Recent zbMATH articles in MSC 35Ahttps://zbmath.org/atom/cc/35A2021-06-15T18:09:00+00:00WerkzeugHardy inequalities for the fractional powers of the Grushin operator.https://zbmath.org/1460.350072021-06-15T18:09:00+00:00"Song, Manli"https://zbmath.org/authors/?q=ai:song.manli"Tan, Jinggang"https://zbmath.org/authors/?q=ai:tan.jinggangSummary: We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in \(\mathbb{R}^N\).https://zbmath.org/1460.351612021-06-15T18:09:00+00:00"Shen, Yaotian"https://zbmath.org/authors/?q=ai:shen.yaotian"Wang, Youjun"https://zbmath.org/authors/?q=ai:wang.youjunSummary: We study the existence of positive solutions of the following degenerate coercive quasilinear elliptic equations:
\[
-\operatorname{div}(g^2(u)\nabla u)+\lambda g(u)g'(u)|\nabla u|^2+V(x)u=\beta u^{(1-\gamma)(2^*-1)}+f(u),\, x\in\mathbb{R}^N,
\]
where \(g(t)\in C(\mathbb{R},\mathbb{R})\), \(V(x)\in C(\mathbb{R}^N,\mathbb{R})\), \(\lambda,\gamma\in\mathbb{R}\), \(\beta\geq 0\) and \(2^*=\frac{2N}{N-2}\), \(N\geq 3\). The novelty of this paper is that \(g(t)\) is non-increasing with respect to \(|t|\) and \(\lim_{|t|\rightarrow+\infty} g(t)=0\). The main results of this paper can be regarded as a supplement to the case that \(g(t)\) is non-decreasing with respect to \(|t|\) which has been extensively studied recently.Existence and nonexistence of positive radial solutions for a class of \(p\)-Laplacian superlinear problems with nonlinear boundary conditions.https://zbmath.org/1460.351722021-06-15T18:09:00+00:00"Alotaibi, Trad"https://zbmath.org/authors/?q=ai:alotaibi.trad"Hai, D. D."https://zbmath.org/authors/?q=ai:hai.dang-dinh"Shivaji, R."https://zbmath.org/authors/?q=ai:shivaji.ratnasinghamSummary: We prove the existence of positive radial solutions to the problem
\[
\begin{cases}
-\Delta_pu=\lambda K(|x|)f(u)\text{ in }|x|>r_0,\\
\dfrac{\partial u}{\partial n}+\tilde{c}(u)u=0\text{ on }|x|=r_0,\quad u(x)\rightarrow 0\text{ as }|x|\rightarrow\infty,
\end{cases}
\]
where
\(\Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\), \(N>p>1\), \(\Omega=\{x\in\mathbb{R}^N:|x|>r_0>0\}\), \(f:(0,\infty)\rightarrow\mathbb{R}\) is \(p\)-superlinear at \(\infty\) with possible singularity at \(0\), and \(\lambda\) is a small positive parameter. A nonexistence result is also established when \(f\) has semipositone structure at \(0\).A hyper-block self-consistent approach to nonlinear Schrödinger equations: breeding, metamorphosis, and killing of Hofstadter butterflies.https://zbmath.org/1460.350972021-06-15T18:09:00+00:00"Solaimani, Mehdi"https://zbmath.org/authors/?q=ai:solaimani.mehdi"Aleomraninejad, S. M. A."https://zbmath.org/authors/?q=ai:aleomraninejad.seyed-m-a|aleomraninejad.s-mohammad-aliSummary: Nonlinear Schrödinger equations play essential roles in different physics and engineering fields. In this paper, a hyper-block finite-difference self-consistent method (HFDSCF) is employed to solve this stationary nonlinear eigenvalue equation and demonstrated its accuracy. By comparing the results with the Sinc self-consistent (SSCF) method and the exact available results, we show that the HFDSCF gives quantum states with high accuracy and can even solve the strongly nonlinear Schrodinger equations. Then, by applying our method to the Hofstadter butterfly problem, we describe the breeding, metamorphosis, and killing of these butterflies by using nonlinear interactions and two constant length multi-well and sinusoidal potentials.On the existence of solutions of nonlinear boundary value problems for inhomogeneous isotropic shallow shells of the Timoshenko type with free edges.https://zbmath.org/1460.353402021-06-15T18:09:00+00:00"Akhmadiev, M. G."https://zbmath.org/authors/?q=ai:akhmadiev.m-g.1"Timergaliev, S. N."https://zbmath.org/authors/?q=ai:timergaliev.samat-n"Uglov, A. N."https://zbmath.org/authors/?q=ai:uglov.a-n"Yakushev, R. S."https://zbmath.org/authors/?q=ai:yakushev.rinat-sAuthors' abstract: The paper deals with the study of solvability to geometrically nonlinear boundary value problem for elastic inhomogeneous isotropic shallow shells with free edges within S. P. Timoshenko shear model. The problem is reduced to one nonlinear equation relative to deflection of shell in Sobolev space. Solvability of equation is proved with the use of contracting mappings principle.
Reviewer: Kaïs Ammari (Monastir)On and beyond propagation of singularities of viscosity solutions.https://zbmath.org/1460.350702021-06-15T18:09:00+00:00"Cannarsa, Piermarco"https://zbmath.org/authors/?q=ai:cannarsa.piermarco"Cheng, Wei"https://zbmath.org/authors/?q=ai:cheng.weiSummary: This is a survey paper for the recent results on and beyond propagation of singularities of viscosity solutions. We also collect some open problems in this topic.
For the entire collection see [Zbl 1454.00056].Existence and behavior of positive solutions for a class of linearly coupled systems with discontinuous nonlinearities in \(\mathbb{R}^N\).https://zbmath.org/1460.351212021-06-15T18:09:00+00:00"de Albuquerque, José Carlos"https://zbmath.org/authors/?q=ai:de-albuquerque.jose-carlos"dos Santos, Gelson G."https://zbmath.org/authors/?q=ai:dos-santos.gelson-g"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcherSummary: In this paper we are concerned with existence and behavior of positive solutions to the following class of linearly coupled elliptic systems with discontinuous nonlinearities
\begin{align*}
\left\{\begin{array}{ll}
-\Delta u+V_1(x)u=H(u-\beta)f_1(u)+a(x)v, & \text{ in }\mathbb{R}^N,\\
-\Delta v+V_2(x)v=H(v-\beta)f_2(v)+a(x)u, & \text{ in }\mathbb{R}^N,\\
u,v\in D^{1,2}(\mathbb{R}^N)\cap W_{\text{loc}}^{2,2}(\mathbb{R}^N),
\end{array}\right.\tag{\(S_\beta\)}
\end{align*}
where \(\beta\geq 0\), \(N\geq 3\), \(V_1,V_2,a:\mathbb{R}^N\rightarrow\mathbb{R}\) are positive potentials, which can vanish at infinity, \(f_1,f_2:\mathbb{R}\rightarrow\mathbb{R}\) are continuous functions and \(H\) is the Heaviside function, i.e, \(H(t)=0\) if \(t\leq 0,H(t)=1\) if \(t>0\). We use a suitable nonsmooth truncation, for systems, to apply a version of the penalization method of \textit{M. A. del Pino} and \textit{P. L. Felmer} [Calc. Var. Partial Differ. Equ. 4, No. 2, 121--137 (1996; Zbl 0844.35032)] combined with the Mountain Pass Theorem for locally Lipschitz functional to obtain a positive solution \((u_{\beta},v_{\beta})\) of \((S)_{\beta}\) in multivalued sense. In addition, we show that \((u_{\beta },v_{\beta})\rightarrow (u,v)\) in \(D^{1,2}(\mathbb{R}^N)\times D^{1,2}(\mathbb{R}^N)\) as \(\beta\rightarrow 0^+\), where \((u, v)\) is a positive solution of the continuous system \((S)_0\) in strong sense.On singular solutions of time-periodic and steady Stokes problems in a power cusp domain.https://zbmath.org/1460.352542021-06-15T18:09:00+00:00"Eismontaite, Alicija"https://zbmath.org/authors/?q=ai:eismontaite.alicija"Pileckas, Konstantin"https://zbmath.org/authors/?q=ai:pileckas.konstantinSummary: The time-periodic and steady Stokes problems with the boundary value having a nonzero flux are considered in the power cusp domains. The asymptotic expansion near the singularity point is constructed in order to reduce the problem to the case where the energy solution exists. The solution of the problem is found then as the sum of the asymptotic expansion and the term with finite dissipation of energy.An accelerated solution for some classes of nonlinear partial differential equations.https://zbmath.org/1460.350762021-06-15T18:09:00+00:00"El-Kalla, Ibrahim L."https://zbmath.org/authors/?q=ai:el-kalla.ibrahim-l"Mohamed, E. M."https://zbmath.org/authors/?q=ai:mohamed.e-m-h|mohamed.emad-m"El-Saka, Hala A. A."https://zbmath.org/authors/?q=ai:el-saka.hala-a-aSummary: In this paper, we apply an accelerated version of the Adomian decomposition method for solving a class of nonlinear partial differential equations. This version is a smart recursive technique in which no differentiation for computing the Adomian polynomials is needed. Convergence analysis of this version is discussed, and the error of the series solution is estimated. Some numerical examples were solved, and the numerical results illustrate the effectiveness of this version.On special regularity properties of solutions of the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation.https://zbmath.org/1460.353132021-06-15T18:09:00+00:00"Nascimento, A. C."https://zbmath.org/authors/?q=ai:nascimento.anderson-c-aSummary: In this paper we study special properties of solutions of the initial value problem (IVP) associated to the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation. We prove that if initial data has some prescribed regularity on the right hand side of the real line, then this regularity is propagated with infinite speed by the flow solution. In other words, the extra regularity on the data propagates in the solutions in the direction of the dispersion. The method of proof to obtain our result uses weighted energy estimates arguments combined with the smoothing properties of the solutions. Hence we need to have local well-posedness for the associated IVP via compactness method. In particular, we establish a local well-posedness in the usual \(L^2(\mathbb{R}^2)\)-based Sobolev spaces \(H^s(\mathbb{R}^2)\) for \(s>\frac{5}{4}\) which coincides with the best available result in the literature proved employing more complicated tools.Existence of solutions for double-phase problems by topological degree.https://zbmath.org/1460.351112021-06-15T18:09:00+00:00"Wang, Bin-Sheng"https://zbmath.org/authors/?q=ai:wang.binsheng"Hou, Gang-Ling"https://zbmath.org/authors/?q=ai:hou.gang-ling"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.binSummary: The double-phase problem with a reaction term depending on the gradient is considered in this paper. Using the topological degree theory for a class of demicontinuous operators, we prove the existence of at least one solution of such problem. Our assumptions are suitable and different from those studied previously.Positive radial solutions for Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space.https://zbmath.org/1460.351302021-06-15T18:09:00+00:00"Ma, Ruyun"https://zbmath.org/authors/?q=ai:ma.ruyun"He, Zhiqian"https://zbmath.org/authors/?q=ai:he.zhiqianSummary: In this paper, we are considered with the Dirichlet problem of quasilinear differential system, involving the mean curvature operator in Minkowski space
\[
\mathcal{M}(w)=\operatorname{div}\biggl(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\biggr),
\]
in a ball in \(\mathbb{R}^N\). Using global bifurcation technique, we obtain the existence of an unbounded branch of positive radial solutions, which is unbounded in positive \(\lambda\)-direction.Fractional Landweber method for an initial inverse problem for time-fractional wave equations.https://zbmath.org/1460.353752021-06-15T18:09:00+00:00"Huynh, Le Nhat"https://zbmath.org/authors/?q=ai:huynh.le-nhat"Zhou, Yong"https://zbmath.org/authors/?q=ai:zhou.yong|zhou.yong.1"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donal"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan.Summary: In this paper, we consider the initial inverse problem (backward problem) for an inhomogeneous time-fractional wave equation in a general bounded domain. We show that the backward problem is ill-posed, and we propose a regularizing scheme using a fractional Landweber regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.Global well-posedness and inviscid limit for the generalized Benjamin-Ono-Burgers equation.https://zbmath.org/1460.353122021-06-15T18:09:00+00:00"Chen, Mingjuan"https://zbmath.org/authors/?q=ai:chen.mingjuan"Guo, Boling"https://zbmath.org/authors/?q=ai:guo.boling"Han, Lijia"https://zbmath.org/authors/?q=ai:han.lijiaSummary: This paper deals with the Cauchy problem for the generalized Benjamin-Ono-Burgers equation \(\partial_tu+\mathcal{H}\partial_x^2u-vu_{xx}+\partial_x(u^{k+1}/(k+1))=0,k\geq 4\), where \(\mathcal{H}\) denotes Hilbert transform. We obtain its global well-posedness results in Besov Spaces if \(k\geq 4\) and the initial data in \(\dot B^{s_k}_{2,1}\) are sufficiently small, where \(s_k:=1/2-1/k\) corresponds to the critical scaling regularity index. Furthermore, we prove its global well-posedness and inviscid limit behavior in Sobolev spaces.The sharp time decay rate of the isentropic Navier-Stokes system in \(\mathbb{R}\).https://zbmath.org/1460.352522021-06-15T18:09:00+00:00"Chen, Yuhui"https://zbmath.org/authors/?q=ai:chen.yuhui"Pan, Ronghua"https://zbmath.org/authors/?q=ai:pan.ronghua"Tong, Leilei"https://zbmath.org/authors/?q=ai:tong.leileiSummary: We investigate the sharp time decay rates of the solution \(U\) for the compressible Navier-Stokes system (1.1) in \(\mathbb{R}^3\) to the constant equilibrium \((\bar\rho>0,0)\) when the initial data is a small smooth perturbation of \((\bar\rho, 0)\). Let \(\widetilde{U}\) be the solution to the corresponding linearized equations with the same initial data. Under a mild non-degenerate condition on initial perturbations, we show that \(\|U-\widetilde{U}\|_{L^2}\) decays at least at the rate of \((1+t)^{-\frac{5}{4}}\), which is faster than the rate \((1+t)^{-\frac{3}{4}}\) for the \(\widetilde{U}\) to its equilibrium \((\bar\rho,0)\). Our method is based on a combination of the linear sharp decay rate obtained from the spectral analysis and the energy estimates.Positive multipeak solutions to a zero mass problem in exterior domains.https://zbmath.org/1460.353622021-06-15T18:09:00+00:00"Clapp, Mónica"https://zbmath.org/authors/?q=ai:clapp.monica"Maia, Liliane A."https://zbmath.org/authors/?q=ai:maia.liliane-a"Pellacci, Benedetta"https://zbmath.org/authors/?q=ai:pellacci.benedettaThe authors establish the existence of positive multipeak solutions to the nonlinear scalar field equation with zero mass, \[- \triangle u = f(u), \quad u \in D_0^{1, 2} (\Omega_R),\] where \(\Omega _R := \{x \in \mathbb R^N : \; \vert u \vert > R \}\) with \(R > 0, \; N \geq 4\), and the nonlinearity \(f\) is subcritical at infinity and supercritical near the origin. They show that the number of positive multipeak solutions becomes arbitrarily large as \(R \to \infty\).
