles are obtained.Non-stationary problems of the dynamics of stepped section plates and rotation cylindrical shells.https://zbmath.org/1449.354122021-01-08T12:24:00+00:00"D'yachenko, Yuriĭ Petrovich"https://zbmath.org/authors/?q=ai:dyachenko.yurii-petrovich"Elenitskiĭ, Éduard Yakovlevich"https://zbmath.org/authors/?q=ai:elenitskii.eduard-yakovlevich"Petrov, Denis Vladislavovich"https://zbmath.org/authors/?q=ai:petrov.denis-vladislavovichSummary: The technique of exact non-stationary dynamic calculation of compound designed systems of in steps-variable thickness a method initial parameters is offered. The settlement scheme considers displacement of median surfaces of interfaced elements. As an example calculation of base plate of a dam GES and matrices of explosive punching is resulted at pulse influence.On solutions of integro-differential equations in dynamic problem of one aeroelastic system ``tandem'' type.https://zbmath.org/1449.353482021-01-08T12:24:00+00:00"Ankilov, Andreĭ Vladimirovich"https://zbmath.org/authors/?q=ai:ankilov.andrey-v"Vel'misov, Pëtr Aleksandrovich"https://zbmath.org/authors/?q=ai:velmisov.petr-aleksandrovich"Semenova, Elizaveta Petrovna"https://zbmath.org/authors/?q=ai:semenova.elizaveta-petrovnaSummary: A mathematical model of the dynamical system of the two elastic ``tandem'' type plates, flowing along of the subsonic flow of gas (fluid) is proposed. The solution of aerohydrodynamic part of problem based on the methods of complex variable functions theory is given. The bound system of the equations allowing to investigate the plates dynamic is obtained. The numerical-analytical method of solution of this equation is offered.Problems with Laplace operator on topological surfaces.https://zbmath.org/1449.354212021-01-08T12:24:00+00:00"Shalaginov, Mikhaĭl Yur'evich"https://zbmath.org/authors/?q=ai:shalaginov.mikhail-yurevich"Ivanov, Mikhaĭl Gennad'evich"https://zbmath.org/authors/?q=ai:ivanov.mikhail-gennadevich"Dolgopolov, Mikhaĭl Vyacheslavovich"https://zbmath.org/authors/?q=ai:dolgopolov.mikhail-vyacheslavovichSummary: This work highlights the problems related to the Laplace operator on topological surfaces such as Möbius strip, Klein bottle and torus. In particular, we discuss oscillations on the surface of the Möbius strip, eigenfunctions and eigenvalues of the Laplace operator on the surface of the Klein bottle, as well as behavior of a charged particle on the torus.On the dynamics of the quantum states set for a system with degenerated Hamiltonian.https://zbmath.org/1449.353972021-01-08T12:24:00+00:00"Sakbaev, Vsevolod Zhanovich"https://zbmath.org/authors/?q=ai:sakbaev.vsevolod-zhSummary: We study the sequence of regularizing Cauchy problem as the elliptic regularization of Cauchy problem for Schrodinger equation with discontinuous and degenerated coefficients. The necessary and sufficient condition of the convergence of the regularizing dynamical semigroups sequence are presented. If the convergence is impossible then divergent sequence of the regularizing quantum states is considered as the stochastic process on the measurable space of regularizing parameter endowing with finite additive measure. The expectation of this stochastic process defines the averaging trajectory in the space of quantum states. It was obtained the condition on the finite additive measure such, that averaging trajectory can be defined by its values in two instants with the help of solving the variational problems.The problem with integral condition for one space analog of hyperbolic type equation degenerated on a coordinate planes.https://zbmath.org/1449.352942021-01-08T12:24:00+00:00"Rodionova, Irina Nikolaevna"https://zbmath.org/authors/?q=ai:rodionova.irina-nikolaevnaSummary: For the full equation of the third order in a three-dimensional Euclidean space the boundary value problem with interface on non-characteristic plane in the area limited by planes of a singularity of the equation factors is solved.Some spectral properties of a generalized Friedrichs model.https://zbmath.org/1449.810232021-01-08T12:24:00+00:00"Rasulov, Tulkin Khusenovich"https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovich"Turdiev, Khalim Khamraevich"https://zbmath.org/authors/?q=ai:turdiev.khalim-khamraevichSummary: We consider self-adjoint generalized Friedrichs model \(h(p)\), \(p \in\mathcal{T}^3\) (\(\mathcal{T}^3\) is the three-dimensional torus), in the case where the parameter functions \(w_1\) and \(w_2\) of this operator has the special forms. These functions has non-degenerate minimum at the several different points. Threshold effects for the considering operator are studied depending on the minimum points of \(w_2\).The Faddeev equation and location of the essential spectrum of a three-particle model operator.https://zbmath.org/1449.810222021-01-08T12:24:00+00:00"Rasulov, Tulkin Khusenovich"https://zbmath.org/authors/?q=ai:rasulov.tulkin-khusenovich"Rakhmonov, Askar Akhmadovich"https://zbmath.org/authors/?q=ai:rakhmonov.askar-akhmadovichSummary: In this paper a model operator \(H\) associated to a system of three-particles on a lattice is considered. The location of the essential spectrum of \(H\) is described by the spectrum of channel operators. The Faddeev type equation for the eigenvectors of \(H\) is obtained.Existence results of self-similar solutions to the Caputo-type's space-fractional heat equation.https://zbmath.org/1449.354292021-01-08T12:24:00+00:00"Basti, Bilal"https://zbmath.org/authors/?q=ai:basti.bilal"Benhamidouche, Noureddine"https://zbmath.org/authors/?q=ai:benhamidouche.noureddineSummary: This paper investigates the problem of existence and uniqueness of solutions under the self-similar forms to the space-fractional heat equation. By applying the properties of Banach's fixed point theorems, Schauder's fixed point theorem and the nonlinear alternative of Leray-Schauder type, we establish several results on the existence and uniqueness of self-similar solutions to this equation.Decay of solutions to a water wave model with a nonlocal viscous term.https://zbmath.org/1449.350682021-01-08T12:24:00+00:00"Dumont, S."https://zbmath.org/authors/?q=ai:dumont.serge"Goubet, O."https://zbmath.org/authors/?q=ai:goubet.olivier"Manoubi, I."https://zbmath.org/authors/?q=ai:manoubi.imenSummary: We update here the results on the decay of solutions to a nonlocal water wave equation that reads
\[ u_t+u_x+\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds+ u u_x= u_{xx}, \]
where \(\frac{1}{\sqrt{\pi}}\frac{\partial}{\partial t} \int_0^t\frac{u(s)}{\sqrt{t-s}} ds\) is the Riemann-Liouville half-order derivative.Singular Schrödinger operators and Robin billiards. Spectral properties and asymptotic expansions.https://zbmath.org/1449.810252021-01-08T12:24:00+00:00"Exner, Pavel"https://zbmath.org/authors/?q=ai:exner.pavelSummary: This paper summarizes the contents of a plenary talk at the Pan African Congress of Mathematics held in Rabat in July 2017. We provide a survey of recent results on spectral properties of Schrödinger operators with singular interactions supported by manifolds of codimension one and of Robin billiards with the focus on the geometrically induced discrete spectrum and its asymptotic expansions in term of the model parameters.Existence of a sign-changing solution for an asymptotically linear Kirchhoff equation.https://zbmath.org/1449.352192021-01-08T12:24:00+00:00"Zhang, Dandan"https://zbmath.org/authors/?q=ai:zhang.dandan"Ding, Ling"https://zbmath.org/authors/?q=ai:ding.lingSummary: In this paper, a class of radially symmetric autonomous Kirchhoff equations with asymptotic linearity at infinity is studied. Under suitable conditions, we prove the Kirchhoff equation to be equivalent to the elliptic system. Furthermore, by using some analysis techniques, the existence result of a sign-changing solution is obtained.On a non-local boundary value problem for loaded differential equation with characteristic form of variable sign.https://zbmath.org/1449.352472021-01-08T12:24:00+00:00"Tokova, Alla Askerbievna"https://zbmath.org/authors/?q=ai:tokova.alla-askerbievnaSummary: Non-local boundary value problem for loaded equation of parabolic type with the sign-variable characteristic form is solved. Common representation of solution is constructed. The theorems of common representation, existence and uniqueness of solution for boundary value problems set in rectangular domain are proved.On solvability of a inverse problem for hyperbolic equation with an integral overdetermination condition.https://zbmath.org/1449.354592021-01-08T12:24:00+00:00"Beĭlina, Natal'ya Viktorovna"https://zbmath.org/authors/?q=ai:beilina.natalya-viktorovnaSummary: In this paper we study an inverse problem with an integral overdetermination condition for a hyperbolic equation with an unknown coefficient in equation. The existence and uniqueness of a solution is proved with the help of an a-priory estimate and Galyorkin procedure.On singular solutions of a multidimensional differential equation of Clairaut-type with power and exponential functions.https://zbmath.org/1449.351722021-01-08T12:24:00+00:00"Ryskina, Liliya Leonidovna"https://zbmath.org/authors/?q=ai:ryskina.liliya-leonidovnaSummary: In the theory of ordinary differential equations, the Clairaut equation is well known. This equation is a non-linear differential equation unresolved with respect to the derivative. Finding the general solution of the Clairaut equation is described in detail in the literature and is known to be a family of integral lines. However, along with the general solution, for such equations there exists a singular (special) solution representing the envelope of the given family of integral lines. Note that the singular solution of the Clairaut equation is of particular interest in a number of applied problems.
In addition to the ordinary Clairaut differential equation, a differential equation of the first order in partial derivatives of the Clairaut type is known. This equation is a multidimensional generalization of the ordinary differential Clairaut equation, in the case when the sought function depends on many variables. The problem of finding a general solution for partial differential equations of the Clairaut is known to be. It is known that the complete integral of the equation is a family of integral (hyper) planes. In addition to the general solution, there may be partial solutions, and, in some cases, it is possible to find a singular solution. Generally speaking, there is no general algorithm for finding a singular solution, since the problem is reduced to solving a system of nonlinear algebraic equations.
The article is devoted to the problem of finding a singular solution of Clairaut type differential equation in partial derivatives for the particular choice of a function from the derivatives in the right-hand side. The work is organized as follows. The introduction provides a brief overview of some of the current results relating to the study of Clairaut-type equations in field theory and classical mechanics. The first part provides general information about differential equations of the Clairaut-type in partial derivatives and the structure of its general solution. In the main part of the paper, we discuss the method for finding singular solutions of the Clairaut-type equations. The main result of the work is to find singular solutions of equations containing power and exponential functions.Solvability of a nonlocal problem for a hyperbolic equation with degenerate integral conditions.https://zbmath.org/1449.352882021-01-08T12:24:00+00:00"Pulkina, Lyadmila Stepanovna"https://zbmath.org/authors/?q=ai:pulkina.lyadmila-stepanovna"Kirichek, Vitaliya Aleksandrovna"https://zbmath.org/authors/?q=ai:kirichek.vitaliya-aleksandrovnaSummary: In this paper, we consider a nonlocal problem with integral conditions for hyperbolic equation. Close attention focuses on degenerate integral conditions, namely, on the second kind integral conditions which degenerate into the first kind conditions at some points. Such kind of nonlocal conditions inevitably involves some specific difficulties when we try to show solvability of the problem. These difficulties can be overcome by a method suggested in our paper. The essence of this method is the reduction of the problem with degenerate conditions to the problem with dynamical conditions. This technique enables to define effectively a generalized solution to the problem, to obtain a priori estimates and to prove the existence of a unique generalized solution to the problem.Construction of analitical solution of 2D stochastically nonlinear boundary value problem of steady creep state with respect to the boundary effects.https://zbmath.org/1449.354142021-01-08T12:24:00+00:00"Kovalenko, Lyudmila Viktorovna"https://zbmath.org/authors/?q=ai:kovalenko.lyudmila-viktorovna"Popov, Nikolaĭ Nikolaevich"https://zbmath.org/authors/?q=ai:popov.n-nSummary: The solution of nonlinear stochastically boundary value problem of creep of a thin plate under plane stress is developed. It is supposed that elastic deformations are insignificant and they can be neglected. Determining equation of creep is taken in accordance with nonlinear theory of viscous flow and is formulated in a stochastic form. By applying the method of small parameter nonlinear stochastic problem reduces to a system of three linear partial differential equations, which is solved about fluctuations of the stress tensor. This system with transition to the stress function has been reduced to a single differential equation solution of which is represented as a sum of two series. The first row gives the solution away from the boundary of the body without boundary effects, the second row represents the solution boundary layer, its members quickly fade as the distance increases from plate boundary. Based on this solution, the statistical analysis random stress fields near the boundary of the plate was taken.On stability of solutions of equations of interaction between elastic walls of channels and affluent liquid.https://zbmath.org/1449.354102021-01-08T12:24:00+00:00"Ankilov, Andreĭ Vladimirovich"https://zbmath.org/authors/?q=ai:ankilov.andrey-v"Vel'misov, P. A."https://zbmath.org/authors/?q=ai:velmisov.petr-aleksandrovichSummary: In this article the dynamical stability of elastic walls of plane channels under flowing of the perfect liquid is investigated on the basis of mathematical models. Either the law of pressure change or the potential of liquid velocity or the longitudinal component of liquid velocity are applied on the input and output from the channels. The sufficient conditions of stability are obtained. These conditions impose a constraint on the velocity of the liquid homogeneous stream, on the compressing (decompressing) force and other parameters of mechanical system.Boltzmann equation and \(H\)-theorem in the functional formulation of classical mechanics.https://zbmath.org/1449.820032021-01-08T12:24:00+00:00"Trushechkin, Anton Sergeevich"https://zbmath.org/authors/?q=ai:trushechkin.anton-sergeevichSummary: We propose a procedure for obtaining the Boltzmann equation from the Liouville equation in a non-thermodynamic limit. It is based on the BBGKY hierarchy, the functional formulation of classical mechanics, and the distinguishing between two scales of space-time, i.e., macro- and microscale. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which have errors. Hence, one can speak about accuracy of the initial probability density function in the Liouville equation. Let's assume that our measuring instruments can observe the variations of physical values only on the macroscale, which is much greater than the characteristic interaction radius (microscale). Then the corresponding initial density function cannot be used as initial data for the Liouville equation, because the last one is a description of the microscopic dynamics, and the particle interaction potential (with the characteristic interaction radius) is contained in it explicitly. Nevertheless, for a macroscopic initial density function we can obtain the Boltzmann equation using the BBGKY hierarchy, if we assume that the initial data for the microscopic density functions are assigned by the macroscopic one. The \(H\)-theorem (entropy growth) is valid for the obtained equation.Solution in explicit form of non-local problem for differential equation with partial fractional derivative of Riemann-Liouville.https://zbmath.org/1449.353202021-01-08T12:24:00+00:00"Saĭganova, Svetlana Aleksandrovna"https://zbmath.org/authors/?q=ai:saiganova.svetlana-aleksandrovnaSummary: A non-local problem for a mixed type equation with partial fractional derivative of Riemann-Liouville is studied, boundary condition of which contains generalized operator of fractional integro-differentiation. Unique solution of the problem is then proved.Nonlocal boundary value problem for a Lykov's type system of first-order.https://zbmath.org/1449.353002021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: In this paper we prove the unique solution of the problem with a shift to a Lykov's type system of differential equations of first order. The proof is given for different values of the generalized operators of fractional integro-differentiation included in the boundary condition.Local existence and global nonexistence theorems for a viscous damped quasi-linear wave equations.https://zbmath.org/1449.353092021-01-08T12:24:00+00:00"Song, Ruili"https://zbmath.org/authors/?q=ai:song.ruili"Wang, Shubin"https://zbmath.org/authors/?q=ai:wang.shubinSummary: In this paper, the existence and uniqueness of the local solution for the initial boundary value problem of a class of quasi-linear viscous damping wave equation in three-dimensional spaces are proved by the Galerkin method and compactness principle. The blow-up of the solution in limited time for this question is proved by means of the energy integral inequality.On local resolvability of a certain class of the first-order partial differential equations.https://zbmath.org/1449.350112021-01-08T12:24:00+00:00"Alekseenko, S. N."https://zbmath.org/authors/?q=ai:alekseenko.sergey-n"Platonova, L. E."https://zbmath.org/authors/?q=ai:platonova.l-eIn this paper the Cauchy problem for some quasilinear PDE of the first order with two independent variables is examined. The curve where initial data are given may be described by parametric or explicit equations; also it may have finite or infinite length. By means of additional argument method for all these cases the Cauchy problem is reduced to the system of integral equations. The authors prove local resolvability of this system and deduce that problem's solution has the same smoothness as the initial data.
Reviewer: Aleksey Syromyasov (Saransk)Special solutions of matrix Gellerstedt equation.https://zbmath.org/1449.353182021-01-08T12:24:00+00:00"Kozlova, Elena Aleksandrovna"https://zbmath.org/authors/?q=ai:kozlova.elena-aleksandrovnaSummary: Fundamental solutions for the Gellerstedt equation and its generalization were obtained in the distribution space using the method applied by I. M. Gelfand and J. Barros-Neto to the studying the Tricomi equation. The degenerating system of the mixed-type partial differential equations was considered, its special solutions were constructed in the regions bounded by the characteristics of these equations (in the hyperbolic half-plane). The elements of the theory of matrices, theory of the generalized functions and the special functions (hypergeometric series) were used for this construction.Existence and uniqueness of mild solutions for nonlinear fractional integro-differential evolution equations.https://zbmath.org/1449.370492021-01-08T12:24:00+00:00"Hou, Mimi"https://zbmath.org/authors/?q=ai:hou.mimi"Xi, Xuanxuan"https://zbmath.org/authors/?q=ai:xi.xuanxuan"Zhou, Xianfeng"https://zbmath.org/authors/?q=ai:zhou.xianfengSummary: In this paper, we study a class of nonlinear fractional integro-differential evolution equations in a Banach space \(X\). We use the fractional power of operators and the theory of analytic semigroups to prove the existence and uniqueness of the solution for the given problem. Furthermore, we give the Hölder continuity of the obtained mild solution.Modeling and simulation of fractional diffusion.https://zbmath.org/1449.354412021-01-08T12:24:00+00:00"Izsák, Ferenc"https://zbmath.org/authors/?q=ai:izsak.ferenc"Szekeres, János"https://zbmath.org/authors/?q=ai:szekeres.janosSummary: A short overview is given on the models of the fractional order diffusion and some numerical methods for the related problems is discussed. We mention more applications and important open problems in this field.Method of general Cole-Hopf substitutions in theory of finite-dimensional dynamical systems.https://zbmath.org/1449.700102021-01-08T12:24:00+00:00"Zhuravlev, Viktor Mikhaĭlovich"https://zbmath.org/authors/?q=ai:zhuravlev.viktor-mikhailovich"Obrubov, Konstantin Sergeevich"https://zbmath.org/authors/?q=ai:obrubov.konstantin-sergeevichSummary: We consider the results of applying the method of generic Cole-Hopf substitutions to integration of finite-dimensional dynamical systems. Dynamical systems are represented in the form of matrix ordinary differential equations with specific matrix algebra of finite dimension. The Cole-Hopf type substitutions are applied to matrix equations by using the differentiation on algebra in the form of commutator with a specific algebra element. Recurrent relations for Cole-Hopf substitutions were found. Particular cases of exactly integrable dynamical systems are presented. The algorithm of calculating the integrals of motion is shown.Global existence and blow-up of solutions in reaction-diffusion system with free boundary.https://zbmath.org/1449.350252021-01-08T12:24:00+00:00"Wang, Zhongqian"https://zbmath.org/authors/?q=ai:wang.zhongqian"Jia, Zhe"https://zbmath.org/authors/?q=ai:jia.zhe"Yuan, Junli"https://zbmath.org/authors/?q=ai:yuan.junli"Yang, Zuodong"https://zbmath.org/authors/?q=ai:yang.zuodongSummary: This paper is concerned with a free boundary problem for the reaction-diffusion system with coupled nonlinear reaction terms. For simplicity, we assume that the conditions and solutions are radially symmetric. At first, we give the local existence and uniqueness of the positive solution. Then, we consider the blow-up property and the long time behavior of the solution. When \({m_2} - {m_1} > -1, {n_1} - {n_2} > -1\), the solution blows up if the initial value is large enough.Accurate solution for sliding Burger fluid flow.https://zbmath.org/1449.353592021-01-08T12:24:00+00:00"Zubair, Muhammad"https://zbmath.org/authors/?q=ai:zubair.muhammad"Hayat, Tasawar"https://zbmath.org/authors/?q=ai:hayat.tasawarSummary: This article addresses the influence of partial slip condition in the hydromagnetic flow of Burgers fluid in a rotating frame of reference. The flows are induced by oscillation of a boundary. Two problems for oscillatory flows are considered. Exact solutions to the resulting boundary value problems are constructed. Analysis has been carried out in the presence of magnetic field. Physical interpretation is made through the plots for various embedded parameters.Weak and strong convergence of solutions to linearized equations of low compressible fluid.https://zbmath.org/1449.351622021-01-08T12:24:00+00:00"Gusev, Nikolaĭ Anatol'evich"https://zbmath.org/authors/?q=ai:gusev.nikolai-anatolevichSummary: Initial-boundary value problem for linearized equations of viscous barotropic low compressible fluid in a bounded domain is considered. Convergence of solutions of this problem at withincompressible limit approaching to zero is studied. Sufficient conditions for the weak and strong convergence of this problem for uncompressible liquid are given.Cauchy problem for the wave equation on non-globally hyperbolic manifolds.https://zbmath.org/1449.352832021-01-08T12:24:00+00:00"Groshev, Oleg Viktorovich"https://zbmath.org/authors/?q=ai:groshev.oleg-viktorovichSummary: We consider Cauchy problem for wave equation on two types of non-global hyperbolic manifolds: Minkowski plane with an attached handle and Misner space. We prove that the classical solution on a plane with a handle exists and is unique if and only if a finite set of point-wise constraints on initial values is satisfied. On the Misner space the existence and uniqueness of a solution is equivalent to much stricter constraints for the initial data.On nonlocal cosmological equations on half-line.https://zbmath.org/1449.830062021-01-08T12:24:00+00:00"Aref'eva, Irina Yaroslavna"https://zbmath.org/authors/?q=ai:arefeva.irina-ya"Volovich, Igor' Vasil'evich"https://zbmath.org/authors/?q=ai:volovich.igor-vSummary: A system of nonlocal cosmological equations where the time variable runs over a half-line is considered. These equations are more suitable for description of the Universe than the previously discussed cosmological equations on the whole line since the Friedmann metric contains a singularity at the beginning of time. Definition of the exponential operator includes a new arbitrary function which is absent in the equations on the whole line. It is shown that this function could be choosen in such a way that one of the slow roll parameters in the chaotic inflation scenario can be made arbitrary small. Solutions of the linearized nonlocal equations on the half-line are constructed.Transmission Robin problem for singular \(p(x)\)-Laplacian equation in a cone.https://zbmath.org/1449.352272021-01-08T12:24:00+00:00"Borsuk, Mikhail"https://zbmath.org/authors/?q=ai:borsuk.mikhail-vSummary: We study the behavior near the boundary angular or conical point of weak solutions to the transmission Robin problem for an elliptic quasilinear second-order equation with the variable \(p(x)\)-Laplacian.Multiple soliton solution of the \( (3+1)\)-dimensional Hirota bilinear soliton equation.https://zbmath.org/1449.351562021-01-08T12:24:00+00:00"Peng, Yali"https://zbmath.org/authors/?q=ai:peng.yali"Taogetusang"https://zbmath.org/authors/?q=ai:taogetusang.Summary: Two methods are used to construct the soliton solution of \( (3+1)\)-dimensional soliton equation. The first method is to transform it into bilinear equation by using the logarithmic function transformation, and to solve the single soliton solution, double soliton solution and N-soliton solution of bilinear equation by using series perturbation method. The second method is to construct the strange wave solution of \( (3+1)\)-dimensional high-dimensional soliton equation by combining generalized rational polynomials with heuristic method.A uniqueness result for a Schrödinger-Poisson system with strong singularity.https://zbmath.org/1449.350082021-01-08T12:24:00+00:00"Yu, Shengbin"https://zbmath.org/authors/?q=ai:yu.shengbin"Chen, Jianqing"https://zbmath.org/authors/?q=ai:chen.jianqingSummary: In this paper, we consider the following Schrödinger-Poisson system with strong singularity \[ -\Delta{u}+\phi u=f(x)u^{-\gamma}, \; x\in \Omega,\] \[ -\Delta{\phi}=u^2, \; x\in\Omega,\] \[ u>0, \; x\in\Omega,\] \[ u=\phi=0, \; x\in\partial\Omega, \] where \(\Omega\subset \mathbb{R}^3\) is a smooth bounded domain, \(\gamma>1\), \(f\in L^1(\Omega)\) is a positive function (i.e. \(f(x)>0\) a.e. in \(\Omega\)). A necessary and sufficient condition on the existence and uniqueness of positive weak solution of the system is obtained. The results supplement the main conclusions in recent literature.Traveling waves for a diffusive SIR-B epidemic model with multiple transmission pathways.https://zbmath.org/1449.350802021-01-08T12:24:00+00:00"Song, Haifeng"https://zbmath.org/authors/?q=ai:song.haifeng"Zhang, Yuxiang"https://zbmath.org/authors/?q=ai:zhang.yuxiangSummary: In this work, we consider a diffusive SIR-B epidemic model with multiple transmission pathways and saturating incidence rates. We first present the explicit formula of the basic reproduction number \(\mathcal{R}_0\). Then we show that if \(\mathcal{R}_0>1\), there exists a constant \(c^*>0\) such that the system admits traveling wave solutions connecting the disease-free equilibrium and endemic equilibrium with speed \(c\) if and only if \(c\geq c^*\). Since the system does not admit the comparison principle, we appeal to the standard Schauder's fixed point theorem to prove the existence of traveling waves. Moreover, a suitable Lyapunov function is constructed to prove the upward convergence of traveling waves.Existence and asymptotics of traveling wave fronts for a coupled nonlocal diffusion and difference system with delay.https://zbmath.org/1449.351422021-01-08T12:24:00+00:00"Chekroun, Abdennasser"https://zbmath.org/authors/?q=ai:chekroun.abdennasserSummary: In this paper, we consider a general study of a recent proposed hematopoietic stem cells model. This model is a combination of nonlocal diffusion equation and difference equation with delay. We deal with the properties of traveling waves for this system such as the existence and asymptotic behavior. By using the Schauder's fixed point theorem combined with the method based on the construction of upper and lower solutions, we obtain the existence of traveling wave fronts for a speed \(c> c^{\star}\). The case \(c= c^{\star}\) is studied by using a limit argument. We prove also that \(c^{\star}\) is the critical value. We finally prove that the nonlocality increases the minimal wave speed.Mixed problem with integral condition for the hyperbolic equation.https://zbmath.org/1449.352902021-01-08T12:24:00+00:00"Golubeva, Natal'ya Dmitrievna"https://zbmath.org/authors/?q=ai:golubeva.natalya-dmitrievnaSummary: In this paper we consider a nonlocal problem with integral condition of the first kind. Existence and uniqueness of a solution of this problem are proved. The proof is based on a priori estimates and auxiliary problem method.Serrin-type blowup criterion of three-dimensional nonhomogeneous heat conducting magnetohydrodynamic flows with vacuum.https://zbmath.org/1449.353582021-01-08T12:24:00+00:00"Zhou, Ling"https://zbmath.org/authors/?q=ai:zhou.lingSummary: We consider an initial boundary value problem for the nonhomogeneous heat conducting magnetohydrodynamic flows. We show that for the initial density allowing vacuum, the strong solution exists globally if the velocity field satisfies Serrin's condition. Our method relies upon the delicate energy estimates and regularity properties of Stokes system and elliptic equations.Temporal and spatial patterns in a diffusive ratio-dependent predator-prey system with linear stocking rate of prey species.https://zbmath.org/1449.920392021-01-08T12:24:00+00:00"Li, Wanjun"https://zbmath.org/authors/?q=ai:li.wanjun"Gao, Xiaoyan"https://zbmath.org/authors/?q=ai:gao.xiaoyan"Fu, Shengmao"https://zbmath.org/authors/?q=ai:fu.shengmaoSummary: The ratio-dependent predator-prey model exhibits rich interesting dynamics due to the singularity of the origin. It is one of prototypical pattern formation models. Stocking in a ratio-dependent predator-prey models is relatively an important research subject from both ecological and mathematical points of view. In this paper, we study the temporal, spatial patterns of a ratio-dependent predator-prey diffusive model with linear stocking rate of prey species. For the spatially homogeneous model, we derive conditions for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solution by the center manifold and the normal form theory. For the reaction-diffusion model, firstly it is shown that Turing (diffusion-driven) instability occurs, which induces spatial inhomogeneous patterns. Then it is demonstrated that the model exhibits Hopf bifurcation which produces temporal inhomogeneous patterns. Finally, the non-existence and existence of positive non-constant steady-state solutions are established. We can see spatial inhomogeneous patterns via Turing instability, temporal periodic patterns via Hopf bifurcation and spatial patterns via the existence of positive non-constant steady state. Moreover, numerical simulations are performed to visualize the complex dynamic behavior.Asymptotic behavior of solutions of a Fisher equation with free boundaries and nonlocal term.https://zbmath.org/1449.352492021-01-08T12:24:00+00:00"Cai, Jingjing"https://zbmath.org/authors/?q=ai:cai.jingjing"Chai, Yuan"https://zbmath.org/authors/?q=ai:chai.yuan"Li, Lizhen"https://zbmath.org/authors/?q=ai:li.lizhen"Wu, Quanjun"https://zbmath.org/authors/?q=ai:wu.quanjunSummary: We study the asymptotic behavior of solutions of a Fisher equation with free boundaries and the nonlocal term (an integral convolution in space). This problem can model the spreading of a biological or chemical species, where free boundaries represent the spreading fronts of the species. We give a dichotomy result, that is, the solution either converges to \(1\) locally uniformly in \(\mathbb{R}\), or to \(0\) uniformly in the occupying domain. Moreover, we give the sharp threshold when the initial data \(u_0=\sigma \phi\), that is, there exists \(\sigma^*>0\) such that spreading happens when \(\sigma>\sigma^*\), and vanishing happens when \(\sigma\leq \sigma^*\).Existence of nontrivial solution for fourth-order semilinear \(\Delta_{\gamma}\)-Laplace equation in \(\mathbb{R}^N\).https://zbmath.org/1449.352122021-01-08T12:24:00+00:00"Luyen, Duong Trong"https://zbmath.org/authors/?q=ai:luyen.duong-trongSummary: In this paper, we study existence of nontrivial solutions for a fourth-order semilinear \(\Delta_{\gamma}\)-Laplace equation in \(\mathbb{R}^N\) \[ \Delta_\gamma^{2}u-\Delta_\gamma u+\lambda b(x)u=f(x, u), \quad x\in \mathbb{R}^N,\quad u\in S_\gamma^{2}(\mathbb{R}^N), \] where \(\lambda >0\) is a parameter and \( \Delta_{\gamma}\) is the subelliptic operator of the type \[ \Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}: =\frac{\partial}{\partial x_{j}},\quad \gamma = (\gamma_1, \gamma_2, \dots, \gamma_N),\quad \Delta^2_\gamma: =\Delta_\gamma(\Delta_\gamma). \] Under some suitable assumptions on \(b(x)\) and \(f(x,\xi)\), we obtain the existence of nontrivial solution for \(\lambda\) large enough.Application of the modified boundary element method for the solution of parabolic problems.https://zbmath.org/1449.652332021-01-08T12:24:00+00:00"Fedotov, Vladimir Petrovich"https://zbmath.org/authors/?q=ai:fedotov.vladimir-petrovich"Nefedova, Ol'ga Anatol'evich"https://zbmath.org/authors/?q=ai:nefedova.olga-anatolevichSummary: An algorithm for finding numerically-analytical solution of parabolic problems (diffusion and heat conduction) is proposed. The problem is solved by the proposed algorithm in three steps. At the first step the one-dimensional problem is solved for a base interval of integration. This problem is of independent significance as well as the basis for the second step. At the second step the two-dimensional parabolic problem is considered. Its solution is performed using the modified boundary elements method. At the third step, the method of step-by-step integration over time is used.Monte-Carlo estimations for powers of Green operator and the first eigenvalue for Dirichlet boundary value problem.https://zbmath.org/1449.650012021-01-08T12:24:00+00:00"Kuznetsov, Andreĭ Nikolavich"https://zbmath.org/authors/?q=ai:kuznetsov.andrei-nikolavich"Rytenkova, Irina Aleksandrovna"https://zbmath.org/authors/?q=ai:rytenkova.irina-aleksandrovna"Sipin, Aleksandr Stepanovich"https://zbmath.org/authors/?q=ai:sipin.aleksandr-stepanovichSummary: In this paper, we examine the algorithm for computing the powers of a Green operator and the first eigenvalue for the Dirichlet boundary value problem using Monte Carlo method. The efficiency of numerical realization of these algorithms is also discussed.Local null controllability for a parabolic-elliptic system with local and nonlocal nonlinearities.https://zbmath.org/1449.930072021-01-08T12:24:00+00:00"Prouvée, Laurent"https://zbmath.org/authors/?q=ai:prouvee.laurent"Limaco, Juan"https://zbmath.org/authors/?q=ai:limaco.juanSummary: This work deals with the null controllability of an initial boundary value problem for a parabolic-elliptic coupled system with nonlinear terms of local and nonlocal kinds. The control is distributed, locally in space and appears only in one PDE. We first prove that, if the initial data is sufficiently small and the linearized system at zero satisfies an appropriate condition, the equations can be driven to zero.Numerical method for MHD flows of fractional viscous equation.https://zbmath.org/1449.652112021-01-08T12:24:00+00:00"Zhang, Jun"https://zbmath.org/authors/?q=ai:zhang.jun.1|zhang.jun|zhang.jun.7|zhang.jun.5|zhang.jun.3|zhang.jun.9|zhang.jun.2|zhang.jun.6|zhang.jun.10Summary: In this paper, the numerical approximation of fractional viscosity MHD equation is discussed. We present an efficient numerical scheme for solving this equation and analyze its stability and error estimates. We prove that the scheme is unconditionally stable and the convergence order of the scheme is \(2 - \beta\) in time and spectral accuracy in space. Finally, numerical examples are given to verify the theoretical results.Nonlocal problem for a equation of mixed type of third order with generalized operators of fractional integro-differentiation of arbitrary order.https://zbmath.org/1449.353252021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: The unique solvability of internally boundary value problem for equation of mixed type of third order with multiple characteristics is investigated. The uniqueness theorem is proved with the restrictions on certain features and different orders of fractional integro-differentiation. The existence of solution is equivalent reduced to a Fredholm integral equation of the second kind.On an estimate in the Sobolev space generated by the second order degenerate elliptic operator defined in the half-plane.https://zbmath.org/1449.352352021-01-08T12:24:00+00:00"Smolkin, G. A."https://zbmath.org/authors/?q=ai:smolkin.g-aThe article considers an elliptic operator that is defined in a half-plane and degenerates along the normal to the boundary of this half-plane. The author shows that the smooth extension of the function to the entire plane, studied in detail by L. N. Slobodetskiy, is enough to get the necessary a priori estimate. The method of proof is based on the Fourier transform with respect to some subset of variables and on Schwarz inequality. It is established that the Sobolev norm of function's second-order derivatives is finite if its restriction to the half-plane boundary and its image belong to Sobolev spaces with exponents 3, 2, respectively.