Reviewer: Anthony D. Osborne (Keele)Multiple positive solutions for coupled Schrödinger equations with perturbations.https://zbmath.org/1460.351292021-06-15T18:09:00+00:00"Li, Haoyu"https://zbmath.org/authors/?q=ai:li.haoyu"Wang, Zhi-Qiang"https://zbmath.org/authors/?q=ai:wang.zhiqiang|wang.zhi-qiangSummary: For coupled Schrödinger equations with nonhomogeneous perturbations we give several results on the existence of multiple positive solutions. In particular in one case we consider perturbations of the permutation symmetry.Weak solutions to the Muskat problem with surface tension via optimal transport.https://zbmath.org/1460.651142021-06-15T18:09:00+00:00"Jacobs, Matt"https://zbmath.org/authors/?q=ai:jacobs.matthew"Kim, Inwon"https://zbmath.org/authors/?q=ai:kim.inwon-christina"Mészáros, Alpár R."https://zbmath.org/authors/?q=ai:meszaros.alpar-richardThe global existence of weak solutions for the Muskat problem with surface tension, based on its gradient flow structure is obtained. The paper is organized as follows. Section 1 is an introduction. In the same section, the statement of the problem and its variational formulation are given. The main theorem of the paper is also formulated in Section 1. In Section 2, the basic properties of the minimizing movements scheme are derived and discrete-time quantities are constructed. The existence of pressure as a Lagrange multiplier for the incompressibility constraint is derived and the Euler-Lagrange equation for the minimization problem is obtained. In Section 3, weak solutions to the Muskat problem are obtained, under the assumption that the internal energy of the discrete solutions converges to the internal energy of the limiting solutions. The main task in Section 3 amounts to showing that one can pass to the limit in the Euler-Lagrange equation obtained in Section 2. In Section 4, several numerical examples with illustrations are given and discussed. Finally, in Appendix A, the results that are used when passing to the limit the weak curvature equation are recalled from the following paper [\textit{T. Laux} and \textit{F. Otto}, Calc. Var. Partial Differ. Equ. 55, No. 5, Paper No. 129, 74 p. (2016; Zbl 1388.35121)].
Reviewer: Temur A. Jangveladze (Tbilisi)Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity.https://zbmath.org/1460.351422021-06-15T18:09:00+00:00"Liu, Chungen"https://zbmath.org/authors/?q=ai:liu.chungen"Zhang, Huabo"https://zbmath.org/authors/?q=ai:zhang.huaboSummary: In this paper, we consider the existence of a ground state nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following fractional critical problem
\[
\begin{cases}
(a+b\int_{\mathbb{R}^3}(|(-\Delta)^{\alpha/2}u|^2)dx)(-\Delta)^{\alpha}u+V(x)u+K(x)\phi u=|u|^{2^{\ast}-2}u+\kappa f(x,u),\\
(-\Delta)^{\beta}\phi=K(x)u^2,\quad x\in\mathbb{R}^3,
\end{cases}
\]
where \(a,b,\kappa\) are positive parameters, \(\alpha\in(\frac{3}{4},1),\beta\in(0,1)\), and \(2^{\ast}_{\alpha}=\frac{6}{3-2\alpha}\), \((-\Delta)^{\alpha}\) stands for the fractional Laplacian. By the nodal Nehari manifold method, for each \(b>0\), we obtain a ground state nodal solution \(u_b\) and a ground-state solution \(v_b\) to this problem when \(\kappa\gg 1\), where the nonlinear function \(f:\mathbb{R}^3\times\mathbb{R}\rightarrow\mathbb{R}\) is a Carathéodory function. We also give an analysis on the behavior of \(u_b\) as the parameter \(b\to 0\).Further regularity and uniqueness results for a non-isothermal Cahn-Hilliard equation.https://zbmath.org/1460.352892021-06-15T18:09:00+00:00"Ipocoana, Erica"https://zbmath.org/authors/?q=ai:ipocoana.erica"Zafferi, Andrea"https://zbmath.org/authors/?q=ai:zafferi.andreaSummary: The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial \(L^{\infty}\) estimate for the temperature, obtained by a suitable Moser iteration scheme. Our results in particular allow us to get a new simplified version of the uniqueness proof for the considered model.Random data theory for the cubic fourth-order nonlinear Schrödinger equation.https://zbmath.org/1460.353232021-06-15T18:09:00+00:00"Dinh, Van Duong"https://zbmath.org/authors/?q=ai:dinh.van-duongSummary: We consider the cubic nonlinear fourth-order Schrödinger equation
\[
i\partial_tu-\Delta^2u+\mu\Delta u=\pm |u|^2u, \quad \mu\geq 0
\]
on \(\mathbb{R}^N\), \(N\geq 5\) with random initial data. We prove almost sure local well-posedness below the scaling critical regularity. We also prove probabilistic small data global well-posedness and scattering. Finally, we prove the global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.Existence of global weak solutions for a 3D Navier-Stokes-Poisson-Korteweg equations.https://zbmath.org/1460.352992021-06-15T18:09:00+00:00"Yang, Jianwei"https://zbmath.org/authors/?q=ai:yang.jianwei"Wang, Zhengyan"https://zbmath.org/authors/?q=ai:wang.zhengyan"Ding, Fengxia"https://zbmath.org/authors/?q=ai:ding.fengxiaSummary: The purpose of this work is to study the global-in-time existence of weak solutions of a viscous capillary model of plasma expressed as a so-called Navier-Stokes-Poisson-Korteweg model for large data in three-dimensional space. Using the compactness argument, we prove the existence of global weak solutions in the classical sense to such system with a cold pressure.On the Cahn-Hilliard equation with mass source for biological applications.https://zbmath.org/1460.353512021-06-15T18:09:00+00:00"Fakih, Hussein"https://zbmath.org/authors/?q=ai:fakih.hussein"Mghames, Ragheb"https://zbmath.org/authors/?q=ai:mghames.ragheb"Nasreddine, Noura"https://zbmath.org/authors/?q=ai:nasreddine.nouraSummary: This article deals with some generalizations of the Cahn-Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn-Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn-Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity.https://zbmath.org/1460.353422021-06-15T18:09:00+00:00"Łazuka, Jarosław"https://zbmath.org/authors/?q=ai:lazuka.jaroslawThe paper is devoted to thermoelasticity of non-simple materials in a three-dimensional space. The mathematical model is described by a system of partial differential equations of fourth order. After defining a suitable evolution equation, the existence of the solution to the Cauchy problem is proven by applying semigroup methods. An asymptotic analysis of the solution is developed. The explicit Fourier representation of the solution is derived. By employing Sobolev, Bessel and Besov spaces and by applying the interpolation method, the author shows the \(L^p-L^q\) time decay estimates for the solution.
Reviewer: Adina Chirila (Braşov)The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field.https://zbmath.org/1460.580092021-06-15T18:09:00+00:00"Alves, Claudianor O."https://zbmath.org/authors/?q=ai:alves.claudianor-oliveira"Nemer, Rodrigo C. M."https://zbmath.org/authors/?q=ai:nemer.rodrigo-c-m"Soares, Sergio H. Monari"https://zbmath.org/authors/?q=ai:soares.sergio-h-monariSummary: Nonlinear Schrödinger equations with an external magnetic field and a power nonlinearity with subcritical exponent \(p\) are considered. It is established a lower bound to the number of nontrivial solutions to these equations in terms of the topology of the domains in which the problem is given if \(p\) is suitably close to the critical exponent \(2^*=2N/(N-2)\), \(N\geq 3\). To prove this lower bound, based on a proof of a result of Benci and Cerami, it is provided an abstract result that establishes Morse relations that are used to count solutions.On the well-posedness of a nonlinear pseudo-parabolic equation.https://zbmath.org/1460.352122021-06-15T18:09:00+00:00"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Au, Vo Van"https://zbmath.org/authors/?q=ai:au.vo-van"Tri, Vo Viet"https://zbmath.org/authors/?q=ai:tri.vo-viet"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper we consider the Cauchy problem for the pseudo-parabolic equation:
\[
\dfrac{\partial}{\partial t}(u+\mu(-\Delta)^{s_1}u)+(-\Delta)^{s_2}u=f(u),\quad x\in\Omega,\,t>0.
\]
Here, the orders \(s_1,s_2\) satisfy \(0<s_1\neq s_2 <1\) (order of diffusion-type terms). We establish the local well-posedness of the solutions to the Cauchy problem when the source \(f\) is globally Lipschitz. In the case when the source term \(f\) satisfies a locally Lipschitz condition, the existence in large time, blow-up in finite time and continuous dependence on the initial data of the solutions are given.Non-local to local transition for ground states of fractional Schrödinger equations on \(\mathbb{R}^N\).https://zbmath.org/1460.353172021-06-15T18:09:00+00:00"Bieganowski, Bartosz"https://zbmath.org/authors/?q=ai:bieganowski.bartosz"Secchi, Simone"https://zbmath.org/authors/?q=ai:secchi.simoneSummary: We consider the nonlinear fractional problem
\[
(-\Delta )^su+V(x)u=f(x,u)\text{ in }\mathbb{R}^N
\]
We show that ground state solutions converge (along a subsequence) in \(L^2_{\text{loc}}(\mathbb{R}^N)\), under suitable conditions on \(f\) and \(V\), to a ground state solution of the local problem as \(s\rightarrow 1^-\).Sparsity-based nonlinear reconstruction of optical parameters in two-photon photoacoustic computed tomography.https://zbmath.org/1460.780112021-06-15T18:09:00+00:00"Gupta, Madhu"https://zbmath.org/authors/?q=ai:gupta.madhu-s"Kumar Mishra, Rohit"https://zbmath.org/authors/?q=ai:mishra.rohit-kumar"Roy, Souvik"https://zbmath.org/authors/?q=ai:roy.souvikVanishing viscosity limit to the 3D Burgers equation in Gevrey class.https://zbmath.org/1460.350172021-06-15T18:09:00+00:00"Selmi, Ridha"https://zbmath.org/authors/?q=ai:selmi.ridha"Chaabani, Abdelkerim"https://zbmath.org/authors/?q=ai:chaabani.abdelkerimSummary: We consider the Cauhcy problem to the 3D diffusive periodic Burgers equation. We prove that a unique solution exists on time interval independent of the viscosity and tends, as the viscosity vanishes, to the solution of the limiting equation, the inviscid periodic three-dimensional Burgers equation, in Gevrey-Sobolev spaces. Compared to Navier-Stokes equations, the main difficulties come from the lack of the divergence-free condition which is essential to handle the nonlinear term. Our alternative tool will be to use a change of functions to estimate nonlinearities. Fourier analysis and compactness methods are widely used.Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity. II: Supercritical blow-up profiles.https://zbmath.org/1460.353282021-06-15T18:09:00+00:00"Holmer, Justin"https://zbmath.org/authors/?q=ai:holmer.justin"Liu, Chang"https://zbmath.org/authors/?q=ai:liu.chang.1Summary: We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity,
\[
i\partial_t\psi+\partial_x^2\psi+\delta|\psi|^{p-1}\psi=0,\tag{(0.1)}
\]
where \(\delta=\delta(x)\) is the delta function supported at the origin. In the \(L^2\) supercritical setting \(p>3\), we construct self-similar blow-up solutions belonging to the energy space \(L_x^\infty\cap\dot H_x^1\). This is reduced to finding outgoing solutions of a certain stationary profile equation. All outgoing solutions to the profile equation are obtained by using parabolic cylinder functions (Weber functions) and solving the jump condition at \(x=0\) imposed by the \(\delta\) term in (0.1). This jump condition is an algebraic condition involving gamma functions, and existence and uniqueness of solutions is obtained using the intermediate value theorem and formulae for the digamma function. We also compute the form of these outgoing solutions in the slightly supercritical case \(0<p-3\ll 1\) using the log Binet formula for the gamma function and steepest descent method in the integral formulae for the parabolic cylinder functions.
For part I, see [the authors, J. Math. Anal. Appl. 483, No. 1, Article ID 123522, 20 p. (2020; Zbl 1436.35290)].Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains.https://zbmath.org/1460.351512021-06-15T18:09:00+00:00"D'Aprile, Teresa"https://zbmath.org/authors/?q=ai:daprile.teresaSummary: We are concerned with the existence of blowing-up solutions to the following boundary value problem
\[
-\Delta u=\lambda V(x)e^u-4\pi N\boldsymbol{\delta}_0\text{ in }\Omega,\quad u=0\text{ on }\partial\Omega,
\]
where \(\Omega\) is a smooth and bounded domain in \(\mathbb{R}^2\) such that \(0\in\Omega\), \(V\) is a positive smooth potential, \(N\) is a positive integer and \(\lambda>0\) is a small parameter. Here \(\boldsymbol{\delta}_0\) defines the Dirac measure with pole at \(0\). We assume that \(\Omega\) is \((N+1)\)-symmetric and we find conditions on the potential \(V\) and the domain \(\Omega\) under which there exists a solution blowing up at \(N+1\) points located at the vertices of a regular polygon with center \(0\).Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation.https://zbmath.org/1460.353152021-06-15T18:09:00+00:00"Ardila, Alex H."https://zbmath.org/authors/?q=ai:ardila.alex-hernandez"Cardoso, Mykael"https://zbmath.org/authors/?q=ai:cardoso.mykaelSummary: Using variational methods we study the stability and strong instability of ground states for the focusing inhomogeneous nonlinear Schrödinger equation (INLS)
\[
i\partial_tu+\Delta u+|x|^{-b}|u|^{p-1}u=0.
\]
We construct two kinds of invariant sets under the evolution flow of (INLS). Then we show that the solution of (INLS) is global and bounded in \(H^1(\mathbb{R}^N)\) in the first kind of the invariant sets, while the solution blow-up in finite time in the other invariant set. Consequently, we prove that if the nonlinearity is \(L^2\)-supercritical, then the ground states are strongly unstable by blow-up.Scattering of the focusing energy-critical NLS with inverse square potential in the radial case.https://zbmath.org/1460.353352021-06-15T18:09:00+00:00"Yang, Kai"https://zbmath.org/authors/?q=ai:yang.kaiSummary: We consider the Cauchy problem of the focusing energy-critical nonlinear Schrödinger equation with an inverse square potential. We prove that if any radial solution obeys the supreme of the kinetic energy over the maximal lifespan is below the kinetic energy of the ground state solution, then the solution exists globally in time and scatters in both time directions.Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: convergence for classical solutions.https://zbmath.org/1460.352942021-06-15T18:09:00+00:00"Luo, Yi-Long"https://zbmath.org/authors/?q=ai:luo.yi-long"Ma, Yangjun"https://zbmath.org/authors/?q=ai:ma.yangjunSummary: In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number \(\epsilon\) for both the compressible system and its differential system with respect to time under uniformly in \(\epsilon\) small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as \(\epsilon\rightarrow 0\), so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of \textit{F. De Anna} and \textit{A. Zarnescu} [J. Differ. Equations 264, No. 2, 1080--1118 (2018; Zbl 1393.35165)]. Moreover, we also obtain the convergence rates associated with \(L^2\)-norm with well-prepared initial data.Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials.https://zbmath.org/1460.351742021-06-15T18:09:00+00:00"Carvalho, Marcos L. M."https://zbmath.org/authors/?q=ai:carvalho.marcos-l-m"Silva, Edcarlos D."https://zbmath.org/authors/?q=ai:da-silva.edcarlos-domingos"Goulart, Claudiney"https://zbmath.org/authors/?q=ai:goulart.claudiney"Santos, Carlos A."https://zbmath.org/authors/?q=ai:santos.carlos-alberto-pSummary: It is established existence of bound and ground state solutions for quasilinear elliptic systems driven by \((\Phi_1,\Phi_2)\)-Laplacian operator. The main feature here is to consider quasilinear elliptic systems involving both nonsingular nonlinearities combined with indefinite potentials and singular cases perturbed by superlinear and subcritical couple terms. These prevent us to use arguments based on Ambrosetti-Rabinowitz condition and variational methods for differentiable functionals. By exploring the Nehari method and doing a fine analysis on the fibering map associated, we get estimates that allow us unify the arguments to show multiplicity of semi-trivial solutions in both cases.Modulation instability and optical solitons of Radhakrishnan-Kundu-Lakshmanan model.https://zbmath.org/1460.780232021-06-15T18:09:00+00:00"Raza, Nauman"https://zbmath.org/authors/?q=ai:raza.nauman"Javid, Ahmad"https://zbmath.org/authors/?q=ai:javid.ahmadSummary: This paper studies the solitons of Radhakrishnan-Kundu-Lakshmanan (RKL) model with power law nonlinearity. The modified simple equation method and \(\exp(-\varphi(q))\) method are presented as integration mechanisms. Dark, bright, singular and periodic soliton solutions are extracted as well as the constraint conditions for their existence. A prized discussion on the stability of these soliton profiles on the basis of index of the power law nonlinearity is also carried out with the help of physical description of solutions. The integration techniques have been proved to be extremely efficient and robust to find new optical solitary wave solutions for various nonlinear evolution equations describing optical pulse propagation. Moreover, using linear stability analysis, modulation instability of the RKL model is studied. Different effects contributing to the modulation instability spectrum gain are analyzed.Existence of solutions for generalized \(p(x)\)-Laplacian systems.https://zbmath.org/1460.351262021-06-15T18:09:00+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 will study the existence of weak solutions for a class of nonlinear \(p(x)\)-Laplace system. The techniques of Young measure valued solutions are used to achieve the needed results.Multiple positive solutions for a class of quasilinear singular elliptic systems.https://zbmath.org/1460.351762021-06-15T18:09:00+00:00"Didi, Hana"https://zbmath.org/authors/?q=ai:didi.hana"Moussaoui, Abdelkrim"https://zbmath.org/authors/?q=ai:moussaoui.abdelkrimSummary: In this paper we establish the existence of two positive solutions for a class of quasilinear singular elliptic systems. The main tools are sub and supersolution method and Leray-Schauder Topological degree.Non-uniform dependence on initial data for the Camassa-Holm equation in the critical Besov space.https://zbmath.org/1460.352912021-06-15T18:09:00+00:00"Li, Jinlu"https://zbmath.org/authors/?q=ai:li.jinlu"Wu, Xing"https://zbmath.org/authors/?q=ai:wu.xing"Yu, Yanghai"https://zbmath.org/authors/?q=ai:yu.yanghai"Zhu, Weipeng"https://zbmath.org/authors/?q=ai:zhu.weipengSummary: Whether or not the data-to-solution map of the Cauchy problem for the Camassa-Holm equation and Novikov equation in the critical Besov space \(B_{2,1}^{3/2}(\mathbb{R})\) is uniformly continuous remains open. In the paper, we aim at solving the open question left in the previous works [the first author et al., J. Differ. Equations 269, No. 10, 8686--8700 (2020; Zbl 1442.35344); J. Math. Fluid Mech. 22, No. 4, Paper No. 50, 10 p. (2020; Zbl 1448.35402)] and giving a negative answer to this problem.Infinitely many solutions for quasilinear elliptic equations without Ambrosetti-Rabinowitz condition and lack of symmetry.https://zbmath.org/1460.351572021-06-15T18:09:00+00:00"Gui, Xue-lin"https://zbmath.org/authors/?q=ai:gui.xue-lin"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.binSummary: In this paper, we consider the existence of solutions for the quasilinear elliptic problem:
\[
\begin{cases}
-\operatorname{div}(A(x,u) \nabla u) + \frac{1}{2} A_t (x,u) |\nabla u|^2 = g(x,u) + h(x), \quad &\text{in } \Omega, \\
u = 0, & \text{on } \partial \Omega,
\end{cases}\tag{\(P_1\)}
\]
where \(\Omega \subset \mathbb{R}^N\) is an open bounded domain, \(N \geq 3\), the real term \(A(x,t)\), \(A_t(x,t) = \frac{\partial A}{\partial t}(x, t)\) and \(g(x,t)\) satisfy Carathéodory condition on \(\Omega \times \mathbb{R}\) and \(h : \Omega \to \mathbb{R}\) is a given measurable function. The intention of the article is to get new results of the existence of infinitely many weak solutions of the problem by weaken the Ambrosetti and Rabinowitz condition. We use a variant of perturbation techniques introduced by \textit{P. H. Rabinowitz} [Trans. Am. Math. Soc. 272, 753--769 (1982; Zbl 0589.35004)] to overcome the lack of symmetry. This extends the previous results.Positive solutions for a critical elliptic problem involving singular nonlinearity.https://zbmath.org/1460.351552021-06-15T18:09:00+00:00"Lei, Chunyu"https://zbmath.org/authors/?q=ai:lei.chunyu"Zheng, Tiantian"https://zbmath.org/authors/?q=ai:zheng.tiantian"Fan, Haining"https://zbmath.org/authors/?q=ai:fan.hainingSummary: We study the existence of positive solutions for a semi-linear elliptic equation involving both critical growth and singular nonlinearity. By applying the Nehari manifold and Lusternik-Schnirelmann category theory to an auxiliary problem, we prove the existence of multiple positive solutions.Existence of a stationary Navier-Stokes flow past a rigid body, with application to starting problem in higher dimensions.https://zbmath.org/1460.352652021-06-15T18:09:00+00:00"Takahashi, Tomoki"https://zbmath.org/authors/?q=ai:takahashi.tomokiSummary: We consider the large time behavior of the Navier-Stokes flow past a rigid body in \(\mathbb{R}^n\) with \(n\geq 3\). We first construct a small stationary solution possessing the optimal summability at spatial infinity, which is the same as that of the Oseen fundamental solution. When the translational velocity of the body gradually increases and is maintained after a certain finite time, we then show that the nonstationary fluid motion converges to the stationary solution corresponding to a small terminal velocity of the body as time \(t\rightarrow \infty\) in \(L^q\) with \(q\in [n,\infty]\). This is called Finn's starting problem and the three-dimensional case was affirmatively solved by \textit{G. P. Galdi} et al. [Arch. Ration. Mech. Anal. 138, No. 4, 307--318 (1997; Zbl 0898.35071)]. The present paper extends the latter cited paper to the case of higher dimensions. Even for the three-dimensional case, our theorem provides new convergence rate, that is determined by the summability of the stationary solution at infinity and seems to be sharp.Dispersionless multi-dimensional integrable systems and related conformal structure generating equations of mathematical physics.https://zbmath.org/1460.170402021-06-15T18:09:00+00:00"Hentosh, Oksana Ye."https://zbmath.org/authors/?q=ai:hentosh.oksana-ye"Prykarpatsky, Yarema A."https://zbmath.org/authors/?q=ai:prykarpatsky.yarema-anatoliyovych"Blackmore, Denis"https://zbmath.org/authors/?q=ai:blackmore.denis-l"Prykarpatski, Anatolij K."https://zbmath.org/authors/?q=ai:prykarpatsky.anatoliy-karolevychSummary: Using diffeomorphism group vector fields on \(\mathbb{C}\)-multiplied tori and the related Lie-algebraic structures, we study multi-dimensional dispersionless integrable systems that describe conformal structure generating equations of mathematical physics. An interesting modification of the devised Lie-algebraic approach subject to spatial-dimensional invariance and meromorphicity of the related differential-geometric structures is described and applied in proving complete integrability of some conformal structure generating equations. As examples, we analyze the Einstein-Weyl metric equation, the modified Einstein-Weyl metric equation, the Dunajski heavenly equation system, the first and second conformal structure generating equations and the inverse first Shabat reduction heavenly equation. We also analyze the modified Plebański heavenly equations, the Husain heavenly equation and the general Monge equation along with their multi-dimensional generalizations. In addition, we construct superconformal analogs of the Whitham heavenly equation.Global solvability results for parabolic equations with \(p\)-Laplacian type diffusion.https://zbmath.org/1460.352062021-06-15T18:09:00+00:00"Chagas, J. Q."https://zbmath.org/authors/?q=ai:chagas.j-q"Guidolin, P. L."https://zbmath.org/authors/?q=ai:guidolin.p-l"Zingano, P. R."https://zbmath.org/authors/?q=ai:zingano.paulo-r-aSummary: We give conditions that assure global existence of bounded weak solutions to the Cauchy problem of general conservative 2nd-order parabolic equations with \(p\)-Laplacian type diffusion \((p>2)\) and initial data \(u_0\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\). Related results of interest are also given.Nonexistence results for systems of parabolic differential inequalities in 2D exterior domains.https://zbmath.org/1460.354012021-06-15T18:09:00+00:00"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamed"Samet, Bessem"https://zbmath.org/authors/?q=ai:samet.bessemSummary: In this paper, we study, for the first time, the nonexistence of solutions to systems of parabolic differential inequalities in 2D exterior domains with Dirichlet and Neumann boundary conditions. Our obtained results complete those derived recently by \textit{Y. Sun} [Pac. J. Math. 293, No. 1, 245--256 (2018; Zbl 1379.35163)] in the \(N\)-dimensional case, \(N\geqslant 3\), under Dirichlet boundary condition.On the solvability of a boundary-value problem for second-order partial differential operator equations.https://zbmath.org/1460.350752021-06-15T18:09:00+00:00"Mirzoev, S. S."https://zbmath.org/authors/?q=ai:mirzoev.sabir-s"Dzhafarov, I. Dzh."https://zbmath.org/authors/?q=ai:dzhafarov.i-dzhFrom the text: In this paper, we indicate sufficient conditions that ensure a regular solvability of the title problem. These conditions are expressed only by the coefficients of the operator differential equation.The role of the mean curvature in a mixed Hardy-Sobolev trace inequality.https://zbmath.org/1460.350082021-06-15T18:09:00+00:00"Thiam, El Hadji Abdoulaye"https://zbmath.org/authors/?q=ai:thiam.el-hadji-abdoulayeSummary: Let \(\Omega\) be a smooth bounded domain of \(\mathbb{R}^{N+1}\) of boundary \(\partial \Omega = \Gamma_1 \cup \Gamma_2\) and such that \(\partial \Omega \cap \Gamma_2\) is a neighborhood of \(0, h \in \mathcal{C}^0(\partial \Omega \cap \Gamma_2)\) and \(s \in [0, 1)\). We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions
\[
\begin{aligned}
\begin{cases}
\Delta u= 0 &\qquad \text{in } \Omega \\
u=0 &\qquad \text{on } \Gamma_1 \\
\frac{\partial u}{\partial \nu} = h(x) u + \frac{u^{q(s)-1}}{d(x)^s} &\qquad \text{on } \Gamma_2,
\end{cases}
\end{aligned}
\tag{\(0.1\)}
\]
where \(q(s):=\frac{2(N-s)}{N-1}\) is the critical Hardy-Sobolev trace exponent and \(\nu\) is the outer unit normal of \(\partial \Omega \). In particular, we prove existence of minimizers when \(N \geq 3\) and the mean curvature is sufficiently below the potential \(h\) at 0.
For the entire collection see [Zbl 1458.00035].Conditional estimates in three-dimensional chemotaxis-Stokes systems and application to a Keller-Segel-fluid model accounting for gradient-dependent flux limitation.https://zbmath.org/1460.353582021-06-15T18:09:00+00:00"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThe goal of this paper is to study effects of the Stokes flow on the fully parabolic chemotaxis system with suitable flux limitation in the equation for the evolution of the density of population. Results on the absence of blowup of solutions obtained in the three-dimensional case are similar to those for the chemotaxis system without fluid. General estimates derived for fluid motion and taxis gradients have also an independent interest for study of global-in-time existence of bounded solutions in related problems.