Reviewer: Pavel A. Shamanaev (Saransk)Coexistence solutions of a Ivlev-type predator-prey model with cross-diffusion and a protection zone.https://zbmath.org/1449.352682021-01-08T12:24:00+00:00"Lin, Na'na"https://zbmath.org/authors/?q=ai:lin.nana"Zhang, Li'na"https://zbmath.org/authors/?q=ai:zhang.linaSummary: We are concerned with the stationary problem of a Ivlev-type predator-prey model with cross-diffusion and a protection zone. The existence of coexistence states is discussed by using the eigenvalue theory and bifurcation theory. As a result, it is shown that the cross-diffusion is beneficial for species coexistence.Monotonicity of input-output mapping related to inverse elastoplastic torsional problem.https://zbmath.org/1449.741302021-01-08T12:24:00+00:00"Tatar, Salih"https://zbmath.org/authors/?q=ai:tatar.salihSummary: In this paper, monotonicity of input-output mapping related to inverse elastoplastic torsional problem is studied. In this context, torsional behavior of power-hardening engineering materials is investigated. The equation of the elastoplastic torsion of a strain hardening bar is given by \(Au:=-\nabla {\cdot}\left(g(| \nabla u|^ 2)\nabla u\right)\). Although this equation is nonlinear in pure plastic case, it is reduced into a linear equation in pure elastic case. The reduced equation is solved to find an analytical formula for the torque in pure elastic case. Using a specific plasticity function \(g=g({\xi}^2)\), \({\xi}^2=| \nabla u|^2\) that corresponds to wide class of materials, namely power-hardening engineering materials as input data, the problem is solved numerically. Moreover a novel comparison principle is proved in elastoplastic case. We obtain some usefull results using this comparison principle. Finally these obtained results are supported by the numerical solution of the nonlinear problem.A decay result of the energy to a viscoelastic equation with memory kernel.https://zbmath.org/1449.350862021-01-08T12:24:00+00:00"Yue, Xiangying"https://zbmath.org/authors/?q=ai:yue.xiangying"Pu, Zhilin"https://zbmath.org/authors/?q=ai:pu.zhilinSummary: In this paper, we study the fading behavior of energy for the solutions of a viscoelastic system with non-dissipative memory kernel. We give some new weaker assumptions on the memory function \(g\), under which we obtain the result that the exponential and polynomial types of decay are only special cases. This work generalizes and improves the known results.A lattice Boltzmann model for Maxwell's equations.https://zbmath.org/1449.760382021-01-08T12:24:00+00:00"Liu, Yanhong"https://zbmath.org/authors/?q=ai:liu.yanhong-a"Yan, Guangwu"https://zbmath.org/authors/?q=ai:yan.guangwuSummary: In this paper, a lattice Boltzmann model for the Maxwell's equations is proposed by taking separate sets of distribution functions for the electric and magnetic fields, and a lattice Boltzmann model for the Maxwell vorticity equations with third order truncation error is proposed by using the higher-order moment method. At the same time, the expressions of the equilibrium distribution function and the stability conditions for this model are given. As numerical examples, some classical electromagnetic phenomena, such as the electric and magnetic fields around a line current source, the electric field and equipotential lines around an electrostatic dipole, the electric and magnetic fields around oscillating dipoles are given. These numerical results agree well with classical ones.Global attractors of Ginzburg-Landau equations in the Fermion-Boson model.https://zbmath.org/1449.351062021-01-08T12:24:00+00:00"Xiong, Chunyan"https://zbmath.org/authors/?q=ai:xiong.chunyan"Chen, Shuhong"https://zbmath.org/authors/?q=ai:chen.shuhongSummary: Based on the Fermion-Boson model, we study longtime behavior of the weak solution of the Ginzburg-Landau theory in Bardeen-Cooper-Schrieffer theory-Bose-Einstein Condensation (BCS-BEC) crossover. Combining Gronwall inequality and other forms of inequalities, we establish a suitable prior estimate, obtain the bounded absorption set of solution semigroups, and further prove the existence of solution semigroups generated by these equations.Global existence of mild solutions for the elastic system with structural damping.https://zbmath.org/1449.351872021-01-08T12:24:00+00:00"Shi, Wei"https://zbmath.org/authors/?q=ai:shi.weiSummary: In this paper, we study the global existence of mild solutions for the semilinear initial-value problems of second order evolution equations, which can model an elastic system with structural damping. This discussion is based on the operator semigroups theory and the Leray-Schauder fixed point theorem. In addition, an example is presented to illustrate our theoretical result.Intermediate regularity results for the solution of a high order parabolic equation.https://zbmath.org/1449.352542021-01-08T12:24:00+00:00"Kheloufi, Arezki"https://zbmath.org/authors/?q=ai:kheloufi.arezkiSummary: In this work we give new intermediate regularity results for the solution of the following \(2m\)-th order parabolic equation \[\partial_{t}u+\left( -1\right)^{m}\sum_{i=1}^{n}\partial_{x_{i}}^{2m}u=0,\] where \(m\) is a positive integer, subject to Dirichlet condition on the lateral boundary of a cylindrical domain and to a non-homogeneous initial Cauchy data.Two problems for three-dimensional space analogue of the third order hyperbolic type equation.https://zbmath.org/1449.352932021-01-08T12:24:00+00:00"Dolgopolov, Vyacheslav Mikhaĭlovich"https://zbmath.org/authors/?q=ai:dolgopolov.vyacheslav-mikhailovich"Rodionova, Irina Nikolaevna"https://zbmath.org/authors/?q=ai:rodionova.irina-nikolaevnaSummary: For a complete hyperbolic equation of the third order with variable coefficients in the infinite rectangle the problem with two integral conditions and conjugation on the characteristic plane (Problem~I) is considered. As auxiliary Darboux problem is solved by Riemann method which is much simplified by the special presentation of one of the boundary conditions. Taking Darboux problem as a basis for the solution, authors reduce the Problem~I to the uniquely solvable integral equation, which gives an explicit solution to the Problem~I.Solutions of homogeneous fractional \(p\)-Kirchhoff equations in \(\mathbb{R}^N\).https://zbmath.org/1449.354392021-01-08T12:24:00+00:00"Ho, Vu"https://zbmath.org/authors/?q=ai:ho.vu-b"Huynh, Nhat Vy"https://zbmath.org/authors/?q=ai:huynh.nhat-vy"Le, Phuong"https://zbmath.org/authors/?q=ai:le.phuong-m|le.phuong-quynhSummary: In this note, we furnish a transformation such that solutions of the fractional \(p\)-Kirchhoff equation in \(\mathbb{R}^N\) are easily obtained from known solutions of the corresponding fractional \(p\)-Laplace equation. As an application, we classify all positive solutions of some (fractional) \(p\)-Kirchhoff equations with sub-critical or critical nonlinearities and Hénon-Hardy potentials. Similar results for Kirchhoff type systems are also discussed.Pullback attractors for the Boussinesq-Beam equation with time delay.https://zbmath.org/1449.351082021-01-08T12:24:00+00:00"Xu, Guigui"https://zbmath.org/authors/?q=ai:xu.guigui"Wang, Libo"https://zbmath.org/authors/?q=ai:wang.libo"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguangSummary: The existence of pullback attractors for the Boussinesq-Beam equation with time delay is handled with the concept of contractive function and some related methods. Firstly, the existence of a pullback absorbing set is verified by taking the inner product and estimating the inequalities. Then the specific energy function is constructed and the method of contractive functions is used to prove that the process \(\{U (t,\tau)\}_{t \ge \tau}\) in \(C_{D (A),V}\) produced by the Boussinesq-Beam equation with time delay possesses compactness. Finally, the existence of pullback attractors in \(C_{D (A),V}\) for the process \(\{U (t,\tau)\}_{t \ge \tau}\) is proved.Existence of nontrival solutions for a class of Schrödinger-Poisson systems.https://zbmath.org/1449.353602021-01-08T12:24:00+00:00"Chen, Lizhen"https://zbmath.org/authors/?q=ai:chen.lizhen"Feng, Xiaojing"https://zbmath.org/authors/?q=ai:feng.xiaojing"Li, Gang"https://zbmath.org/authors/?q=ai:li.gang.8Summary: We investigate a class of Schrödinger-Poisson systems by means of variational method and critical point theory. Here, the Poisson term is a more general form. By adding quasi-critical growth and AR conditions to the nonlinear term, we prove the existence of nontrival solution of the system. The results supplement and promote the previous results on the Schrödinger-Poisson systems.Existence of positive solution for a class of \(N\)-Kirchhoff type equation.https://zbmath.org/1449.350192021-01-08T12:24:00+00:00"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.5|chen.lin.1|chen.lin.4|chen.lin.2|chen.lin.6|chen.lin|chen.lin.3Summary: This paper studies the existence of positive solution for a class of \(N\)-Kirchhoff type problem whose nonlinearity depends on the gradient of the solution. Applying a variational method and an iterative technique, the analysis proves that the problem has at least one positive weak solution.Pullback attractor of \(p\)-Laplacian equation with time-dependent parameters on the entire space.https://zbmath.org/1449.351042021-01-08T12:24:00+00:00"Wang, Yi"https://zbmath.org/authors/?q=ai:wang.yi.5|wang.yi.7|wang.yi.8|wang.yi.1|wang.yi.10|wang.yi.2|wang.yi.4|wang.yi.3|wang.yi.9|wang.yi.6"Ma, Qiaozhen"https://zbmath.org/authors/?q=ai:ma.qiaozhenSummary: Long time dynamical behavior of the \(p\)-Laplacian equation \({u_t} - {\mathrm{div}} (\varepsilon (t)|\nabla u|^{p-2}\nabla u) + f (x,u) = g (x,t)\) is considered. We prove that the process associated with the equation is asymptotically compact under the condition that the forcing term satisfies certain integral condition. By using the tail estimates of solution, the existence of pullback attractor is proved as well.Existence of solutions for non-coercivity quasilinear elliptic equations with Hardy potential.https://zbmath.org/1449.352302021-01-08T12:24:00+00:00"Xiawu, Jimao"https://zbmath.org/authors/?q=ai:xiawu.jimao"Huang, Shuibo"https://zbmath.org/authors/?q=ai:huang.shuibo"Deng, Dejie"https://zbmath.org/authors/?q=ai:deng.dejieSummary: This paper studies the existence and regularity of solutions to non-coercivity quasilinear elliptic problems with lower order terms and the Hardy potential, and focuses on the regularizing effect of lower order terms and the influence of the Hardy potential.Analysis of higher order difference method for a pseudo-parabolic equation with delay.https://zbmath.org/1449.652222021-01-08T12:24:00+00:00"Amirali, Ilhame"https://zbmath.org/authors/?q=ai:amirali.ilhameSummary: In this paper, the author considers the one dimensional initial-boundary problem for a pseudo-parabolic equation with time delay in second spatial derivative. To solve this problem numerically, the author constructs higher order difference method and obtain the error estimate for its solution. Based on the method of energy estimates the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Some numerical examples illustrate the convergence and effectiveness of the numerical method.Consistent Riccati expansion solvability and interaction solutions of modified Boussinesq system.https://zbmath.org/1449.353902021-01-08T12:24:00+00:00"Xia, Yarong"https://zbmath.org/authors/?q=ai:xia.yarongSummary: The modified Boussinesq system is investigated by using the consistent Riccati expansion (CRE) method. It is proved that the modified Boussinesq system is CRE solvable. Soliton, soliton-elliptic periodic wave interaction solutions are obtained explicitly from different special solutions of the Schwarzian equation.Dynamics analysis of cannibalistic model with density dependence.https://zbmath.org/1449.352752021-01-08T12:24:00+00:00"Zhu, Xue"https://zbmath.org/authors/?q=ai:zhu.xue"Lin, Xiaolin"https://zbmath.org/authors/?q=ai:lin.xiaolin"Li, Jianquan"https://zbmath.org/authors/?q=ai:li.jianquanSummary: Based on the density dependence of juvenile individuals, a cannibalism model with two stages (juvenile and adult) is established, and the existence and stability of equilibria of the model are completely analyzed. For the case without cannibalism, the global stability is proved by constructing the appropriate Lyapunov functions. It is found that, for the case with cannibalism, the saddle-node bifurcation can occur, and the global dynamic behavior of the model is investigated by applying the Dulac function to rule out the existence of periodic solutions. Finally, the obtained results are verified by numerical simulations.Qualitative analysis of a modified Leslie-Gower model with Allee effect in predator.https://zbmath.org/1449.352742021-01-08T12:24:00+00:00"Zhao, Dan"https://zbmath.org/authors/?q=ai:zhao.dan"Yang, Wenbin"https://zbmath.org/authors/?q=ai:yang.wenbin"Li, Yanling"https://zbmath.org/authors/?q=ai:li.yanlingSummary: The stability and existence of positive solutions of a modified Leslie-Gower model with Allee effect in predator are considered, subject to homogeneous Neumann boundary condition. Firstly, the priori estimates of positive solutions are obtained by using the maximum principle and the Harnack inequality, then the asymptotic stability of positive constant solution is acquired by means of stability theory. Secondly, the non-existence of the non-constant positive solutions is verified through the integral property and Poincaré inequality. Finally, based on the methods of Leray-Schauder degree theory, sufficient conditions for the existence of non-constant positive solutions are derived. The fact shows that the two species can be co-existed under some conditions.Singular perturbation problems in diffusion model of chemical reactions.https://zbmath.org/1449.352042021-01-08T12:24:00+00:00"Song, Ying"https://zbmath.org/authors/?q=ai:song.ying"Sun, Ning"https://zbmath.org/authors/?q=ai:sun.ningSummary: We consider the first exit time problem in a diffusion model of chemical reaction. Firstly, we transform the problem of stochastic differential equation into a boundary value problem for a partial differential equation. Secondly, we use the coordinate transformation and asymptotic expansion of singular perturbation to obtain the first order approximate solution. Finally, we give the results combined with the asymptotic method of Laplace integral.3-D transient analytical solution based on Green's function to temperature field in friction stir welding.https://zbmath.org/1449.800112021-01-08T12:24:00+00:00"Haghpanahi, Mohammad"https://zbmath.org/authors/?q=ai:haghpanahi.mohammad"Salimi, Solaleh"https://zbmath.org/authors/?q=ai:salimi.solaleh"Bahemmat, Pouya"https://zbmath.org/authors/?q=ai:bahemmat.pouya"Sima, Sadaf"https://zbmath.org/authors/?q=ai:sima.sadafSummary: Friction stir welding (FSW) is a relatively modern welding process, which not only provides the advantages offered by fusion welding methods, but also improves mechanical properties as well as metallurgical transformations due to the pure solid-state joining of metals. The FSW process is composed of three main stages; penetrating or preheating stage, welding stage and cooling stage. The thermal history and cooling rate during and after the FSW process are decisive factors, which dictate the weld characteristics. In the current paper, a novel transient analytical solution based on the Green's function method is established to obtain the three-dimensional temperature field in the welding stage by considering the FSW tool as a circular heat source moving in a finite rectangular plate with cooling surface and non-uniform and non-homogeneous boundary and initial conditions. The effect of penetrating/preheating stage is also taken into account by considering the temperature field induced by the preheating stage to be the non-uniform initial condition for the welding stage. Similarly, cooling rate can be calculated in the cooling stage. Furthermore, the simulation of the FSW process via FEM commercial software showed that the analytical and the numerical results are in good agreement, which validates the accuracy of the developed analytical solution.The existence of solutions for the critical Kirchhoff type problem in \(\mathbb{R}^4\).https://zbmath.org/1449.352052021-01-08T12:24:00+00:00"Zhao, Yaru"https://zbmath.org/authors/?q=ai:zhao.yaru"Sun, Yan"https://zbmath.org/authors/?q=ai:sun.yan"Luan, Shixia"https://zbmath.org/authors/?q=ai:luan.shixiaSummary: We study the following Kirchhoff type problem
\[- (a + b\int_{\mathbb{R}^4}|\nabla u|^2\,dx)\Delta u + V (x)u = K (x){u^3} + \mu f(u)\text{ in }\mathbb{R}^4, \tag{*}\]
where \(a, b > 0\) are constants, suitable conditions are imposed on \(V, f, K\). By using the generalized mountain pass theorem, the existence and non-existence results of solutions of (*) are obtained. By using the Nehari manifold, a ground state solution for equation (*) is obtained.Lie symmetry analysis for the space-time fractional porous medium equations.https://zbmath.org/1449.350172021-01-08T12:24:00+00:00"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.ying"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhenSummary: In this paper, we study the space-time fractional porous medium equation, the space-time fractional porous medium equation with a nonlinear convection term, the space-time fractional dual porous medium equation using Lie symmetry analysis. The corresponding symmetry groups of these three types of porous medium equations are obtained. Based on the above results we perform the similarity reduction and obtain the group-invariant solutions to these equations.The formation of singularity for the classical solutions to the quasilinear hyperbolic systems with characteristics with constant multiplicity.https://zbmath.org/1449.350052021-01-08T12:24:00+00:00"Wu, Xiaojie"https://zbmath.org/authors/?q=ai:wu.xiaojie"Xu, Yumei"https://zbmath.org/authors/?q=ai:xu.yumeiSummary: This paper considered the Cauchy problem of quasilinear hyperbolic systems with constant multiplicity characteristics. Under the assumption that constant multiplicity characteristics are linear degeneracy in \(u=0\), this paper weakened decaying initial data and obtained a blow-up result and the life-span of the solution.Blow-up phenomena for parabolic equation with variable frequency heat source.https://zbmath.org/1449.351202021-01-08T12:24:00+00:00"Liu, Luyan"https://zbmath.org/authors/?q=ai:liu.luyan"Du, Guangwei"https://zbmath.org/authors/?q=ai:du.guangwei"Li, Fushan"https://zbmath.org/authors/?q=ai:li.fushanSummary: In this paper, the sufficient conditions to ensure the blow-up phenomena occurring of the solution to the semilinear heat equation with time dependent coefficients under inhomogeneous Neumann boundary conditions and the upper and lower bounds for the blow-up time \({t^*}\) by constructing some suitable auxiliary functionals are given.Parabolic system related to the \(p\)-Laplacian with degeneracy on the boundary.https://zbmath.org/1449.352792021-01-08T12:24:00+00:00"Ou, Qitong"https://zbmath.org/authors/?q=ai:ou.qitong"Zhan, Huashui"https://zbmath.org/authors/?q=ai:zhan.huashuiSummary: In this article, we study the system with boundary degeneracy
\[u_{it}-\operatorname{div} (a(x) |\nabla u_i|^{p_i-2}\nabla u_i) = f_i(x, t, u_1, u_2), \quad (x,t) \in \Omega_T.\]
Applying the monotone iteration technique and the regularization method, we get the existence of solution for a regularized system. Moreover, under an integral condition on the coefficient function \(a(x)\), the existence and the uniqueness of the local solutions of the system are obtained by using a standard limiting process. Finally, the stability of the solutions is proved without any boundary value condition, provided that \(a(x)\) satisfies another restriction.A second-order hyperbolic chemotaxis model.https://zbmath.org/1449.352872021-01-08T12:24:00+00:00"Wu, Shaohua"https://zbmath.org/authors/?q=ai:wu.shaohua"Chen, Haiying"https://zbmath.org/authors/?q=ai:chen.haiyingSummary: In this paper, we study a hyperbolic type chemotaxis model in one space dimension. We assume that the speed is constant, the production and degradation of the external signal \(s\) are given by \(-\beta s + f ({u^+} + {u^-})\), where \(f ({u^+} + {u^-})\) is the general form and \({u^+}, {u^-}\) depend on \(s\). The existence of the weak solution of the model considered in the paper is obtained by the method of characteristics and the contraction mapping principle.Structural stability of \(p (x)\)-Laplace problems with Fourier type boundary condition.https://zbmath.org/1449.350512021-01-08T12:24:00+00:00"Kansie, Kpe"https://zbmath.org/authors/?q=ai:kansie.kpe"Ouaro, Stanislas"https://zbmath.org/authors/?q=ai:ouaro.stanislasSummary: We study the continuous dependence on coefficients of solutions of the nonlinear nonhomogeneous Fourier boundary value problems involving the \(p (x)\)-Laplace operator.Blow up of solutions to semilinear wave equations with logarithmic decay initial data.https://zbmath.org/1449.351142021-01-08T12:24:00+00:00"Cai, Chunling"https://zbmath.org/authors/?q=ai:cai.chunling"Huang, Shoujun"https://zbmath.org/authors/?q=ai:huang.shoujunSummary: This paper considers the blow up of solutions to a class of semilinear wave equations with logarithmic decay initial data. By utilizing the method of an iteration argument, we obtain the blow up and the lower bound of lifespan of solutions to the Cauchy problem for the semilinear wave equations, which generalize the existing facts on semilinear wave equations. In addition, some applications are also given.The estimates of the solution of the Dirichlet problem with boundary function from \(L_p\) for a second-order elliptic equation.https://zbmath.org/1449.351632021-01-08T12:24:00+00:00"Gushchin, Anatoliĭ Konstantinovich"https://zbmath.org/authors/?q=ai:gushchin.anatolii-konstantinovichSummary: We study the solvability of the Dirichlet problem for a second-order elliptic equation with measurable and bounded coefficients. Assuming that coefficients of equation are Dini-continued on the boundary, it is established that there is the unique solution of the Dirichlet problem with boundary function from \(L_p\), \(p>1\). We prove the estimate of the analogue of area integral.Boundary value problem for the flexible axially loaded compound shells of revolution and beams systems.https://zbmath.org/1449.354132021-01-08T12:24:00+00:00"Elenitskiĭ, Éduard Yashevich"https://zbmath.org/authors/?q=ai:elenitskii.eduard-yashevichSummary: The effective algorithm of static calculation of geometrically nonlinear compound thin structures is offered. Linear differential equations of moment theory are used. Nonlinearity is considered by assigning unknown initial angular displacement of each segment retaining the form of the generating line. Unknown values of algebraic equations resolving system are the arbitrary constants of the general solution and the initial angles of generating lines rotation. Linearization is realized by Newton-Raphson iterative method and provides the high precision of results.Sharp estimates for eigenvalues of bi-drifting Laplacian.https://zbmath.org/1449.353332021-01-08T12:24:00+00:00"Li, Yanli"https://zbmath.org/authors/?q=ai:li.yanli"Du, Feng"https://zbmath.org/authors/?q=ai:du.fengSummary: In this paper, we study four types of eigenvalue problems for the bi-drifting Laplacian. By using the weighted Reilly formula, we get some sharp lower bounds of the first nonzero eigenvalue for these eigenvalue problems on compact smooth metric measure spaces with boundary and under some condition on the \(m\)-weighted Ricci curvature, which generalize the corresponding results for the eigenvalues of biharmonic operator.Variation inequality for heat semigroup related to Schrödinger operator on the weighted Morrey spaces.https://zbmath.org/1449.471002021-01-08T12:24:00+00:00"Yu, Jinxia"https://zbmath.org/authors/?q=ai:yu.jinxia"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.5|zhang.jing.9|zhang.jing.11|zhang.jing.7|zhang.jing.10|zhang.jing.12|zhang.jing.6|zhang.jing.8|zhang.jing.1|zhang.jing.3|zhang.jing.2Summary: This paper is devoted to investigate variation inequality of heat semigroup related to Schrödinger operator on the weighted Morrey spaces. By estimating the kernel function and using weights and weight's properties, it is proved that the variation operator is bounded on the weighted Morrey spaces.Hardy type unique continuation properties for abstract Schrödinger equations and applications.https://zbmath.org/1449.353632021-01-08T12:24:00+00:00"Shakhmurov, Veli"https://zbmath.org/authors/?q=ai:shakhmurov.veli-bSummary: In this paper, Hardy's uncertainty principle and unique continuation properties of Schrödinger equations with operator potentials in Hilbert space-valued \(L^{2}\) classes are obtained. Since the Hilbert space \(H\) and linear operators are arbitrary, by choosing the appropriate spaces and operators we obtain numerous classes of Schrödinger type equations and its finite and infinite many systems which occur in a wide variety of physical systems.The Monge-Ampère equation for strictly \( (n-1)\)-convex functions with Neumann condition.https://zbmath.org/1449.352412021-01-08T12:24:00+00:00"Deng, Bin"https://zbmath.org/authors/?q=ai:deng.binSummary: A \({C^2}\) function on \({\mathbb{R}^n}\) is called strictly \( (n-1)\)-convex if the sum of any \(n-1\) eigenvalues of its Hessian is positive. In this paper, we establish a global \({C^2}\) estimate to the Monge-Ampère equation for strictly \( (n-1)\)-convex functions with Neumann condition. By the method of continuity, we prove an existence theorem for strictly \( (n-1)\)-convex solutions of the Neumann problems.Well-posedness for the 2D non-autonomous incompressible fluid flow in Lipschitz-like domain.https://zbmath.org/1449.760152021-01-08T12:24:00+00:00"Yang, Xinguang"https://zbmath.org/authors/?q=ai:yang.xinguang"Wang, Shubin"https://zbmath.org/authors/?q=ai:wang.shubinSummary: This paper is concerned with the global well-posedness and regularity of weak solutions for the 2D non-autonomous incompressible Navier-Stokes equation with a inhomogeneous boundary condition in Lipschitz-like domain. Using the estimate for governing steady state equation and Hardy's inequality, the existence and regularity of global unique weak solution can be proved. Moreover, these results also hold for 2D Navier-Stokes equation with Rayleigh's friction and Navier-Stokes-Voigt flow, but invalid for three dimension.The existence of ground state solution for a Kirchhoff type problem involving nonlocal operator.https://zbmath.org/1449.352212021-01-08T12:24:00+00:00"Zhou, Dizhi"https://zbmath.org/authors/?q=ai:zhou.dizhi"Chu, Changmu"https://zbmath.org/authors/?q=ai:chu.changmu"Cai, Zhipeng"https://zbmath.org/authors/?q=ai:cai.zhipengSummary: In this paper, a class of Kirchhoff nonlocal operators involving concave-convex nonlinearities and sign-changing weight functions are studied. The existence of ground state solution is obtained by variational methods.On the regularity criteria for 3-D liquid crystal flows in terms of the horizontal derivative components of the pressure.https://zbmath.org/1449.351412021-01-08T12:24:00+00:00"Zhao, Lingling"https://zbmath.org/authors/?q=ai:zhao.lingling"Li, Fengquan"https://zbmath.org/authors/?q=ai:li.fengquanSummary: This paper is devoted to investigating regularity criteria for the 3-D nematic liquid crystal flows in terms of horizontal derivative components of the pressure and gradient of the orientation field. More precisely, we mainly proved that the strong solution \( (u, d)\) can be extended beyond \(T\), provided that the horizontal derivative components of the pressure \({\nabla_h}P = ({\partial_{x_1}}P, {\partial_{x_2}}P)\) and gradient of the orientation field satisfy \[{\nabla_h}P \in {L^s} (0, T; {L^q} (\mathbb{R}^3)),\; \frac{2}{s} + \frac{3}{q} \le \frac{5}{2},\; \frac{18}{13} \le q \le 6\] and \[\nabla d \in {L^\beta} (0, T; {L^\gamma} (\mathbb{R}^3)),\; \frac{2}{\gamma} + \frac{3}{\beta} \le \frac{3}{4},\; \frac{36}{7} \le \beta \le 12.\]Stability of smooth solutions of Cauchy problem for Timoshenko equations.https://zbmath.org/1449.350502021-01-08T12:24:00+00:00"Hou, Hongle"https://zbmath.org/authors/?q=ai:hou.hongle"Liu, Cunming"https://zbmath.org/authors/?q=ai:liu.cunmingSummary: The global stability of Cauchy problem for Timoshenko equations with small perturbation initial smooth solutions was studied. The hyperbolic equations were transformed into general symmetric hyperbolic equations. Under the Sobolev space framework, the energy estimated value of the solutions and the dissipated energy estimates of the corresponding variables were obtained by using the energy integral method and the anti-symmetry technique. The global existence of the solution was obtained by continuous extension method. Compared with the discussion of this stability problem under the Besove space framework, the proof method here was more concise.Fractional Schrödinger equations with singular potentials and bounded discontinuous nonlinearities.https://zbmath.org/1449.353912021-01-08T12:24:00+00:00"Du, Yanhong"https://zbmath.org/authors/?q=ai:du.yanhongSummary: In this article, we study the fractional Schrödinger equations with singular potentials and bounded discontinuous nonlinearities. For the first time, we prove a new compact embedding from \(H_r^s (R^N)\) into \({L^1} (R^N, Q)\), and prove the existence of multiple solutions by using non-smooth critical point theory.Principal eigenvalue for cooperative \((p,q)\)-biharmonic systems.https://zbmath.org/1449.353342021-01-08T12:24:00+00:00"Leadi, Liamidi A."https://zbmath.org/authors/?q=ai:leadi.liamidi-a"Toyou, Robert L."https://zbmath.org/authors/?q=ai:toyou.robert-lSummary: In this article, we are interested in the simplicity and the existence of the first strictly principal eigenvalue or semitrivial principal eigenvalue of the \( (p, q)\)-biharmonic systems with Navier boundary conditions.The least energy solution for \(k\)-coupled Schrödinger equations with critical exponent.https://zbmath.org/1449.350442021-01-08T12:24:00+00:00"Hu, Xiaona"https://zbmath.org/authors/?q=ai:hu.xiaona"Yue, Xiaorui"https://zbmath.org/authors/?q=ai:yue.xiaorui"Li, Shengjun"https://zbmath.org/authors/?q=ai:li.shengjunSummary: In this paper, we consider the \(k\)-coupled Schrödinger system: \[ (P)\begin{cases}-\Delta {u_j} + {\lambda_j}{u_j} = \sum\limits_{i = 1}^k {\beta_{ji}}{u_j}{u_i^2},\; x \in \Omega\\{u_j} = 0,\; j = 1, 2, \cdots, k,\; x \in \partial\Omega \end{cases}.\] We prove the existence of positive least energy solution of the system \( (P)\) and the non-existence of the nontrivial solution of system the \( (P)\) when the coefficients meet certain conditions.Brezis-Lieb lemma of \(k\)-coupled form.https://zbmath.org/1449.350132021-01-08T12:24:00+00:00"Yin, Lijie"https://zbmath.org/authors/?q=ai:yin.lijie"Yue, Xiaorui"https://zbmath.org/authors/?q=ai:yue.xiaoruiSummary: As a generalization of Brezis-Lieb lemma (single variable), we prove that the \(k\)-coupled form also satisfies the similar theorem in the current paper. Let \(\Omega\) be an open subset of \({R^N}\) and let \(\{u_{ni}\} \subset L^{p_i} (\Omega)\), \(N \ge 2\), \(2 \le {p_i} < \infty, i = 1, 2, \cdots, k, k \ge 2\). If \(\{u_{ni}\}\) is bounded in \(L^{p_i} (\Omega)\) and \({u_{ni}} \to {u_i}\) almost everywhere in \(\Omega\), then we have \[\mathop {\lim}\limits_{x \to \infty}\left[\int_\Omega\sum\limits_{i,j = 1}^k {u_{ni}^{\frac{p_i}{2}}} {u_{nj}^{\frac{p_j}{2}}}{\mathrm{d}}x - \int_\Omega\sum\limits_{i,j = 1}^k ({u_{ni}} - {u_i})^{\frac{p_i}{2}} ({u_{nj}} - {u_j})^{\frac{p_j}{2}}{\mathrm{d}}x \right] = \int_\Omega\sum\limits_{i,j = 1}^k {u_i^{\frac{p_i}{2}}} {u_j^{\frac{p_j}{2}}}{\mathrm{d}}x.\] This conclusion can be used in dealing with \(k\)-coupled system.The local solvability of a problem of determining the spatial part of a multidimensional kernel in the integro-differential equation of hyperbolic type.https://zbmath.org/1449.354612021-01-08T12:24:00+00:00"Durdiev, Durdimurod Kalandarovich"https://zbmath.org/authors/?q=ai:durdiev.durdimurod-kalandarovich"Safarov, Zhurabek Shakarovich"https://zbmath.org/authors/?q=ai:safarov.zhurabek-shakarovichSummary: The multidimensional inverse problem of determining spatial part of integral member kernel in integro-differential wave equation is considered. Herein, the direct problem is represented by the initial-boundary problem for this with zero initial data and Neyman's boundary condition as Dirac's delta-function concentrated on the boundary of the domain \((x,t)\in \mathbb{R}^{n+1}\), \(z>0\). As information in order to solve the inverse problem on the boundary of the considered domain the traces of direct problem solution are given. The significant moment of the problem setup is such a circumstance that all given functions are real analytical functions of variables \(x\in \mathbb{R}^n\). The main result of the work is concluded in obtaining the local unique solvability of the inverse problem in the class of continuous functions on variable \(z\) and analytical on other spatial variables. For this, by means of singularity separation method, the inverse problem is replaced by the initial-boundary problem for the regular part of the solution of this problem. Further, direct and inverse problems are reduced to the solution of equivalent system of Volterra type integro-differential equations. For the solution of the latter, the method of Banach space scale of real analytical functions is used.Blow-up and effectiveness analysis in a parabolic equation with dissipative gradient function.https://zbmath.org/1449.351182021-01-08T12:24:00+00:00"Ling, Zhengqiu"https://zbmath.org/authors/?q=ai:ling.zhengqiu"He, Bing"https://zbmath.org/authors/?q=ai:he.bing.4Summary: By establishing an appropriate Sobolev inequality, we solved the blow-up problem of solutions of a class of parabolic equations with dissipative gradient functions under mixed boundary conditions. Furthermore, two methods were used to obtain estimates of the lower bound of blow-up time for blow-up solutions, and the effectiveness of the these methods was analyzed.Lattice Boltzmann method for Ross-Macdonald system with drift property.https://zbmath.org/1449.652852021-01-08T12:24:00+00:00"Li, Ting"https://zbmath.org/authors/?q=ai:li.ting"Yan, Guangwu"https://zbmath.org/authors/?q=ai:yan.guangwuSummary: We used the lattice Boltzmann method to study the Ross-Macdonald equations with drift property which described the Malaria-Malaria mosquito system. We first constructed a lattice Boltzmann model for the Ross-Macdonald equations, then used this model to simulate the Malaria-Malaria mosquito system, and compared the numerical solutions of the lattice Boltzmann method with those of the finite difference method. The results show that this method can be used to simulate the Malaria-Malaria mosquito system.Existence and regularity of a weak solution to a class of systems in a multi-connected domain.https://zbmath.org/1449.351332021-01-08T12:24:00+00:00"Aramaki, Junichi"https://zbmath.org/authors/?q=ai:aramaki.junichiSummary: We consider the existence and regularity of a weak solution to a class of systems containing a \(p\)-curl system in a multi-connected domain. This paper extends the result of the regularity theory for a class containing a \(p\)-curl system that is given in the author's previous paper. The optimal \({C^{1+\alpha}}\)-regularity of a weak solution is shown in a multi-connected domain.Pseudodifferential operators in infinite dimensional spaces: a survey of recent results.https://zbmath.org/1449.354682021-01-08T12:24:00+00:00"Jager, Lisette"https://zbmath.org/authors/?q=ai:jager.lisetteThe author gathers previously published results concerning the quantization of pseudodifferential analysis in abstract Wiener spaces, in Fock spaces and in Gaussian Hilbert spaces. She starts briefly recalling the Weyl calculus in finite dimensions. Moving to the infinite dimensional cases, she first defines abstract Wiener spaces. Taking a real, separable and infinite dimensional Hilbert space \(\mathcal{H}\), a finite dimensional subspace \(E\) of \(\mathcal{H}\) and the orthogonal projection \(\pi_{E}\) on \(E\), she introduces the pseudomeasure \(\mu_{\mathcal{H},s}(C)\) on a cylinder \(C=\{x\in H:\pi_{E}(x)\in A\}\) where \(A\) is a Borel set of \(E\) through \(\mu_{\mathcal{H},s}(C)=\int_{A}\exp (-\frac{\left\vert y\right\vert ^{2}}{2s}) (2\pi s)^{-\dim (E)/2}d\lambda_{E}(y)\), where \(\lambda_{E}\) is the Lebesgue measure on \(E\). If \(\mathcal{H}\) can be endowed with a norm \(\left\Vert \cdot \right\Vert\) which satisfies a measurability property, let \(B\) be the completion of \(\mathcal{H}\) with respect to this norm and \(i\) the injection from \(\mathcal{H}\) to \(B\). Then \((i,\mathcal{H},B)\) is an abstract Wiener space. She presents the conditions for a continuous function on \(\mathcal{H}\) to admit a stochastic extension on \(L^{2}(B,\mu_{B,h})\) and proves that measures \(\mu_{B,s}(x,\cdot)\) and \(\mu_{B,t}(y,\cdot)\), with \(\mu_{B,s}(x,A)=\mu_{B,s}(A-x)\) for every Borel subset \(A\) of \(B\), are absolutely continuous with respect to one another if and only if \(s=t\) and \(x-y\in \mathcal{H}\).