Reviewer: Piotr Biler (Wrocław)On the cutoff approximation for the Boltzmann equation with long-range interaction.https://zbmath.org/1460.352492021-06-15T18:09:00+00:00"He, Ling-Bing"https://zbmath.org/authors/?q=ai:he.lingbing"Jiang, Jin-Cheng"https://zbmath.org/authors/?q=ai:jiang.jin-cheng"Zhou, Yu-Long"https://zbmath.org/authors/?q=ai:zhou.yulongSummary: The Boltzmann collision operator for long-range interactions is usually employed in its ``weak form'' in the literature. However the weak form utilizes the symmetry property of the spherical integral and thus should be understood more or less in the \textit{principle value} sense especially for strong angular singularity. To study the integral in the Lebesgue sense, it is natural to define the collision operator via the cutoff approximation. In this way, we give a rigorous proof to the local well-posedness of the Boltzmann equation with the long-range interactions. The result has the following main features and innovations: (1). The initial data is not necessarily a small perturbation around equilibrium but satisfies \textit{compatible conditions}. (2). A quasi-linear method instead of the standard linearization method is used to prove existence and non-negativity of the solution in a suitably designed energy space depending heavily on the initial data. In such space, we derive the first uniqueness result for the equation in particular for hard potential case.Long time existence of solutions to an elastic flow of networks.https://zbmath.org/1460.350362021-06-15T18:09:00+00:00"Garcke, Harald"https://zbmath.org/authors/?q=ai:garcke.harald"Menzel, Julia"https://zbmath.org/authors/?q=ai:menzel.julia-h"Pluda, Alessandra"https://zbmath.org/authors/?q=ai:pluda.alessandraSummary: The \(L^2\)-gradient flow of the elastic energy of networks leads to a Willmore type evolution law with non-trivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition, we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.Existence and uniqueness of nontrivial radial solutions for \(k\)-Hessian equations.https://zbmath.org/1460.351492021-06-15T18:09:00+00:00"Zhang, Xuemei"https://zbmath.org/authors/?q=ai:zhang.xuemeiSummary: We prove existence and uniqueness of nontrivial radial solutions to the \(k\)-Hessian problem
\[
\begin{cases}
S_k (D^2u) = \lambda f (-u) \quad &\text{in } \Omega,\\
u = 0 &\text{on } \partial \Omega,
\end{cases}
\]
where \(S_k( D^2 u)\) is the \(k\)-Hessian operator of \(u\), \(\Omega\) is the open unit ball in \(\mathbb{R}^n\), \(f\) is a continuous function and may have \(k\)-sublinear growth at \(\infty\) with possible \(k\)-superlinear growth at 0, and \(\lambda\) is a large parameter.A mixed-primal finite element method for the coupling of Brinkman-Darcy flow and nonlinear transport.https://zbmath.org/1460.651382021-06-15T18:09:00+00:00"Alvarez, Mario"https://zbmath.org/authors/?q=ai:alvarez.mario-m"Gatica, Gabriel N."https://zbmath.org/authors/?q=ai:gatica.gabriel-n"Ruiz-Baier, Ricardo"https://zbmath.org/authors/?q=ai:ruiz-baier.ricardoSummary: This paper is devoted to the mathematical and numerical analysis of a model describing the interfacial flow-transport interaction in a porous-fluidic domain. The medium consists of a highly permeable material, where the flow of an incompressible viscous fluid is governed by Brinkman equations (written in terms of vorticity, velocity and pressure), and a porous medium where Darcy's law describes fluid motion using filtration velocity and pressure. Gravity and the local fluctuations of a scalar field (representing for instance, the solids volume fraction or the concentration of a contaminant) are the main drivers of the fluid patterns on the whole domain, and the Brinkman-Darcy equations are coupled to a nonlinear transport equation accounting for mass balance of the scalar concentration. We introduce a mixed-primal variational formulation of the problem and establish existence and uniqueness of solution using fixed-point arguments and small-data assumptions. A family of Galerkin discretizations that produce divergence-free discrete velocities is also presented and analysed using similar tools to those employed in the continuous problem. Convergence of the resulting mixed-primal finite element method is proven, and some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported.On a stationary Schrödinger equation with periodic magnetic potential.https://zbmath.org/1460.353162021-06-15T18:09:00+00:00"Bégout, Pascal"https://zbmath.org/authors/?q=ai:begout.pascal"Schindler, Ian"https://zbmath.org/authors/?q=ai:schindler.ianSummary: We prove existence results for a stationary Schrödinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.Sturm-Picone theorem for fractional nonlocal equations.https://zbmath.org/1460.351462021-06-15T18:09:00+00:00"Tyagi, J."https://zbmath.org/authors/?q=ai:tyagi.jagmohan|tyagi.jai-kishoreSummary: We establish a generalization of Sturm-Picone comparison theorem for a pair of fractional nonlocal equations:
\[
\begin{aligned}
(-\operatorname{div}. (A_1(x)\nabla))^s u &= C_1(x) u \quad \text{in }\Omega, \\
u &=0\quad\text{on }\partial \Omega,
\end{aligned}
\]
and
\[
\begin{aligned}
(-\operatorname{div}. (A_2(x)\nabla))^s v &= C_2(x) v \quad\text{in }\Omega \\
v &=0 \quad\text{on }\partial \Omega,
\end{aligned}
\]
where \(\Omega \subset \mathbb{R}^n\) is an open bounded subset with smooth boundary, \(0<s<1\), \(A_1\), \(A_2\) are smooth, real symmetric and positive definite matrices on \(\overline{\Omega}\) and \(C_1\), \(C_2\in C^\alpha (\overline{\Omega})\).Periodic solutions to Navier-Stokes equations on non-compact Einstein manifolds with negative curvature.https://zbmath.org/1460.352632021-06-15T18:09:00+00:00"Nguyen, Thieu Huy"https://zbmath.org/authors/?q=ai:nguyen-thieu-huy."Pham, Truong Xuan"https://zbmath.org/authors/?q=ai:pham.truong-xuan"Vu, Thi Ngoc Ha"https://zbmath.org/authors/?q=ai:vu-thi-ngoc-ha."Vu, Thi Mai"https://zbmath.org/authors/?q=ai:vu.thi-maiSummary: Consider the Navier-Stokes Equations (NSE) for viscous incompressible fluid flows on a non-compact, smooth, simply-connected and complete Einstein manifold \((\mathbf{M},g)\) with negative Ricci curvature tensor. We prove the existence and uniqueness of a time-periodic solution to NSE for vector fields on \((\mathbf{M},g)\). Our method is based on the dispersive and smoothing properties of the semigroup generated by the linearized Stokes equations to construct a bounded (in time) solution of the nonhomogeneous Stokes equation and on the ergodic method to obtain the periodic solution to Stokes equation. Then, using the fixed point arguments, we can pass from the Stokes equations to Navier-Stokes equations to obtain periodic solutions to NSE on the Einstein manifold \((\mathbf{M},g)\). We also prove the stability of the periodic solution.Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces \(M_{p,q}^s(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\).https://zbmath.org/1460.353182021-06-15T18:09:00+00:00"Chaichenets, Leonid"https://zbmath.org/authors/?q=ai:chaichenets.leonid"Hundertmark, Dirk"https://zbmath.org/authors/?q=ai:hundertmark.dirk"Kunstmann, Peer Christian"https://zbmath.org/authors/?q=ai:kunstmann.peer-christian"Pattakos, Nikolaos"https://zbmath.org/authors/?q=ai:pattakos.nikolaosSummary: We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in [\textit{M. Sugimoto} et al., Integral Transforms Spec. Funct. 22, No. 4--5, 351--358 (2011; Zbl 1221.44007)], of the intersection \(M^s_{p,q}(\mathbb{R}^d)\cap M_{\infty,1}(\mathbb{R}^d)\) for \(d\in\mathbb{N}\), \(p,q\in [1,\infty]\) and \(s\geq 0\). We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves a theorem by \textit{Á. Bényi} and \textit{K. A. Okoudjou} [Bull. Lond. Math. Soc. 41, No. 3, 549--558 (2009; Zbl 1173.35115)], where only the case \(q=1\) is considered, and closes a gap in the literature. If \(q>1\) and \(s>d\left(1-\frac{1}{q}\right)\) or if \(q=1\) and \(s\geq 0\) then \(M^s_{p,q}(\mathbb{R}^d)\hookrightarrow M_{\infty,1}(\mathbb{R}^d)\) and the above intersection is superfluous. For this case we also reobtain a Hölder-type inequality for modulation spaces.
For the entire collection see [Zbl 1457.35005].Solvability of some integro-differential equations with anomalous diffusion in higher dimensions.https://zbmath.org/1460.353672021-06-15T18:09:00+00:00"Vougalter, Vitali"https://zbmath.org/authors/?q=ai:vougalter.vitaliSummary: The article deals with the studies of the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in \(\mathbb{R}^d\), \(d=4,5\). The proof of the existence of solutions relies on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.Multiplicity of positive solutions for an anisotropic problem via sub-supersolution method and mountain pass theorem.https://zbmath.org/1460.351352021-06-15T18:09:00+00:00"dos Santos, Gelson C. G."https://zbmath.org/authors/?q=ai:dos-santos.gelson-c-g"Figueiredo, Giovany"https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcher"Silva, Julio R. S."https://zbmath.org/authors/?q=ai:silva.julio-r-sSummary: We use the sub-supersolution method and the Mountain Pass Theorem in order to show existence and multiplicity of solution for an anisotropic problem given by
\[\begin{cases}-\left[\sum\limits^{N}_{i=1}\frac{\partial}{\partial x_{i}} \left(\left\vert \frac{\partial u}{\partial x_{i}}\right\vert^{pi-2} \frac{\partial u}{\partial x_{i}}\right)\ \right]=a(x)u+ h(x,u) \text{ in } \Omega,\\ \ u>0\text{ in }\Omega,\quad u=0\text{ on } \partial\Omega. \end{cases} \]
We also prove the uniqueness of the solution for the linear anisotropic problem, a Comparison Principle for the anisotropic operator and a regularity result.Well-posedness of a family of degenerate parabolic mixed equations.https://zbmath.org/1460.780262021-06-15T18:09:00+00:00"Acevedo, Ramiro"https://zbmath.org/authors/?q=ai:acevedo.ramiro"Gómez, Christian"https://zbmath.org/authors/?q=ai:gomez.christian"López-Rodríguez, Bibiana"https://zbmath.org/authors/?q=ai:lopez-rodriguez.bibianaSummary: In this study, we present an abstract framework for analyzing a family of linear degenerate parabolic mixed equations. We combine the theory of degenerate parabolic equations with the classical Babuška-Brezzi theory for linear mixed stationary equations to deduce sufficient conditions to prove the well-posedness of the problem. Finally, we illustrate the application of the abstract framework based on examples from physical science applications, including fluid dynamics models and electromagnetic problems.Stochastic PDEs via convex minimization.https://zbmath.org/1460.354032021-06-15T18:09:00+00:00"Scarpa, Luca"https://zbmath.org/authors/?q=ai:scarpa.luca"Stefanelli, Ulisse"https://zbmath.org/authors/?q=ai:stefanelli.ulisseSummary: We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a parameter-dependent convex functional on entire trajectories. Its unique minimizers correspond to elliptic-in-time regularizations of the stochastic differential problem. As the regularization parameter tends to zero, solutions of the limiting problem are recovered. This in particular provides a direct approach via convex optimization to the approximation of nonlinear stochastic partial differential equations.Wave and Klein-Gordon equations on certain locally symmetric spaces.https://zbmath.org/1460.353362021-06-15T18:09:00+00:00"Zhang, Hong-Wei"https://zbmath.org/authors/?q=ai:zhang.hongweiSummary: This paper is devoted to study the dispersive properties of the linear Klein-Gordon and wave equations on a class of locally symmetric spaces. As a consequence, we obtain the Strichartz estimate and prove global well-posedness results for the corresponding semilinear equation with low regularity data as on real hyperbolic spaces.Non-uniqueness of delta shocks and contact discontinuities in the multi-dimensional model of Chaplygin gas.https://zbmath.org/1460.352692021-06-15T18:09:00+00:00"Březina, Jan"https://zbmath.org/authors/?q=ai:brezina.jan"Kreml, Ondřej"https://zbmath.org/authors/?q=ai:kreml.ondrej"Mácha, Václav"https://zbmath.org/authors/?q=ai:macha.vaclavSummary: We study the Riemann problem for the isentropic compressible Euler equations in two space dimensions with the pressure law describing the Chaplygin gas. It is well known that there are Riemann initial data for which the 1D Riemann problem does not have a classical \textit{BV} solution, instead a \(\delta\)-shock appears, which can be viewed as a generalized measure-valued solution with a concentration measure in the density component. We prove that in the case of two space dimensions there exist infinitely many bounded admissible weak solutions starting from the same initial data. Moreover, we show the same property also for a subset of initial data for which the classical 1D Riemann solution consists of two contact discontinuities. As a consequence of the latter result we observe that any criterion based on the principle of maximal dissipation of energy will not pick the classical 1D solution as the physical one. In particular, not only the criterion based on comparing dissipation rates of total energy but also a stronger version based on comparing dissipation measures fails to pick the 1D solution.On the global well-posedness of the quadratic NLS on \(H^1(\mathbb{T}) + L^2(\mathbb{R})\).https://zbmath.org/1460.353192021-06-15T18:09:00+00:00"Chaichenets, L."https://zbmath.org/authors/?q=ai:chaichenets.leonid"Hundertmark, D."https://zbmath.org/authors/?q=ai:hundertmark.dirk"Kunstmann, P."https://zbmath.org/authors/?q=ai:kunstmann.peer-christian"Pattakos, N."https://zbmath.org/authors/?q=ai:pattakos.nikolaosSummary: We study the one dimensional nonlinear Schrödinger equation with power nonlinearity \(|u|^{\alpha-1} u\) for \(\alpha \in [1,5]\) and initial data \(u_0 \in H^1(\mathbb{T}) + L^2 (\mathbb{R})\). We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity \((\alpha=2)\) we obtain \textit{global} well-posedness in the space \(C(\mathbb{R}, H^1(\mathbb{T}) + L^2 (\mathbb{R}))\) via Gronwall's inequality.Strong solutions to the Cauchy problem of two-dimensional nonhomogeneous micropolar fluid equations with nonnegative density.https://zbmath.org/1460.353032021-06-15T18:09:00+00:00"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: We consider the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity on the whole space \(\mathbb{R}^2\). By weighted energy method, we show the local existence and uniqueness of strong solutions provided that the initial density decays not too slowly at infinity.Uniqueness and stability of the saddle-shaped solution to the fractional Allen-Cahn equation.https://zbmath.org/1460.351522021-06-15T18:09:00+00:00"Felipe-Navarro, Juan Carlos"https://zbmath.org/authors/?q=ai:felipe-navarro.juan-carlos"Sanz-Perela, Tomás"https://zbmath.org/authors/?q=ai:sanz-perela.tomasSummary: In this paper we prove the uniqueness of the saddle-shaped solution \(u\colon \mathbb{R}^{2m} \to \mathbb{R}\) to the semilinear nonlocal elliptic equation \((-\Delta)^\gamma u = f(u)\) in \(\mathbb{R}^{2m} \), where \(\gamma \in (0,1)\) and \(f\) is of Allen-Cahn type. Moreover, we prove that this solution is stable if \(2m\geq 14\). As a consequence of this result and the connection of the problem with nonlocal minimal surfaces, we show that the Simons cone \(\{(x', x'') \in \mathbb{R}^m\times \mathbb{R}^m: |x'| = |x''|\}\) is a stable nonlocal \((2\gamma)\)-minimal surface in dimensions \(2m\geq 14\). Saddle-shaped solutions of the fractional Allen-Cahn equation are doubly radial, odd with respect to the Simons cone, and vanish only in this set. It was known that these solutions exist in all even dimensions and are unstable in dimensions 2, 4, and 6. Thus, after our result, the stability remains an open problem only in dimensions 8, 10, and 12.
The importance of studying this type of solution is due to its relation with the fractional version of a conjecture by De Giorgi. Saddle-shaped solutions are the simplest non 1D candidates to be global minimizers in high dimensions, a property not yet established in any dimension.Nehari-type ground state solutions for superlinear elliptic equations with variable exponent in \(\mathbb{R}^N\).https://zbmath.org/1460.351772021-06-15T18:09:00+00:00"Ge, Bin"https://zbmath.org/authors/?q=ai:ge.bin"Zhang, Bei-Lei"https://zbmath.org/authors/?q=ai:zhang.bei-lei"Hou, Gang-Ling"https://zbmath.org/authors/?q=ai:hou.gang-lingSummary: We deal with the existence of Nehari-type ground state solutions for the superlinear \(p(x)\)-Laplacian equation
\[
-\triangle_{p(x)} u+V(x)|u|^{p(x)-2}u= f(x,u),\; x\in \mathbb{R}^N,\; u\in W^{1,p(x)}(\mathbb{R}^N).
\]
Under a weaker Nehari condition, we establish some existence criteria to guarantee that the above problem has Nehari-type ground state solutions using Non-Nehari manifold method.Multiple solutions for nonhomogeneous Schrödinger equations.https://zbmath.org/1460.351582021-06-15T18:09:00+00:00"Liang, Ruixi"https://zbmath.org/authors/?q=ai:liang.ruixi"Shang, Tingting"https://zbmath.org/authors/?q=ai:shang.tingtingSummary: In this article, we study the following nonhomogeneous Schrödinger equation
\[
-\Delta u+V(x)u-\Delta (u^2)u=f(x,u)+k(x),\quad x\in \mathbb{R}^N,
\]
where \(V(x)\) is a given positive potential. Under some suitable assumptions on \(V\), \(f\) and \(k\), the existence of multiple solutions is proved using the Ekeland's variational principle and the Mountain Pass Theorem in critical point theory.Verified computations for solutions to semilinear parabolic equations using the evolution operator.https://zbmath.org/1460.352162021-06-15T18:09:00+00:00"Takayasu, Akitoshi"https://zbmath.org/authors/?q=ai:takayasu.akitoshi"Mizuguchi, Makoto"https://zbmath.org/authors/?q=ai:mizuguchi.makoto"Kubo, Takayuki"https://zbmath.org/authors/?q=ai:kubo.takayuki"Oishi, Shin'ichi"https://zbmath.org/authors/?q=ai:oishi.shinichiSummary: This article presents a theorem for guaranteeing existence of a solution for an initial-boundary value problem of semilinear parabolic equations. The sufficient condition of our main theorem is derived by a fixed-point formulation using the evolution operator. We note that the sufficient condition can be checked by verified numerical computations.
For the entire collection see [Zbl 1334.68018].Steady asymmetric vortex pairs for Euler equations.https://zbmath.org/1460.352862021-06-15T18:09:00+00:00"Hassainia, Zineb"https://zbmath.org/authors/?q=ai:hassainia.zineb"Hmidi, Taoufik"https://zbmath.org/authors/?q=ai:hmidi.taoufikSummary: In this paper, we study the existence of co-rotating and counter-rotating unequal-sized pairs of simply connected patches for Euler equations. In particular, we prove the existence of curves of steadily co-rotating and counter-rotating asymmetric vortex pairs passing through a point vortex pairs with unequal circulations. We also provide a careful study of the asymptotic behavior for the angular velocity and the translating speed close to the point vortex pairs.Semilinear elliptic equations with Hardy potential and gradient nonlinearity.https://zbmath.org/1460.351542021-06-15T18:09:00+00:00"Gkikas, Konstantinos"https://zbmath.org/authors/?q=ai:gkikas.konstantinos-t"Nguyen, Phuoc-Tai"https://zbmath.org/authors/?q=ai:nguyen-phuoc-tai.The paper deals with a semilinear elliptic equation with Hardy potential and gradient nonlinearity, subject to a boundary trace condition. Results on existence, boundary behavior, a priority estimates and certain qualitative properties are given. Linear problems associated to the original one constitute technical tools in the proofs.
Reviewer: Dumitru Motreanu (Perpignan)On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density.https://zbmath.org/1460.351242021-06-15T18:09:00+00:00"Mercuri, Carlo"https://zbmath.org/authors/?q=ai:mercuri.carlo"Tyler, Teresa Megan"https://zbmath.org/authors/?q=ai:tyler.teresa-meganSummary: In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system
\[\begin{cases}
- \Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, & x\in \mathbb{R}^3, \\
-\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3,
\end{cases}\]
under different assumptions on \(\rho\colon \mathbb{R}^3\rightarrow \mathbb{R}_+\) at infinity. Our results cover the range \(p\in(2,3)\) where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a `limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem
\[\begin{cases}
- \epsilon^2\Delta u+ u + \rho (x) \phi u = |u|^{p-1} u, & x\in \mathbb{R}^3, \\
-\Delta \phi=\rho(x) u^2, & x\in \mathbb{R}^3,
\end{cases}\]
in various functional settings which are suitable for both variational and perturbation methods.Nonexistence for hyperbolic problems on Riemannian manifolds.https://zbmath.org/1460.353642021-06-15T18:09:00+00:00"Monticelli, Dario D."https://zbmath.org/authors/?q=ai:monticelli.dario-daniele"Punzo, Fabio"https://zbmath.org/authors/?q=ai:punzo.fabio"Squassina, Marco"https://zbmath.org/authors/?q=ai:squassina.marcoSummary: We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.Local well-posedness for the Klein-Gordon-Zakharov system in 3D.https://zbmath.org/1460.353322021-06-15T18:09:00+00:00"Pecher, Hartmut"https://zbmath.org/authors/?q=ai:pecher.hartmutSummary: We study the Cauchy problem for the Klein-Gordon-Zakharov system in 3D with low regularity data. We lower down the regularity to the critical value with respect to scaling up to the endpoint. The decisive bilinear estimates are proved by means of methods developed by \textit{I. Bejenaru} and \textit{S. Herr} [J. Funct. Anal. 261, No. 2, 478--506 (2011; Zbl 1228.42027)] for the Zakharov system and already applied by \textit{S. Kinoshita} [Discrete Contin. Dyn. Syst. 38, No. 3, 1479--1504 (2018; Zbl 1397.35281)] to the Klein-Gordon-Zakharov system in 2D.\(\mathbb{C}\)-constructible enhanced ind-sheaves.https://zbmath.org/1460.320092021-06-15T18:09:00+00:00"Ito, Yohei"https://zbmath.org/authors/?q=ai:ito.yoheiSummary: \textit{A. D'Agnolo} and \textit{M. Kashiwara} [Publ. Math., Inst. Hautes Étud. Sci. 123, 69--197 (2016; Zbl 1351.32017)] proved that their enhanced solution functor induces a fully faithful embedding of the triangulated category of holonomic \(\mathcal{D}\)-modules into the one of \(\mathbb{R}\)-constructible enhanced ind-sheaves. In this paper, we introduce a notion of \(\mathbb{C}\)-constructible enhanced ind-sheaves and show that the triangulated category of them is equivalent to its essential image. Moreover we show that there exists a t-structure on it whose heart is equivalent to the abelian category of holonomic \(\mathcal{D}\)-modules.Well-posedness for the three dimensional stochastic planetary geostrophic equations of large-scale ocean circulation.https://zbmath.org/1460.353482021-06-15T18:09:00+00:00"You, Bo"https://zbmath.org/authors/?q=ai:you.boSummary: The objective of this paper is to study the well-posedness of solutions for the three dimensional planetary geostrophic equations of large-scale ocean circulation with additive noise. Since strong coupling terms and the noise term create some difficulties in the process of showing the existence of weak solutions, we will first show the existence of weak solutions by the monotonicity methods when the initial data satisfies some ``regular'' condition. For the case of general initial data, we will establish the existence of weak solutions by taking a sequence of ``regular'' initial data and proving the convergence in probability as well as some weak convergence of the corresponding solution sequences. Finally, we establish the existence of weak \(\mathcal{D}\)-pullback mean random attractors in the framework developed in [\textit{P. E. Kloeden} and \textit{T. Lorenz}, J. Differ. Equations 253, No. 5, 1422--1438 (2012; Zbl 1267.37018); \textit{B. Wang}, J. Dyn. Differ. Equations 31, No. 4, 2177--2204 (2019; Zbl 1428.35052)].The local well-posedness to the density-dependent magnetic Bénard system with nonnegative density.https://zbmath.org/1460.353012021-06-15T18:09:00+00:00"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: We study the Cauchy problem of density-dependent magnetic Bénard system with zero density at infinity on the whole two-dimensional (2D) space. Despite the degenerate nature of the problem, we show the local existence of a unique strong solution in weighted Sobolev spaces by energy method.Uniqueness of positive radial solutions of superlinear elliptic equations in annuli.https://zbmath.org/1460.350052021-06-15T18:09:00+00:00"Shioji, Naoki"https://zbmath.org/authors/?q=ai:shioji.naoki"Tanaka, Satoshi"https://zbmath.org/authors/?q=ai:tanaka.satoshi"Watanabe, Kohtaro"https://zbmath.org/authors/?q=ai:watanabe.kohtaroSummary: We present a generalized comparison identity for a two point boundary value problem, and we apply it to show the uniqueness of positive radial solutions of
\[\left\{\begin{aligned} \Delta u ( x ) + f ( u ( x ) ) & = 0 && \quad\text{in } A_{a , b} , \\
u ( x ) & = 0 && \quad\text{on } \partial A_{a , b} , \end{aligned}
\right.\]
where \(n \in \mathbb{N}\) with \(n \geq 2\), \(0 < a < b < \infty\), \(A_{a , b} = \{x \in \mathbb{R}^N : a < | x | < b \}\) and \(f \in C^1 [0, \infty)\) is a superlinear function.Global large solutions and optimal time-decay estimates to the Korteweg system.https://zbmath.org/1460.353002021-06-15T18:09:00+00:00"Zhai, Xiaoping"https://zbmath.org/authors/?q=ai:zhai.xiaoping"Li, Yongsheng"https://zbmath.org/authors/?q=ai:li.yongshengSummary: We prove the global solutions to the Korteweg system without smallness condition imposed on the vertical component of the incompressible part of the velocity. The weighted Chemin-Lerner-norm technique which is well-known for the incompressible Navier-Stokes equations is introduced to derive the a priori estimates. As a byproduct, we obtain the optimal time decay rates of the solutions by using the pure energy argument (independent of spectral analysis). In contrast to the compressible Navier-Stokes system, the time-decay estimates are more accurate owing to smoothing effect from the Korteweg tensor.Darcy's law with a source term.https://zbmath.org/1460.350102021-06-15T18:09:00+00:00"Jacobs, Matt"https://zbmath.org/authors/?q=ai:jacobs.matthew"Kim, Inwon"https://zbmath.org/authors/?q=ai:kim.inwon-christina"Tong, Jiajun"https://zbmath.org/authors/?q=ai:tong.jiajunSummary: We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform \(L^1\)-equicontinuity. Many of these properties are new, even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.Nonlinear diffusion equations with degenerate fast-decay mobility by coordinate transformation.https://zbmath.org/1460.352102021-06-15T18:09:00+00:00"Ansini, Nadia"https://zbmath.org/authors/?q=ai:ansini.nadia"Fagioli, Simone"https://zbmath.org/authors/?q=ai:fagioli.simoneSummary: We prove an existence and uniqueness result for solutions to nonlinear diffusion equations with degenerate mobility posed on a bounded interval for a certain density \(u\). In case of \textit{fast-decay} mobilities, namely mobilities functions under an Osgood integrability condition, a suitable coordinate transformation is introduced and a new nonlinear diffusion equation with linear mobility is obtained. We observe that the coordinate transformation induces a mass-preserving scaling on the density and the nonlinearity, described by the original nonlinear mobility, is included in the diffusive process. We show that the rescaled density \(\rho\) is the unique weak solution to the nonlinear diffusion equation with linear mobility. Moreover, the results obtained for the density \(\rho\) allow us to motivate the aforementioned change of variable and to state the results in terms of the original density \(u\) without prescribing any boundary conditions.Positive vector solutions for a Schrödinger system with external source terms.https://zbmath.org/1460.351232021-06-15T18:09:00+00:00"Long, Wei"https://zbmath.org/authors/?q=ai:long.wei"Peng, Shuangjie"https://zbmath.org/authors/?q=ai:peng.shuangjieAuthors' abstract: This paper is concerned with the existence of many synchronized vector solutions for the following Schrödinger system with external source terms \[\begin{cases} - \Delta u + u=a(x) u^3+\beta u v^2+f(x), & \quad x \; \in \; {\mathbb{R}}^3,\\ - \Delta v + v=b(x) v^3+\beta u^2 v+g(x), & \quad x \; \in \; {\mathbb{R}}^3,\\ u,v >0, & \quad x\in{\mathbb{R}}^3, \end{cases}\] where \(\beta \in{\mathbb{R}}\) is a coupling constant, \(a, b\in C({\mathbb{R}}^3)\) and \(f, g\in L^2({\mathbb{R}}^3)\cap L^\infty ({\mathbb{R}}^3)\). This type of system arises in Bose-Einstein condensates and Kerr-like photo refractive media. This paper tries to reveal the influence of the external source terms \(f\) and \(g\) on the number of the solutions. It is shown that the level set of the corresponding functional has a quite rich topology and the system admits \(k\) spikes synchronized vector solutions for any \(k\in \mathbb{Z}^+\) when \(f\) and \(g\) are small and \(a(x), b(x)\) satisfy some additional assumptions at infinity. The proof is based on the Lyapunov-Schmidt reduction scheme and the main ingredient is to improve the estimate on the remainder term obtained in the reduction process.