The author then defines a Gaussian Hilbert space as a real vector \(\mathcal{M}\) of random variables \(\xi \) defined on a probability space \((\Omega,\mathcal{F},P)\), such that every random variable is centered and Gaussian. She also recalls the definition of a Fock space and the notions of coherent state and of Wick and anti-Wick symbols. She recalls the notion of symbol class in this infinite dimensional case and their properties. She then defines the heat operator on an abstract Wiener space as \(\widetilde{H}_{t}f(x)=\int_{B}f(x+y)\,d\mu_{B,t}(y)\) for every Borel bounded function \(f\) on the Wiener space \((B,\mathcal{B}(B))\) and she recalls its properties. She defines the space \(\mathcal{D}\) which replaces the Schwartz space \(\mathcal{S}(\mathbb{R}^{n})\) in finite dimension, the Wigner function \(W_{h}(f,g)\) attached to functions \(f,g\) in \(\mathcal{D}\), the quadratic form \(Q_{h}^{W}(\widetilde{F})\) associated to a bounded Borel function \(\widetilde{F}\) on \(B^{2}\), the Calderon-Vaillancourt classes and the pseudo-differential operators in this infinite dimensional case. She presents a Beals characterization, composition results and properties of the Wick and Weyl symbols. The paper ends with some applications.
Reviewer: Alain Brillard (Riedisheim)Initial value randomization of nonlinear evolution equations.https://zbmath.org/1449.370502021-01-08T12:24:00+00:00"Huang, Jianhua"https://zbmath.org/authors/?q=ai:huang.jianhua"Yan, Wei"https://zbmath.org/authors/?q=ai:yan.weiSummary: This paper aims to introduce some nonlinear evolution equations. Firstly, we present Schrödinger equations with random data, KdV equation with random data, and wave equation with random data. Then we give the harmonic analysis tools which are used to solve random data problem. At last, some unsolved problems related to random data are presented.Existence of uniform random attractor for non-autonomous stochastic strongly damped wave equation on unbounded domains.https://zbmath.org/1449.351112021-01-08T12:24:00+00:00"Zhang, Jie"https://zbmath.org/authors/?q=ai:zhang.jie|zhang.jie.5|zhang.jie.2|zhang.jie.4|zhang.jie.1|zhang.jie.3"Li, Xiaojun"https://zbmath.org/authors/?q=ai:li.xiaojunSummary: In this paper, we study the existence of uniform attractors for a class of nonautonomous stochastic strongly damped wave equations with additive white noise on unbounded domains. Firstly, by using the uniform estimates of solutions of transformed system, we prove that the stochastic dynamical system corresponding to the original equation has a uniformly pullback absorbing set. Second, by asymptotic tail estimation, we obtain that the solution is uniformly pullback asymptotically compact. The existence of uniform random attractor of the original system is obtained.An efficient two-level method for solving incompressible Navier-Stokes equations.https://zbmath.org/1449.652882021-01-08T12:24:00+00:00"Du, Binbin"https://zbmath.org/authors/?q=ai:du.binbin"Huang, Jianguo"https://zbmath.org/authors/?q=ai:huang.jianguoSummary: This paper proposes a two-level Arrow-Hurwicz (A-H) method (simply called the \(m\)-A-H-1-Oseen method) for solving incompressible Navier-Stokes (N-S)equations. The incompressible N-S equations are first solved by the A-H method to obtain a numerical solution on a coarse mesh. Next, the desired numerical solution is obtained by solving the Oseen scheme, which is derived on a fine mesh by linearizing the original equations using the coarse solution, leading to the required two-level method. The convergence analysis is developed systematically.Mellin transform method for European option pricing under sub-fractional stochastic interest rate model.https://zbmath.org/1449.911592021-01-08T12:24:00+00:00"Sun, Jiaojiao"https://zbmath.org/authors/?q=ai:sun.jiaojiaoSummary: The pricing formula of European call option in financial market is given when the spot interest rate is driven by sub-fractional Vašíček stochastic interest rate model. The Mellin transform method is used to solve the Black-Scholes partial differential equation satisfied by European option value, and the pricing formula is got in simple integral form of European call option. The analytical solution of European call option is obtained through the convolution formula of Mellin transform. A numerical example is given to verify the convergence of the Mellin transform method. Finally we analyze the influence of various parameters on the value of European call options, thus promoting the option pricing method.Exact and similarity solutions of variable coefficient nonlinear Schrödinger equation.https://zbmath.org/1449.353952021-01-08T12:24:00+00:00"Ma, Lin"https://zbmath.org/authors/?q=ai:ma.lin"Wei, Huan"https://zbmath.org/authors/?q=ai:wei.huanSummary: The nonlinear Schrödinger equation is one of the most basic mathematical models for describing light wave/pulse propagation in nonlinear fiber systems. In this paper, a variable-coefficient nonlinear Schrödinger equation is studied. Using the \(G'/G\) expansion method, we obtain abundant exact solutions of the equation. By comparing with the standard Schrödinger equation, we obtain the similarity solutions of the variable-coefficient nonlinear Schrödinger equation.Existence of solutions for viscous Cahn-Hilliard equation with inertial term.https://zbmath.org/1449.353212021-01-08T12:24:00+00:00"Xu, Hongmei"https://zbmath.org/authors/?q=ai:xu.hongmei"Wang, Yiping"https://zbmath.org/authors/?q=ai:wang.yipingSummary: Cauchy problem of viscous Cahn-Hilliard equation with inertial term is studied. By detailed analysis of the Green's function in different frequency, based on fixed point theorem, global existence of classical solution with large initial data is obtained, which extends the formers work which focused on weak solution or quasi-strong solution or classical solution with small initial data.Backward-compact dynamics for non-autonomous Navier-Stokes equations on unbounded domains.https://zbmath.org/1449.353402021-01-08T12:24:00+00:00"She, Lianbing"https://zbmath.org/authors/?q=ai:she.lianbing"Gao, Yunlong"https://zbmath.org/authors/?q=ai:gao.yunlongSummary: This paper is devoted to the backward-compact dynamics for a non-autonomous Navier-Stokes equation. Assume that the time-dependent force is weakly backward-tempered, then we can show that the system has an increasing and bounded pullback absorbing set in an energy space. In order to overcome the loss of compact Sobolev embedding, Ball's method of energy equation is used to show the backward asymptotical compactness of the system. Finally, it is proved that the non-autonomous Navier-Stokes equation has a backward-compact pullback attractor in the energy space.Optimality of the boundary knot method for numerical solutions of 2D Helmholtz-type equations.https://zbmath.org/1449.653372021-01-08T12:24:00+00:00"Wang, Fuzhang"https://zbmath.org/authors/?q=ai:wang.fuzhang"Zheng, Kehong"https://zbmath.org/authors/?q=ai:zheng.kehong"Li, Congcong"https://zbmath.org/authors/?q=ai:li.congcong"Zhang, Juan"https://zbmath.org/authors/?q=ai:zhang.juanSummary: The boundary knot method (BKM) is a boundary-type meshfree method. Only non-singular general solutions are used during the whole solution procedures. The effective condition number (ECN), which depends on the right-hand side vector of a linear system, is considered as an alternative criterion to the traditional condition number. In this paper, the effective condition number is used to help determine the position and distribution of the collocation points as well as the quasi-optimal collocation point numbers. During the solution process, we propose an NMN-search algorithm. Numerical examples show that the ECN is reliable to measure the feasibility of the BKM.The solution of equations of ideal gas that describes Galileo invariant motion with helical level lines, with the collapse in the helix.https://zbmath.org/1449.350632021-01-08T12:24:00+00:00"Yulmukhametova, Yuliya Valer'evna"https://zbmath.org/authors/?q=ai:yulmukhametova.yu-vSummary: We consider the equations of ideal gas dynamics in a cylindrical coordinate system with the arbitrary equation of state and one two-dimensional subalgebra from the optimum system of an 11-dimensional Lie algebra of differentiation operators of the first order. The basis of the subalgebra operators consists of the operator of Galilean transfer and the operator of movement on spiral lines. Invariants of operators set representation: type of speed, density and entropy. After substitution of the solution representation into the equations of gas dynamics the assumption of the linear relation of a radial component of speed and spatial coordinate is entered. Transformations of equivalence which are allowed by a set of equations of gas dynamics after substitution of the solution representation are written down. For the state equation of polytropic gas all four solutions depending on an isentropic exponent are found. For each case the equations of world lines of gas particles motion are written down. The transition Jacobian from Eulerian variables to Lagrangian is found. The instants of collapse of gas particles are determined by value of the Jacobian. As a result the solutions describe movement on straight lines from a helicoid surface. Movements of the particles on equiangular spirals lying on a paraboloid and on hyperbolic spirals, lying on a cone.The conditions of existence with probability one of generalized solutions of Cauchy problem for the heat equation with a random right part.https://zbmath.org/1449.352462021-01-08T12:24:00+00:00"Tylyshchak, A. I."https://zbmath.org/authors/?q=ai:tylyshchak.a-iSummary: The subject of this work is at the intersection of two branches of mathematics: mathematical physics and stochastic processes. The influence of random factors should often be taken into account in solving problems of mathematical physics. The heat equation with random conditions is a classical problem of mathematical physics. In this paper we consider a Cauchy problem for the heat equations with a random right part. We study the inhomogeneous heat equation on a line with a random right part. We consider the right part as a random function of the space \(\text{Sub}_{\varphi}(\Omega)\). The conditions of existence with probability one generalized solution of the problem are investigated. Using this results one can construct modeless, which approximate solutions of such equations with given accuracy and reliability in the uniform metric.The modified local Crank-Nicolson schemes for Rosenau-Burgers equation.https://zbmath.org/1449.651962021-01-08T12:24:00+00:00"Muyassar, Ahmat"https://zbmath.org/authors/?q=ai:muyassar.ahmat"Abdurishit, Abduwali"https://zbmath.org/authors/?q=ai:abduwali.abdurishit"Abdugeni, Abduxkur"https://zbmath.org/authors/?q=ai:abdugeni.abduxkurSummary: Two classes of modified local Crank-Nicolson schemes for Rosenau-Burgers equation are proposed. Firstly, we obtain the exact solution of the ODE which is obtained from the original PDE by using central finite difference discretization in space direction. Next, the exponential coefficient matrix of this equation is approximated by using matrix splitting technique by line and element. Finally, two types of methods are achieved by using the modified local Crank-Nicolson schemes. The stability, convergence and priori error estimation of the two schemes are discussed. The accuracy of theoretical proof and efficiency of both schemes are demonstrated by numerical results. The proposed methods possess the advantages of simple structure and high accuracy.The Bitsadze-Samarskii problem for some characteristically loaded hyperbolic-parabolic equation.https://zbmath.org/1449.353172021-01-08T12:24:00+00:00"Khubiev, Kazbek Uzeirovich"https://zbmath.org/authors/?q=ai:khubiev.kazbek-uzeirovichSummary: The paper considers a characteristically loaded equation of a mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity part of the domain. In the hyperbolic part of the domain, we have a loaded one-velocity transport equation, known in mathematical biology as the Mac Kendrick Von Forester equation, in the parabolic part we have a loaded diffusion equation. The purpose of the paper is to study the uniqueness and existence of the solution of the nonlocal inner boundary value problem with Bitsadze-Samarskii type boundary conditions and the continuous conjugation conditions in the parabolic domain; the hyperbolic domain is exempt from the boundary conditions.
The problem under investigation is reduced to a non-local problem for an ordinary second-order differential equation with respect to the trace of the unknown function in the line of the type changing. The existence and uniqueness theorem for the solution of the problem has been proved; the solution is written out explicitly in the hyperbolic part of the domain. In the parabolic part, the problem under study is reduced to the Volterra integral equation of the second kind, and the solution representation has been found.Multiplicity of positive solutions for quasi-linear elliptic equations involving concave-convex nonlinearity and Sobolev-Hardy term.https://zbmath.org/1449.350202021-01-08T12:24:00+00:00"Du, Ming"https://zbmath.org/authors/?q=ai:du.ming"Liu, Xiaochun"https://zbmath.org/authors/?q=ai:liu.xiaochunSummary: In this paper, we investigate the quasi-linear elliptic equations involving concave-convex nonlinearity and Sobolev-Hardy term. By using the theory of the Lusternik-Schnirelmann category and the relationship between the Nehari manifold and fibering maps, we get some improvement on existence and multiplicity of positive solution.A domain decomposition method for linearized Boussinesq-type equations.https://zbmath.org/1449.652372021-01-08T12:24:00+00:00"Steinstraesser, Joao Guilherme Caldas"https://zbmath.org/authors/?q=ai:steinstraesser.joao-guilherme-caldas"Kemlin, Gaspard"https://zbmath.org/authors/?q=ai:kemlin.gaspard"Rousseau, Antoine"https://zbmath.org/authors/?q=ai:rousseau.antoineSummary: In this paper, we derive discrete transparent boundary conditions for a class of linearized Boussinesq equations. These conditions happen to be non-local in time and we test numerically their accuracy with a Crank-Nicolson time-discretization on a staggered grid. We use the derived transparent boundary conditions as interface conditions in a domain decomposition method, where they become local in time. We analyze numerically their efficiency thanks to comparisons made with other interface conditions.A linearized difference method for generalized SRLW equation with damping term.https://zbmath.org/1449.652022021-01-08T12:24:00+00:00"Wang, Xi"https://zbmath.org/authors/?q=ai:wang.xi"Zhang, Hong"https://zbmath.org/authors/?q=ai:zhang.hong.4|zhang.hong|zhang.hong.3|zhang.hong.5|zhang.hong.2|zhang.hong.1"Hu, Jinsong"https://zbmath.org/authors/?q=ai:hu.jinsongSummary: In this paper, numerical solution of the initial-boundary value problem of dissipative generalized SRLW equation with damping term is considered. A three-level linearized difference scheme with second order accuracy is proposed. It is proved that the difference scheme is convergent and stable by using mathematical induction and discrete functional analysis. Efficiency of the method is demonstrated by some numerical examples.Two conservative compact finite difference schemes for the long-wave short-wave interaction equation.https://zbmath.org/1449.651852021-01-08T12:24:00+00:00"Jiang, Jiaping"https://zbmath.org/authors/?q=ai:jiang.jiaping"Wang, Tingchun"https://zbmath.org/authors/?q=ai:wang.tingchunSummary: This paper focuses on numerical simulation of the long-wave short-wave interaction equation. Two fourth-order compact finite difference schemes are proposed and proved to preserve the total mass and energy in the discrete sense. Numerical results show the good stability of the schemes and fourth-order and second-order convergence of the numerical solutions in space and time, respectively. Simulation results also show that the schemes preserve well the total mass and energy.A new application of the extended tanh-function method and new solutions of the Riccati equation and sine-Gordon equation.https://zbmath.org/1449.350092021-01-08T12:24:00+00:00"Lin, Fubiao"https://zbmath.org/authors/?q=ai:lin.fubiao"Zhang, Qianhong"https://zbmath.org/authors/?q=ai:zhang.qianhongSummary: The explicit analytical solutions of nonlinear partial differential equations, in particular, the traveling wave solutions, contain rich information about the equations, and they are very important for describing the development of various phenomena. In the paper, many types of new explicit traveling wave solutions are presented for the KdV equation. First, many new explicit analytical solutions of the Riccati equation are given by using the trial function method and Matlab software. Second, many types of new explicit analytical solutions of the sine-Gordon equation are obtained by using the extended tanh-function method and new solutions of Riccati equation. Finally, as a new application, many new traveling wave solutions of the KdV equation are provided by using the sine-cosine function method, new solutions of the sine-Gordon equation and its simplified transformation forms. The obtained results extend and complement some relevant existing works. In particular, these methods and obtained new results can be applied to find explicit new traveling wave solutions of many nonlinear partial differential equations.Numerical analysis of the Allen-Cahn equation with coarse meshes.https://zbmath.org/1449.651872021-01-08T12:24:00+00:00"Kemmochi, Tomoya"https://zbmath.org/authors/?q=ai:kemmochi.tomoyaSummary: In this paper, we consider the finite difference semi-discretization of the Allen-Cahn equation with the diffuse interface parameter \(\varepsilon\). While it is natural to make the mesh size parameter \(h\) smaller than \(\varepsilon\), it is desirable that \(h\) is as big as possible in view of computational costs. In fact, when \(h\) is bigger than \(\varepsilon\) (i.e., the mesh is relatively coarse), it is observed that the numerical solution does not move at all. The purpose of this paper is to clarify the mechanism of this phenomenon. We prove that the numerical solution converges to that of the ordinary equation without the diffusion term if \(h\) is bigger than \(\varepsilon\). Numerical examples are presented to support the result.Convective flow of blood in square and circular cavities.https://zbmath.org/1449.760702021-01-08T12:24:00+00:00"Senel, P."https://zbmath.org/authors/?q=ai:senel.pelin"Tezer-Sezgin, M."https://zbmath.org/authors/?q=ai:tezer-sezgin.munevverSummary: In this study, the fully developed, steady, laminar flow of blood is studied in a long pipe with square and circular cross-sections subjected to a magnetic field generated by an electric wire. Temperature difference between the walls causes heat transfer within the fluid by the displacement of the magnetizable fluid particles in the cavity. The governing equations are the coupled Navier-Stokes and energy equations including magnetization terms. The axial velocity is also computed with the obtained plane velocity. The dual reciprocity boundary element method (DRBEM) is used by taking all the terms other than Laplacian as inhomogeneity which transforms the partial differential equations into the boundary integral equations. Numerical results are given for increasing values of magnetic $(Mn)$ and Rayleigh $(Ra)$ numbers. The numerical results reveal that an increase in $Mn$ accelerates the plane velocity in the cavity but decelerates the axial velocity around the magnetic source. Pressure increases through the channel starting from the magnetic source. Isotherms show the cooling of the channel with high $Mn$ and $Ra$ only leaving a thin hot layer near the top heated wall. As $Ra$ increases viscous effect is reduced leaving its place to convection in the channel. The use of DRBEM has considerably small computational expense compared to domain type methods.The modified weak Galerkin finite element method for solving Brinkman equations.https://zbmath.org/1449.653242021-01-08T12:24:00+00:00"Sun, Li'na"https://zbmath.org/authors/?q=ai:sun.lina"Feng, Yue"https://zbmath.org/authors/?q=ai:feng.yue"Liu, Yuanyuan"https://zbmath.org/authors/?q=ai:liu.yuanyuan.3|liu.yuanyuan|liu.yuanyuan.2|liu.yuanyuan.1"Zhang, Ran"https://zbmath.org/authors/?q=ai:zhang.ran.1|zhang.ran.2|zhang.ranSummary: A modified weak Galerkin (MWG) finite element method is introduced for the Brinkman equations in this paper. We approximate the model by the variational formulation based on two discrete weak gradient operators. In the MWG finite element method, discontinuous piecewise polynomials of degree \(k\) and \(k-1\) are used to approximate the velocity \(u\) and the pressure \(p\), respectively. The main idea of the MWG finite element method is to replace the boundary functions by the average of the interior functions. Therefore, the MWG finite element method has fewer degrees of freedom than the WG finite element method without loss of accuracy. The MWG finite element method satisfies the stability conditions for any polynomial with degree no more than \(k-1\). The MWG finite element method is highly flexible by allowing the use of discontinuous functions on arbitrary polygons or polyhedra with certain shape regularity. Optimal order error estimates are established for the velocity and pressure approximations in \(H^1\) and \(L^2\) norms. Some numerical examples are presented to demonstrate the accuracy, convergence and stability of the method.New conditions for pattern solutions of a Brusselator model.https://zbmath.org/1449.352512021-01-08T12:24:00+00:00"Tong, Chang-Qing"https://zbmath.org/authors/?q=ai:tong.changqing"Lin, Jia-Yun"https://zbmath.org/authors/?q=ai:lin.jiayun"Ma, Man-Jun"https://zbmath.org/authors/?q=ai:ma.manjun"Tao, Ji-Cheng"https://zbmath.org/authors/?q=ai:tao.jichengSummary: This paper is devoted to establishing a critical value of the concentration of one intermediary reactant which determines whether pattern solutions of a class of Brusselator models exist or not. We introduce a new method to compute the degree index of the related linear operator so that the obtained sufficient conditions are easier to verify than those in the known references. The proofs mainly rely on Leray-Schauder degree theory, implicit function theorem and analytical techniques.A semilinear pseudo-parabolic equation in exterior domains.https://zbmath.org/1449.352762021-01-08T12:24:00+00:00"Cao, Yang"https://zbmath.org/authors/?q=ai:cao.yang"Yin, Jingxue"https://zbmath.org/authors/?q=ai:yin.jingxueSummary: This paper is concerned with large time behavior of solutions for the semilinear pseudo-parabolic equation in exterior domains. It is revealed that the inhomogeneous boundary condition may develop large variation of solutions with the evolution of time.The EM algorithm for ML estimators under nonlinear inequalities restrictions on the parameters.https://zbmath.org/1449.350792021-01-08T12:24:00+00:00"Shen, Qi-Xia"https://zbmath.org/authors/?q=ai:shen.qixia"Miao, Peng"https://zbmath.org/authors/?q=ai:miao.peng"Liang, Yin-Shuang"https://zbmath.org/authors/?q=ai:liang.yinshuangSummary: One of the most powerful algorithms for obtaining maximum likelihood estimates for many incomplete-data problems is the EM algorithm. However, when the parameters satisfy a set of nonlinear restrictions, It is difficult to apply the EM algorithm directly. In this paper, we propose an asymptotic maximum likelihood estimation procedure under a set of nonlinear inequalities restrictions on the parameters, in which the EM algorithm can be used. Essentially this kind of estimation problem is a stochastic optimization problem in the M-step. We make use of methods in stochastic optimization to overcome the difficulty caused by nonlinearity in the given constraints.Progress of the study on Landau-Ginzburg A-model.https://zbmath.org/1449.354042021-01-08T12:24:00+00:00"Fan, Huijun"https://zbmath.org/authors/?q=ai:fan.huijun"Jiang, Wenfeng"https://zbmath.org/authors/?q=ai:jiang.wenfeng"Yang, Dingyu"https://zbmath.org/authors/?q=ai:yang.dingyuSummary: A brief introduction of Landau-Ginzburg A-model (LG A-model) in homological mirror symmetry is given. Firstly, a short review of the general picture of the homological mirror symmetry is given. Then the background of Landau-Ginzburg model and its role in homological mirror symmetry are discussed. Finally, a brief introduction of our recent work on the Fukaya category of LG model is included. Both mathematical and physical backgrounds are discussed in this introduction as much as possible.Approximate controllability of boundary control problems for a class of degenerate parabolic equations.https://zbmath.org/1449.930092021-01-08T12:24:00+00:00"Xie, Chunlei"https://zbmath.org/authors/?q=ai:xie.chunlei"Du, Runmei"https://zbmath.org/authors/?q=ai:du.runmei"Yuan, Yuan"https://zbmath.org/authors/?q=ai:yuan.yuan.1|yuan.yuan|yuan.yuan.3|yuan.yuan.2Summary: By using Carleman estimates of the adjoint equations, we investigated the boundary control problem of degenerate parabolic equations, and obtained its approximate controllability. The result shows that for any objective function, there exists a control function such that the solution of the problem can be fully close to the objective function in a finite time.High precise linearized difference scheme for Benjamin-Bona-Mahony equation.https://zbmath.org/1449.652102021-01-08T12:24:00+00:00"Zhang, Hong"https://zbmath.org/authors/?q=ai:zhang.hong.3|zhang.hong|zhang.hong.2|zhang.hong.5|zhang.hong.1|zhang.hong.4"Wang, Xi"https://zbmath.org/authors/?q=ai:wang.xi"Hu, Jinsong"https://zbmath.org/authors/?q=ai:hu.jinsongSummary: In this paper, numerical solution for the initial-boundary value problem of Benjamin-Bona-Mahony equation under homogeneous boundary is considered. A linearized three-level difference scheme is introduced. The difference scheme simulates a conservative quantity of the problem. Furthermore, convergence and stability of the difference scheme are proved by energy method. Numerical experiments indicate the reliability of this method.New extended direct algebraic method for the Tzitzéica type evolution equations arising in nonlinear optics.https://zbmath.org/1449.354072021-01-08T12:24:00+00:00"Mirhosseini-Alizamini, Seyed Mehdi"https://zbmath.org/authors/?q=ai:mirhosseini-alizamini.seyed-mehdi"Rezazadeh, Hadi"https://zbmath.org/authors/?q=ai:rezazadeh.hadi"Eslami, Mostafa"https://zbmath.org/authors/?q=ai:eslami.mostafa"Mirzazadeh, Mohammad"https://zbmath.org/authors/?q=ai:mirzazadeh.mohammad"Korkmaz, Alpert"https://zbmath.org/authors/?q=ai:korkmaz.alpertSummary: In this study, the new extended direct algebraic method is exerted for constructing more general exact solutions of the three nonlinear evolution equations with physical interest namely, the Tzitzéica equation, the Dodd-Bullough-Mikhailor equation and the Liouville equation. By using of an appropriate traveling wave transformation reduces these equations to ODE. We state that this method is excellently a generalized form to obtain solitary wave solutions of the nonlinear evolution equations that are widely used in theoretical physics. The method appears to be easier and faster by means of symbolic computation system.A shifted Chebyshev-tau method for finding a time-dependent heat source in heat equation.https://zbmath.org/1449.354582021-01-08T12:24:00+00:00"Akbarpour, Samaneh"https://zbmath.org/authors/?q=ai:akbarpour.samaneh"Shidfar, Abdollah"https://zbmath.org/authors/?q=ai:shidfar.abdollah"Saberi Najafi, Hashem"https://zbmath.org/authors/?q=ai:saberi-najafi.hashemSummary: This paper investigates the inverse problem of determining the time-dependent heat source and the temperature for the heat equation with Dirichlet boundary conditions and an integral over determination conditions. The numerical method is presented for solving the inverse problem. Shifted Chebyshev polynomial is used to approximate the solution of the equation as a base of the tau method which is based on the Chebyshev operational matrices. The main advantage of this method is based upon reducing the partial differential equation into a system of algebraic equations of the solution. Numerical results are presented and discussed.Galerkin finite element method for 2D Riesz fractional differential equation based on unstructured meshes.https://zbmath.org/1449.653112021-01-08T12:24:00+00:00"Bu, Weiping"https://zbmath.org/authors/?q=ai:bu.weipingSummary: Galerkin finite element method is developed for the two-dimensional Riesz fractional diffusion equations based on Dirichlet boundary conditions. The Lagrange linear piecewise polynomial is employed as the basic function. Based on triangle unstructured meshes, the implementation of the finite element method for fractional differential equations is described in detail. Compared with the existing methods, the developed method efficiently reduces the computational cost and increases the accuracy of the stiffness matrix. Finally, some numerical tests are given to verify the effectiveness of the devised method.Numerical study on the modulational instability of space fractional Schrödinger equation.https://zbmath.org/1449.652752021-01-08T12:24:00+00:00"Li, Wenbin"https://zbmath.org/authors/?q=ai:li.wenbin"Wang, Dongling"https://zbmath.org/authors/?q=ai:wang.donglingSummary: Modulational instability is widely used in mathematics and physics. In this work, we mainly used splitting Fourier spectral method to numerically calculate the space fractional Schrödinger equation and deduced the modulational instability condition of space fractional Schrödinger equation by the Benjamin-Feir-Lighthill criterion. Then we studied the different modulational instability behavior of space fractional Schrödinger equation in different initial conditions, and we also compared it with the integer order Schrödinger equation. The comparison results show that the modulational instability behavior of the integer order Schrödinger equation can be applied to fractional Schrödinger equations as well.Continuous dependence on boundary parameters for three-dimensional viscous primitive equation of large-scale ocean atmospheric dynamics.https://zbmath.org/1449.350362021-01-08T12:24:00+00:00"Li, Yuanfei"https://zbmath.org/authors/?q=ai:li.yuanfeiSummary: By using a priori bounds of the solutions of the equations and the technique of differential inequality, the author proved that the solution of the three-dimensional viscous primitive equation of large-scale ocean atmospheric dynamics was continuously dependent on the boundary parameters.High energy solutions for a class of fractional Kirchhoff-type equation.https://zbmath.org/1449.352092021-01-08T12:24:00+00:00"Zhang, Shengui"https://zbmath.org/authors/?q=ai:zhang.shenguiSummary: By using the fountain theorem in critical point theory and the theory of fractional variable exponent Sobolev space, the author gave the existence of infinitely many high energy solutions of fractional Kirchhoff-type equation with \(p (x)\)-Laplacian operator, without assuming that the (AR) type superlinear condition held.Asymptotic behavior of solutions for classical reaction-diffusion equation with nonlinear boundary condition and weak memory.https://zbmath.org/1449.350892021-01-08T12:24:00+00:00"Zhao, Tao"https://zbmath.org/authors/?q=ai:zhao.tao.1"Wang, Xuan"https://zbmath.org/authors/?q=ai:wang.xuanSummary: We considered the asymptotic behavior of solutions for the classical reaction-diffusion equation with weak memory. When the internal and boundary nonlinearities adhered to supercritical exponential growth and the equilibrium condition, we proved the existence of a global attractor in \({L^2} (\Omega) \times {L_\mu^2} (\mathbb{R}^+; {L^2} (\Omega))\) by using contractive function method and semigroup theory.Local energy decay of coupled wave equations in exterior domain.https://zbmath.org/1449.350832021-01-08T12:24:00+00:00"Wu, Jieqiong"https://zbmath.org/authors/?q=ai:wu.jieqiong"Wu, Jiangtao"https://zbmath.org/authors/?q=ai:wu.jiangtaoSummary: The local energy decay of wave equations in an exterior domain coupled by the displacement was studied. Assume that the two wave systems were coupled in a neighborhood of the boundary and the damping of the system was exerted only in a bounded domain. The decay estimate of local energy for the coupled system was obtained by using multiplier method, weighted function method and the cut-off technology.An \(hp\)-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels.https://zbmath.org/1449.652742021-01-08T12:24:00+00:00"Jia, Hongli"https://zbmath.org/authors/?q=ai:jia.hongli"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.5|yang.yang.2|yang.yang.1|yang.yang.3|yang.yang.4"Wang, Zhongqing"https://zbmath.org/authors/?q=ai:wang.zhongqingSummary: This paper presents an \(hp\)-version Chebyshev spectral collocation method for nonlinear Volterra integro-differential equations with weakly singular kernels. The \(hp\)-version error bound of the collocation method under the \({H^1}\)-norm is established on an arbitrary mesh. Numerical experiments demonstrate the effectiveness of the proposed method.A high order operator splitting method for the Degasperis-Procesi equation.https://zbmath.org/1449.651792021-01-08T12:24:00+00:00"Guo, Yunrui"https://zbmath.org/authors/?q=ai:guo.yunrui"Zhang, Hong"https://zbmath.org/authors/?q=ai:zhang.hong.5|zhang.hong|zhang.hong.2|zhang.hong.3|zhang.hong.4|zhang.hong.1"Yang, Wenjing"https://zbmath.org/authors/?q=ai:yang.wenjing"Wang, Ji"https://zbmath.org/authors/?q=ai:wang.ji"Song, Songhe"https://zbmath.org/authors/?q=ai:song.songheSummary: The Degasperis-Procesi equation is split into a system of a hyperbolic equation and an elliptic equation. For the hyperbolic equation, we use the high order finite difference WENO-Z scheme to approximate the nonlinear flux. For the elliptic equation, the wavelet collocation method is employed to discretize the high order derivative. Due to the combination of the WENO-Z reconstruction and the wavelet collocation, the splitting method shows an excellent ability in capturing the formation and propagation of shock-peakon solutions. The numerical simulations for different solutions of the Degasperis-Procesi equation are conducted to illustrate high accuracy and capability of the proposed method.Finite difference schemes for the variable coefficients single and multi-term time-fractional diffusion equations with non-smooth solutions on graded and uniform meshes.https://zbmath.org/1449.651752021-01-08T12:24:00+00:00"Cui, Mingrong"https://zbmath.org/authors/?q=ai:cui.mingrongSummary: A finite difference scheme for the variable coefficients subdiffusion equations with non-smooth solutions is constructed and analyzed. The spatial derivative is discretized on a uniform mesh, and an \(L1\) approximation is used for the discretization of the fractional time derivative on a possibly graded mesh. The stability of the proposed scheme is given using the discrete energy method. The numerical scheme is \(\mathcal{O} (N^{-\min \{2-\alpha, r\alpha\}})\) accurate in time, where \(\alpha\) \((0 < \alpha < 1)\) is the order of the fractional time derivative, \(r\) is an index of the mesh partition, and it is second order accurate in space. The extension to multi-term time-fractional problems with nonhomogeneous boundary conditions is also discussed, with the stability and error estimate proved both in the discrete \({l^2}\)-norm and the \({l^\infty}\)-norm on the nonuniform temporal mesh. Numerical results are given for both the two-dimensional single and multi-term time-fractional equations.Convergence analysis of stochastic collocation methods for Maxwell equations with random inputs.https://zbmath.org/1449.652712021-01-08T12:24:00+00:00"Cheng, Lizheng"https://zbmath.org/authors/?q=ai:cheng.lizheng"Man, Jiaju"https://zbmath.org/authors/?q=ai:man.jiaju"Xie, Ziqing"https://zbmath.org/authors/?q=ai:xie.ziqingSummary: In this paper, we consider a stochastic collocation method for the Maxwell equations with random inputs. We first verify the regularity of the solutions for the model problem, which depends on the random dielectric constant, random magnetic permeability and the initial and boundary data. Then the convergence of our numerical approach is proved. Further some numerical examples are presented to support the analysis.Time-stepping error bound for a stochastic parabolic Volterra equation disturbed by fractional Brownian motions.https://zbmath.org/1449.652562021-01-08T12:24:00+00:00"Qi, Ruisheng"https://zbmath.org/authors/?q=ai:qi.ruisheng"Lin, Qiu"https://zbmath.org/authors/?q=ai:lin.qiuSummary: In this paper, we consider a stochastic parabolic Volterra equation driven by the infinite dimensional fractional Brownian motion with Hurst parameter \(H \in \left[ {\frac{1}{2}, 1} \right)\). We apply the piecewise constant, discontinuous Galerkin method to discretize this equation in the temporal direction. Based on the explicit form of the scalar resolvent function and the refined estimates for the Mittag-Leffler function, we derive sharp mean-square regularity results for the mild solution. The sharp regularity results enable us to obtain the optimal error bound of the time discretization. These theoretical findings are finally accompanied by several numerical examples.Solution of nonlinear creep problem for stochastically inhomogeneous plane on the basis of the second approximation for small parameter method.https://zbmath.org/1449.354152021-01-08T12:24:00+00:00"Popov, Nikolaĭ Nikolaevich"https://zbmath.org/authors/?q=ai:popov.n-n"Chernova, Ol'ga Olegovna"https://zbmath.org/authors/?q=ai:chernova.olga-olegovnaSummary: The analytical method for nonlinear stochastic creep problem solving for a plane stressed state was developed. Stochasticity was introduced into the determinative creep equation, which was taken in accordance with the nonlinear theory of viscous flow, through a homogeneous random function of coordinates. The problem was solved on the basis of the second approximation for small parameter method in stress tensor components. The main statistical characteristics of the random stress field were calculated. The analysis of the results in the first and second approximations was obtained.An unconditionally stable Laguerre based finite difference method for transient diffusion and convection-diffusion problems.https://zbmath.org/1449.651762021-01-08T12:24:00+00:00"De Sousa, Wescley T. B."https://zbmath.org/authors/?q=ai:de-sousa.wescley-t-b"Matt, Carlos F. T."https://zbmath.org/authors/?q=ai:matt.carlos-frederico-trottaSummary: This paper describes an application of weighted Laguerre polynomial functions to produce an unconditionally stable and accurate finite-difference scheme for the numerical solution of transient diffusion and convection-diffusion problems. The unconditionally stability of Laguerre-FDM (L-FDM) is guaranteed by expanding the time dependency of the unknown potential as a series of orthogonal functions in the domain \( (0,\infty)\), avoiding thus any time integration scheme. The L-FDM is a marching-on-in-degree scheme instead of traditional marching-on-in-time methods. For the two heat-transfer problems, we demonstrated the accuracy, numerical stability and computational efficiency of the proposed L-FDM by comparing its results against closed-form analytical solutions and numerical results obtained from classical finite-difference schemes as, for instance, the alternating direction implicit method.A splitting scheme for the numerical solution of the KWC system.https://zbmath.org/1449.652892021-01-08T12:24:00+00:00"Hoppe, R. H. W."https://zbmath.org/authors/?q=ai:hoppe.ronald-h-w"Winkle, J. J."https://zbmath.org/authors/?q=ai:winkle.j-jSummary: We consider a splitting method for the numerical solution of the regularized Kobayashi-Warren-Carter (KWC) system which describes the growth of single crystal particles of different orientations in two spatial dimensions. The KWC model is a system of two nonlinear parabolic PDEs representing gradient flows associated with a free energy in two variables. Based on an implicit time discretization by the backward Euler method, we suggest a splitting method and prove the existence as well as the energy stability of a solution. The discretization in space is taken care of by Lagrangian finite elements with respect to a geometrically conforming, shape regular, simplicial triangulation of the computational domain and requires the successive solution of two individual discrete elliptic problems. Viewing the time as a parameter, the fully discrete equations represent a parameter dependent nonlinear system which is solved by a predictor corrector continuation strategy with an adaptive choice of the time step size. Numerical results illustrate the performance of the splitting method.Nonexistence and long time behavior of solutions to a class of nonlinear degenerate parabolic variational inequalities.https://zbmath.org/1449.352802021-01-08T12:24:00+00:00"Sun, Yudong"https://zbmath.org/authors/?q=ai:sun.yudong"Qiu, Mingxue"https://zbmath.org/authors/?q=ai:qiu.mingxueSummary: In this text, the nonlinear variational inequalities based on degenerate parabolic operators are studied. The nonexistence of the solutions is proved by a differential inequality technique. In addition, we also obtain the convergence with respect to time.Wong-ZaKai approximations for a stochastic reaction-diffusion equation on unbounded domains.https://zbmath.org/1449.352722021-01-08T12:24:00+00:00"Wang, Xiuxiu"https://zbmath.org/authors/?q=ai:wang.xiuxiu"Li, Xiaojun"https://zbmath.org/authors/?q=ai:li.xiaojunSummary: In this paper, we study the relationship between the stochastic reaction-diffusion equation and its corresponding Wong-ZaKai approximation system. First, we prove the existence of the pullback attractor of the original system and the Wong-ZaKai approximation system. Then the upper semi-continuity of the random attractor is illustrated.Optimal rate convergence analysis of a second order numerical scheme for the Poisson-Nernst-Planck system.https://zbmath.org/1449.651772021-01-08T12:24:00+00:00"Ding, Jie"https://zbmath.org/authors/?q=ai:ding.jie"Wang, Cheng"https://zbmath.org/authors/?q=ai:wang.cheng.1"Zhou, Shenggao"https://zbmath.org/authors/?q=ai:zhou.shenggaoSummary: In this work, we propose and analyze a second-order accurate numerical scheme, both in time and space, for the multi-dimensional Poisson-Nernst-Planck system. Linearized stability analysis is developed, so that the second order accuracy is theoretically justified for the numerical scheme, in both temporal and spatial discretization. In particularly, the discrete \({W^{1, 4}}\) estimate for the electric potential field, which plays a crucial role in the proof, is rigorously established. In addition, various numerical tests have confirmed the anticipated numerical accuracy, and further demonstrated the effectiveness and robustness of the numerical scheme in solving problems of practical interest.An implicit scheme for solving unsteady Boltzmann model equation.https://zbmath.org/1449.651932021-01-08T12:24:00+00:00"Li, Xiaowei"https://zbmath.org/authors/?q=ai:li.xiaowei"Li, Chunxin"https://zbmath.org/authors/?q=ai:li.chunxin"Zhang, Dan"https://zbmath.org/authors/?q=ai:zhang.dan"Li, Zhihui"https://zbmath.org/authors/?q=ai:li.zhihuiSummary: When solving hyperbolic Boltzmann model equation with discrete velocity models (DVM), the strong discontinuity of the velocity distribution function can be captured well by utilizing the non-oscillatory and non-free parameter dissipation (NND) finite difference scheme. However, most NND scheme solvers march in time explicitly, which compromise the computation efficiency due to the limitation of stability condition, especially when solving unsteady problems. In order to improve the efficiency, an implicit scheme based on NND is presented in this paper. Linearization factors are introduced to construct the implicit scheme and to reduce the stencil size. With the help of dual time-stepping method, the convergence rate of unsteady rarefied flow simulation can be massively improved. Numerical tests of steady and unsteady supersonic flow around cylinders are computed in different flow regimes. Results are shown to prove the validity and efficiency of the implicit scheme.Finite difference schemes for the tempered fractional Laplacian.https://zbmath.org/1449.652132021-01-08T12:24:00+00:00"Zhang, Zhijiang"https://zbmath.org/authors/?q=ai:zhang.zhijiang"Deng, Weihua"https://zbmath.org/authors/?q=ai:deng.weihua"Fan, Hongtao"https://zbmath.org/authors/?q=ai:fan.hongtaoSummary: The second and all higher order moments of the \(\beta \)-stable Lévy process diverge, the feature of which is sometimes referred to as shortcoming of the model when applied to physical processes. So, a parameter \(\lambda \) is introduced to exponentially temper the Lévy process. The generator of the new process is the tempered fractional Laplacian \({\left ({\Delta + \lambda} \right)^{\beta /2}}\) In this paper, we first design the finite difference schemes for the tempered fractional Laplacian equation with the generalized Dirichlet type boundary condition, their accuracy depends on the regularity of the exact solution on \({\bar \Omega}\). Then the techniques of effectively solving the resulting algebraic equation are presented, and the performances of the schemes are demonstrated by several numerical examples.Bifurcation of traveling wave solutions for the \( (2+1)\)-dimensional generalized dissipative Ablowitz-Kaup-Newell-Segur equation.https://zbmath.org/1449.350432021-01-08T12:24:00+00:00"Zhou, Yuqian"https://zbmath.org/authors/?q=ai:zhou.yuqian"Fan, Feiting"https://zbmath.org/authors/?q=ai:fan.feiting"Liu, Qian"https://zbmath.org/authors/?q=ai:liu.qianSummary: This paper employs the bifurcation method of dynamical system to study the \( (2+1)\)-dimensional generalized dissipative Ablowitz-Kaup-Newell-Segur (ANKS) equation. By qualitative analysis, phase portraits of the dynamic system corresponding to the equation are derived under different parameter conditions. Finally, according to the discussion of all bounded orbits in the obtained phase portraits, the exact expressions of three types of bounded traveling wave solutions for the \( (2+1)\)-dimensional generalized dissipative AKNS equation are given by calculation of complex elliptic integrals.Antiplane strain of a cylindrically anisotropic elastic bar.https://zbmath.org/1449.741232021-01-08T12:24:00+00:00"Bogan, Yuriĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:bogan.yurii-aleksandrovichSummary: The problem of antiplane deformation of general cylindrical anisotropic material is studied in this paper. Explicit solutions of Dirichlet and Neumann problems are given for a circular domain. The existence of unique weak solution of the Dirichlet problem in a bounded region with a piece-wise smooth boundary is proved.Curved domain walls dynamics driven by magnetic field and electric current in hard ferromagnets.https://zbmath.org/1449.740882021-01-08T12:24:00+00:00"Consolo, Giancarlo"https://zbmath.org/authors/?q=ai:consolo.giancarlo"Currò, Carmela"https://zbmath.org/authors/?q=ai:curro.carmela"Valenti, Giovanna"https://zbmath.org/authors/?q=ai:valenti.giovannaSummary: The propagation of curved domain walls in hard ferromagnetic materials is studied by applying a reductive perturbation method to the generalized Landau-Lifshitz-Gilbert equation. The extended model herein considered explicitly takes into account the effects of a spin-polarized current as well as those arising from a nonlinear dissipation.
Under the assumption of steady regime of propagation, the domain wall velocity is derived as a function of the domain wall curvature, the nonlinear damping coefficient, the magnetic field and the electric current. Threshold and Walker-like breakdown conditions for the external sources are also determined. The analytical results are evaluated numerically for different domain wall surfaces (planes, cylinders and spheres) and their physical implications are discussed.A finite difference method for a class of nonlinear Schrödinger equations involving quintic terms.https://zbmath.org/1449.652092021-01-08T12:24:00+00:00"Zhang, Fayong"https://zbmath.org/authors/?q=ai:zhang.fayong"Jiang, Xue"https://zbmath.org/authors/?q=ai:jiang.xue|jiang.xue.1Summary: In this paper, we study the finite difference method for the approximate solution of the initial-boundary value problem for the nonlinear Schrödinger equation involving quintic terms as follows \[i{u_t} + \alpha {u_{xx}} + qc{\left| u \right|^2}u + {q_q}{\left| u \right|^4}u = 0, \;\left ({x, t} \right) \in \left ({{x_l}, {x_r}} \right) \times \left ({0, T} \right],\] \[u\left ({{x_l}, t} \right) = u\left ({{x_r}, t} \right) = 0,\; 0 \le t \le T,\] \[u\left ({x, 0} \right) = {u_0}\left (x \right), \;{x_l} \le x \le {x_r}.\] Here \(\alpha > 0\), \({q_q} \le 0\), \({q_c}\) is any real number. Due to this problem, a new conservative difference scheme is proposed, the priori estimates of the finite difference solutions on \({L^\infty}\) norm are obtained. On this basis, error estimates of optimal order on \({L^2}\) norm of the finite difference solutions are obtained.Quasilinear elliptic systems in perturbed form.https://zbmath.org/1449.352262021-01-08T12:24:00+00:00"Azroul, Elhoussine"https://zbmath.org/authors/?q=ai:azroul.elhoussine"Balaadich, Farah"https://zbmath.org/authors/?q=ai:balaadich.farahSummary: In this paper, we consider the boundary value problem of a quasilinear elliptic system in degenerate form with data belongs to the dual of Sobolev spaces. The existence result is proved by means of Young measures and mild monotonicity assumptions.A factorization method for nonlinear telegraph equation.https://zbmath.org/1449.353102021-01-08T12:24:00+00:00"Guo, Peng"https://zbmath.org/authors/?q=ai:guo.peng"Tang, Rongan"https://zbmath.org/authors/?q=ai:tang.rongan"Sun, Xiaowei"https://zbmath.org/authors/?q=ai:sun.xiaoweiSummary: Several types of exact solutions of nonlinear telegraph equation are obtained by using a factorization method based on theory of Cornejo-Pérez and Rosu group. It is shown that the Cornejo-Pérez and Rosu method is simple, straightforward and can be used for many other nonlinear mathematical physics equations.Nontrivial solutions of Schrödinger-Poisson systems involving critical Sobolev exponent.https://zbmath.org/1449.354012021-01-08T12:24:00+00:00"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.3|zhang.jing.1|zhang.jing.12|zhang.jing.2|zhang.jing.7|zhang.jing.6|zhang.jing.9|zhang.jing.5|zhang.jing.10|zhang.jing.8|zhang.jing.11Summary: In this paper, we are concerned with the following Schrödinger-Poisson systems involving critical Sobolev exponent \[\begin{cases} -\Delta u + \varepsilon K (x)\varphi (x)u + \varepsilon a (x){|u|^{p-2}}u = {|u|^4}u, & x \in {\mathbb{R}}^3, \\ -\Delta \varphi = K (x){u^2}, & x \in {\mathbb{R}}^3, \end{cases}\] with \(3 < p < 6\). Assuming that \(K > 0\), \(K (x) \in {L^\infty} (\mathbb{R}^3) \cap {L^q} (\mathbb{R}^3)\) with \(\frac{6}{5} < q < 2\), \(a (x) \ge 0\) and \(a (x) \in {L^\infty} (\mathbb{R}^3) \cap {L^r} (\mathbb{R}^3)\) with \(r < \frac{6}{6-p}\). We use an abstract perturbation method in critical point theory to obtain the existence of nontrivial solutions of the above equations for small value of \(|\varepsilon|\).Oscillation of certain neutral hyperbolic equations with continuous distributed deviating arguments and damped terms.https://zbmath.org/1449.350142021-01-08T12:24:00+00:00"Liu, Caiyun"https://zbmath.org/authors/?q=ai:liu.caiyun"Zhang, Zhiyu"https://zbmath.org/authors/?q=ai:zhang.zhiyuSummary: In this paper, we consider certain hyperbolic equation with continuous distributed deviating arguments and damped terms. Some new sufficient conditions are presented for every solution of Dirichlet boundary value problem to be oscillatory by using differential inequality and calculus technique. The results generalize and improve some recent results.Exponential stability for the defocusing semilinear Schrödinger equation with locally distributed damping on a bounded domain.https://zbmath.org/1449.350452021-01-08T12:24:00+00:00"Bortot, César Augusto"https://zbmath.org/authors/?q=ai:bortot.cesar-augusto"Corrêa, Wellington José"https://zbmath.org/authors/?q=ai:correa.wellington-joseSummary: In this paper, we study the exponential stability for the semilinear defocusing Schrödinger equation with locally distributed damping on a bounded domain \(\Omega\subset\mathbb{R}^n\) with smooth boundary \(\partial\Omega\). The proofs are based on a result of unique continuation property due to \textit{M. M. Cavalcanti} et al. [Differ. Integral Equ. 22, No. 7--8, 617--636 (2009; Zbl 1240.35509)] and on a forced smoothing effect due to \textit{L. Aloui} [Asymptotic Anal. 59, No. 3--4, 179--193 (2008; Zbl 1173.35392)] combined with ideas from \textit{M. M. Cavalcanti} et. al. [loc. cit.; J. Differ. Equations 248, No. 12, 2955--2971 (2010; Zbl 1190.35206)] adapted to the present context.Nodal solutions for Lane-Emden problems in almost-annular domains.https://zbmath.org/1449.352362021-01-08T12:24:00+00:00"Amadori, Anna Lisa"https://zbmath.org/authors/?q=ai:amadori.anna-lisa"Gladiali, Francesca"https://zbmath.org/authors/?q=ai:gladiali.francesca"Grossi, Massimo"https://zbmath.org/authors/?q=ai:grossi.massimoSummary: In this paper, we prove an existence result to the problem
\[ \begin{cases} -\Delta u=|u|^{p-1}u\quad & \text{in }\Omega, \\ u=0\qquad &\text{on }\partial\Omega, \end{cases} \]
where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) which is a perturbation of the annulus. Then there exists a sequence \(p_1<p_2<\cdots\) with \(\lim\limits_{k\to +\infty}p_k=+\infty\) such that for any real number \(p>1\) and \(p\neq p_k\) there exist at least one solution with \(m\) nodal zones. In doing so, we also investigate the radial nodal solution in an annulus: we provide an estimate of its Morse index and analyze the asymptotic behavior as \(p\to 1\).Perturbation solution for a solitary wave of the nonlinear higher dimensional disturbed Klein-Gordon equation.https://zbmath.org/1449.350332021-01-08T12:24:00+00:00"Xu, Jianzhong"https://zbmath.org/authors/?q=ai:xu.jianzhong"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqiSummary: In this paper, a class of nonlinear forced disturbed Klein-Gordon equations are considered using the method of generalized variational iteration. Firstly, the solitary waves of an undisturbed Klein-Gordon equation are solved using the method of undetermined coefficients for hyperbolic functions. Then, perturbed approximate solutions for a soliton of a nonlinear forced disturbed Klein-Gordon equation are obtained using the functional variational iterative principle. Finally, the uniform validity for the approximate solutions is proved. The obtained approximate solution is an analytic expression. So it can be used for carrying out analytic operations. However, these cannot be obtained via a simple simulation.A high-order compact difference method for solving the Helmholtz equation with variable wave number.https://zbmath.org/1449.653012021-01-08T12:24:00+00:00"Wang, Zhi"https://zbmath.org/authors/?q=ai:wang.zhi"Ge, Yongbin"https://zbmath.org/authors/?q=ai:ge.yongbinSummary: The numerical calculation of Helmholtz equation for large wave number and variable wave number problem is a subject that needs further study. The numerical methods of Helmholtz equation have important theoretical value and practical significance. With the help of the idea of combining the Talyor series expansions and blended compact difference, for the one-dimensional and two-dimensional Helmholtz equations, a sixth-order compact difference method is proposed. The scheme involves the values of unknown functions and their first and second derivatives. In order to ensure the global accuracy of the present scheme, the sixth-order compact difference schemes are also used for the computation of the first and second derivatives. The scheme has the sixth order accuracy in the case of small wave number and variable wave number, and it can still keep the accuracy above the third order in the case of large wave number. Numerical experiments are given to show the efficiency and dependability of the present scheme.New exact solutions and soliton evolvements for the \( (2+1)\)-dimensional dispersive long wave equation.https://zbmath.org/1449.353712021-01-08T12:24:00+00:00"Yang, Juan"https://zbmath.org/authors/?q=ai:yang.juan"Zeng, Chunhua"https://zbmath.org/authors/?q=ai:zeng.chunhuaSummary: With the help of the symbolic computation system Mathematica and an extended \(G'/G\)-expansion method and a variable separation method, new exact solutions of the \( (2+1)\)-dimensional dispersive long wave equation are derived. By selecting the appropriate function, we can construct dromion solution, solitoff solution, periodic solitary wave solution, and further study the evolution of soliton with time.Long-time dynamics of solutions for a class of coupling beam equations with nonlinear boundary conditions.https://zbmath.org/1449.351052021-01-08T12:24:00+00:00"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.4|wang.yu.5|wang.yu.2|wang.yu|wang.yu.8|wang.yu.3|wang.yu.1"Zhang, Jianwen"https://zbmath.org/authors/?q=ai:zhang.jianwenSummary: In this paper, we study long-time dynamics of solutions for a class of coupling beam equations with strong damping under nonlinear boundary conditions. Firstly, we prove the existence and uniqueness of global solutions by some inequalities and prior estimates methods. Secondly, by an absorbing set and asymptotic compactness of the related solution semigroup, we prove the existence of a global attractor.New exact solutions for a class of nonlinear fractional evolution equations.https://zbmath.org/1449.471302021-01-08T12:24:00+00:00"Yang, Juan"https://zbmath.org/authors/?q=ai:yang.juan"Zeng, Chunhua"https://zbmath.org/authors/?q=ai:zeng.chunhua"Feng, Qingjiang"https://zbmath.org/authors/?q=ai:feng.qingjiangSummary: Using \(\exp(-\Phi (\xi))\)-expansion method, the new exact solutions of the nonlinear fractional Phi-4 equation, the nonlinear fractional order foam drainage equation and the nonlinear fractional SRLW equation are obtained. The practice proves that this method is simple and convenient, it has very important significance for the research of nonlinear fractional evolution equations.Existence of exponential attractors in Plate equation.https://zbmath.org/1449.350982021-01-08T12:24:00+00:00"Su, Xiaohu"https://zbmath.org/authors/?q=ai:su.xiaohu"Jiang, Jinping"https://zbmath.org/authors/?q=ai:jiang.jinpingSummary: In this paper, we study the Plate equation's exponential attractor on bounded region \(\partial \Omega\) with smooth boundary. On the basis of previous studies, the existence of Plate equation's exponential attractor is proved by Lipschitz continuous and discrete squeezing conditions in phase space.The existence of solutions for a class of quasilinear elliptic equations.https://zbmath.org/1449.352292021-01-08T12:24:00+00:00"Gao, Fang"https://zbmath.org/authors/?q=ai:gao.fang"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.5|chen.lin.4|chen.lin.1|chen.lin.2|chen.lin.3|chen.lin.6|chen.linSummary: In this paper, we consider the quasilinear elliptic equation
\[ -\operatorname{div}(|x|^{-ap}|\nabla u|^{p - 2}\nabla u) + V (x)|u|^{p - 2}u = f (u, |\nabla u|^{p - 2}\nabla u)\text{ in }\mathbb{R}^N, \]
where \(1 < p < N\) and \(f\) is a nonlinearity depending also on the gradient of the solution. The existence of a positive solution is stated through an iterative method based on mountain pass techniques.Dynamic analysis of three species of plankton ecosystem with harvesting term.https://zbmath.org/1449.350492021-01-08T12:24:00+00:00"Guo, Yanfen"https://zbmath.org/authors/?q=ai:guo.yanfen"Jian, Zhichao"https://zbmath.org/authors/?q=ai:jian.zhichao"Liu, Shaonan"https://zbmath.org/authors/?q=ai:liu.shaonan"Zhang, Jing"https://zbmath.org/authors/?q=ai:zhang.jing.2|zhang.jing.5|zhang.jing.1|zhang.jing.12|zhang.jing.6|zhang.jing.9|zhang.jing.3|zhang.jing.11|zhang.jing.8|zhang.jing.10|zhang.jing.7"Bi, Xiaohua"https://zbmath.org/authors/?q=ai:bi.xiaohuaSummary: In this paper, a three species NP-P-Z model of plankton ecosystem is investigated considering the effects of harvesting and diffusion. The local and global asymptotic stability of the positive equilibrium of the non-diffusion system and optimal harvesting policy are investigated. For the diffusion system, the conditions of Turing instability are given. At last, numerical simulations are used to verify the correctness of the theory.The long-term dynamic behavior of a class of nonlinear evolution equations.https://zbmath.org/1449.350912021-01-08T12:24:00+00:00"Gao, Qingpei"https://zbmath.org/authors/?q=ai:gao.qingpei"Chai, Yuzhen"https://zbmath.org/authors/?q=ai:chai.yuzhenSummary: The existence of bounded absorption set for coupled beam equations is obtained by using Galerkin method combined with a priori estimate and some inequality techniques. lt is proved that the solution semigroup \(S (t)\) is asymptotically compact. The global attractor of the equation in space \(H_0^2 (\Omega) \times {L^2} (\Omega) \times {L^2} (\Omega)\) is obtained.Numerical method of value boundary problem decision for 2D equation of heat conductivity with fractional derivatives.https://zbmath.org/1449.651722021-01-08T12:24:00+00:00"Beĭbalaev, Vetlugin Dzhabrilovich"https://zbmath.org/authors/?q=ai:beibalaev.vetlugin-dzhabrilovich"Shabanova, Mumina Ruslanovna"https://zbmath.org/authors/?q=ai:shabanova.mumina-ruslanovnaSummary: In this work a solution is obtained for the boundary problem for two-dimensional thermal conductivity equation with derivatives of fractional order on time and space variables by grid method. Explicit and implicit difference schemes are developed. Stability criteria of these difference schemes are proven. It is shown that the approximation order by time is equal one but by space variables it is equal two. A solution method is suggested using fractional steps. It is proved that the transition module, corresponding to two half-steps, approximates the transition module for the given equation.Additive and multiplicative noise excitability of stochastic partial differential equations.https://zbmath.org/1449.354652021-01-08T12:24:00+00:00"Guo, Zhongkai"https://zbmath.org/authors/?q=ai:guo.zhongkai"Cheng, Shuilin"https://zbmath.org/authors/?q=ai:cheng.shuilin"Wang, Weifeng"https://zbmath.org/authors/?q=ai:wang.weifengSummary: In this paper, noise excitability of energy solution to stochastic partial differential equation with additive and multiplicative cases is considered. By using Itô's formula and energy estimate method, we obtain excitation indices of \(u\) at different cases with different results. Thus, from this point of view, the effects of additive noise and multiplicative noise on the system are different.Control and prediction of a reaction-diffusion epidemic model with media effect and time delay.https://zbmath.org/1449.490082021-01-08T12:24:00+00:00"Zhang, Lijuan"https://zbmath.org/authors/?q=ai:zhang.lijuan"Wang, Fuchang"https://zbmath.org/authors/?q=ai:wang.fuchang"Zhao, Yibin"https://zbmath.org/authors/?q=ai:zhao.yibin"Zhang, Ximing"https://zbmath.org/authors/?q=ai:zhang.ximingSummary: In this paper, a reaction diffusion model with media effect is established. The stability of the equilibrium, Hopf bifurcation and important parameters such as time delay, infection rate and media effects on the Turing structure of the model are studied. The Turing structure and the parameters of the exact Turing instability are given with the corresponding numerical simulation. The theoretical analysis and numerical simulation reveal the complexity mechanism of spatial dynamics and provide a powerful theoretical basis for the control of the spread of disease.The implicit midpoint method for two-side space fractional diffusion equation with a nonlinear source term.https://zbmath.org/1449.651822021-01-08T12:24:00+00:00"Hu, Dongdong"https://zbmath.org/authors/?q=ai:hu.dongdong"Cao, Xuenian"https://zbmath.org/authors/?q=ai:cao.xuenian"Jiang, Huiling"https://zbmath.org/authors/?q=ai:jiang.huilingSummary: In this paper, the numerical scheme was constructed for solving the space fractional diffusion equation with a nonlinear source term where the implicit midpoint method was applied to discretize the first order time partial derivative, and the quasi-compact difference operator was utilized to approximate Riemann-Liouville space fractional partial derivative. Stability and convergence analysis of this numerical method were given. Numerical experiments show that the numerical method is effective.A high order difference scheme for two-dimensional linear hyperbolic equation with Neumann boundary conditions.https://zbmath.org/1449.651992021-01-08T12:24:00+00:00"Sheng, Xiulan"https://zbmath.org/authors/?q=ai:sheng.xiulan"Zhao, Runmiao"https://zbmath.org/authors/?q=ai:zhao.runmiao"Wu, Hongwei"https://zbmath.org/authors/?q=ai:wu.hongweiSummary: A high order difference scheme is established for two-dimensional linear hyperbolic equation with Neumann boundary conditions. The third and fifth derivatives of solution at the boundary can be got by using the boundary conditions and the equation, then the nine points, six points and four points compact difference schemes are respectively established at the inner points of the region, inner points and corner points of the boundary by using the finite difference method. To obtain the convergence and stability of the numerical solution in maximum norm, a new norm is introduced to estimate the maximum norm. Then two priori estimates of the difference scheme are shown and convergence and stability are derived. The convergence order of the difference scheme in maximum norm is \(O ({\tau^2} + {h^4})\) where \(\tau\) and \(h\) are temporal and spatial step size, respectively. Some numerical examples illustrate the convergence of the high order difference schemes presented in this paper.Regularity for weak solutions of anisotropic double obstacle problems of elliptic equations.https://zbmath.org/1449.351382021-01-08T12:24:00+00:00"Xie, Suying"https://zbmath.org/authors/?q=ai:xie.suying"Yang, Chao"https://zbmath.org/authors/?q=ai:yang.chao.3|yang.chao.2|yang.chao.1Summary: Under some suitable assumptions, a local regularity of weak solution of anisotropic double obstacle problem for quasi-linear elliptic equation \(-{\mathrm{div}}A (x, \nabla u) = B (x,u, \nabla u)\) is given by using an anisotropic reverse Hölder's inequality and Sobolev inequality, and the corresponding results of some single obstacle problems are generalized.Global stability of rarefaction waves for the one-dimensional nonisothermal compressible Navier-Stokes-Korteweg system.https://zbmath.org/1449.350482021-01-08T12:24:00+00:00"Guo, Qidong"https://zbmath.org/authors/?q=ai:guo.qidong"Chen, Zhengzheng"https://zbmath.org/authors/?q=ai:chen.zhengzhengSummary: This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional nonisothermal compressible Navier-Stokes-Korteweg system with density-dependent capillarity coefficient and temperature-dependent viscosity and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. If the corresponding Riemann problem of the compressible Euler system can be solved by a rarefaction wave, we prove that the 1D compressible Navier-Stokes-Korteweg system admits a unique global strong solution which tends to the rarefaction wave as time goes to infinity. Here both the initial perturbation and the strength of the rarefaction wave can be arbitrarily large. The proof is given by an elementary \({L^2}\) energy method.The existence of weak solutions of a higher order nonlinear eilliptic equation.https://zbmath.org/1449.351662021-01-08T12:24:00+00:00"Liu, Mingji"https://zbmath.org/authors/?q=ai:liu.mingji"Liu, Xu"https://zbmath.org/authors/?q=ai:liu.xu"Cai, Hua"https://zbmath.org/authors/?q=ai:cai.huaSummary: In this paper, we show the existence of weak solutions for a higher order nonlinear elliptic equation. Our main method is to show that the evolution operator satisfies the fixed point theorem for Banach semilattice.Asymptotic properties of solutions for a degenerate parabolic system.https://zbmath.org/1449.350762021-01-08T12:24:00+00:00"Qin, Siqian"https://zbmath.org/authors/?q=ai:qin.siqian"Zhou, Zewen"https://zbmath.org/authors/?q=ai:zhou.zewen"Ling, Zhengqiu"https://zbmath.org/authors/?q=ai:ling.zhengqiuSummary: This paper deals with a degenerate parabolic system with the help of the regularized techniques and the super- and sub-solution methods. The asymptotic properties, such as the conditions of global existence and blow-up of solutions, are determined.On the global existence and stability of 3-D viscous cylindrical circulatory flows.https://zbmath.org/1449.353422021-01-08T12:24:00+00:00"Yin, Huicheng"https://zbmath.org/authors/?q=ai:yin.huicheng"Lin, Zhang"https://zbmath.org/authors/?q=ai:lin.zhangThe main result in this paper is a global existence and uniqueness theorem of cylindrical symmetric circulatory flows for the three-dimensional compressible Navier-Stokes equations. It is also shown that the flow is globally stable in time when the corresponding initial states are perturbed suitably small. The proof follows from the local existence result of classical solutions, continuity arguments, and is essentially based on uniform weighted energy estimates.