Reviewer: Jiří Rákosník (Praha)A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source.https://zbmath.org/1460.353592021-06-15T18:09:00+00:00"Xie, Jianing"https://zbmath.org/authors/?q=ai:xie.jianingA version of the doubly parabolic Keller-Segel system with nonlinear (but nondegenerate) diffusion and logistic source term is studied in bounded domains of \(\mathbb R^N\), \(N\ge 2\). Using a new energy-type inequality global-in-time and uniform boundedness of solutions is studied under suitable assumptions on the diffusion term.
Reviewer: Piotr Biler (Wrocław)Quasi linear parabolic PDE posed on a network with non linear Neumann boundary condition at vertices.https://zbmath.org/1460.351962021-06-15T18:09:00+00:00"Ohavi, Isaac"https://zbmath.org/authors/?q=ai:ohavi.isaacSummary: The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.Global diffeomorphism of the Lagrangian flow-map for a Pollard-like internal water wave.https://zbmath.org/1460.353472021-06-15T18:09:00+00:00"Kluczek, Mateusz"https://zbmath.org/authors/?q=ai:kluczek.mateusz"Rodríguez-Sanjurjo, Adrián"https://zbmath.org/authors/?q=ai:rodriguez-sanjurjo.adrianSummary: In this article we provide an overview of a rigorous justification of the global validity of the fluid motion described by a new exact and explicit solution prescribed in terms of Lagrangian variables of the nonlinear geophysical equations. More precisely, the three-dimensional Lagrangian flow-map describing this exact and explicit solution is proven to be a global diffeomorphism from the labelling domain into the fluid domain. Then, the flow motion is shown to be dynamically possible.
For the entire collection see [Zbl 1432.35003].Existence and multiplicity results for the fractional Schrödinger equations with indefinite potentials.https://zbmath.org/1460.353712021-06-15T18:09:00+00:00"Cui, Na"https://zbmath.org/authors/?q=ai:cui.na"Sun, Hong-Rui"https://zbmath.org/authors/?q=ai:sun.hongruiSummary: In this paper, we consider the fractional Schrödinger equations
\[
(-\Delta)^su+V(x)u=f(x,u)\text{ in }\mathbb{R}^N,
\]
where s \(\in (0,1)\) and \(N>2s\). Firstly, we prove that the problem admits a nontrivial solution and infinitely many nontrivial solutions under the assumptions that \(V\) is allowed to be indefinite potential and \(f\) is superlinear and subcritical. In addition, we establish an existence criterion of infinitely many nontrivial solutions for the aforementioned problem with concave and critical nonlinearities as well as indefinite potential, which improves the result of \textit{M. Du} and \textit{L. Tian} [Discrete Contin. Dyn. Syst., Ser. B 21, No. 10, 3407--3428 (2016; Zbl 1353.35305), Theorem 1.6].On entropy solutions of anisotropic elliptic equations with variable nonlinearity indices in unbounded domains.https://zbmath.org/1460.351082021-06-15T18:09:00+00:00"Kozhevnikova, L. M."https://zbmath.org/authors/?q=ai:kozhevnikova.larisa-mikhailovnaSummary: For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.Multiplicity of weak solutions to a fourth-order elliptic equation with sign-changing weight function.https://zbmath.org/1460.351142021-06-15T18:09:00+00:00"Han, Yuzhu"https://zbmath.org/authors/?q=ai:han.yuzhu"Cao, Chunling"https://zbmath.org/authors/?q=ai:cao.chunling"Li, Jiankang"https://zbmath.org/authors/?q=ai:li.jiankangSummary: In this paper, a fourth-order elliptic equation with a sign-changing weight function is investigated. By the modified logarithmic Sobolev inequality and Nehari manifold it is shown that the problem admits at least two nontrivial weak solutions, which shows how the sign-changing weight function affect the existence and multiplicity of weak solutions to the problem.The zilch electromagnetic conservation law revisited.https://zbmath.org/1460.780052021-06-15T18:09:00+00:00"Aghapour, Sajad"https://zbmath.org/authors/?q=ai:aghapour.sajad"Andersson, Lars"https://zbmath.org/authors/?q=ai:andersson.lars-erik|andersson.lars-l|andersson.lars-ake"Rosquist, Kjell"https://zbmath.org/authors/?q=ai:rosquist.kjellSummary: It is shown that the zilch conservation law arises as the Noether current corresponding to a variational symmetry of a duality-symmetric Maxwell Lagrangian. The action of the corresponding symmetry generator on the duality-symmetric Lagrangian, while non-vanishing, is a total divergence as required by the Noether theory. The variational nature of the zilch conservation law was previously known only for some of the components of the zilch tensor, notably the optical chirality. By contrast, our analysis is fully covariant and is, therefore, valid for all components of the zilch tensor. The analysis is presented here for both the real and complex versions of duality-symmetric Maxwell Lagrangians.
{\copyright 2020 American Institute of Physics}A uniqueness result for 3D incompressible fluid-rigid body interaction problem.https://zbmath.org/1460.352962021-06-15T18:09:00+00:00"Muha, Boris"https://zbmath.org/authors/?q=ai:muha.boris"Nečasová, Šárka"https://zbmath.org/authors/?q=ai:necasova.sarka"Radošević, Ana"https://zbmath.org/authors/?q=ai:radosevic.anaSummary: We study a 3D nonlinear moving boundary fluid-structure interaction problem describing the interaction of the fluid flow with a rigid body. The fluid flow is governed by 3D incompressible Navier-Stokes equations, while the motion of the rigid body is described by a system of ordinary differential equations called Euler equations for the rigid body. The equations are fully coupled via dynamical and kinematic coupling conditions. We consider two different kinds of kinematic coupling conditions: no-slip and slip. In both cases we prove a generalization of the well-known weak-strong uniqueness result for the Navier-Stokes equations to the fluid-rigid body system. More precisely, we prove that weak solutions that additionally satisfy the Prodi-Serrin \(\mathrm{L}^r-\mathrm{L}^s\) condition are unique in the class of Leray-Hopf weak solutions.Tube structures of co-rank 1 with forms defined on compact surfaces.https://zbmath.org/1460.580142021-06-15T18:09:00+00:00"Hounie, J."https://zbmath.org/authors/?q=ai:hounie.j"Zugliani, G."https://zbmath.org/authors/?q=ai:zugliani.giuliano-angeloSummary: We study the global solvability of a locally integrable structure of tube type and co-rank 1 by considering a linear partial differential operator \(\mathbb{L}\) associated to a general complex smooth closed 1-form \(c\) defined on a smooth closed \(n\)-manifold. The main result characterizes the global solvability of \(\mathbb{L}\) when \(n=2\) in terms of geometric properties of a primitive of a convenient exact pullback of the form \(\mathfrak{Im}(c)\) as well as in terms of homological properties of \(\mathfrak{Re}(c)\) related to small divisors phenomena. Although the full characterization is restricted to orientable surfaces, some partial results hold true for compact manifolds of any dimension, in particular, the necessity of the conditions, and the equivalence when \(\mathfrak{Im}(c)\) is exact. We also obtain informations on the global hypoellipticity of \(\mathbb{L}\) and the global solvability of \(\mathbb{L}^{n-1}\) -- the last non-trivial operator of the complex when \(M\) is orientable.On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials.https://zbmath.org/1460.351032021-06-15T18:09:00+00:00"Yang, Jianfu"https://zbmath.org/authors/?q=ai:yang.jianfu"Yang, Jinge"https://zbmath.org/authors/?q=ai:yang.jingeThe existence and concentration of solutions to a supercritical elliptic equation with ellipse-shaped potential in the plane are studied through a variational approach.
Reviewer: Dumitru Motreanu (Perpignan)Multiple solutions for a class of \(p(x)\)-curl systems arising in electromagnetism.https://zbmath.org/1460.780222021-06-15T18:09:00+00:00"Nguyen Thanh Chung"https://zbmath.org/authors/?q=ai:nguyen-thanh-chung.Summary: In this paper, we study the existence of solutions for a class of of \(p(x)\)-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities, we obtain at least two non-trivial solutions for the problem by using the mountain pass theorem combined with the Ekeland variational principle. Our main result in this paper complements and extends some earlier ones concerning the \(p(x)\)-curl operator in [\textit{A. Bahrouni} and \textit{D. Repovš}, Complex Var. Elliptic Equ. 63, No. 2, 292--301 (2018; Zbl 1423.35124); \textit{D. Medková}, J. Differ. Equations 261, No. 10, 5670--5689 (2016; Zbl 1356.35181)].Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers.https://zbmath.org/1460.351092021-06-15T18:09:00+00:00"Kuchta, Miroslav"https://zbmath.org/authors/?q=ai:kuchta.miroslav"Laurino, Federica"https://zbmath.org/authors/?q=ai:laurino.federica"Mardal, Kent-Andre"https://zbmath.org/authors/?q=ai:mardal.kent-andre"Zunino, Paolo"https://zbmath.org/authors/?q=ai:zunino.paoloSampling in thermoacoustic tomography.https://zbmath.org/1460.354052021-06-15T18:09:00+00:00"Mathison, Chase"https://zbmath.org/authors/?q=ai:mathison.chaseSummary: We explore the effect of sampling rates when measuring data given by \(Mf\) for special operators \(M\) arising in Thermoacoustic Tomography. We start with sampling requirements on \(Mf\) given \(f\) satisfying certain conditions. After this we discuss the resolution limit on \(f\) posed by the sampling rate of \(Mf\) without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when \(Mf\) is known to be under sampled in one or more of its variables. Finally, we discuss averaging of measurement data and resulting aliasing and artifacts, along with a scheme for anti-aliasing.Optimal decay to the non-isentropic compressible micropolar fluids.https://zbmath.org/1460.352922021-06-15T18:09:00+00:00"liu, Lvqiao"https://zbmath.org/authors/?q=ai:liu.lvqiao"Zhang, Lan"https://zbmath.org/authors/?q=ai:zhang.lanSummary: In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic compressible micropolar fluid. For the initial data near the given equilibrium we prove the global well-posedness of classical solutions and obtain the optimal algebraic rate of convergence in the three-dimensional whole space. Moreover, it turns out that the density, the velocity and the temperature tend to the corresponding equilibrium state with rate \((1+t)^{-3/4}\) in \(L^2\) norm and the micro-rotational velocity tends to the equilibrium state with the faster rate \((1+t)^{-5/4}\) in \(L^2\) norm. The proof is based on the detailed analysis of the Green function and time-weighted energy estimates.Existence theory for the Boussinesq equation in modulation spaces.https://zbmath.org/1460.352762021-06-15T18:09:00+00:00"Banquet, Carlos"https://zbmath.org/authors/?q=ai:banquet-brango.carlos"Villamizar-Roa, Élder J."https://zbmath.org/authors/?q=ai:villamizar-roa.elder-jesusSummary: In this paper we study the Cauchy problem for the generalized Boussinesq equation with initial data in modulation spaces \(M^s_{p',q}(\mathbb{R}^n)\), \(n\geq 1\). After a decomposition of the Boussinesq equation in a \(2\times 2\)-nonlinear system, we obtain the existence of global and local solutions in several classes of functions with values in \(M^s_{p,q}\times D^{-1}JM^s_{p,q}\)-spaces for suitable \(p,q\) and \(s\), including the special case \(p=2\), \(q=1\) and \(s=0\). Finally, we prove some results of scattering and asymptotic stability in the framework of modulation spaces.Initial data and black holes for matter models.https://zbmath.org/1460.830042021-06-15T18:09:00+00:00"Burtscher, Annegret Y."https://zbmath.org/authors/?q=ai:burtscher.annegret-ySummary: To observe the dynamic formation of black holes in general relativity, one essentially needs to prove that closed trapped surfaces form during evolution from initial data that do not already contain trapped surfaces. We discuss the recent development of the construction of such admissible initial data for matter models. In addition, we extend known results for the Einstein equations coupled to perfect fluids in spherical symmetry and with linear equation of state to unbounded domains. Polytropic equations of state and regularity issues with the direct application of the singularity theorems in general relativity are discussed briefly.