Reviewer: Radu Precup (Cluj-Napoca)Superconvergence analysis of Hermite-type rectangular element method for two-dimensional time fractional diffusion equations.https://zbmath.org/1449.652622021-01-08T12:24:00+00:00"Wang, Pingli"https://zbmath.org/authors/?q=ai:wang.pingli"Niu, Yuqi"https://zbmath.org/authors/?q=ai:niu.yuqi"Zhao, Yanmin"https://zbmath.org/authors/?q=ai:zhao.yanmin"Wang, Fenling"https://zbmath.org/authors/?q=ai:wang.fenling"Shi, Yanhua"https://zbmath.org/authors/?q=ai:shi.yanhuaSummary: Based on the classical \(L1\) approximation scheme, a Hermite-type rectangular element method is proposed for two-dimensional time fractional diffusion equations under the fully-discrete scheme. Firstly, unconditional stability analysis of the approximate scheme is provided. Secondly, by use of the integral identity result of the Hermite-type rectangular element, a superconvergence estimate in \(H^1\)-norm is established between the interpolation and Ritz projection. Moreover, combining with the relationship between the interpolation operator and Ritz projection, and by dealing with fractional derivatives skillfully, superclose and superconvergence results are obtained, which cannot be deduced by interpolation or Ritz projection alone. Finally, the global superconvergence property is derived by the technique of the postprocessing operator.Exact solutions to the bidirectional SK-Ramani equation.https://zbmath.org/1449.353742021-01-08T12:24:00+00:00"Zou, Jiahui"https://zbmath.org/authors/?q=ai:zou.jiahui"Wang, Yunhu"https://zbmath.org/authors/?q=ai:wang.yunhuSummary: In this paper, the bidirectional SK-Ramani equation is investigated by means of the extended homoclinic test approach and Riemann theta function method, respectively. Based on the Hirota bilinear method, exact solutions including one-soliton wave solution are obtained by using the extended homoclinic approach and one-periodic wave solution is constructed by using the Riemann theta function method. A limiting procedure is presented to analyze in detail the relations between the one periodic wave solution and one-soliton solution.Generating new super dynamical systems in \( (2 + 1)\)-dimensional space.https://zbmath.org/1449.353892021-01-08T12:24:00+00:00"Wei, Hanyu"https://zbmath.org/authors/?q=ai:wei.hanyu"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.4|zhang.yan.3|zhang.yan.2"Xia, Tiecheng"https://zbmath.org/authors/?q=ai:xia.tie-chengSummary: In the article, we make use of the binomial-residue-representation (BRR) to generate \( (2+1)\)-dimensional super dynamical systems. By using these systems, a new \( (2+1)\)-dimensional super NLS-MKdV hierarchy is deduced, which can be reduced to super nonlinear Schrödinger equation. Especially, two main results are obtained which have important physical applications. One of them is a set of \( (2+1)\)-dimensional super integrable couplings, the other one is a \( (2+1)\)-dimensional diffusion equation. Furthermore, Super trace identity is used to furnish the super Hamiltonian structures for the new \( (2+1)\)-dimensional super integrable system.Global attractors for a class of coupled equations with memorizing terms.https://zbmath.org/1449.351122021-01-08T12:24:00+00:00"Zhang, Liyuan"https://zbmath.org/authors/?q=ai:zhang.liyuan"Ren, Yonghua"https://zbmath.org/authors/?q=ai:ren.yonghuaSummary: In this paper, we study the problem of global attractors for a class of coupled equations with memory term. By using Faedo-Galerkin method, we obtain the existence of the solutions of the equations. By proving the existence of the system absorption set and the asymptotic compactness of the semigroup of \(S (t)\), we further prove the existence of the global attractor for the equations.Stability of age-structured epidemiological model with hepatitis B.https://zbmath.org/1449.350532021-01-08T12:24:00+00:00"Liu, Jixuan"https://zbmath.org/authors/?q=ai:liu.jixuan"Wang, Gaixia"https://zbmath.org/authors/?q=ai:wang.gaixia"Li, Xuezhi"https://zbmath.org/authors/?q=ai:li.xuezhiSummary: An age-structured hepatitis B infectious disease model is discussed, and the expression of basic reproductive number \({\mathfrak{R}_0}\) is obtained. It is proved that when \({\mathfrak{R}_0} < 1\), the disease free equilibrium is locally asymptotically stable and globally asymptotically stable. When \({\mathfrak{R}_0} > 1\), there is a unique endemic equilibrium, and the local asymptotic stability condition of endemic equilibrium is given. These conditions have important theoretical and practical significance in controlling the spread of diseases.Existence and multiplicity of solutions to a class of Klein-Gordon-Maxwell system.https://zbmath.org/1449.354092021-01-08T12:24:00+00:00"Chen, Lizhen"https://zbmath.org/authors/?q=ai:chen.lizhen"Li, Anran"https://zbmath.org/authors/?q=ai:li.anran"Li, Gang"https://zbmath.org/authors/?q=ai:li.gang.8Summary: The Klein-Gordon-Maxwell system has strong physical backgrounds and it can describe the ``binary model'' between the charged particle matter and the electromagnetic field produced by itself. According to this model, the particle matter is the solitary wave solution to a nonlinear field equation, and the effect of the electromagnetic field is determined by the coupling of the field equation with the Maxwell equation. In this paper, we use the variational method and critical point theory to study the existence and multiplicity of solutions for a class of Klein-Gordon-Maxwell systems. We first investigate the existence of non-trivial solutions to the above system by using mountain pass lemma. One of the solutions is non-negative and the other one is non-positive. Secondly, under some assumptions on the nonlinear term, we establish the existence of infinitely many high energy solutions by using the fountain theorem. Our results generalize the previous conclusions.The long-time behavior of ratio-dependent competition systems.https://zbmath.org/1449.350812021-01-08T12:24:00+00:00"Wei, Xi"https://zbmath.org/authors/?q=ai:wei.xi"Li, Yanling"https://zbmath.org/authors/?q=ai:li.yanling"Yan, Xiao"https://zbmath.org/authors/?q=ai:yan.xiaoSummary: In this paper, we are concerned with the long-time behavior of competition systems between two species with ratio-dependent functional responses subject to the homogeneous Robin boundary condition. First, we establish the existence of positive solutions to these systems by using the fixed point index theory in cone and comparison principle. Second, we discuss the relationships between positive equilibria and positive solutions of these systems over a large domain. Finally, we study the extinction and permanence of time-dependent positive solutions to these systems and obtain the conditions under which two species can coexist.Two-dimensional Müntz-Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations.https://zbmath.org/1449.652782021-01-08T12:24:00+00:00"Sabermahani, Sedigheh"https://zbmath.org/authors/?q=ai:sabermahani.sedigheh"Ordokhani, Yadollah"https://zbmath.org/authors/?q=ai:ordokhani.yadollah"Yousefi, Sohrab-Ali"https://zbmath.org/authors/?q=ai:yousefi.sohrab-aliSummary: In this manuscript, we present a new numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Müntz-Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann-Liouville operational matrix of 2D Müntz-Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works.About one method of obtaining of the exact analytical decision of the hyperbolic equation of heat conductivity on the basis of use of orthogonal methods.https://zbmath.org/1449.800052021-01-08T12:24:00+00:00"Kudinov, Vasiliĭ Aleksandrovich"https://zbmath.org/authors/?q=ai:kudinov.vasilii-aleksandrovich"Kudinov, Igor' Vasil'evich"https://zbmath.org/authors/?q=ai:kudinov.igor-vasilevichSummary: Using a method of division of variables and an orthogonal method of Bubnov-Galerkin the exact analytical solution of the hyperbolic equation of heat conductivity for an infinite plate under boundary conditions of the first sort is obtained. It is shown that having warmed up (or cooled) a body depends on the movement front of shock thermal waves on which there is a breakage temperature curve (temperature jump). The front of a thermal wave divides the investigated area on two subareas -- one where the temperature changes from the wall temperature (a boundary condition of the first sort) to the temperature at the front of waves, and one where the temperature is equal to the reference temperature.Weak Galerkin finite-element method for time-fractional nonlinear integro-differential equations.https://zbmath.org/1449.653252021-01-08T12:24:00+00:00"Wang, Haifeng"https://zbmath.org/authors/?q=ai:wang.haifeng"Xu, Da"https://zbmath.org/authors/?q=ai:xu.da"Guo, Jing"https://zbmath.org/authors/?q=ai:guo.jingSummary: In this article, a fully discrete scheme for one-dimensional (1D) time-fractional nonlinear integro-differential equation is established based on the weak Galerkin finite-element method. The stability and convergence of this scheme are proved. Several numerical experiments are presented to illustrate the theoretical analysis and to show the strong potential of this method.An alternating band parallel difference method for time fractional diffusion equation.https://zbmath.org/1449.652082021-01-08T12:24:00+00:00"Yang, Xiaozhong"https://zbmath.org/authors/?q=ai:yang.xiaozhong"Wu, Lifei"https://zbmath.org/authors/?q=ai:wu.lifeiSummary: The fractional anomalous diffusion equation has profound physical background and rich theoretical connotation, and its numerical methods are of important scientific significance and engineering application value. For the two-dimensional time fractional anomalous diffusion equation, an alternating band Crank-Nicolson difference parallel computing method (ABdC-N method) is studied in this paper. Based on the alternating segment technology, the ABdC-N scheme is constructed from the classic explicit scheme, implicit scheme and Crank-Nicolson difference scheme. It can be seen from both theoretical analyses and numerical experiments that the ABdC-N method is unconditionally stable and convergent. This method has good characteristics of parallel computing, and its computation efficiency is much higher than the classical serial differential method. Our results thus show that the ABdC-N difference method is effective for solving the two-dimensional time fractional anomalous diffusion equation.Coexistence solutions of the unstirred chemostat model with internal storage.https://zbmath.org/1449.352702021-01-08T12:24:00+00:00"Li, Xingxing"https://zbmath.org/authors/?q=ai:li.xingxing"Nie, Hua"https://zbmath.org/authors/?q=ai:nie.huaSummary: The existence of positive steady state solutions of an unstirred chemostat model with internal storage is studied. Due to the singularity arising from the ratio terms in this model, standard technique such as linearization and bifurcation can not be applied here. In order to overcome the mathematical difficulties caused by the singularity in the ratio terms, the sharp priori estimates for positive steady state solutions of the system are established firstly, which ensure that all positive steady state solutions of the system belong to a special cone. Secondly, some sufficient conditions for the existence of positive steady state solutions are given by using the principle eigenvalues of associated nonlinear eigenvalue problems and the degree theory in the special cone.Analytic solutions to the one dimensional time fractional Keller-Segel model.https://zbmath.org/1449.353812021-01-08T12:24:00+00:00"Hou, Jie"https://zbmath.org/authors/?q=ai:hou.jie"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhenSummary: In this paper, the generalized separation variable method, homogenous balanced principle and invariant subspace method are developed to study the generalized Burgers equation with fractional derivatives. With the help of the relation between the parabolic-elliptic Keller-Segel equations and Burgers equation, several exact solutions of the one-dimensional fractional Keller-Segel equations are established.Existence of positive solutions for a predator-prey model with cross-diffusion.https://zbmath.org/1449.350222021-01-08T12:24:00+00:00"Lv, Yang"https://zbmath.org/authors/?q=ai:lv.yang"Guo, Gaihui"https://zbmath.org/authors/?q=ai:guo.gaihui"Yuan, Hailong"https://zbmath.org/authors/?q=ai:yuan.hailong"Li, Shuxuan"https://zbmath.org/authors/?q=ai:li.shuxuanSummary: The existence of positive solutions to a predator-prey model with cross-diffusion is considered. First, we derive some estimates which are independent of cross-diffusion by the maximum principle. Second, the limiting behavior of positive solutions for large cross-diffusion is established. Finally, we show the existence of local bifurcation solutions of the limiting system near the semi-trivial solution by the local bifurcation theorem, and we extend the local bifurcation solutions to the global bifurcation solutions by the global bifurcation theorem, and we show that the global bifurcation solutions can extend to infinity as the bifurcation parameter approaches infinity. The results demonstrate that the two species can coexist when the cross-diffusion is large.An efficient multigrid method for semilinear elliptic equation.https://zbmath.org/1449.653472021-01-08T12:24:00+00:00"Xie, Hehu"https://zbmath.org/authors/?q=ai:xie.hehu"Xie, Manting"https://zbmath.org/authors/?q=ai:xie.manting"Zhang, Ning"https://zbmath.org/authors/?q=ai:zhang.ning.1|zhang.ning.2|zhang.ningSummary: A full multigrid method is proposed to solve the semilinear elliptic problem by the finite element method based on the combination of multilevel correction method and multigrid method for the linear elliptic problems. In the proposed method, solving the semilinear problem is decomposed into solutions of the linear elliptic problem by the multigrid method, and the semilinear problem is defined in a very low dimension space. With the help of parallel computing technique, the overfull efficiency can be improved clearly. Furthermore, when the nonlinear term is a polynomial function, an efficient full multigrid method is designed such that the asymptotically computational work is absolutely optimal. One numerical example is provided to validate the efficiency of the proposed method in this paper.Numerical examples for solving a class of nonlinear eigenvalue problems.https://zbmath.org/1449.653092021-01-08T12:24:00+00:00"Yang, Qingzhi"https://zbmath.org/authors/?q=ai:yang.qingzhi"Huang, Pengfei"https://zbmath.org/authors/?q=ai:huang.pengfei.1"Liu, Yajun"https://zbmath.org/authors/?q=ai:liu.yajunSummary: The Gross-Pitaevskii equation, which depicts the Bose-Einstein condensate state (BEC), is discretized by a differential method and transformed into a class of nonlinear eigenvalue problems (BEC problem). This article discusses the solution to the BEC problem and gives numerical examples. A bound of the minimum nonlinear eigenvalue of the BEC problem is calculated by the semidefinite programming relaxation method (SDP relaxation method) and the alternating direction multiplier method (ADMM). All real nonlinear eigenvalues of the BEC problem can be calculated sequentially by Lasserre semidefinite programming relaxation. In the numerical example, we compare the SDP relaxation method and ADMM from the scale of the problem and the speed of solving the problem. At the same time, we use the fmincon method that comes with Matlab to solve the problem and compare their results preliminarily.The multiscale algorithms for the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients in a bounded convex Lipschitz domain under the Weyl gauge.https://zbmath.org/1449.351602021-01-08T12:24:00+00:00"Fu, Yaoyao"https://zbmath.org/authors/?q=ai:fu.yaoyao"Cao, Liqun"https://zbmath.org/authors/?q=ai:cao.liqun"Ma, Chupeng"https://zbmath.org/authors/?q=ai:ma.chupengSummary: The Maxwell-Dirac system has wide applications in materials science such as topological insulators, graphene, superconductors and so on. In this paper, we first present the homogenization method and the multiscale asymptotic method for the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients in a bounded convex Lipschitz domain under the Weyl gauge. Based on the multiscale asymptotic expansions of the solution of the Maxwell-Dirac system, combining the time-splitting spectral method and the adaptive edge element method, we develop multiscale algorithms for solving the Maxwell-Dirac system with rapidly oscillating discontinuous coefficients and dealing with the multiscale problems with multiple time and space scales. Numerical examples are then carried out to validate the method presented in this paper.On the steady state bifurcation of the binary alloys system.https://zbmath.org/1449.350422021-01-08T12:24:00+00:00"Yan, Dongming"https://zbmath.org/authors/?q=ai:yan.dongmingSummary: In this paper, the steady state bifurcation of the binary alloys system is investigated. By using the nonlinear bifurcation theory, we prove that this system bifurcates from the trivial solution to the nontrivial solution branch when the parameter crosses certain critical value, which is related to the first eigenvalue of the relevant linear problem. In this case phase separation will occur in binary alloys. Furthermore, we obtain that lattice spacing and the shape of the domain are the key factors that affect the phase separation in binary alloys.Pohožaev identities of elliptic equations and systems with variable exponents and some applications.https://zbmath.org/1449.352282021-01-08T12:24:00+00:00"Feng, Tingfu"https://zbmath.org/authors/?q=ai:feng.tingfuSummary: In this paper, we obtain Pohožaev identities for the elliptic equation and system with variable exponent, respectively. Moreover, nonexistence of nontrivial solutions is derived under some suitable conditions, which extends the results in the literature.On the existence of multiple solutions for a three-point nonlinear boundary value problem of \(p\)-Laplacian type.https://zbmath.org/1449.352372021-01-08T12:24:00+00:00"Abaspour, S."https://zbmath.org/authors/?q=ai:abaspour.s"Khademloo, S."https://zbmath.org/authors/?q=ai:khademloo.somayeh|khademloo.somaye"Rasouli, S. H."https://zbmath.org/authors/?q=ai:rasouli.sayyed-hahsem|rasouli.sayyed-hasem|rasouli.sayyed-hashem|rasouli.sayyyed-h|rasouli.seyed-h|rasouli.sayed-hashemSummary: Using variational methods, we establish the existence of at least three solutions for a three-point nonlinear boundary value problem. Our technical approach is based on critical point theory.Jacobi collocation methods for solving generalized space-fractional Burgers' equations.https://zbmath.org/1449.652802021-01-08T12:24:00+00:00"Wu, Qingqing"https://zbmath.org/authors/?q=ai:wu.qingqing"Zeng, Xiaoyan"https://zbmath.org/authors/?q=ai:zeng.xiaoyanSummary: The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplified Jacobi operational matrix, we produce the differentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary differential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.Heteroclinic and traveling wave solutions for an SIR epidemic model with nonlocal response.https://zbmath.org/1449.341252021-01-08T12:24:00+00:00"Wang, Zongyi"https://zbmath.org/authors/?q=ai:wang.zongyiSummary: The existence of positive heteroclinic solutions is proved for a class of SIR epidemic model with nonlocal interaction and non-monotone property. Applying the theory of Fredholm operator decomposition and nonlinear perturbation, we study a connection between traveling wave solutions for the reaction-diffusion system and heteroclinic solutions of the associated differential equations. Existence and dynamics of wavefront profile are obtained as a consequence.The 3D compressible Euler equations with damping in the general unbounded domain.https://zbmath.org/1449.353462021-01-08T12:24:00+00:00"Yang, Jiaqi"https://zbmath.org/authors/?q=ai:yang.jiaqi"Yuan, Meng"https://zbmath.org/authors/?q=ai:yuan.mengSummary: In this paper, the authors consider the 3D damped compressible Euler equations in the general unbounded domain with slip boundary condition. The authors obtain the global existence and uniqueness when the initial data are near the equilibrium. Meanwhile, they also investigate the decay rates of the system in the half space. The authors show that the classical solution decays in the \({L^2}\)-norm to the constant background state at the rate of \( (1+t)^{-\frac{3}{4}}\).Fully diagonalized spectral methods for solving Neumann boundary value problems.https://zbmath.org/1449.653352021-01-08T12:24:00+00:00"Liu, Fujun"https://zbmath.org/authors/?q=ai:liu.fujun"Lu, Jing"https://zbmath.org/authors/?q=ai:lu.jingSummary: The fully diagonalized spectral methods using generalized Laguerre functions are proposed for the second-order elliptic problems with Neumann boundary conditions on the half line. Some Fourier-like Sobolev orthogonal basis functions are constructed for the diagonalized Laguerre spectral methods of Neumann boundary value problems. Numerical experiments demonstrate the effectiveness and the spectral accuracy.Multiple positive solutions for a inhomogeneous Neumann problem with critical weight Hardy-Sobolev exponent and boundary singularities.https://zbmath.org/1449.350242021-01-08T12:24:00+00:00"Shang, Yanying"https://zbmath.org/authors/?q=ai:shang.yanying"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.congSummary: In this paper, the authors study an inhomogeneous Neumann elliptic equation with boundary singularities. By Ekeland's variational principle, mountain pass lemma and some analysis technology, the existence of multiple positive solutions is established.The localization principle for formal Fourier series summarized by Gauss-Weierstrass method.https://zbmath.org/1449.352612021-01-08T12:24:00+00:00"Gorodets'kyĭ, V. V."https://zbmath.org/authors/?q=ai:gorodetskij.v-v|gorodetskyi.vasyl-v"Martynyuk, O. V."https://zbmath.org/authors/?q=ai:martynyuk.olga-vSummary: For formal series identifying linear continuous functionals given on the space of trigonometric polynomials and summarized by Gauss-Weierstrass methods, we prove an analog of the Riemann localization principle: if \(\{f_1, f_2\}\subset L_1 [0, 2\pi]\) are converge on the interval \( (a, b)\subset [0, 2\pi]\), then at each segment \([a + \varepsilon, b-\varepsilon]\subset (a, b)\) their difference of Fourier series uniformly converges to zero. Generally speaking, the principle of localization for Fourier series of \(2\pi\)-periodic generalized functions is not fulfilled. When studying various problems of mathematical physics and analysis, it is often used not the Fourier series itself, but the series summarized by one or another regular method, so it is natural to fulfill the principle of localization for such series. For example, the solution of the Dirichlet problem for the Laplace equation in a unit circle is represented by the Fourier series of the boundary function summarized by the Abel-Poisson method; the solution of the Cauchy periodic problem for the equation of thermal conductivity and the initial condition in the space of generalized periodic functions is treated as a formal Fourier series of the initial function summarized by the Gauss-Weierstrass method. The paper investigates multiple Fourier series of periodic hyperfunctions and ultradistributions.The symmetry of solutions for a class of Baouendi-Grushin equations.https://zbmath.org/1449.350162021-01-08T12:24:00+00:00"Qian, Hongli"https://zbmath.org/authors/?q=ai:qian.hongli"Huang, Xiaotao"https://zbmath.org/authors/?q=ai:huang.xiaotaoSummary: To study the symmetry for solutions of a class of Baouendi-Grushin equations, firstly, the symmetry is converted into a constrained minimization problem. Then, by Sobolev embedding theorem and some priori estimates, it is proved that the solutions after Schwarz rearrangement are also the solutions of its Lagrangian minimization functional. Thus the existence and symmetry results of the solutions of Baouendi-Grushin equations are obtained.Parallel finite-difference algorithms for three-dimensional space-fractional diffusion equation with \(\psi\)-Caputo derivatives.https://zbmath.org/1449.354302021-01-08T12:24:00+00:00"Bohaienko, V. O."https://zbmath.org/authors/?q=ai:bohaienko.v-oSummary: The paper deals with the issues of parallel computations' organization while solving three-dimensional space-fractional diffusion equation with the \(\psi\)-Caputo derivatives using finite difference schemes. For an implicit scheme and locally one-dimensional splitting scheme, we present parallel algorithms for distributed memory systems that use one-dimensional block and red-black data partitioning. To reduce the order of algorithms' computational complexity, we use an approach based on the expansion of integral operator's kernel into series. We present the theoretical estimates of parallel algorithms' performance and the results of computational experiments conducted on a testing problem that has an analytical solution for the case of the Caputo-Katugampola derivative. The results of the experiments show close-to-linear parallelization efficiency of one-dimensional splitting scheme with block partitioning and inefficiency of red-black partitioning in this case. For the implicit scheme, the scalability of parallel algorithms is weak and the use of red-black partitioning is more efficient than the use of block partitioning when running on a small number of computational resources.Parallel computation method of mixed difference schemes for time fractional reaction-diffusion equation.https://zbmath.org/1449.652972021-01-08T12:24:00+00:00"Dang, Xu"https://zbmath.org/authors/?q=ai:dang.xu"Yang, Xiaozhong"https://zbmath.org/authors/?q=ai:yang.xiaozhongSummary: The fractional reaction-diffusion equation has profound physical and engineering background, and its numerical methods are of great scientific significance and application value. A parallel computation method of mixed difference schemes for time fractional reaction-diffusion equation is proposed, and a class of alternative segment explicit-implicit scheme (ASE-I) and alternative segment implicit-explicit scheme (ASI-E) are constructed. This kind of parallel difference scheme is based on the effective combination of the Saul'yev asymmetric scheme, classical explicit difference scheme and classical implicit difference scheme. Theoretical analysis shows that the solution of ASE-I (ASI-E) scheme is uniquely solvable, unconditionally stable and convergent. Numerical experiments verify the theoretical analysis, which shows that the ASE-I scheme and the ASI-E scheme have ideal calculation accuracy and obvious parallel computing properties. It is proved that this kind of parallel difference method is effective for solving the time fractional reaction-diffusion equation.Structural stability on boundary reaction terms in a porous medium of Brinkman-Forchheimer type.https://zbmath.org/1449.350542021-01-08T12:24:00+00:00"Li, Yuanfei"https://zbmath.org/authors/?q=ai:li.yuanfei"Guo, Lianhong"https://zbmath.org/authors/?q=ai:guo.lianhongSummary: A saturated porous medium of Brinkman-Forchheimer type with an exothermic reaction occurring on the domain boundary is studied. The continuous dependence of the equation on the boundary reaction term and Soret coefficient is obtained by using energy estimation, differential inequality technique and the prior bounds of velocity, temperature and salt concentration.Liouville type theorem for an integral system with the poly-harmonic extension operator.https://zbmath.org/1449.351322021-01-08T12:24:00+00:00"Tang, Sufang"https://zbmath.org/authors/?q=ai:tang.sufangSummary: Liouville type theorem is considered for a system of integral equations with the poly-harmonic extension operator on the upper half Euclidean space in this paper. Under the natural structure conditions, we classify the positive solutions to the system based on the method of moving sphere in integral form and some integral inequality.The Riemann problem with delta initial data for Chaplygin nonsymmetric Keyfitz-Kranzer system with a source term.https://zbmath.org/1449.353382021-01-08T12:24:00+00:00"Song, Yun"https://zbmath.org/authors/?q=ai:song.yun"Guo, Lihui"https://zbmath.org/authors/?q=ai:guo.lihuiSummary: This paper is concerned with the Riemann problem with delta initial data for Chaplygin nonsymmetric Keyfitz-Kranzer system with a Coulomb-like friction term. It is interesting to see that the source term makes the Riemann solutions no longer self-similar. Delta contact discontinuities appear in some situations. Under generalized Rankine-Hugoniot conditions and entropy condition, we obtain the propagation speed, position and strength of delta shock wave. Furthermore, under delta initial data, stability of generalized solutions are obtained.Ground-state solutions for Schrödinger-Maxwell equations in the critical growth.https://zbmath.org/1449.353922021-01-08T12:24:00+00:00"Fang, Liwan"https://zbmath.org/authors/?q=ai:fang.liwan"Huang, Wennian"https://zbmath.org/authors/?q=ai:huang.wennian"Wang, Minqing"https://zbmath.org/authors/?q=ai:wang.minqingSummary: In this paper, we study the existence of the ground state solutions for the following Schrodinger-Maxwell equations \[\begin{cases}-\Delta u + V (x)u - (K (x)+\alpha)\phi u = \beta{|u|^4} u + b (x){|u|^{p-1}}u, & (x,u)\in (\mathbb{R}^3, \mathbb{R}), \\ \Delta \phi = (K (x)+\alpha){u^2}, & (x,u)\in (\mathbb{R}^3, \mathbb{R}),\end{cases}\] where \(\beta\) is a positive constant. Under some assumptions on \(V,K\) and \(b (x)\), by using the variational method and critical point theorem, we prove that such a class of equations has at least a ground state solution for \(\alpha < 0\) and \(p\in (3,4)\).A PDE approach to the existence of the critical value in time-periodic Hamiltonian system.https://zbmath.org/1449.351732021-01-08T12:24:00+00:00"Zhu, Haijiao"https://zbmath.org/authors/?q=ai:zhu.haijiao"Li, Xia"https://zbmath.org/authors/?q=ai:li.xiaSummary: We are devoted to prove that there exists a constant \(c\) such that \(u (x, t) - ct\) is bounded with PDE method, where \(u (x, t)\) is the viscosity solution of Hamilton-Jacobi equation \({u_t} + H (x, t, D_x u) = 0\), if the time 1-periodic Hamiltonian \(H (x, t, p)\) is continuous in \( (x, t, p)\), convex and coercive in \(p\), and linear in \(t\).Low Mach number limit of strong solutions to 3-D full Navier-Stokes equations with Dirichlet boundary condition.https://zbmath.org/1449.351702021-01-08T12:24:00+00:00"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.boling"Zeng, Lan"https://zbmath.org/authors/?q=ai:zeng.lan"Ni, Guoxi"https://zbmath.org/authors/?q=ai:ni.guoxiSummary: In this paper, we consider the low Mach number limit of the full compressible Navier-Stokes equations in a three-dimensional bounded domain where the velocity field and the temperature satisfy the Dirichlet boundary condition and the Neumann boundary condition, respectively. The uniform estimates in the Mach number for the strong solutions are derived in a short time interval, provided that the initial density and temperature are close to the constant states. Thus the solutions of the full compressible Navier-Stokes equations converge to the isentropic incompressible Navier-Stokes equations, as the Mach number tends to zero.New analytical and numerical results for fractional Bogoyavlensky-Konopelchenko equation arising in fluid dynamics.https://zbmath.org/1449.354452021-01-08T12:24:00+00:00"Kurt, Ali"https://zbmath.org/authors/?q=ai:kurt.aliSummary: In this article, \((2+1)\)-dimensional time fractional Bogoyavlensky-Konopelchenko (BK) equation is studied, which describes the interaction of wave propagating along the \(x\) axis and \(y\) axis. To acquire the exact solutions of BK equation we employed sub equation method that is predicated on Riccati equation, and for numerical solutions the residual power series method is implemented. Some graphical results that compares the numerical and analytical solutions are given for different values of \(\mu\). Also comparative table for the obtained solutions is presented.Propagation of traveling wave solutions for nonlinear evolution equation through the implementation of the extended modified direct algebraic method.https://zbmath.org/1449.351482021-01-08T12:24:00+00:00"Yaro, David"https://zbmath.org/authors/?q=ai:yaro.david"Seadawy, Aly"https://zbmath.org/authors/?q=ai:seadawy.aly-r"Lu, Dian-chen"https://zbmath.org/authors/?q=ai:lu.dianchenSummary: In this work, different kinds of traveling wave solutions and uncategorized soliton wave solutions are obtained in a three dimensional (3-D) nonlinear evolution equations (NEEs) through the implementation of the modified extended direct algebraic method. Bright-singular and dark-singular combo solitons, Jacobi's elliptic functions, Weierstrass elliptic functions, constant wave solutions and so on are attained beside their existing conditions. Physical interpretation of the solutions to the 3-D modified KdV-Zakharov-Kuznetsov equation are also given.The solitary wave solution in quantum plasma model.https://zbmath.org/1449.351522021-01-08T12:24:00+00:00"Feng, Yihu"https://zbmath.org/authors/?q=ai:feng.yihu"Wang, Weigang"https://zbmath.org/authors/?q=ai:wang.weigang"Mo, Jiaqi"https://zbmath.org/authors/?q=ai:mo.jiaqiSummary: The quantum plasma system is discussed. A nonlinear dynamic disturbed model is studied. Using the undetermined coefficients method for the hyperbolic functions, perturbation theory and method, the solitary wave solution of the corresponding model is solved. The characteristics of the corresponding physical quantity are expounded.Initial boundary value problem for higher-order nonlinear Kirchhoff-type equation.https://zbmath.org/1449.351842021-01-08T12:24:00+00:00"Ye, Yaojun"https://zbmath.org/authors/?q=ai:ye.yaojun"Tao, Xiangxing"https://zbmath.org/authors/?q=ai:tao.xiangxingSummary: In this paper, the initial boundary value problem for some nonlinear higher-order Kirchhoff-type equations with damping and source terms in a bounded domain is studied. We prove the existence of global solutions for this problem by constructing a stable set and establish the energy decay estimate by applying a difference inequality. Meanwhile, under the condition of the positive initial energy, it is shown that the solution blows up in finite time and the lifespan estimate of solution is also given.A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers' equation.https://zbmath.org/1449.653002021-01-08T12:24:00+00:00"Pan, Kejia"https://zbmath.org/authors/?q=ai:pan.kejia"Wu, Xiaoxin"https://zbmath.org/authors/?q=ai:wu.xiaoxin"Yue, Xiaoqiang"https://zbmath.org/authors/?q=ai:yue.xiaoqiang"Ni, Runxin"https://zbmath.org/authors/?q=ai:ni.runxinSummary: In this paper, a high-order compact difference scheme is proposed for solving multidimensional nonlinear Burgers' equation. The three-stage third-order total variation diminishing (TVD) Runge-Kutta scheme is employed in time, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. The proposed TVD-CCD method is free of using Hopf-Cole transformation, and treats the nonlinear term explicitly. Thus it is very efficient and easy to implement. Our method is effective to capture shock wave, third-order accurate in time, and sixth-order accurate in space. In addition, we show the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments are given to demonstrate the high efficiency and accuracy of the proposed method.Global dissipative solutions of the Dullin-Gottwald-Holm equation with a forcing.https://zbmath.org/1449.352852021-01-08T12:24:00+00:00"Li, Bin"https://zbmath.org/authors/?q=ai:li.bin.1|li.bin"Zhu, Shihui"https://zbmath.org/authors/?q=ai:zhu.shihuiSummary: Based on the new characteristic method, by exploiting the balance law and some estimates, we prove the existence of global dissipative solutions for the Dullin-Gottwald-Holm equation with a forcing term in \({H^1} (\mathbb{R})\).On the existence results for \((p,q)\)-Kirchhoff type systems with multiple parameters.https://zbmath.org/1449.352142021-01-08T12:24:00+00:00"Shakeri, Saleh"https://zbmath.org/authors/?q=ai:shakeri.saleh"Hadjian, Armin"https://zbmath.org/authors/?q=ai:hadjian.arminSummary: In this paper, we are interested in the existence of positive solutions for the following nonlocal \(p\)-Kirchhoff problem of the type
\[ \begin{cases}
-M_1(\int_\Omega|\nabla u|^p\,dx)\Delta_pu=\lambda a(x)v^\alpha-\mu&\text{ in }\;\Omega, \\
-M_2(\int_\Omega|\nabla v|^q\,dx)\Delta_qv=\lambda b(x)u^\beta-\mu& \text{ in }\;\Omega, \\
u=v=0&\text{ on }\; \partial\Omega. \end{cases} \]
where \(\Omega\) is a bounded smooth domain of \(\mathbb{R}^N\), \(p,q > 1\), \(0<\alpha < p-1\), \(0 <\beta<q-1\), \(M_i:\mathbb{R}_0^+\to\mathbb{R}^+\), \(i=1,2\), are two continuous and increasing functions, \(\lambda\), \(\mu\) are two positive parameters, and \(a,b\in C(\overline{\Omega})\).Blow-up phenomenon for a coupled diffusion system with exponential reaction terms and space-dependent coefficients.https://zbmath.org/1449.351232021-01-08T12:24:00+00:00"Ma, Danni"https://zbmath.org/authors/?q=ai:ma.danni"Fang, Zhongbo"https://zbmath.org/authors/?q=ai:fang.zhongboSummary: Blow-up phenomena for the Dirichlet initial boundary value problem of a coupled diffusion system with exponential reaction terms and space-dependent coefficients is considered. By virtue of the Bernoulli equation, the method of super-and-sub solutions and the modified differential inequality techniques, we find the influence of space-dependent coefficients on the existence of global solution or blow-up solution at finite time. Moreover, upper and lower bounds for the blow-up time of the solution are derived under different measures in whole dimensional spaces \( (N \ge 1)\).Decay estimates of the global solution for the Landau-Lifshitz-Gilbert equation in three dimensions.https://zbmath.org/1449.352552021-01-08T12:24:00+00:00"Lin, Junyu"https://zbmath.org/authors/?q=ai:lin.junyu"Wan, Minglian"https://zbmath.org/authors/?q=ai:wan.minglian"Cao, Muhua"https://zbmath.org/authors/?q=ai:cao.muhuaSummary: In this paper, the authors consider the Cauchy problem for the Landau-Lifshitz-Gilbert equation in \({\mathbb{R}^3}\). By energy method and standard continuity argument, the authors obtain the global existence and uniqueness of smooth solution under suitable small initial data firstly. After establishing a monotone inequality for this solution, the authors build the time decay rates for this solution by the Fourier splitting method.Pointwise estimates for systems of wave equations with viscosity.https://zbmath.org/1449.350842021-01-08T12:24:00+00:00"Wu, Zhigang"https://zbmath.org/authors/?q=ai:wu.zhigang"Miao, Xiaofang"https://zbmath.org/authors/?q=ai:miao.xiaofangSummary: The Cauchy problem for two systems of wave equations with viscosity in dimension three is considered. By using the long wave and short wave decomposition method together with energy method and Green function, the pointwise estimates of the time-asymptotic shape of the solution are given, which exhibit two kinds of generalized Huygens' waves. As a byproduct, the optimal \({L^p}\)-decay rates with \(p \ge 1\) of the solutions of these systems are also established.Lorentz estimates for nondivergence parabolic equations with partially BMO coefficients.https://zbmath.org/1449.352482021-01-08T12:24:00+00:00"Zhang, Junjie"https://zbmath.org/authors/?q=ai:zhang.junjie"Zheng, Shenzhou"https://zbmath.org/authors/?q=ai:zheng.shenzhou"Yu, Haiyan"https://zbmath.org/authors/?q=ai:yu.haiyanSummary: In this paper, we prove an interior Lorentz estimate for Hessian of the strong solutions to nondivergence linear parabolic equations \({u_t} - {a_{ij}} (x, t){D_{ij}}u (x, t) = f (x, t)\). Here, the leading coefficients \({a_{ij}} (x, t)\) are assumed to be merely measurable in one spatial variable and have small BMO semi-norms with respect to the remaining variables.An indirect finite element method for variable-coefficient space-fractional diffusion equations and its optimal-order error estimates.https://zbmath.org/1449.653312021-01-08T12:24:00+00:00"Zheng, Xiangcheng"https://zbmath.org/authors/?q=ai:zheng.xiangcheng"Ervin, V. J."https://zbmath.org/authors/?q=ai:ervin.vincent-j"Wang, Hong"https://zbmath.org/authors/?q=ai:wang.hong.1Summary: We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions \(u(x)\) to the proposed variable coefficient models in terms of \(v(x)\), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations \(v_h(x)\) to \(v(x)\) and then obtain the approximations \(u_h(x)\) of \(u(x)\) by plugging \(v_h(x)\) into the representation of \(u(x)\). Optimal-order convergence estimates of \(u(x)-u_h(x)\) are proved in both \(L^2\) and \(H^{\alpha /2}\) norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.A finite-difference approximation for the one- and two-dimensional tempered fractional Laplacian.https://zbmath.org/1449.354532021-01-08T12:24:00+00:00"Yan, Yaoqiang"https://zbmath.org/authors/?q=ai:yan.yaoqiang"Deng, Weihua"https://zbmath.org/authors/?q=ai:deng.weihua"Nie, Daxin"https://zbmath.org/authors/?q=ai:nie.daxinSummary: This paper provides a finite-difference discretization for the one- and two-dimensional tempered fractional Laplacian and solves the tempered fractional Poisson equation with homogeneous Dirichlet boundary conditions. The main ideas are to, respectively, use linear and quadratic interpolations to approximate the singularity and non-singularity of the one-dimensional tempered fractional Laplacian and bilinear and biquadratic interpolations to the two-dimensional tempered fractional Laplacian. Then, we give the truncation errors and prove the convergence. Numerical experiments verify the convergence rates of the order \(O(h^{2-2s})\).A high-accuracy analysis of unconventional Hermite-type rectangular element for nonlinear parabolic integro-differential equations.https://zbmath.org/1449.652512021-01-08T12:24:00+00:00"Li, Xianzhi"https://zbmath.org/authors/?q=ai:li.xianzhi"Fan, Zhongguang"https://zbmath.org/authors/?q=ai:fan.zhongguangSummary: An unconventional Hermite-type rectangular element approximation is discussed for a class of nonlinear parabolic integro-differential equations under a semi-discrete scheme. The superclose property with order \(O (h^3)\) in \({H^1}\) norm is obtained by means of the interpolation theory, a high-accuracy analysis and the derivative transfer techniques for the time \(t\). Furthermore, the global superconvergence result is derived with the interpolated post-processing technique. At the same time, the high-accuracy extrapolation solution with order \(O (h^4)\) is deduced through constructing a suitable extrapolation scheme.Numerical algorithm for the time-Caputo and space-Riesz fractional diffusion equation.https://zbmath.org/1449.652122021-01-08T12:24:00+00:00"Zhang, Yuxin"https://zbmath.org/authors/?q=ai:zhang.yuxin"Ding, Hengfei"https://zbmath.org/authors/?q=ai:ding.hengfeiSummary: In this paper, we develop a novel finite-difference scheme for the time-Caputo and space-Riesz fractional diffusion equation with convergence order \(\mathcal{O}(\tau^{2-\alpha}+h^2)\). The stability and convergence of the scheme are analyzed by mathematical induction. Moreover, some numerical results are provided to verify the effectiveness of the developed difference scheme.A high order formula to approximate the Caputo fractional derivative.https://zbmath.org/1449.651952021-01-08T12:24:00+00:00"Mokhtari, R."https://zbmath.org/authors/?q=ai:mokhtari.reza"Mostajeran, F."https://zbmath.org/authors/?q=ai:mostajeran.fSummary: We present here a high-order numerical formula for approximating the Caputo fractional derivative of order \(\alpha\) for \(0<\alpha<1\). This new formula is on the basis of the third degree Lagrange interpolating polynomial and may be used as a powerful tool in solving some kinds of fractional ordinary/partial differential equations. In comparison with the previous formulae, the main superiority of the new formula is its order of accuracy which is \(4-\alpha\), while the order of accuracy of the previous ones is less than 3. It must be pointed out that the proposed formula and other existing formulae have almost the same computational cost. The effectiveness and the applicability of the proposed formula are investigated by testing three distinct numerical examples. Moreover, an application of the new formula in solving some fractional partial differential equations is presented by constructing a finite difference scheme. A PDE-based image denoising approach is proposed to demonstrate the performance of the proposed scheme.The solvability of dual Minkowski problem in \(\mathbb{R}^2\).https://zbmath.org/1449.351982021-01-08T12:24:00+00:00"Wei, Na"https://zbmath.org/authors/?q=ai:wei.naSummary: In this paper, we study the existence of minimum of a constrained variational problem in the Sobolev space \(W^{1,4} (\mathbb{S})\). If \(\int_\mathbb{S} g (\theta)\,d\theta > 0\), the minimum is a positive solution to the related Euler-Lagrange equation \[u'' + u = \frac{g (\theta)}{u (u^2 + u'^2)}, \quad\theta \in \mathbb{S}.\]
Based on this, we prove the solvability of the dual Minkowski problem in \(\mathbb{R}^2\).The generalized Riemann problem for chromatography equations with delta shock wave.https://zbmath.org/1449.353372021-01-08T12:24:00+00:00"Pan, Lijun"https://zbmath.org/authors/?q=ai:pan.lijun"Han, Xinli"https://zbmath.org/authors/?q=ai:han.xinli"Li, Tong"https://zbmath.org/authors/?q=ai:li.tongSummary: This paper is concerned with the generalized Riemann problem for the nonlinear chromatography equations, where the delta shock wave occurs in the corresponding Riemann solution. It is quite different from the previous generalized Riemann problems which focus on classical elementary waves. We constructively solve the generalized Riemann problem in a neighborhood of the origin on the \(x-t\) plane. In solutions, we find that the generalized Riemann solutions have a structure similar to the solution of the corresponding Riemann problem for most of cases. However, a delta shock wave in the corresponding Riemann solution may turn into a shock wave followed by a contact discontinuity, which provides us with a detailed method for analyzing the internal mechanism of a delta shock wave.Gradient flow method in nonlinear optical lattices.https://zbmath.org/1449.354022021-01-08T12:24:00+00:00"Zhang, Ruifeng"https://zbmath.org/authors/?q=ai:zhang.ruifeng"Liu, Nan"https://zbmath.org/authors/?q=ai:liu.nanSummary: In this paper, we study the existence of the steady state solutions for a classical Schrödinger equation in nonlinear optical lattices by means of gradient flow method. We first establish the existence of a global solution of the governing parabolic equation. Then we prove the convergence of the global solution to an equilibrium (i.e., a steady state solution in optical lattices model) as time goes to infinity. Furthermore, we provide an estimate on the convergence rate by using the Lojasiewicz-Simon inequality.Convergence analysis of Wilson element for parabolic integro-differential equation.https://zbmath.org/1449.653172021-01-08T12:24:00+00:00"Liang, Conggang"https://zbmath.org/authors/?q=ai:liang.conggang"Yang, Xiaoxia"https://zbmath.org/authors/?q=ai:yang.xiaoxia"Shi, Dongyang"https://zbmath.org/authors/?q=ai:shi.dongyangSummary: In this paper, with the help of the Wilson element, new semi-discrete and fully-discrete schemes are proposed for parabolic integro-differential equation. Based on the properties of the element, through defining a new bilinear form, without using the technique of extrapolation and interpolated postprocessing, in the norm which is stronger than the usual \({H^1}\)-norm, the convergence results with order \(O (h^2)/O (h^2 + \tau)\) for the primitive solution are obtained for the corresponding schemes, respectively. The above results are just one order higher than the usual error estimates for the Wilson element. Here, \(h\) and \(\tau\) are parameters of the subdivision in space and time step, respectively. Finally, numerical results are provided to confirm the theoretical analysis.Lifespan estimation of solutions to Cauchy problem of semilinear wave equation.https://zbmath.org/1449.353072021-01-08T12:24:00+00:00"Jiang, Hongbiao"https://zbmath.org/authors/?q=ai:jiang.hongbiao"Wang, Haihang"https://zbmath.org/authors/?q=ai:wang.haihangSummary: In this paper, the lifespan estimate to the Cauchy problem of the semi-linear wave equation
\[ u_{tt} - \Delta u = (1+|x|^2)^\alpha|u|^p\quad\text{in }\mathbb{R}^n\]
is studied. The upper bound of the lifespan is improved for the cases \(n = 2\), \(1 < p \le 2\) and \(n = 1\), \(p > 1\), by using the improved Kato's type lemma.Regularity criteria for the NS and MHD equations in terms of horizontal components.https://zbmath.org/1449.351392021-01-08T12:24:00+00:00"Zhang, Hui"https://zbmath.org/authors/?q=ai:zhang.hui.9|zhang.hui.7|zhang.hui.3|zhang.hui.5|zhang.hui|zhang.hui.2|zhang.hui.6|zhang.hui.8|zhang.hui.1|zhang.hui.11|zhang.hui.10|zhang.hui.4"Xu, Juan"https://zbmath.org/authors/?q=ai:xu.juan.1|xu.juanSummary: In this paper, we consider the regularity of weak solutions to the incompressible Navier-Stokes (NS) equations and MHD equations in the Triebel-Lizorkin space and multiplier space respectively. By using Littlewood-Paley decomposition and energy estimate methods, we proved that if horizontal velocity \(\tilde u = ({u_1}, {u_2}, 0)\) satisfies \[{\nabla_h}\tilde u \in {L^p} (0,T;\dot F_{q, \frac{{2q}}{3}}^0 (\mathbb{R}^3)),\;\;\; \frac{2}{p} + \frac{3}{q} = 0,\;\;\; \frac{3}{2} < q \le \infty,\] then the weak solution is actually the unique strong solution on \([0,T)\). For MHD equations, we prove that if horizontal velocity and magnetic field satisfy \[ (\tilde u, \tilde b) \in L^{\frac{2}{1-r}} (0,T;{\dot X}_r), r \in [0,1),\] or horizontal gradient satisfies \[ ({\nabla_h}\tilde u, {\nabla_h}\tilde b) \in L^{\frac{2}{2-r}} (0, T;{\dot X}_r), r \in [0,1),\] then the weak solution is actually unique strong solution on \([0,T)\).Global regularity for 3D generalized Oldroyd-B type models with fractional dissipation.https://zbmath.org/1449.351402021-01-08T12:24:00+00:00"Zhang, Qiuyue"https://zbmath.org/authors/?q=ai:zhang.qiuyueSummary: In this paper, we consider the 3D generalized Oldroyd-B type models with fractional Laplacian dissipation \( (-\Delta)^{\eta_1}u\) and \( (-\Delta)^{\eta_2}\tau\) in the corotational case. By using energy method, for \({\eta_1} \ge \frac{5}{4}\) and \({\eta_2} \ge \frac{5}{4}\), we obtain the global regularity of classical solutions when the initial data (\({u_0}, {\tau_0}\)) are sufficiently smooth.Homogenization of the Neumann boundary value problem: the sharper \(W^{1,p}\) estimate.https://zbmath.org/1449.350352021-01-08T12:24:00+00:00"Wang, Juan"https://zbmath.org/authors/?q=ai:wang.juan"Zhao, Jie"https://zbmath.org/authors/?q=ai:zhao.jieSummary: In this paper, we shall strengthen our results on the \(W^{1, p}\) convergence rates for homogenization problems of solutions of partial differential equations with rapidly oscillating Neumann boundary data. Such a problem raised due to its importance for higher order approximation in homogenization theory, which gives rise to the so-called boundary layer phenomenon. Our techniques are based on integral representation of the solutions as well as analysis of oscillatory integrals, in conjunction with Fourier expansion of the oscillating periodic function.The global solution and asymptotic behavior of parabolic-parabolic Keller-Segel type model.https://zbmath.org/1449.350822021-01-08T12:24:00+00:00"Wu, Jie"https://zbmath.org/authors/?q=ai:wu.jie.4|wu.jie.6|wu.jie.3|wu.jie.1|wu.jie.2|wu.jie.5|wu.jie"Lin, Hongxia"https://zbmath.org/authors/?q=ai:lin.hongxiaSummary: This paper concerns the parabolic-parabolic Keller-Segel type model. By making use of the Neumann heat semigroup, the asymptotic inequality on the gradient of \(\rho\) depending on the chemotaxis signal \(\chi\) is derived. Meanwhile, the convergence results on \(\| \rho (\cdot, t)\|_{L^1 (\Omega)}, \|n (\cdot, t)\|_{C^\theta (\bar{\Omega})}\) and \(\|c (\cdot, t)\|_{L^\infty (\Omega)}\) have also been obtained. It is revealed that mass of sperm will tend to the initial difference between sperm and eggs mass in the process of evolution, eggs are all fertilized and the concentration of the chemical substance will be also exhausted eventually. At the same time, the depletion of chemical substance is accompanied by complete fertilization of eggs. It is illustrated that concentration of the chemical substance plays a relevant role in the fertilization process of corals.Existence of the second positive solution for a class of nonhomogeneous Kirchhoff type problems with critical exponent.https://zbmath.org/1449.350212021-01-08T12:24:00+00:00"Ji, Lei"https://zbmath.org/authors/?q=ai:ji.lei"Liao, Jiafeng"https://zbmath.org/authors/?q=ai:liao.jiafengSummary: The following nonhomogeneous Kirchhoff type equations with critical exponent \[\begin{cases}- (a+b\int_\Omega|\nabla u|^2 \,dx) \Delta u = |u|^4 u + \lambda f (x), & x \in \Omega, \\ u = 0, & x \in \partial\Omega,\end{cases}\] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^3\), \(a, b, \lambda > 0\) are parameters and \(f\in L^{\frac{6}{5}} (\Omega)\) is nonzero and nonnegative, are considered. By the variational method, the second positive solution is obtained which completes and improves the results in the literature.Three types of solutions for a class of nonlinear Schrödinger equations.https://zbmath.org/1449.353962021-01-08T12:24:00+00:00"Mei, Yanfang"https://zbmath.org/authors/?q=ai:mei.yanfang"Wang, Youjun"https://zbmath.org/authors/?q=ai:wang.youjunSummary: In this paper, the existence of ground state, oscillation solution and soliton solution of a class of nonlinear Schrödinger equations in plasma are considered.Weak solutions for the systems of multifluid flows.https://zbmath.org/1449.351672021-01-08T12:24:00+00:00"Liu, Shujun"https://zbmath.org/authors/?q=ai:liu.shujunSummary: In this paper, we study the weak solutions for the systems of multifluid flows, which include the system of isentropic gas dynamics in Eulerian coordinates and a system arising from river flows. There are more linearly degenerate fields compared with single-component system, and singularities in these linearly degenerate fields emerge when considering the corresponding vanishing viscosity system. we obtain the existence of global solutions for the system of multifluid flows by analyzing the uniform BV estimates in linearly degenerate fields, coupled with the compensated compactness method and the vanishing viscosity method.New exact periodic solitary wave solutions for the \( (3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation.https://zbmath.org/1449.350282021-01-08T12:24:00+00:00"Li, Ying"https://zbmath.org/authors/?q=ai:li.ying.2|li.ying|li.ying.1|li.ying.3"Liu, Jianguo"https://zbmath.org/authors/?q=ai:liu.jian-guo|liu.jian-guo.1"Yang, Lianwu"https://zbmath.org/authors/?q=ai:yang.lianwuSummary: In this paper, we investigate the generalized Kadomtsev-Petviashvili equation for the evolution of nonlinear, long waves of small amplitude with slow dependence on the transverse coordinate. By virtue of the Hirota's bilinear form and the extended homoclinic test approach, new exact periodic solitary wave solutions for the \( (3+1)\)-dimensional generalized Kadomtsev-Petviashvili equation are obtained, which are different from those in previous literatures. With the aid of symbolic computation, the properties and characteristics for these new exact periodic wave solutions are presented with some figures.Application of the generalized integral Laplace transform to solving differential equations.https://zbmath.org/1449.350102021-01-08T12:24:00+00:00"Zaikina, Svetlana Mikhaĭlovna"https://zbmath.org/authors/?q=ai:zaikina.svetlana-mikhailovnaSummary: We consider some boundary value problems for parabolic equations. The solutions were obtained with the generalized Laplace integral transform.Some methods for determining the lower bound of blow-up time in a parabolic problem and effectiveness analysis.https://zbmath.org/1449.351252021-01-08T12:24:00+00:00"Qin, Siqian"https://zbmath.org/authors/?q=ai:qin.siqian"Ling, Zhengqiu"https://zbmath.org/authors/?q=ai:ling.zhengqiu"Zhou, Zewen"https://zbmath.org/authors/?q=ai:zhou.zewenSummary: In this paper, we consider the blow-up phenomenon to a type of Newtonian filtration equation with variable source subject to homogeneous Neumann boundary condition. We give two methods to determine the lower bound for blow-up time of solution in \(\Omega \subset {\mathbb{R}^3}\) if the solutions blow up by energy estimation method and differential inequality technique. Moreover, the effectiveness of these methods is also discussed.Convergence analysis of the lowest nonconforming mixed finite element for nonlinear Sobolev-Galpern type equations of moisture migration.https://zbmath.org/1449.653302021-01-08T12:24:00+00:00"Zhang, Houchao"https://zbmath.org/authors/?q=ai:zhang.houchao"Wang, An"https://zbmath.org/authors/?q=ai:wang.anSummary: Based on the nonconforming linear triangular finite element, the lowest nonconforming mixed finite element approximate scheme is established for nonlinear Sobolev-Galpern type equations of moisture migration. The existence and uniqueness of approximation solution are proved. At the same time, without the conventional Ritz projection, the optimal error estimates of exact solution \(u\) in \({H^1}\)-norm and intermediate variable \(\boldsymbol{P} = - (a (u)\nabla {u_t} + b (u)\nabla u)\) in \({L^2}\)-norm are deduced by some special properties of the elements.The explicit solution of fifth order KdV equation.https://zbmath.org/1449.353872021-01-08T12:24:00+00:00"Wang, Hongwei"https://zbmath.org/authors/?q=ai:wang.hongwei.1|wang.hongwei.2|wang.hongwei"Li, Tingxian"https://zbmath.org/authors/?q=ai:li.tingxian"Yuan, Wei"https://zbmath.org/authors/?q=ai:yuan.weiSummary: In this paper, we study the initial boundary value problem of fifth order KdV equation on the half-line. By the unified transform method of Fokas, the explicit solution is obtained with deforming the proper integral contour, Green formula and Jordan's Lemma. The conclusions obtained will lay a theoretical foundation for the study of the well-posedness of this equation.Stochastic fractional non-autonomous Ginzburg-Landau equations with multiplicative noise in weighted space.https://zbmath.org/1449.354662021-01-08T12:24:00+00:00"Wang, Yunxiao"https://zbmath.org/authors/?q=ai:wang.yunxiao"Shu, Ji"https://zbmath.org/authors/?q=ai:shu.ji"Yang, Yuan"https://zbmath.org/authors/?q=ai:yang.yuan"Li, Qian"https://zbmath.org/authors/?q=ai:li.qian"Wang, Chunjiang"https://zbmath.org/authors/?q=ai:wang.chunjiangSummary: In this paper, we consider the asymptotic dynamic for random attractors of stochastic fractional non-autonomous Ginzburg-Landau equations with multiplicative noise in \(L_\rho^2 (\text bf{R}^n)\). Firstly, we transform the partial differential equation into the random equation that only induces the random parameters. Then, using estimates for far-field values of solutions and a cut-off technique, asymptotic compactness is proved. At last, the existence of a random attractor in \(L_\rho^2(\text bf{R}^n)\) for the random dynamical system is established.Existence of nontrivial solutions for a class of nonlinear equations.https://zbmath.org/1449.351802021-01-08T12:24:00+00:00"Yang, Wenping"https://zbmath.org/authors/?q=ai:yang.wenping"Chen, Zhihui"https://zbmath.org/authors/?q=ai:chen.zhihuiSummary: In this paper, the nonlinear problem is transformed into a semilinear problem by a variable transformation, and the original problem discussed in Orlicz space is put into Sobolev space for discussion. Through the improved AR condition and the application of mountain pass lemma, it is proved that there is a nontrivial solution to this problem.Global attractor family and its dimension estimation for higher-order Kirchhoff type equation with strong damping.https://zbmath.org/1449.350942021-01-08T12:24:00+00:00"Lin, Guoguang"https://zbmath.org/authors/?q=ai:lin.guoguang"Guan, Liping"https://zbmath.org/authors/?q=ai:guan.lipingSummary: The initial-boundary value problem for higher-order Kirchhoff equation with strong nonlinear damping term is studied. Under the appropriate assumption of Kirchhoff stress term and second-order nonlinear source term, the existence and uniqueness of global solutions are obtained by using prior estimation and Galerkin method, and the bounded absorption set and solution semigroup are constructed by prior estimation. The existence of the global attractor family is proved by its complete continuity. Secondly, the Hausdorff dimension and the fractal dimension of the global attractor family are estimated by linearization equation and Frechet differentiability of the solution semigroup.Existence of infinitely many solutions for a class of Kirchhoff-type problems.https://zbmath.org/1449.351992021-01-08T12:24:00+00:00"Ye, Hongyan"https://zbmath.org/authors/?q=ai:ye.hongyan"Suo, Hongmin"https://zbmath.org/authors/?q=ai:suo.hong-min"An, Yucheng"https://zbmath.org/authors/?q=ai:an.yuchengSummary: In this paper, we use a variational method and a truncation technique to study a class of Kirchhoff-type equations for Neumann boundary value problems. Firstly, the equivalence condition of trivial solutions is obtained by the definition of energy functionals and solutions corresponding to the equations. Secondly, the condition of compactness is proved by the odd hypothesis of the nonlinear term. Finally, the existence of infinite solutions is obtained based on spatial decomposition, and the corresponding energy functionals have zero gradation.Ground state solutions of an elliptic problem with nonlocal operators.https://zbmath.org/1449.354462021-01-08T12:24:00+00:00"Liu, Jian"https://zbmath.org/authors/?q=ai:liu.jian.5|liu.jian.6|liu.jian"Wang, Jianing"https://zbmath.org/authors/?q=ai:wang.jianingSummary: In this paper, we study the following elliptic problems with nonlocal operators \[\begin{cases}\mathcal{L}_K u = f (x, u), & {\mathrm{in}}\;\Omega, \\ u = 0, & {\mathrm{on}}\;{\mathbb{R}^N}\backslash \Omega, \end{cases}\] where \(\Omega \subset {\mathbb{R}^N} (N > ps)\) is an open bounded set with Lipschitz boundary, \(s \in (0, 1), 1 < p < \frac{N}{s}\), the nonlocal operator \({\mathcal{L}_K}\) is defined by
\[{\mathcal{L}_K}u (x) = 2{\mathrm{P}}.{\mathrm{V}}. \int_{\mathbb{R}^N} {|u (x) - u (y)|^{p-2}} (u (x) - u (y))K (x - y)\,dy.\]