For the entire collection see [Zbl 1453.35003].Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator.https://zbmath.org/1460.351562021-06-15T18:09:00+00:00"Luyen, Duong Trong"https://zbmath.org/authors/?q=ai:luyen.duong-trong"Tri, Nguyen Minh"https://zbmath.org/authors/?q=ai:tri.nguyen-minhSummary: In this paper, we study the multiplicity of weak solutions to the boundary value problem
\begin{align*}
-G_\alpha u & =f(x,y,u)+g(x,y,u)\text{ in }\Omega,\\
u & =0\text{ on }\partial\Omega
\end{align*}
where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N\) \((N\geq 2)\), \(\alpha\in\mathbb{N}\), \(f(x,y,\xi)\) is odd in \(\xi\) and \(g(x,y,\xi)\) is a perturbation term. Under some growth conditions on \(f\) and \(g\), we show that there are infinitely many weak solutions to the problem. Here we do not require that \(f\) satisfies the Ambrosetti-Rabinowitz (AR) condition. The conditions on \(f\) and \(g\) are relatively weak and our result is new even in the case \(\alpha=0\), i.e. for the classical Laplace equation with the Dirichlet boundary condition.The Boussinesq system revisited.https://zbmath.org/1460.352952021-06-15T18:09:00+00:00"Molinet, Luc"https://zbmath.org/authors/?q=ai:molinet.luc"Talhouk, Raafat"https://zbmath.org/authors/?q=ai:talhouk.raafat"Zaiter, Ibtissam"https://zbmath.org/authors/?q=ai:zaiter.ibtissamOn the existence of solution of the boundary-domain integral equation system derived from the 2D Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix.https://zbmath.org/1460.351062021-06-15T18:09:00+00:00"Fresneda-Portillo, C."https://zbmath.org/authors/?q=ai:fresneda-portillo.carlos"Woldemicheal, Z. W."https://zbmath.org/authors/?q=ai:woldemicheal.zenebe-wSummary: A system of boundary-domain integral equations is derived from the bidimensional Dirichlet problem for the diffusion equation with variable coefficient using a novel parametrix different from the one widely used in the literature by the authors Chkadua, Mikhailov and Natroshvili. Mapping properties of the surface and volume parametrix-based potential-type operators are analysed. Invertibility of the single layer potential is also studied in detail in appropriate Sobolev spaces. We show that the system of boundary-domain integral equations derived is equivalent to the Dirichlet problem prescribed and we prove the existence and uniqueness of solution in suitable Sobolev spaces of the system obtained by using arguments of compactness and Fredholm Alternative theory. A discussion of the possible applications of this new parametrix is included.Existence of entropy solutions for anisotropic elliptic nonlinear problem in weighted Sobolev space.https://zbmath.org/1460.351322021-06-15T18:09:00+00:00"Abbassi, Adil"https://zbmath.org/authors/?q=ai:abbassi.adil"Allalou, Chakir"https://zbmath.org/authors/?q=ai:allalou.chakir"Kassidi, Abderrazak"https://zbmath.org/authors/?q=ai:kassidi.abderrazakSummary: In this paper, we will study the existence of an entropy solution to the unilateral problem for a class of nonlinear anisotropic elliptic equation, with second term being an element of \(L^1(\varOmega)\). Our technical approach is based on a monotony method and the truncation techniques in the framework of the weighted anisotropic Sobolev space.
For the entire collection see [Zbl 1459.35003].Signed-distance function based non-rigid registration of image series with varying image intensity.https://zbmath.org/1460.650272021-06-15T18:09:00+00:00"Škardová, Kateřina"https://zbmath.org/authors/?q=ai:skardova.katerina"Oberhuber, Tomáš"https://zbmath.org/authors/?q=ai:oberhuber.tomas"Tintěra, Jaroslav"https://zbmath.org/authors/?q=ai:tintera.jaroslav"Chabiniok, Radomír"https://zbmath.org/authors/?q=ai:chabiniok.radomirSummary: In this paper we propose a method for locally adjusted optical flow-based registration of multimodal images, which uses the segmentation of object of interest and its representation by the signed-distance function (OF\(^{dist}\) method). We deal with non-rigid registration of the image series acquired by the Modiffied Look-Locker Inversion Recovery (MOLLI) magnetic resonance imaging sequence, which is used for a pixel-wise estimation of \(T_1\) relaxation time. The spatial registration of the images within the series is necessary to compensate the patient's imperfect breath-holding. The evolution of intensities and a large variation of image contrast within the MOLLI image series, together with the myocardium of left ventricle (the object of interest) typically not being the most distinct object in the scene, makes the registration challenging. The paper describes all components of the proposed OF\(^{dist}\) method and their implementation. The method is then compared to the performance of a standard mutual information maximization-based registration method, applied either to the original image (MIM) or to the signed-distance function (MIM\(^{dist}\)). Several experiments with synthetic and real MOLLI images are carried out. On synthetic image with a single object, MIM performed the best, while OF\(^{dist}\) and MIM\(^{dist}\) provided better results on synthetic images with more than one object and on real images. When applied to signed-distance function of two objects of interest, MIM\(^{dist}\) provided a larger registration error, but more homogeneously distributed, compared to OF\(^{dist}\). For the real MOLLI image series with left ventricle pre-segmented using a level-set method, the proposed OF\(^{dist}\) registration performed the best, as is demonstrated visually and by measuring the increase of mutual information in the object of interest and its neighborhood.Backward self-similar solutions for compressible Navier-Stokes equations.https://zbmath.org/1460.352562021-06-15T18:09:00+00:00"Germain, Pierre"https://zbmath.org/authors/?q=ai:germain.pierre"Iwabuchi, Tsukasa"https://zbmath.org/authors/?q=ai:iwabuchi.tsukasa"Léger, Tristan"https://zbmath.org/authors/?q=ai:leger.tristanLocal existence and nonexistence for fractional in time weakly coupled reaction-diffusion systems.https://zbmath.org/1460.353792021-06-15T18:09:00+00:00"Suzuki, Masamitsu"https://zbmath.org/authors/?q=ai:suzuki.masamitsuSummary: We study a fractional in time weakly coupled reaction-diffusion system in a bounded domain with the Dirichlet boundary condition. The domain is imbedded in an \(N\)-dimensional space and it has \(C^2\) boundary, and fractional derivatives are meant in a generalized Caputo sense. The system can be referred to as a standard reaction-diffusion system in two components with polynomial growth. We obtain integrability conditions on the initial state functions which determine the existence/nonexistence of a local in time mild solution.Robin spectral rigidity of the ellipse.https://zbmath.org/1460.352432021-06-15T18:09:00+00:00"Vig, Amir"https://zbmath.org/authors/?q=ai:vig.amirSummary: In this paper, we investigate \(C^1\) isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. If the deformation is in fact smooth, reparametrizing allows us to show that the first variation actually vanishes to infinite order. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard's variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we in fact construct an explicit parametrix for the wave propagator in the interior, microlocally near geodesic loops.Existence of stable standing waves for the Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation.https://zbmath.org/1460.351532021-06-15T18:09:00+00:00"Feng, Binhua"https://zbmath.org/authors/?q=ai:feng.binhua"Cao, Leijin"https://zbmath.org/authors/?q=ai:cao.leijin"Liu, Jiayin"https://zbmath.org/authors/?q=ai:liu.jiayinSummary: In this paper, we study existence of stable standing waves for the following Lee-Huang-Yang corrected dipolar Gross-Pitaevskii equation with a partial harmonic confine
\[
i \partial_t \psi = - \Delta \psi + (x_1^2 + x_2^2) \psi + \lambda_1 | \psi |^2 \psi + \lambda_2 (K \ast | \psi |^2) \psi + \lambda_3 | \psi |^p \psi, \quad (t, x) \in [0, T^\ast) \times \mathbb{R}^3.
\]
When \(0 < p < 4\) and \(\lambda_3 < 0\), we can prove the existence of stable standing waves for this equation. Our results are a complementary to the ones of \textit{Y. Luo} and \textit{A. Stylianou} [Discrete Contin. Dyn. Syst. Ser. B 26(6), 3455-3477 (2021, \url{http://dx.doi.org/10.3934/dcdsb.2020239})], where existence and nonexistence of standing waves have been studied for the complete harmonic potential.Reduction of a damped, driven Klein-Gordon equation into a discrete nonlinear Schrödinger equation: justification and numerical comparison.https://zbmath.org/1460.353302021-06-15T18:09:00+00:00"Muda, Yuslenita"https://zbmath.org/authors/?q=ai:muda.yuslenita"Akbar, Fiki T."https://zbmath.org/authors/?q=ai:akbar.fiki-taufik"Kusdiantara, Rudy"https://zbmath.org/authors/?q=ai:kusdiantara.rudy"Gunara, Bobby E."https://zbmath.org/authors/?q=ai:gunara.bobby-eka"Susanto, Hadi"https://zbmath.org/authors/?q=ai:susanto.hadiSummary: We consider a discrete nonlinear Klein-Gordon equation with damping and external drive. Using a small amplitude ansatz, one usually approximates the equation using a damped, driven discrete nonlinear Schrödinger equation. Here, we show for the first time the justification of this approximation by finding the error bound using energy estimate. Additionally, we prove the local and global existence of the Schrödinger equation. Numerical simulations are performed that describe the analytical results. Comparisons between discrete breathers of the Klein-Gordon equation and discrete solitons of the discrete nonlinear Schrödinger equation are presented.A stationary heat conduction problem in low dimensional sets in \({\mathbb{R}}^N\).https://zbmath.org/1460.351102021-06-15T18:09:00+00:00"Rybka, Piotr"https://zbmath.org/authors/?q=ai:rybka.piotr"Zatorska-Goldstein, Anna"https://zbmath.org/authors/?q=ai:zatorska-goldstein.annaThe authors use the variational approach to study the existence and uniqueness of weak solutions to a linear elliptic equation related to the problem of heat distribution in a conductor \(S\) in the ambient space \(\Omega\subset \mathbb R^N\). The conductor \(S\) is insulated at the boundary of \(\Omega\), the set \(\Omega\setminus S\) is neither conducting nor it contains any heat source, and the dimension of \(S\) is smaller than \(N\). Using the measure theoretic tools they follow the approach of [\textit{G. Bouchitté} et al., Calc. Var. Partial Differ. Equ. 5, No. 1, 37--54 (1997; Zbl 0934.49011]). Contrary to the previous approaches they do not assume a Poincaré-type inequality to hold globally on \(S\) and they consider the Neumann boundary condition.
Reviewer: Jiří Rákosník (Praha)Construction of a solitary wave solution of the nonlinear focusing Schrödinger equation outside a strictly convex obstacle in the \(L^2\)-supercritical case.https://zbmath.org/1460.353292021-06-15T18:09:00+00:00"Landoulsi, Oussama"https://zbmath.org/authors/?q=ai:landoulsi.oussamaSummary: We consider the focusing \(L^2\)-supercritical Schrödinger equation in the exterior of a smooth, compact, strictly convex obstacle \(\Theta\subset\mathbb{R}^3\). We construct a solution behaving asymptotically as a solitary wave on \(\mathbb{R}^3,\) for large times. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by \textit{F. Merle} [Commun. Math. Phys. 129, No. 2, 223--240 (1990; Zbl 0707.35021)] to construct solutions of the NLS equation blowing up at several points together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at the soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of \textit{R. Killip} et al. [AMRX, Appl. Math. Res. Express 2016, No. 1, 146--180 (2016; Zbl 1345.35102)], which is the same as the one on the whole Euclidean space given by \textit{T. Duyckaerts} et al. [Math. Res. Lett. 15, No. 5--6, 1233--1250 (2008; Zbl 1171.35472)] and \textit{J. Holmer} and \textit{S. Roudenko} [Commun. Math. Phys. 282, No. 2, 435--467 (2008; Zbl 1155.35094)] in the radial case and by the previous authors with T. Duyckaerts in the non-radial case.Normalized solutions for nonlinear Schrödinger systems with linear couples.https://zbmath.org/1460.353202021-06-15T18:09:00+00:00"Chen, Zhen"https://zbmath.org/authors/?q=ai:chen.zhen"Zou, Wenming"https://zbmath.org/authors/?q=ai:zou.wenmingSummary: In this paper, we study the normalized solutions to the following system
\[
\begin{cases}
- \Delta u + ( V_1 ( x ) + \lambda_1 ) u = \mu_1 | u |^{p - 2} u + \beta v \quad & \text{in } \mathbb{R}^N, \\
- \Delta v + ( V_2 ( x ) + \lambda_2 ) v = \mu_2 | v |^{q - 2} v + \beta u & \text{in } \mathbb{R}^N, \\
\int_{\mathbb{R}^N} u^2 = a, \quad \int_{\mathbb{R}^N} v^2 = b,
\end{cases}
\]
with the mass-subcritical condition \(2 < p, q < 2 + \frac{ 4}{ N} \), where \(\mu_1, \mu_2, a, b > 0\), \(\beta \in \mathbb{R} \setminus \{0 \}\) are prescribed; \( \lambda_1, \lambda_2 \in \mathbb{R}\) are to be determined. We prove the existence of a solution with prescribed \(L^2\)-norm under some various conditions on the potential \(V_1, V_2 : \mathbb{R}^N \to \mathbb{R} \). The proof is based on the refined energy estimates.One-dimensional multicomponent hemodynamics.https://zbmath.org/1460.353552021-06-15T18:09:00+00:00"Mamontov, A. E."https://zbmath.org/authors/?q=ai:mamontov.alexander-e"Prokudin, D. A."https://zbmath.org/authors/?q=ai:prokudin.dmitry-alexeyevichA one-dimensional model of blood flow in an artery has been developed in many works (in particular in [\textit{J. R. Womersley}, Philos. Mag., VII. Ser. 46, 199--221 (1955; Zbl 0064.43903); \textit{J. W. Lambert}, ``On the nonlinearities of fluid flow in nonrigid tubes'', J. Franklin Inst. 266, No. 2, 83--102 (1958); \textit{T. J. R. Hughes} and \textit{J. Lubliner}, Math. Biosci. 18, 161--170 (1973; Zbl 0262.92004); \textit{D. Bessems} et al., J. Fluid Mech. 580, 145--168 (2007; Zbl 1175.76171); \textit{R. Raghu} et al., ``Comparative study of viscoelastic arterial wall models in nonlinear one-dimensional finite element simulations of blood flow'', J. Biomech. Eng. 133, No. 8, 5--32 (2011); \textit{G. Mulder} et al., ``Patient-specific modeling of cerebral blood flow: geometrical variations in a 1D model'', Cardiovasc. Eng. Tech. 2, 334--348 (2011)]).
Under certain assumptions, the generalization of this model to the multicomponent case is studied.
After the transition from the Euler coordinate to the mass Lagrangian coordinates, a system of differential equations is obtained, which coincides in formulation with the system of differential equations of one-dimensional polytropic flows of viscous compressible multicomponent fluid. [\textit{A. E. Mamontov} and \textit{D. A. Prokudin}, Sib. Zh. Chist. Prikl. Mat. 17, No. 2, 52--68 (2017; Zbl 1438.76030); translation in J. Math. Sci., New York 231, No. 2, 227--242 (2018)].
The problem of initial-boundary value for this system is investigated and the conditions of existence and uniqueness of the classical solution are established.