\(f (x, u)\) is asymptotically linear with respect to \({u^{p-1}}\) at infinity. By using the variational methods and mountain pass lemma, we obtain that there exists at least one ground state positive solution for the above elliptic problem with nonlocal operators.Existence of mild solutions for a class of fractional stochastic evolution equations with nonlocal initial conditions.https://zbmath.org/1449.354332021-01-08T12:24:00+00:00"Chen, Pengyu"https://zbmath.org/authors/?q=ai:chen.pengyu"Ma, Weifeng"https://zbmath.org/authors/?q=ai:ma.weifeng"Ahmed, Abdelmonem"https://zbmath.org/authors/?q=ai:ahmed.abdelmonemSummary: This paper obtains the existence results of mild solutions to a class of fractional stochastic evolution equations with nonlocal conditions by applying stochastic analysis theory, Schauder fixed point theorem and approximation method, and assuming that the nonlinear term is Caretheodory continuous and satisfies some weak growth condition, the nonlocal term depends on all the value of independent variable on the whole interval and satisfies some weak growth condition. This work may be viewed as an attempt to develop a general existence theory for fractional stochastic evolution equations with general nonlocal conditions. Finally, as a sample of application, the results are applied to a fractional stochastic partial differential equation with nonlocal integral condition.Neumann's problem for one equation fourth order.https://zbmath.org/1449.351792021-01-08T12:24:00+00:00"Utkina, E. A."https://zbmath.org/authors/?q=ai:utkina.e-aSummary: Neumann's problem for pseudoparabolic equation of the 4th order in a rectangular domain is considered in the paper. The existence and uniqueness of solution are proved.Nonconstant periodic solutions of discrete \(p\)-Laplacian system via Clark duality and computations of the critical groups.https://zbmath.org/1449.390112021-01-08T12:24:00+00:00"Zheng, Bo"https://zbmath.org/authors/?q=ai:zheng.bo|zheng.bo.1Summary: We study the existence of periodic solutions to a discrete \(p\)-Laplacian system. By using the Clark duality method and computing the critical groups, we find a simple condition that is sufficient to ensure the existence of nonconstant periodic solutions to the system.Dynamic behavior for non-autonomous stochastic semi-linear degenerate parabolic equations.https://zbmath.org/1449.351092021-01-08T12:24:00+00:00"Yang, Xiao"https://zbmath.org/authors/?q=ai:yang.xiao"Li, Xiaojun"https://zbmath.org/authors/?q=ai:li.xiaojunSummary: In this paper, we study the existence of random attractors for stochastic semi-linear degenerate parabolic equations, where the nonlinear term satisfies the arbitrary growth order, and the stochastic term is driven by a Wiener multiplicative noise. By estimating the solutions of the transformed system, we get the existence of the asymptotically \(\mathcal{D}\)-pullback absorbing set and then get the existence of the random attractor.The second order semi-implicit asymmetric iteration scheme for solving one-dimensional fractional convection diffusion equations.https://zbmath.org/1449.652172021-01-08T12:24:00+00:00"Zhu, Lin"https://zbmath.org/authors/?q=ai:zhu.linSummary: In this paper, we construct a second order semi-implicit finite difference scheme for solving one-dimensional fractional convection diffusion equations combining with the asymmetric iteration technique. The second order weighted and shifted Grünwald-Letnikov operator is used to discretize the Riemann-Liouville fractional derivative and the central-difference operator is used to discretize the convection term. The presented scheme is formally implicit, but it can be computed explicitly by choosing unknowns in different nodal-point sequences at the odd time level and the even time level, respectively. The stability is proved by Fourier analysis method and the error estimate between the discrete and analytical solution in discrete \({l^2}\) norm is presented. At last, a numerical example is given for confirming the theoretical conclusions.Existence of nontrivial radial convex solutions for a kind of Monge-Ampère systems.https://zbmath.org/1449.352422021-01-08T12:24:00+00:00"Yang, Yang"https://zbmath.org/authors/?q=ai:yang.yang.3|yang.yang.5|yang.yang.2|yang.yang.1|yang.yang.4"Xue, Chunyan"https://zbmath.org/authors/?q=ai:xue.chunyanSummary: In this paper, we investigate the solution of a kind of Monge-Ampère systems composed of \(n\) equations. The existence of nontrivial radial convex solution for Monge-Ampère systems with general nonlinear terms is obtained. Firstly, the Monge-Ampère system can be transformed into an equivalent ordinary differential equation system by an ingenious transformation under the support of the radial solution. Secondly, we construct suitable nonnegative cone and complete continuous operators in Banach space. Finally, the existence of positive solutions of the ordinary differential equations in unit sphere is studied by using the index theory of fixed points. Then the existence of nontrivial radial convex solutions of the original Monge-Ampère system can be obtained. We can prove that there is at least one nontrivial radial convex solution of the original Monge-Ampère system when the nonlinear term is superlinear or sublinear.\(N\)-soliton solutions for a class of nonlinear partial differential equations.https://zbmath.org/1449.351552021-01-08T12:24:00+00:00"Li, Wei"https://zbmath.org/authors/?q=ai:li.wei.10|li.wei|li.wei-wayne|li.wei.7|li.wei.8|li.wei.5|li.wei.9Summary: Differential equations contain linear and nonlinear differential equations. Research of the nonlinear differential equations is the subject of differential equations, especially nonlinear partial differential equations. Many significant natural science and engineering problems can be attributed to nonlinear partial differential equation. In addition, with the development of research, some problems which can be approximated by linear partial differential equations must also consider nonlinear effects. From the traditional point of view, the solution of partial differential equation is very difficult. After several decades of research and exploration, we have found some tectonic solution methods. In this paper, with the help of Cole-Hope transform, one of the conditions for the equation \(Af + B = 0\) to be true if \(A = 0\) and \(B = 0\), \(N\)-soliton solutions of \( (2+1)\)-dimensional Burgers equation and KdV equation have been presented. This method could solve a series of partial differential equations.Application and numerical simulation of fractional reaction diffusion model in Turing pattern.https://zbmath.org/1449.352732021-01-08T12:24:00+00:00"Zhang, Rongpei"https://zbmath.org/authors/?q=ai:zhang.rongpei"Wang, Yu"https://zbmath.org/authors/?q=ai:wang.yu.8Summary: Patterns are non-uniform macrostructures with some regularity in space or time, which can describe the pattern formation by a reaction diffusion system. The stable state will be unstable under certain conditions and spontaneously produce the spatial stationary pattern, namely Turing pattern. We can describe anomalous diffusion motion through the fractional reaction diffusion system. The paper obtains the Turing instability of two-dimensional fractional order Gierer-Meinhardt model by spectral decomposition of fractional Laplacian operator. An efficient high-precision numerical scheme is used in the numerical simulation. The Fourier spectral method is used in the spatial discretization, which has spectral accuracy. The Runge-Kutta exponential time difference method is applied to the time discretization. Numerical simulations in Gierer-Meinhardt model show that the system can generate patterns by controlling the value of fractional order in the system, and verify the theoretical results in the previous stability analysis.The bivariate Müntz wavelets composite collocation method for solving space-time-fractional partial differential equations.https://zbmath.org/1449.652772021-01-08T12:24:00+00:00"Rahimkhani, Parisa"https://zbmath.org/authors/?q=ai:rahimkhani.parisa"Ordokhani, Yadollah"https://zbmath.org/authors/?q=ai:ordokhani.yadollahSummary: Herein, we consider an effective numerical scheme for numerical evaluation of three classes of space-time-fractional partial differential equations (FPDEs). We are going to solve these problems via composite collocation method. The procedure is based upon the bivariate Müntz-Legendre wavelets (MLWs). The bivariate Müntz-Legendre wavelets are constructed for first time. The bivariate MLWs operational matrix of fractional-order integral is constructed. The proposed scheme transforms FPDEs to the solution of a system of algebraic equations which these systems will be solved using the Newton's iterative scheme. Also, the error analysis of the suggested procedure is determined. To test the applicability and validity of our technique, we have solved three classes of FPDEs.Convergence theory for the exact interpolation scheme with approximation vector as the first column of the prolongator and Rayleigh quotient iteration nonlinear smoother.https://zbmath.org/1449.653462021-01-08T12:24:00+00:00"Vaněk, Petr"https://zbmath.org/authors/?q=ai:vanek.petr"Pultarová, Ivana"https://zbmath.org/authors/?q=ai:pultarova.ivanaThis paper is concerned with a convergence analysis of the nonlinear two-level method with a nonlinear Rayleigh quotient iteration smoother, applied to the partial eigenvalue problem of the following article by \textit{R. Kužel} and \textit{P. Vaněk} [Numer. Linear Algebra Appl. 22, No. 6, 950--964 (2015; Zbl 1374.65203)]. The matrix is assumed to be symmetric and positive definite with a simple minimal eigenvalue. The authors seek the minimum eigenvalue and the corresponding eigenvector. The paper is organized as follows. Section 1 is an Introduction. In Section 2, the algorithm and the convergence result of the following article are presented -- [\textit{P. Fraňková, M. Hanuš, H. Kopincová, R. Kužel, I. Marek, I. Pultarová, P. Vaněk,} and \textit{Z. Vastl}, ``Convergence theory for the exact interpolation scheme with approximation vector as the first column of the prolongator: the partial eigenvalue problem'' (submitted to Numer. Math.)]. In the key Section 3, the nonlinear speed up of the Rayleigh quotient iteration smoother compared to the linear inverse power method is analyzed. Section 4 contains the final convergence theorem. The numerical tests with tables and graphical illustrations are presented in the final Section 5.
Reviewer: Temur A. Jangveladze (Tbilisi)Multiplicity of solutions for nonlocal fractional equations with nonsmooth potentials.https://zbmath.org/1449.354542021-01-08T12:24:00+00:00"Yuan, Ziqing"https://zbmath.org/authors/?q=ai:yuan.ziqingSummary: This paper is concerned with a class of nonlocal fractional Laplacian problems with nonsmooth potentials. By exploiting an abstract three critical points theorem for nonsmooth functionals, combining with an analytical context on fractional Sobolev spaces, we obtain the existence of at least three weak solutions for nonlocal fractional problems.Extended transformed rational function method for abundant solutions of extended \( (3+1)\)-dimensional Jimbo-Miwa equations.https://zbmath.org/1449.353662021-01-08T12:24:00+00:00"Ha, Jinting"https://zbmath.org/authors/?q=ai:ha.jinting"Li, Xinyue"https://zbmath.org/authors/?q=ai:li.xinyue"Zhang, Huiqun"https://zbmath.org/authors/?q=ai:zhang.huiqunSummary: In this paper, we study two extended \( (3+1)\)-dimensional Jimbo-Miwa equations by using the extended transformed rational function method. Consequently, we generate abundant exact solutions, especially the resonant multiple wave solutions, composite solutions and combining trigonometric, hyperbolic and exponential function solutions. Moreover, the detailed structures of the generated solutions are presented analytically and graphically.Novel methods for time-space fractional diffusion equation.https://zbmath.org/1449.354522021-01-08T12:24:00+00:00"Wang, Kexin"https://zbmath.org/authors/?q=ai:wang.kexin"Yan, Xingjie"https://zbmath.org/authors/?q=ai:yan.xingjie"Yin, Kun"https://zbmath.org/authors/?q=ai:yin.kunSummary: In this article, we suggest three methods to address the time-space fractional diffusion equations with Caputo fractional derivative, the separation of variables and the series expansion method for numerical solution, Fourier transform and Laplace transform for analytical solution. The orders of derivative are restricted by \(0 < \gamma \le 1\), \(1 < \beta \le 2\) for the time and space domains, respectively.Reconstruction of unknown surface heat flux from an internal temperature history.https://zbmath.org/1449.352452021-01-08T12:24:00+00:00"Ren, Jianlong"https://zbmath.org/authors/?q=ai:ren.jianlongSummary: An inverse heat conduction problem is discussed by using the temperature observed at a certain point in the interior of the heater to recover the surface heat flux of heater wall. The forward problem is solved with the method of variable separation, then the original problem is transformed into Volterra integral equation of the second kind and the uniqueness of the solution of integral equation is proved. Numerical simulation is conducted with the direct solving method. The numerical results are carried out to confirm the feasible and effective of the aforementioned method, and the heat flux is recovered very well.Theoretical understanding of unsteady flow separation for shear flow past three square cylinders in vee shape using structural bifurcation analysis.https://zbmath.org/1449.652982021-01-08T12:24:00+00:00"Kumar, Atendra"https://zbmath.org/authors/?q=ai:kumar.atendra"Ray, Rajendra K."https://zbmath.org/authors/?q=ai:ray.rajendra-kSummary: The unsteady flow separation of two-dimensional (2-D) incompressible shear flow past three identical square cylinders arranged in vee shape is studied in this paper, using theoretical structural bifurcation analysis based on topological equivalence. Through this analysis, the exact location and time of occurrence of bifurcation points (flow separation points) associated with secondary and tertiary vortices on all cylinders are studied. The existence of saddle points is also studied during primary flow separation. Different gap ratios between the downstream cylinders, \(s/d = 0.6-3.0\) (where \(s\) is the gap between cylinders, \(d\) is the length of cylinder side) with fixed gap \(2d\) between upstream and downstream cylinders for different shear parameter \((K)\) values ranging from \(K=0.0\) to 0.4 are considered at Reynolds number (Re) 100. In this process, the instantaneous vorticity contours and streakline patterns, center-line velocity fluctuation, phase diagram, lift and drag coefficients are studied to confirm the theoretical results. Computations are carried out by using higher order compact finite difference scheme. Present study mainly investigates the effect of \(K\) and gap ratio on unsteady flow separation and vortex-shedding phenomenon. All the computed results very efficiently and very accurately reproduce the complex flow phenomenon. Through this study, many noticeable and interesting results are reported for the first time for this problem.Qualitative analysis of a diffusive predator-prey model with density dependence.https://zbmath.org/1449.352672021-01-08T12:24:00+00:00"Li, Haixia"https://zbmath.org/authors/?q=ai:li.haixiaSummary: The uniqueness of coexistence solutions and asymptotic behavior for a diffusive predator-prey model with density dependence are studied. The sufficient conditions of the existence of coexistence solutions are given by means of the fixed point index theory. Then, by making use of the perturbation theory for linear operators, we discuss the stability and uniqueness of coexistence solutions. Finally, the conditions of the extinction and permanence for the system by the comparison principle for parabolic equations, and the theoretical results of asymptotic behavior for the system are verified by some numerical simulations. The results show that the two species can coexist and the system has a unique coexistence solution under certain conditions.Existence of ground states for linear coupled systems of lower critical Choquard type.https://zbmath.org/1449.350072021-01-08T12:24:00+00:00"Wang, Peiting"https://zbmath.org/authors/?q=ai:wang.peiting"Li, Anran"https://zbmath.org/authors/?q=ai:li.anran"Wei, Chongqing"https://zbmath.org/authors/?q=ai:wei.chongqingSummary: To the best of our knowledge, there is few result about linearly coupled systems of Choquard type with the lower critical up to now. The existence of ground state solutions for a class of Choquard-type linear coupled systems with lower critical exponents is studied by variational methods. It is a promotion and supplement to the previous research results.Non-uniform dependence on initial data for the periodic two-coupled Camassa-Holm system.https://zbmath.org/1449.353862021-01-08T12:24:00+00:00"Wang, Haiquan"https://zbmath.org/authors/?q=ai:wang.haiquanSummary: Considered herein is the initial value problem for the periodic two-coupled Camassa-Holm system. It is shown that the solution map of this problem is not uniformly continuous in Besov spaces \(B_{2, 1}^{3/2} (\text bf{T}) \times B_{2, 1}^{3/2} (\text bf{T})\). Based on the well-posedness result and the lifespan for this problem, the method of approximate solutions is utilized. The same approach can be used to discuss this property of the solutions for the other nonlinear partial differential equations.Resolution of nonlinear convection-diffusion-reaction equations of Cauchy kind by the Laplace SBA method.https://zbmath.org/1449.652922021-01-08T12:24:00+00:00"Paré, Youssouf"https://zbmath.org/authors/?q=ai:pare.youssouf"Youssouf, Minoungou"https://zbmath.org/authors/?q=ai:minoungou.youssouf"Nébie, Abdoul Wassiha"https://zbmath.org/authors/?q=ai:nebie.abdoul-wassihaSummary: In this paper, we propose the solution of a few nonlinear partial differential equations modelling diffusion, convection and reaction problems Cauchy type. The Laplace SBA method based on combination of Laplaces transform, Adomian Decomposition Method (ADM), Picard principle and successive approximations is used for solving these equations.Existence of infinitely many high energy solutions of a class of fourth-order elliptic equations with nonlocal terms.https://zbmath.org/1449.352072021-01-08T12:24:00+00:00"Zhang, Nian"https://zbmath.org/authors/?q=ai:zhang.nian"Jia, Gao"https://zbmath.org/authors/?q=ai:jia.gaoSummary: We study a class of fourth-order elliptic equations with nonlocal term,
\[\begin{cases}\Delta^2u - \left (a+ b{\int_{\mathbb{R}^N}}|\nabla u|^2\,dx \right)\Delta u + V(x)u - \frac{1}{2}\Delta (u^2)u = f(x,u),\quad x \in\mathbb{R}^N, \\ u(x) \in H^2(\mathbb{R}^N),\end{cases}\]
where \(N \le 5\), constants \(a > 0\), \(b \ge 0\), \(\Delta^2 = \Delta (\Delta)\) is the biharmonic operator, the nonlinearity \(f(x,u)\) does not satisfy AR condition and the potential function \(V(x)\) is also allowed to be sign-changing. We establish the existence of a sequence of high energy weak solutions for this class of elliptic equations via variational methods.Uniqueness of positive solutions of singular \(p\)-biharmonic equations with Hardy terms.https://zbmath.org/1449.350232021-01-08T12:24:00+00:00"Sang, Yanbin"https://zbmath.org/authors/?q=ai:sang.yanbin"Chen, Juan"https://zbmath.org/authors/?q=ai:chen.juan"Ren, Yan"https://zbmath.org/authors/?q=ai:ren.yanSummary: We study a class of singular \(p\)-biharmonic equations with Hardy terms. The existence and uniqueness of the positive solution for the above problem is obtained by minimization method.Asymptotic analysis to the singularly perturbed Korteweg-de Vries equation.https://zbmath.org/1449.353842021-01-08T12:24:00+00:00"Samoĭlenko, V.Hr."https://zbmath.org/authors/?q=ai:samoylenko.v-g"Samoĭlenko, Yu. I."https://zbmath.org/authors/?q=ai:samoilenko.yu-i"Vovk, V. S."https://zbmath.org/authors/?q=ai:vovk.v-sSummary: The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.Asymptotic periodic solutions for second-order evolution equation.https://zbmath.org/1449.350292021-01-08T12:24:00+00:00"Shi, Wei"https://zbmath.org/authors/?q=ai:shi.wei"Fan, Hongxia"https://zbmath.org/authors/?q=ai:fan.hongxiaSummary: This article is concerned with the existence and uniqueness of pseudo \(S\)-asymptotically \(\omega\)-periodic mild solutions for the second-order evolution equations. Moreover, one application is given to illustrate the feasibility of our theoretical results.Cauchy problem for the hyperbolic system with mixed derivative.https://zbmath.org/1449.353012021-01-08T12:24:00+00:00"Kozlova, Elena Aleksandrovna"https://zbmath.org/authors/?q=ai:kozlova.elena-aleksandrovnaSummary: Cauchy problem for the hyperbolic system with mixed derivative is considered. The given system is transformed to the triangular or diagonal form for the further equations separation. The Cauchy problem for each equation (homogeneous or inhomogeneous) is obtained.On the Cauchy problem for the periodic fifth-order KP-I equation.https://zbmath.org/1449.353552021-01-08T12:24:00+00:00"Robert, Tristan"https://zbmath.org/authors/?q=ai:robert.tristanSummary: The aim of this paper is to investigate the Cauchy problem for the periodic fifth order KP-I equation \[\partial_tu-\partial_x^5u-\partial_x^{-1}\partial_y^2u+u\partial_xu=0,\ (t,x,y)\in\mathbb{R}\times\mathbb{T}^2.\] We prove global well-posedness for constant \(x\) mean value initial data in the space \(\mathbf{E}=\{u\in L^2,\ \partial_x^2u\in L^2,\ \partial_x^{-1}\partial_yu\in L^2\}\) which is the natural energy space associated with this equation.Convergence of attractors and invariant measures for a \(p\)-Laplace equation in \(\mathbb{R}^n\).https://zbmath.org/1449.350962021-01-08T12:24:00+00:00"Miao, Fahe"https://zbmath.org/authors/?q=ai:miao.fahe"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.2|liu.hui.4|liu.hui.1|liu.hui.3"Xin, Jie"https://zbmath.org/authors/?q=ai:xin.jieSummary: Using the conditions of uniform boundedness about the pullback attractor, the convergence of attractors for a \(p\)-Laplace equation in the whole space \(\mathbb{R}^n\) is studied. Then, the existence of a unique family of Borel invariant probability measures for the pullback attractor is established.Global existence and asymptotic stability for the initial boundary value problem of the linear Bresse system with a time-varying delay term.https://zbmath.org/1449.350562021-01-08T12:24:00+00:00"Mahdi, Fatima Z."https://zbmath.org/authors/?q=ai:mahdi.fatima-zohra"Hakem, Ali"https://zbmath.org/authors/?q=ai:hakem.aliSummary: The authors of this paper study the Bresse system in bounded domain with delay terms. First, they prove the global existence of its solutions in Sobolev spaces by means of semigroup theory. Furthermore, the asymptotic stability is given by using an appropriate Lyapunov functional.A blow-up result with arbitrary positive initial energy for nonlinear wave equations with degenerate damping terms.https://zbmath.org/1449.351272021-01-08T12:24:00+00:00"Su, Xiao"https://zbmath.org/authors/?q=ai:su.xiao"Wang, Shubin"https://zbmath.org/authors/?q=ai:wang.shubinSummary: This article is concerned with the finite time blow-up of weak solutions to the wave equations with nonlinear damping and source terms. We provide the sufficient conditions of finite time blow-up of weak solutions with arbitrary positive initial energy by constructing a energy perturbation function.Convergence in wavelet collocation methods for parabolic problems.https://zbmath.org/1449.652822021-01-08T12:24:00+00:00"Zhao, Jianbin"https://zbmath.org/authors/?q=ai:zhao.jianbin"Li, Siwen"https://zbmath.org/authors/?q=ai:li.siwen"Li, Hua"https://zbmath.org/authors/?q=ai:li.huaSummary: This paper studies the second-generation interpolating wavelet collocation methods in space and different Euler time stepping methods for parabolic problems. The convergence and stability are investigated. The operators are formulated using an efficient and exact formulation. The numerical results verify the efficiency of the methods.Existence of solutions to elliptic equation with exponential nonlinearities and singular term.https://zbmath.org/1449.352182021-01-08T12:24:00+00:00"Xue, Yimin"https://zbmath.org/authors/?q=ai:xue.yimin"Chen, Shouting"https://zbmath.org/authors/?q=ai:chen.shoutingSummary: In this paper, we consider an elliptic equation with exponential nonlinearities and singular term. By constructing the corresponding variational framework, and using a singular Trudinger-Moser inequality, mountain-pass theorem and Ekeland's variational principle, we get a nontrivial positive weak solution.A novel proof on the existence of the solution of fractional control problem governed by Burgers equations.https://zbmath.org/1449.354362021-01-08T12:24:00+00:00"Georgiev, S. G."https://zbmath.org/authors/?q=ai:georgiev.svetlin-georgiev"Mohammadizadeh, F."https://zbmath.org/authors/?q=ai:mohammadizadeh.fatemeh"Tehrani, H. A."https://zbmath.org/authors/?q=ai:tehrani.hojjat-ahsani"Skandari, M. H. N."https://zbmath.org/authors/?q=ai:skandari.mohammad-hadi-nooriSummary: In this study, first we give some conditions to prove that a fractional Burgers equation has a unique solution. For this aim, we define a special operator to deduce the uniqueness of the solution. Then we prove that the optimal control problem under fractional Burgers equation has at least one optimal solution.Approximation of smooth stable invariant manifolds for stochastic partial differential equations.https://zbmath.org/1449.370342021-01-08T12:24:00+00:00"Guo, Zhongkai"https://zbmath.org/authors/?q=ai:guo.zhongkai"Yan, Xingjie"https://zbmath.org/authors/?q=ai:yan.xingjie"Yang, Xinguang"https://zbmath.org/authors/?q=ai:yang.xinguangSummary: Invariant manifolds are complicated random sets used for describing and understanding the qualitative behavior of nonlinear dynamical systems. The purpose of the present paper is to try to approximate smooth stable invariant manifolds for a type of stochastic partial differential equations with multiplicative white noise near the fixed point. Two examples are given to illustrate our results.Method of pure shear problem solving for stochastically inhomogeneous plane in a steady-state creep.https://zbmath.org/1449.354162021-01-08T12:24:00+00:00"Popov, Nikolaĭ Nikolaevich"https://zbmath.org/authors/?q=ai:popov.n-n"Chernova, Ol'ga Olegovna"https://zbmath.org/authors/?q=ai:chernova.olga-olegovnaSummary: The analytical method for nonlinear problem of steady-state creep solving for pure shear of stochastically inhomogeneous plane on the basis of the second approximation method of small parameter was developed. It is supposed that elastic deformations are insignificant and they can be neglected. Stochasticity was introduced into the determinative creep equation, which was taken in accordance with the nonlinear theory of viscous flow, through a homogeneous random function of coordinates. By using the decomposition technique of stress tensor components in a small parameter to the members of the second order of smallness, partial differential system of the first and the second approximation of stress was obtained. This system was solved by the introduction of the stress function. The mathematical expectation and variances of the random stress field were calculated. The analysis of the results in the first and second approximations was obtained.Properties for nonlinear fractional subLaplace equations on the Heisenberg group.https://zbmath.org/1449.354232021-01-08T12:24:00+00:00"Wang, Xinjing"https://zbmath.org/authors/?q=ai:wang.xinjing"Niu, Pengcheng"https://zbmath.org/authors/?q=ai:niu.pengchengSummary: The aim of the paper is to study properties of solutions to the nonlinear fractional subLaplace equations on the Heisenberg group. Based on the method of moving planes to the Heisenberg group, we prove the Liouville property of solutions on a half space and the symmetry and monotonicity of the solutions on the whole group respectively.Entire large solutions to semilinear elliptic systems of competitive type.https://zbmath.org/1449.350182021-01-08T12:24:00+00:00"Lair, Alan V."https://zbmath.org/authors/?q=ai:lair.alan-vSummary: We consider the elliptic system \(\Delta u = p\left({\left| x \right|} \right){u^a}{v^b}, \Delta v = q\left({\left| x \right|} \right){u^c}{v^d}\) on \({\mathbb{R}^n}\left({n \ge 3} \right)\), where \(a, b, c, d\) are nonnegative constants with max\(\left\{ {a, d} \right\} \le 1\), and the functions \(p\) and \(q\) are nonnegative, continuous, and the support of min\(\left\{ {p\left(r \right), q\left(r \right)} \right\}\) is not compact. We establish conditions on \(p\) and \(q\), along with the exponents \(a, b, c, d\), which ensure the existence of a positive entire solution satisfying \({\lim_{\left| x \right| \to \infty}}u\left(x \right){\lim_{\left| x \right| \to \infty}}v\left(x \right) = \infty\).The well-posedness of the local boundary value problem in a cylindric domain for the multi-dimensional wave equation.https://zbmath.org/1449.352812021-01-08T12:24:00+00:00"Aldashev, Serik Aĭmurzaevich"https://zbmath.org/authors/?q=ai:aldashev.serik-aimurzaevichSummary: This paper proves the unique solvability of the local boundary value problem in a cylindric domain for the multi-dimensional wave equation, which is the generalization of the Dirichlet and Poincaré problems. We also obtain the criterion for the uniqueness of the regular solution.On a nonlinear heat equation with degeneracy on the boundary.https://zbmath.org/1449.352782021-01-08T12:24:00+00:00"Zhan, Huashui"https://zbmath.org/authors/?q=ai:zhan.huashuiSummary: The paper studies the stability of weak solutions of a nonlinear heat equation with degenerate on the boundary. A new kind of weak solutions are introduced. By the new weak solutions, the stability of weak solutions is proved only dependent on the initial value.The problem with shift for the Bitsadze-Lykov equation.https://zbmath.org/1449.353132021-01-08T12:24:00+00:00"Arlanova, Ekaterina Yur'evna"https://zbmath.org/authors/?q=ai:arlanova.ekaterina-yurevnaSummary: The Bitsadze-Lykov equation is considered. The problem with shift containing the Kober-Erdélyi and M. Saigo operators in boundary condition is set for this equation. The questions of uniqueness (ununiqueness) of this problem solution with different functions and constants in boundary condition are investigated. The number of theorems is formulated and proved.Problem with shift for the third-order equation with discontinuous coefficients.https://zbmath.org/1449.353262021-01-08T12:24:00+00:00"Repin, Oleg Aleksandrovich"https://zbmath.org/authors/?q=ai:repin.oleg-aleksandrovich"Kumykova, Svetlana Kanshubievna"https://zbmath.org/authors/?q=ai:kumykova.svetlana-kanshubievnaSummary: The unique solvability of boundary value problem with Saigo operators for the third-order equation with multiple characteristics was investigated. The uniqueness theorem with constraints of inequality type on the known functions and different orders of generalized fractional integro-differentiation was proved. The existence of solution is equivalently reduced to the solvability of Fredholm integral equation of the second kind.A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation.https://zbmath.org/1449.354322021-01-08T12:24:00+00:00"Cheng, Xiao-liang"https://zbmath.org/authors/?q=ai:cheng.xiaoliang"Yuan, Le-le"https://zbmath.org/authors/?q=ai:yuan.lele"Liang, Ke-wei"https://zbmath.org/authors/?q=ai:liang.keweiSummary: In this paper, we consider a Cauchy problem of the time fractional diffusion equation (TFDE) in \(x \in [0, L]\). This problem is ubiquitous in science and engineering applications. The illposedness of the Cauchy problem is explained by its solution in frequency domain. Furthermore, the problem is formulated into a minimization problem with a modified Tikhonov regularization method. The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem. The error estimates for the regularized solutions are obtained under \(H^p\) norm priori bound assumptions. Finally, numerical examples illustrate the effectiveness of the proposed method.Modeling and analysis of macroeconomic system dynamics in Matlab/simulink environment.https://zbmath.org/1449.354192021-01-08T12:24:00+00:00"Lubentsova, Vera Stepanovna"https://zbmath.org/authors/?q=ai:lubentsova.vera-stepanovna"Andreyanov, Denis Anatol'evich"https://zbmath.org/authors/?q=ai:andreyanov.denis-anatolevichSummary: The general theory of cyclic processes is examined in an economy. The simulation model of solutions of differential equations describing the cyclic vibrations of business activity and the trend curve of GDP growth is built in MATLAB/Simulink environment. Stability of the system is investigated. The graphic image of solutions of differential equations is brought, the influence of accelerator power on the dynamics of the economic system is analyzed.Explicit solution of Cauchy problem for the linearized system of phase field equations.https://zbmath.org/1449.351752021-01-08T12:24:00+00:00"Umarov, Khasan Galsanovich"https://zbmath.org/authors/?q=ai:umarov.khasan-galsanovichSummary: The explicit solution of Cauchy problem for the linearized system of phase field equations is received by reduction it to the abstract Cauchy problem in Banach space.On number of solutions in eigenvalue problems for elliptic equations with discontinuous nonlinearities.https://zbmath.org/1449.352032021-01-08T12:24:00+00:00"Potapov, Dmitriĭ Konstantinovich"https://zbmath.org/authors/?q=ai:potapov.dmitrii-konstantinovichSummary: We study the existence of solutions of eigenvalue problems for elliptic equations of the second order with nonlinearity discontinuous with respect to a phase variable. Using the variational method, we receive the theorems on number of solutions for investigated problems. M. A. Gol'dshtik's problem on separated flows of incompressible fluid is considered as an appendix.The analogue of d'Alembert formula for hyperbolic differential equation of the third order with nonmultiple characteristics.https://zbmath.org/1449.352982021-01-08T12:24:00+00:00"Yakovleva, Yuliya Olegovna"https://zbmath.org/authors/?q=ai:yakovleva.yuliya-olegovnaSummary: The Cauchy problem for the third order hyperbolic differential equation with nonmultiple characteristics is considered. The analogue of d'Alembert formula is obtained as a solution that allows describing the propagation of initial displacement, initial velocity and initial acceleration.Hopf bifurcation of a diffusive predator-prey model with fear factors and strong Allee effects.https://zbmath.org/1449.350382021-01-08T12:24:00+00:00"Fu, Shengmao"https://zbmath.org/authors/?q=ai:fu.shengmao"Su, Faru"https://zbmath.org/authors/?q=ai:su.faruSummary: The Hopf bifurcation of a diffusive predator-prey model with fear factors and strong Allee effects is considered. Firstly, the local asymptotic stability of the non-negative equilibrium points is given. Secondly, by choosing the predator's natural growth rate as a bifurcation parameter, the existence conditions of Hopf bifurcation for the model are obtained. Next, the Hopf branch direction of diffusive system and the conditions for the stability of periodic solutions are discussed by using the center manifold theory and the normal form method. Finally, some numerical simulations are presented to verify these theoretical results.Applications of invariant subspace method in the space-time fractional partial differential equations.https://zbmath.org/1449.350122021-01-08T12:24:00+00:00"Hou, Jie"https://zbmath.org/authors/?q=ai:hou.jie"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhenSummary: This paper introduces the invariant subspace method and its main steps. Applying this method, we investigate six classes of time-space fractional partial differential equations with Caputo derivative and construct the analytic solutions of these equations or present the determining equations.A note on global \({C^2}\) estimates of 2-Hessian equations.https://zbmath.org/1449.351832021-01-08T12:24:00+00:00"Wan, Xin"https://zbmath.org/authors/?q=ai:wan.xin"Yuan, Mengjie"https://zbmath.org/authors/?q=ai:yuan.mengjieSummary: The global \({C^2}\) estimate of the solution of Dirichlet problem of the general 2-Hessian equation \({\sigma_2} (\lambda ({\boldsymbol{D}^2}u + \boldsymbol{B})) = f (x,u,\boldsymbol{D}u)\) is investigated by using the test function. The purpose of this paper is to give an extensional generalization of the form of 2-Hessian equation's \({C^2}\) estimate, establish the global \({C^2}\) estimate of the equation's solution and obtain the existence of the solution of equation's Dirichlet problem.Existence of solution for two classes of Schrödinger equations in \(\mathbb{R}^N\) with magnetic field and zero mass.https://zbmath.org/1449.354002021-01-08T12:24:00+00:00"Yin, Zhao"https://zbmath.org/authors/?q=ai:yin.zhao"Ji, Chao"https://zbmath.org/authors/?q=ai:ji.chaoSummary: In this paper, we consider the existence of a nontrivial solution for the following Schrödinger equations with a magnetic potential \(A\)
\[
-\Delta_A u = K(x)f(\vert u\vert ^2)u\quad \text{in } \mathbb{R}^N,
\]
where \(N\geq 3\), \(K\) is a nonnegative function verifying two kinds of conditions and \(f\) is continuous with subcritical growth.Existence of global attractors for a class of reaction-diffusion system.https://zbmath.org/1449.350932021-01-08T12:24:00+00:00"Li, Junyan"https://zbmath.org/authors/?q=ai:li.junyan"Wu, Ruili"https://zbmath.org/authors/?q=ai:wu.ruiliSummary: By using the theory and method of partial differential equations, the uniform compact conditions for semigroups were reduced to an easily verifiable condition: C-condition, the existence of a global attractor for a kind of prey-predator model was investigated. The existence of the strong continuously semigroup and the bounded attracting set, as well as the verification of C-condition were discussed. Finally, the conclusion was drawn that the model possesses a global attractor in \({L^2} (\Omega, {R^+})^2\), which attracts all bounded subsets of \({L^2} (\Omega, {R^+})^2\) in the appropriate norm.The nonlocal conjugation problem for one-dimensional parabolic equation with discontinuous coefficients and associated Feller semigroup.https://zbmath.org/1449.601242021-01-08T12:24:00+00:00"Kopytko, B. I."https://zbmath.org/authors/?q=ai:kopytko.bogdan-i"Shevchuk, R. V."https://zbmath.org/authors/?q=ai:shevchuk.r-vSummary: By the boundary integral equations method we establish the classical solvability of the conjugation problem for one-dimensional linear parabolic equation of the second order (backward Kolmogorov equation) with nonlocal Feller-Wentzell conjugation condition. Using the solution of this problem, we construct the two-parameter Feller semigroup associated with the inhomogeneous diffusion process in bounded domain with moving membrane.Multiplicity of solutions for \(p\)-Kirchhoff equation with logarithmic nonlinearity.https://zbmath.org/1449.352152021-01-08T12:24:00+00:00"Duan, Bixiao"https://zbmath.org/authors/?q=ai:duan.bixiao"Wang, Shuli"https://zbmath.org/authors/?q=ai:wang.shuli"Guo, Zuji"https://zbmath.org/authors/?q=ai:guo.zujiSummary: Multiple solutions for \(p\)-Kirchhoff equation with logarithmic nonlinearity were investigated. Multiplicity of nontrivial solutions were discussed on a bounded domain by mountain pass theorem, Ekeland's variational principle and logarithmic Sobolev inequality. It was proved that the functional satisfied conditions of mountain pass theorem and combined with Ekeland's variational principle. A conclusion was drawn that the problem has at least two nontrivial solutions.Invariant subspaces and exact solutions for a system of fractional PDEs in higher dimensions.https://zbmath.org/1449.354352021-01-08T12:24:00+00:00"Choudhary, Sangita"https://zbmath.org/authors/?q=ai:choudhary.sangita"Prakash, P."https://zbmath.org/authors/?q=ai:prakash.pankaj|prakash.pradyot|prakash.p-v|prakash.prem|prakash.periasamy"Daftardar-Gejji, Varsha"https://zbmath.org/authors/?q=ai:daftardar-gejji.varshaSummary: In this article, we develop an invariant subspace method for a system of time-fractional nonlinear partial differential equations in \((1+2)\) dimensions. Efficacy of the method is demonstrated by solving coupled system of nonlinear time-fractional diffusion equations and coupled system of time-fractional Burger's equations in higher dimensions. Furthermore, the algorithmic approach to find more than one invariant subspace is proposed and corresponding exact solutions are constructed.On one problem in an infinity half-strip for biaxisimmetric Helmholtz equation.https://zbmath.org/1449.351892021-01-08T12:24:00+00:00"Abashkin, Anton Aleksandrovich"https://zbmath.org/authors/?q=ai:abashkin.anton-aleksandrovichSummary: Boundary value problem in an infinity half-strip for biaxisymmetric Helmholtz equation is explored. Existence conditions of this problem are gotten with help of Fourier-Bessel series expansion. Uniqueness of solution of this boundary value problem is proved for some parameters values. Lack of uniqueness of solution is proved for some other parameters values.On Cauchy problem for system of \(n\) Euler-Poisson-Darboux equations in the plane.https://zbmath.org/1449.352992021-01-08T12:24:00+00:00"Maksimova, Ekaterina Alekseevna"https://zbmath.org/authors/?q=ai:maksimova.ekaterina-alekseevnaSummary: The system of Euler-Poisson-Darboux equations is considered, the Cauchy problem is solved for the case of real \(n\times n\) matrix-coefficient with one real eigenvalue or two complex conjugate eigenvalues with real part in the interval \((-1/2, 1/2)\).Three-dimensional integro-multipoint boundary value problem for loaded Volterra-hyperbolic integro-differential equations of Bianchi type.https://zbmath.org/1449.352972021-01-08T12:24:00+00:00"Mamedov, Il'gar Gurbam"https://zbmath.org/authors/?q=ai:mamedov.ilgar-gurbamSummary: In this paper the combined three-dimensional non-local boundary value problem with integro-multipoint conditions for loaded volterra-hyperbolic integro-differential equation of Bianchi type is explored. The matter of principle is the fact, that the considered equation has discontinuous coefficients which satisfy only some conditions of \(P\)-integrability type and boundedness, i.e. the considered hyperbolic differential operator has no traditional conjugate operator. In particular, for example, Riemann function under Goursat conditions for such equation cannot be constructed by classical method of characteristics.The Fourier transform on 2-step Lie groups.https://zbmath.org/1449.430052021-01-08T12:24:00+00:00"Lévy, Guillaume"https://zbmath.org/authors/?q=ai:levy.guillaumeThe author develops harmonic analysis on a nilpotent Lie group of step 2. The Fourier transform is expressed in terms of the canonical bilinear form and its matrix coefficients. The parameter space of these matrix coefficients and its completion with respect to a natural distance are computed explicitly, as well as the integral kernel of the matrix coefficients Fourier transform, the analogue for the above framework of the classical Fourier kernel \((x, \xi)\mapsto e^{i(x\cdot \xi)}\).
Reviewer: Anatoly N. Kochubei (Kyïv)Problem of determining a multidimensional thermal memory in a heat conductivity equation.https://zbmath.org/1449.354262021-01-08T12:24:00+00:00"Durdiev, D. K."https://zbmath.org/authors/?q=ai:durdiev.durdimurod-kalandarovich"Zhumaev, Zh. Zh."https://zbmath.org/authors/?q=ai:zhumaev.zh-zhThe authors study the problem of determining the functions \(u(x,t)\), \(K(x',t)\), \(x=(x_1,x_2,\ldots ,x_{n-1},x_n)=(x',x_n)\in \mathbb{R}^n\), \(t>0\), from the equation \[ u_t-\Delta u=\int\limits_0^t K(x',t-\tau)\Delta u(x, \tau)\,d\tau,\quad x\in \mathbb{R}^n,t\in [0,T], \] with appropriate initial and boundary conditions. The existence and uniqueness results are obtained.
Reviewer: Anatoly N. Kochubei (Kyïv)Euler-Lagrange equations for DNA chain by an elastic rod model.https://zbmath.org/1449.354182021-01-08T12:24:00+00:00"Xiao, Ye"https://zbmath.org/authors/?q=ai:xiao.ye"Kong, Bin"https://zbmath.org/authors/?q=ai:kong.bin"Li, Chun"https://zbmath.org/authors/?q=ai:li.chun.2|li.chun.1|li.chun|li.chun.3Summary: Based on the analytical mechanics, the Euler-Lagrange equations of DNA chain with elastic rod model are derived in detail by the variation of the free energy functional, which depends on the curvature \(\kappa\), torsion \(\tau\), twisting angle \(\chi\) and its derivative with respect to the arc-length of central axis curve of rod. With the different shapes of rod, we obtain the equilibrium equations of DNA with circular and noncircular cross sections, which provide an approach to describe the physical behaviors of A-, B-, Z-DNA. The results show that the elastic rod model with circular cross section can accurately characterize the equilibrium configurations of A- and B-DNA, while the model with elliptical cross sections (\(k = 0.141\)) is more suitable for Z-DNA. This study might be helpful to characterize the mechanical properties of DNA chains or design DNA-template devices in a wide range of applications.High-order compact alternating direction implicit scheme for complex Ginzburg-Landau equations in two dimensions.https://zbmath.org/1449.652162021-01-08T12:24:00+00:00"Zhu, Chenyi"https://zbmath.org/authors/?q=ai:zhu.chenyi"Wang, Tingchun"https://zbmath.org/authors/?q=ai:wang.tingchunSummary: In the paper, we propose a time-splitting high-order compact alternating direction implicit (ADI) finite difference scheme for two-dimensional complex Ginzburg-Landau (GL) equation. The GL equation is split into a nonlinear sub-problem and two linear sub-problems. The nonlinear sub-problem and one of the linear subproblems are solved exactly. Then a compact alternating direction implicit difference scheme is constructed for another linear subproblem. In practical computation, a family of constant coefficient tri-diagonal linear algebraic equations by using the catch-up method at each time step is solved to make the algorithm get high accuracy and efficiency. Numerical experiments show that the algorithm has second-order and fourth-order accuracy in time and space direction, respectively. Finally, some dynamics behaviors of the equation are simulated.About Green's vector functions of Dirichlet and Neumann semi-space problems for second-order parabolic equations with specificities and degenerations.https://zbmath.org/1449.352522021-01-08T12:24:00+00:00"Turchyna, N. I."https://zbmath.org/authors/?q=ai:turchyna.n-iSummary: In this paper, we consider the Dirichlet and Neumann problems in the domain \(\Pi^+_{(0 ,T]} :=\{( t, x):t\in (0, T], x\in \mathbb{R}^n_+\}\) for two second-order parabolic equations. In both equations, coefficients at the second-order derivatives with respect to \(x\) are constant, and the coefficients at the first-order derivatives with respect to \(x_j\) are functions \(bx_j + a_ j\), \(j=1,\ldots, n,\) where \(b \ne 0\) and \(a_n = 0\). The second equation also contains degeneration at \(t = 0\). For such problems, Green's vector functions are constructed, estimates of the components of these functions and their derivatives are obtained. In order to construct the Green's vector functions we use the fundamental solutions of the Cauchy problem for the equations and parabolic potentials of the simple and double layers. The obtained results could be used for establishing the correct solvability of the boundary value problems, integral representation and the properties of their solutions.Two-point problem for linear systems of partial differential equations.https://zbmath.org/1449.351782021-01-08T12:24:00+00:00"Symotyuk, M. M."https://zbmath.org/authors/?q=ai:symotyuk.m-mSummary: We introduce classes \(H, H_{\delta}, S_{\delta}\) of linear systems of partial differential equations. Some symmetric polynomials in the roots of the characteristic equations of the systems for these classes are nontrivial and its allow power estimates from below. Based on the metric approach and theory of symmetric polynomials we show that almost all systems of partial differential equations with constants coefficients (with respect to the Lebesgue measure in the space spanned by system coefficients) belong to the introduced classes. The problem with two multiple nodes on the selected variable \(t\) and periodicity conditions in other coordinates \(x_1,\dots ,x_p\) for linear systems of partial differential equations belonging to the described classes \(H, H_{\delta}, S_{\delta}\) is investigated. The conditions of solvability problem in the spaces of smooth vector-functions with exponential behavior of Fourier vector-coefficients are established. It is proved that estimates for small denominators provided the existence of the solution of the problem are performed for almost all (respect to the Lebesgue measure and the Hausdorff fractal measure) of the values of the second interpolation node for linear systems from the classes \(H, H_{\delta}, S_{\delta}\).Positive solutions of indefinite semipositone problems via sub-super solutions.https://zbmath.org/1449.352002021-01-08T12:24:00+00:00"Kaufmann, Uriel"https://zbmath.org/authors/?q=ai:kaufmann.uriel"Ramos Quoirin, Humberto"https://zbmath.org/authors/?q=ai:ramos-quoirin.humbertoSummary: Let \(\Omega\subset\mathbb{R}^N\), \(N\geq 1\) be a smooth bounded domain, and let \(m:\Omega\rightarrow\mathbb{R}\) be a possibly sign-changing function. We investigate the existence of positive solutions for the semipositone problem \[ \begin{cases} -\Delta u=\lambda m(x)(f(u)-k) &\text{in}\quad\Omega,\\ \qquad u=0 &\text{on}\quad\partial\Omega,\end{cases} \] where \(\lambda,k>0\) and \(f\) is either sublinear at infinity with \(f(0)=0\), or \(f\) has a singularity at \(0\). We prove the existence of a positive solution for certain ranges of \(\lambda\) provided that the negative part of \(m\) is suitably small. Our main tool is the sub-supersolutions method, combined with some rescaling properties.Mixed exact solutions of \( (2+1)\)-dimensional extended Kadomtsev-Petviashvili equation.https://zbmath.org/1449.353682021-01-08T12:24:00+00:00"Wang, Meineng"https://zbmath.org/authors/?q=ai:wang.meinengSummary: The Kadomtsev-Petviashvili equations are rich in type and they describe many mathematical and physical phenomena. It is necessary to study the exact solution of the Kadomtsev-Petviashvili equation. In this paper, we mainly discuss the \( (2+1)\)-dimensional extended Kadomtsev-Petviashvili equation. Based on the Hirota bilinear form and the symbolic computing software Mathematica, and considering the mixing of exponential functions, trigonometric functions and hyperbolic functions, we obtain some new mixed exact solutions of the \( (2+1)\)-dimension extended Kadomtsev-Petviashvili equation. The physical structure and characteristics of these solutions are illustrated by using some three-dimensional graphics.Numerical computations of nonlocal Schrödinger equations on the real line.https://zbmath.org/1449.820042021-01-08T12:24:00+00:00"Yan, Yonggui"https://zbmath.org/authors/?q=ai:yan.yonggui"Zhang, Jiwei"https://zbmath.org/authors/?q=ai:zhang.jiwei"Zheng, Chunxiong"https://zbmath.org/authors/?q=ai:zheng.chunxiongSummary: The numerical computation of nonlocal Schrödinger equations (SEs) on the whole real axis is considered. Based on the artificial boundary method, we first derive the exact artificial nonreflecting boundary conditions. For the numerical implementation, we employ the quadrature scheme proposed in [\textit{X. Tian} and \textit{Q. Du}, SIAM J. Numer. Anal. 51, No. 6, 3458--3482 (2013; Zbl 1295.82021)] to discretize the nonlocal operator, and apply the \(z\)-transform to the discrete nonlocal system in an exterior domain, and derive an exact solution expression for the discrete system. This solution expression is referred to our exact nonreflecting boundary condition and leads us to reformulate the original infinite discrete system into an equivalent finite discrete system. Meanwhile, the trapezoidal quadrature rule is introduced to discretize the contour integral involved in exact boundary conditions. Numerical examples are finally provided to demonstrate the effectiveness of our approach.Multigrid methods for time-fractional evolution equations: a numerical study.https://zbmath.org/1449.652482021-01-08T12:24:00+00:00"Jin, Bangti"https://zbmath.org/authors/?q=ai:jin.bangti"Zhou, Zhi"https://zbmath.org/authors/?q=ai:zhou.zhiSummary: In this work, we develop an efficient iterative scheme for a class of nonlocal evolution models involving a Caputo fractional derivative of order \(\alpha (0,1)\) in time. The fully discrete scheme is obtained using the standard Galerkin method with conforming piecewise linear finite elements in space and corrected high-order BDF convolution quadrature in time. At each time step, instead of solving the linear algebraic system exactly, we employ a multigrid iteration with a Gauss-Seidel smoother to approximate the solution efficiently. Illustrative numerical results for nonsmooth problem data are presented to demonstrate the approach.Bifurcation and stability analysis in complex cross-diffusion mathematical model of phytoplankton-fish dynamics.https://zbmath.org/1449.350392021-01-08T12:24:00+00:00"Ouedraogo, Hamidou"https://zbmath.org/authors/?q=ai:ouedraogo.hamidou"Ouedraogo, Wendkouni"https://zbmath.org/authors/?q=ai:ouedraogo.wendkouni"Sangare, Boureima"https://zbmath.org/authors/?q=ai:sangare.boureimaSummary: In this paper, we propose a nonlinear reaction-diffusion system describing the interaction between toxin-producing phytoplankton and fish population. We analyze the effect of cross-diffusion on the dynamics of the system. The mathematical study of the model leads us to have an idea on the existence of a solution, the existence of equilibria and the stability of the stationary equilibria. Finally, numerical simulations performed at two-dimensions allowed us to establish the formation of spatial patterns and a threshold of release of the toxin, above which we talk about the phytoplankton blooms.Lump solutions of \( (3+1)\)-dimensional Kadomtsev-Petviashvili equation with variable coefficients.https://zbmath.org/1449.353732021-01-08T12:24:00+00:00"Zhou, Xiaohong"https://zbmath.org/authors/?q=ai:zhou.xiaohong"Deng, Changrui"https://zbmath.org/authors/?q=ai:deng.changruiSummary: The Kadomtsev-Petviashvili equation with variable coefficients is often used to describe long and small amplitude surface waves with weak nonlinearity, weak dispersion and weak perturbation in fluid mechanics. In this paper, some new lump solutions of the \( (3+1)\)-dimensional Kadomtsev-Petviashvili equation with variable coefficients are obtained by using Hirota bilinear form and symbolic calculation software Mathematica. The dynamic behavior of these solutions is analyzed by using some three-dimensional graphs.Mean-square stability of stochastic age-dependent delay population systems with Poisson jumps.https://zbmath.org/1449.350522021-01-08T12:24:00+00:00"Li, Qiang"https://zbmath.org/authors/?q=ai:li.qiang.3"Kang, Ting"https://zbmath.org/authors/?q=ai:kang.ting"Chen, Feifei"https://zbmath.org/authors/?q=ai:chen.feifei"Zhang, Qimin"https://zbmath.org/authors/?q=ai:zhang.qiminSummary: This paper deals with the mean-square stability problem of stochastic age-dependent delay population systems with Poisson jumps. Under certain conditions, the definition of mean-square stability of the numerical solution is given. By utilizing the compensated stochastic \(\theta\) methods, the mean-square stability of the numerical solution is investigated and a sufficient condition for mean-square stability of the numerical solution is presented. It is shown that the compensated stochastic \(\theta\) methods are mean-square stable for any stepsize \(\Delta \tau /m\) when \(1/2 \le \theta \le 1\), and they are exponentially mean-square stable if the stepsize \(\Delta t \in (0, \Delta {t_0})\) when \(0 \le \theta < 1\). Finally, the theoretical results are also confirmed by a numerical experiment.Multiplicity of solutions for a fractional \(p (x)\)-Laplacian equation.https://zbmath.org/1449.352402021-01-08T12:24:00+00:00"Zhang, Jinguo"https://zbmath.org/authors/?q=ai:zhang.jinguo"Jiao, Hongying"https://zbmath.org/authors/?q=ai:jiao.hongying"Liu, Qiuyun"https://zbmath.org/authors/?q=ai:liu.qiuyunSummary: In this paper, we investigate the existence and multiplicity of solutions to a class of fractional \(p (x)\)-Laplacian equation. By means of fountain theorem and the theory of fractional variable exponent Sobolev space, we show that the equation has a sequence of nontrivial solutions with high energies, which generalizes the results of classical variable exponent problem.A new compact finite difference scheme for the quintic nonlinear Schrödinger equation.https://zbmath.org/1449.652062021-01-08T12:24:00+00:00"Xue, Xiang"https://zbmath.org/authors/?q=ai:xue.xiang"Wang, Tingchun"https://zbmath.org/authors/?q=ai:wang.tingchunSummary: In this paper, we study the initial boundary value problem of the nonlinear Schrödinger equation with a quintic term. By using the finite difference method to construct a fourth-order compact finite difference scheme, we prove that the scheme preserves the total mass and energy, respectively. By introducing the lifting technique, the optimal error estimate of the proposed scheme is established by using the standard energy method and the mathematical induction. It is proved that the numerical solution has accuracy of fourth-order and second-order in space and time, respectively. Numerical experiments are given to verify the theoretical results and compared with the existing results, which show that the proposed scheme has higher computational efficiency under the condition of maintaining high accuracy.Asymptotic behavior of compressible Navier-Stokes fluid in porous medium.https://zbmath.org/1449.350852021-01-08T12:24:00+00:00"Yuan, Guozhi"https://zbmath.org/authors/?q=ai:yuan.guozhi"Zhao, Hongxing"https://zbmath.org/authors/?q=ai:zhao.hongxingSummary: We study the asymptotic behavior of the solution to the full compressible Navier-Stokes fluid in porous medium. By using standard energy and two-scale convergence, we prove the strong convergence of the density and the temperature with characteristic size of the pores \(\varepsilon\) in \({R^n}\) for \(n = 2\) or 3 and obtain the homogenized result for this model, when \(\varepsilon \to 0\), which gives another explanation to the results in references.Concentration in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations.https://zbmath.org/1449.353062021-01-08T12:24:00+00:00"Zhang, Qingling"https://zbmath.org/authors/?q=ai:zhang.qingling.1|zhang.qinglingSummary: In this paper, two kinds of occurrence mechanism on the phenomenon of concentration and the formation of delta shock waves in the flux approximation limit of Riemann solutions to the extended Chaplygin gas equations are analyzed. By phase plane analysis and generalized characteristic analysis, we construct the Riemann solution to the extended Chaplygin gas equations completely and obtain two results: on one hand, as the pressure vanishes, any two-shock Riemann solution to the extended Chaplygin gas equations tends to a \(\delta\)-shock solution to the transportation equation; on the other hand, as the pressure approaches the generalized Chaplygin pressure, any two-shock Riemann solution tends to a \(\delta\)-shock solution to the generalized Chaplygin gas equations, which are generalized to the extended Chaplygin gas.Analytic regularity of solutions to spatially homogeneous Landau equation.https://zbmath.org/1449.351372021-01-08T12:24:00+00:00"Wang, Yanlin"https://zbmath.org/authors/?q=ai:wang.yanlin"Xu, Weiman"https://zbmath.org/authors/?q=ai:xu.weimanSummary: In this paper, we investigate the smoothness effect for the Cauchy problem of Landau equation with \(\gamma \in [0,1]\). Analytic estimate involving time and analytic smoothness effect of the solutions are established under some weak assumptions on the initial data.Wave breaking in the periodic integrable Hunter-Saxton equation with a dispersive term.https://zbmath.org/1449.351302021-01-08T12:24:00+00:00"Zhang, Ying"https://zbmath.org/authors/?q=ai:zhang.ying.5|zhang.ying.4|zhang.ying"Pei, Ruichang"https://zbmath.org/authors/?q=ai:pei.ruichang"Cui, Dewang"https://zbmath.org/authors/?q=ai:cui.dewangSummary: Considered here is the periodic Cauchy problem for an integrable Hunter-Saxton equation with a dispersive term. Firstly, we derive a precise blow-up criterion of strong solutions to the equation. Secondly, sufficient conditions guaranteeing the development of breaking waves in finite time are demonstrated by applying some conservative quantities and the method of characteristics, respectively. Finally, the exact blow-up rate is determined.Multiplicity of solutions for Klein-Gordon-Maxwell systems with sign-changing potential.https://zbmath.org/1449.353762021-01-08T12:24:00+00:00"Duan, Yu"https://zbmath.org/authors/?q=ai:duan.yu"Sun, Xin"https://zbmath.org/authors/?q=ai:sun.xin"An, Yucheng"https://zbmath.org/authors/?q=ai:an.yuchengSummary: In this paper, we establish the multiplicity of solutions for a class of Klein-Gordon-Maxwell system with sign-changing potential. When the nonlinearity involves a combination of convex and concave terms and convex terms satisfy general superlinear growth at infinity, the multiplicity result of nontrivial solutions for the system is obtained via variational methods.Instabilities of periodic waves for the Lugiato-Lefever equation.https://zbmath.org/1449.354062021-01-08T12:24:00+00:00"Delcey, Lucie"https://zbmath.org/authors/?q=ai:delcey.lucie"Haragus, Mariana"https://zbmath.org/authors/?q=ai:haragus.marianaThe purpose of this article is to study the existence and stability of all steady periodic solution bifurcations of the nonlinear Lugiato-Lefever equation. First bifurcations of steady periodic solutions occurring in Turing instability are studied. The different cases are illustrated by color graphs. A first long theorem is stated. Then Turing instability for minimal wavelength is studied and the conditions for resonance are computed. A similar graph is shown and a similar theorem is stated. In Section 4 steady periodic solutions in two extreme cases are studied, graphs are shown and a similar theorem is stated. Finally, normal dispersion is studied along similar lines. The proofs use Hilbert and Banach spaces, phase space.
Reviewer: Thomas Ernst (Uppsala)Differential invariant equation of Whitham-Broer-Kaup equation.https://zbmath.org/1449.353792021-01-08T12:24:00+00:00"Han, Guotao"https://zbmath.org/authors/?q=ai:han.guotao"Lv, Zhiyi"https://zbmath.org/authors/?q=ai:lv.zhiyiSummary: The research object of this paper is Whitham-Broer-Kaup equation which is an important model in oceanic science, nonlinear dynamics, mathematical physics and other fields. The differential invariant equation and normalized differential invariants of the Whitham-Broer-Kaup equation play a very important role in solving the Whitham-Broer-Kaup equation with differential invariants. Due to the nonlinear complexity of this equation and the limitation of classic moving frames, it is difficult to find the differential invariant equation and normalized differential invariants of this equation. The new equivariant moving frames theory and Maple software are used to solve differential invariant equation and normalized differential invariants of Whitham-Broer-Kaup equation in this paper. This algorithm breaks the limitation of previous methods. Only the infinitesimal determining equations and suitable cross-section are used. It is not limited to the geometry range of classic moving frame and is efficient and operative in computation. These conclusions are useful to find the solutions and the invariant properties of Whitham-Broer-Kaup equation by using differential invariant and to analyze the nature of nonlinear motion in oceanic science and atmospheric science.Global attractor for stochastic strongly damped semilinear wave equations on unbounded domain.https://zbmath.org/1449.350922021-01-08T12:24:00+00:00"Han, Yinghao"https://zbmath.org/authors/?q=ai:han.yinghao"Pei, Tong"https://zbmath.org/authors/?q=ai:pei.tong"Yang, Yutong"https://zbmath.org/authors/?q=ai:yang.yutong"Chang, Yifang"https://zbmath.org/authors/?q=ai:chang.yifangSummary: In this paper we study the asymptotic dynamics for strongly damped wave equations with nonlinear weak damping and additive noise defined on unbounded domains. We prove the existence of the global attractor for the random dynamical system associated with the equations. For this end, we first prove the existence of weak solutions and bounded absorbing sets, and then prove the asymptotic compactness by using the decomposition method of appropriate cut-off functions. The main difficulty of this paper is that, due to the unboundedness of the domain, some compactness results are not available. To overcome this difficulty we use the decomposition method of the solution to the equations.Inversion of initial-value problem by means of quasi-reversibility regularization method combined with discrete random noise.https://zbmath.org/1449.652282021-01-08T12:24:00+00:00"Yang, Fan"https://zbmath.org/authors/?q=ai:yang.fan.1"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.4|zhang.yan.3|zhang.yan.2"Li, Xiaoxiao"https://zbmath.org/authors/?q=ai:li.xiaoxiaoSummary: The inversion of initial value problem of fractional diffusion equation is explored with discrete random noise. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the measured data. The quasi-reversibility regularization method is used to obtain a regularized approximate solution and the convergence estimate is given under a priori parameter choice rule. Numerical results show that this method will be effective and stable.