Reviewer: Yaroslav Baranetskij (Lviv)Perturbation method in the theory of propagation of two-frequency electromagnetic waves in a nonlinear waveguide. I: TE-TE waves.https://zbmath.org/1460.780102021-06-15T18:09:00+00:00"Valovik, D. V."https://zbmath.org/authors/?q=ai:valovik.dmitry-vSummary: The article deals with the problem of propagation of a two-frequency electromagnetic wave in a waveguide filled with a nonlinear medium. A two-frequency wave is the sum of two monochromatic TE waves with different frequencies. The permittivity of the waveguide is characterized by a very general nonlinearity function corresponding to self-action effects. It is shown that, under certain conditions, the two-frequency wave is an eigenmode of the waveguide. From a mathematical point of view, the problem reduces to a nonlinear two-parameter eigenvalue problem for the system of (nonlinear) Maxwell's equations. The main result of the article is the proof of the existence of nonlinearizable solutions of the problem.Optimal feedback control for a model of motion of a nonlinearly viscous fluid.https://zbmath.org/1460.353042021-06-15T18:09:00+00:00"Zvyagin, V. G."https://zbmath.org/authors/?q=ai:zvyagin.viktor-grigorevich|zvyagin.victor-g"Zvyagin, A. V."https://zbmath.org/authors/?q=ai:zvyagin.alexander-v|zvyagin.andrey-v"Nguyen Minh Hong"https://zbmath.org/authors/?q=ai:nguyen-minh-hong.Summary: We consider an optimal feedback control problem for an initial-boundary value problem describing the motion of a nonlinearly viscous fluid. We prove the existence of an optimal solution minimizing a given performance functional. To prove the existence of an optimal solution, we use a topological approximation method for studying hydrodynamic problems.Quasilinear nonlocal elliptic problems with variable singular exponent.https://zbmath.org/1460.351382021-06-15T18:09:00+00:00"Garain, Prashanta"https://zbmath.org/authors/?q=ai:garain.prashanta"Mukherjee, Tuhina"https://zbmath.org/authors/?q=ai:mukherjee.tuhinaSummary: In this article, we provide existence results to the following nonlocal equation
\[
\begin{cases}
(-\Delta)_p^su=g(x,u),\;u>0\text{ in }\Omega,\\
u=0\text{ in }\mathbb{R}^N\setminus\Omega,
\end{cases}\tag{\(P_\lambda\)}
\]
where \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian operator. Here \(\Omega\subset\mathbb{R}^N\) is a smooth bounded domain, \(s\in(0,1)\) \(p>1\) and \(N>sp\). We establish existence of at least one weak solution for \((P_\lambda)\) when \(g(x,u) = f(x)u^{-q(x)}\) and existence of at least two weak solutions when \(g(x,u)=\lambda u^{-q(x)}+u^r\) for a suitable range of \(\lambda>0\). Here \(r\in(p-1,p_s^*-1)\) where \(p_s^*\) is the critical Sobolev exponent and \(0<q\in C^1(\bar{\Omega})\).Well-posedness for a system of quadratic derivative nonlinear Schrödinger equations in almost critical spaces.https://zbmath.org/1460.353272021-06-15T18:09:00+00:00"Hirayama, Hiroyuki"https://zbmath.org/authors/?q=ai:hirayama.hiroyuki"Kinoshita, Shinya"https://zbmath.org/authors/?q=ai:kinoshita.shinya"Okamoto, Mamoru"https://zbmath.org/authors/?q=ai:okamoto.mamoruSummary: In this paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations introduced by \textit{M. Colin} and \textit{T. Colin} [Differ. Integral Equ. 17, No. 3--4, 297--330 (2004; Zbl 1174.35528)]. We determine an almost optimal Sobolev regularity where the smooth flow map of the Cauchy problem exists, except for the scaling critical case. This result covers a gap left open in [the first author, Commun. Pure Appl. Anal. 13, No. 4, 1563--1591 (2014; Zbl 1294.35139); with the second author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 178, 205--226 (2019; Zbl 1406.35357)].On a class of fractional Schrödinger equations in \(\mathbb{R}^N\) with sign-changing potential.https://zbmath.org/1460.351342021-06-15T18:09:00+00:00"de Souza, Manassés"https://zbmath.org/authors/?q=ai:de-souza.manasses"Araújo, Yane Lísley"https://zbmath.org/authors/?q=ai:araujo.yane-lisleySummary: We are interested in finding solutions to a class of problems involving the fractional Laplacian operator. Specifically, we study the equation
\[
(-\Delta)^su+V(x)u=f(x,u)\quad\text{in}\quad\mathbb{R}^N,
\]
where \(0< s< 1\), \((-\Delta)^s\) denotes the fractional Laplacian of order \(s\), \(N\geq 1\), \(V(x)\) is a continuous and unbounded potential which may change sign, and the nonlinearity \(f(x,\xi)\) is a continuous function which may be unbounded in \(x\) since its growth is controlled by \(V(x)\) and has subcritical growth in \(\xi\) in the sense of the Sobolev embedding. Assuming suitable conditions under \(V(x)\) and \(f(x,\xi)\) and applying a approach variational, we prove the existence of the multiplicity of solution for this equation.Fast discrete finite Hankel transform for equations in a thin annulus.https://zbmath.org/1460.350802021-06-15T18:09:00+00:00"Budzinskiy, S. S."https://zbmath.org/authors/?q=ai:budzinskiy.stanislav-s"Romanenko, T. E."https://zbmath.org/authors/?q=ai:romanenko.t-eSummary: An algorithm is proposed for a fast discrete finite Hankel transform of a function in a thin annulus. The transform arises in a natural way in the Neumann boundary-value problem for the Poisson equation in an annulus when spectral methods are applied for its numerical solution. The proposed algorithm uses the limiting properties of eigenvalues and eigenfunctions of the Laplace operator as the annulus thickness goes to zero.Non-linear bi-harmonic Choquard equations.https://zbmath.org/1460.353332021-06-15T18:09:00+00:00"Saanouni, Tarek"https://zbmath.org/authors/?q=ai:saanouni.tarekSummary: This note studies the fourth-order Choquard equation
\[
i\dot u+\Delta^2 u\pm(I_\alpha *|u|^p)|u|^{p-2}u=0.
\]
In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.Sign-changing solutions for some class of elliptic system.https://zbmath.org/1460.351222021-06-15T18:09:00+00:00"Gan, Lu"https://zbmath.org/authors/?q=ai:gan.luSummary: This paper is concerned with the existence of multiple non-radial sign-changing solutions for
\begin{align*}
&-\Delta u+u+\alpha K(|x|)\Phi(x)u=|u|^{p-2}u, &\quad & x\in\mathbb{R}^3,\\
&-\Delta\Phi=K(|x|)u^2, &\quad & x\in\mathbb{R}^3,
\end{align*}
where \(2<p<6\), \(\alpha<0\) can be regarded as a parameter and \(K(r)(r=|x|)\) is a positive continuous function. We proved that the above equation possesses a non-radial sign-changing solutions with exactly \(k\) maximum points and \(k\) minimum points for suitable range of \(\alpha\).Low perturbations of \(p\)-biharmonic equations with competing nonlinearities.https://zbmath.org/1460.310202021-06-15T18:09:00+00:00"Alsaedi, R."https://zbmath.org/authors/?q=ai:alsaedi.ramzi-s-n"Dhifli, A."https://zbmath.org/authors/?q=ai:dhifli.abdelwaheb"Ghanmi, A."https://zbmath.org/authors/?q=ai:ghanmi.abdejabbar|ghanmi.allal|ghanmi.ahmedSummary: In the present paper, by using a variational approach combined with the Nehari manifold method and fibreing maps, the existence of two non-trivial solutions for a class of \(p\)-biharmonic problems is obtained.Existence of solutions for critical fractional \(p\)\&\(q\)-Laplacian system.https://zbmath.org/1460.351332021-06-15T18:09:00+00:00"Chen, Wenjing"https://zbmath.org/authors/?q=ai:chen.wenjingSummary: In this article, we are concerned with the existence of solutions for the following critical fractional \(p\)\&\(q\)-Laplacian system
\begin{align*}
(-\Delta)^{s_1}_pu+(-\Delta)^{s_2}_qu & =\lambda Q(x)|u|^{r-2}u+\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^\beta, & \text{ in } & \Omega;\\
(-\Delta)^{s_1}_pv+(-\Delta)^{s_2}_qv & = \mu Q(x)|v|^{r-2}v+\frac{2\beta}{\alpha+\beta}|u|^\alpha |v|^{\beta-2}v & \text{ in } &\Omega;\tag{1}\\
u=v & =0, & \text{ in } & \mathbb{R}^n\backslash\Omega
\end{align*}
where \(\Omega\) is a smooth bounded set in \(\mathbb{R}^n\), \(\lambda,\mu>0\) are two parameters, \(0<s_2<s_1<1\), \(1<q<p<r<p^\ast_{s_1}\), \(n>ps_1\), and \(\alpha>1\), \(\beta>1\) satisfy \(\alpha+\beta=p^\ast_{s_1}\) with \(p^\ast_{s_1}=\frac{np}{n-ps_1}\) is the fractional Sobolev critical exponent, and \((-\Delta)^s_t\) is the fractional \(t\)-Laplacian operator. By using variational methods, we show that there exists a nontrivial weak solution for system (1).Ground state solutions for a quasilinear elliptic equation with general critical nonlinearity.https://zbmath.org/1460.351602021-06-15T18:09:00+00:00"Shang, Tingting"https://zbmath.org/authors/?q=ai:shang.tingting"Liang, Ruixi"https://zbmath.org/authors/?q=ai:liang.ruixiSummary: In this article, we study the quasilinear elliptic equation:
\[
-\Delta u+V(x)u-\Delta[(1+u^2)^{1/2}]\frac{u}{2(1+u^2)^{1/2}}=h(u),\quad x\in\mathbb{R}^N,
\]
where \(N\geq 3\), \(V:\mathbb{R}^N\to\mathbb{R}\) satisfies suitable assumptions. Unlike \(h\in C^1(\mathbb{R},\mathbb{R})\), we only need to assume that \(h\in C(\mathbb{R},\mathbb{R})\). By using a change of variable, we obtain the existence of ground state solutions with general critical growth.On the existence of ground states of an equation of Schrödinger-Poisson-Slater type.https://zbmath.org/1460.350842021-06-15T18:09:00+00:00"Lei, Chunyu"https://zbmath.org/authors/?q=ai:lei.chunyu"Lei, Yutian"https://zbmath.org/authors/?q=ai:lei.yutianSummary: We study the existence of ground states of a Schrödinger-Poisson-Slater type equation with pure power nonlinearity. By carrying out the constrained minimization on a special manifold, which is a combination of the Pohozaev manifold and Nehari manifold, we obtain the existence of ground state solutions of this system.Bubbling solutions for the gravitational Maxwell gauged \(O(3)\) model in \(\mathbb{R}^2\).https://zbmath.org/1460.350812021-06-15T18:09:00+00:00"Choi, Nari"https://zbmath.org/authors/?q=ai:choi.nari"Han, Jongmin"https://zbmath.org/authors/?q=ai:han.jongminSummary: In this paper, we construct type II nontopological bubbling solutions of the self-dual equation for the Maxwell gauged \(O(3)\) sigma model coupled with gravity in \(\mathbb{R}^2\). Our solutions blow up at some of the string points or the anti-string points as the coupling parameter tends to zero and they are asymptotically radial near each blow-up point.Nonradial solutions of elliptic weighted superlinear problems in bounded symmetric domains.https://zbmath.org/1460.351652021-06-15T18:09:00+00:00"Aduén, Hugo"https://zbmath.org/authors/?q=ai:aduen.hugo"Herrón, Sigifredo"https://zbmath.org/authors/?q=ai:herron.sigifredoSummary: The present work has two objectives. First, we prove that a weighted superlinear elliptic problem has infinitely many nonradial solutions in the unit ball. Second, we obtain the same conclusion in annuli for a more general nonlinearity which also involves a weight. We use a lower estimate of the energy level of radial solutions with \(k - 1\) zeros in the interior of the domain and a simple counting. Uniqueness results due to \textit{S. Tanaka} [Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 6, 1331--1343 (2008; Zbl 1155.35377); Differ. Integral Equ. 20, No. 1, 93-104 (2007 Zbl 1212.34040] are very useful in our approach.Instability and nonordering of localized steady states to a classs of reaction-diffusion equations in \(\mathbb{R}^N\).https://zbmath.org/1460.351712021-06-15T18:09:00+00:00"Sourdis, Christos"https://zbmath.org/authors/?q=ai:sourdis.christosSummary: We show that the elliptic problem \(\Delta u+f(u)=0\) in \(\mathbb{R}^N\), \(N\ge 1\), with \(f\in C^1(\mathbb{R})\) and \(f(0)=0\) does not have nontrivial stable solutions that decay to zero at infinity, provided that \(f\) is nonincreasing near the origin. As a corollary, we can show that any two nontrivial solutions that decay to zero at infinity must intersect each other, provided that at least one of them is sign-changing. This property was previously known only in the case where both solutions are positive with a different approach. We also discuss implications of our main result on the existence of monotone heteroclinic solutions to the corresponding reaction-diffusion equation.Ground state solution of \(p\)-Laplacian equation with finite many critical nonlinearities.https://zbmath.org/1460.351802021-06-15T18:09:00+00:00"Su, Yu"https://zbmath.org/authors/?q=ai:su.yu.4|su.yu.1|su.yu|su.yu.2|su.yu.3"Chen, Haibo"https://zbmath.org/authors/?q=ai:chen.haibo.3|chen.haibo"Liu, Senli"https://zbmath.org/authors/?q=ai:liu.senli"Che, Guofeng"https://zbmath.org/authors/?q=ai:che.guofengSummary: In this paper, we consider the following problem:
\[
-\Delta_pu-\zeta\frac{|u|^{p-2}u}{|x|^p}=\sum\limits^k_{i=1}\left(I_{\alpha i}\ast|u|^{p^\ast_{\alpha_i}}\right)|u|^{p^\ast_{\alpha_i}-2}u+|u|^{p^\ast-2}u,\text{ in }\mathbb{R}^N,
\]
where \(N=3,4,5\), \(p\in(1,2]\), \(\zeta\in[0,\Lambda)\), \(\Lambda=\left(\frac{N-p}{p}\right)^p\), \(\Delta_p:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator, \(p^\ast=\frac{Np}{N-p}\) is the critical Sobolev exponent, \(p^\ast\frac{p}{2}\left(\frac{N+\alpha_i}{N-p}\right)\) are the Hardy-Littlewood-Sobolev critical upper exponents, the parameters \(\alpha_i\) satisfy some assumptions. First, we establish the refined Sobolev inequality with Coulomb norm, and show the corresponding best constant is achieved in \(\mathbb{R}^N\) by a nonnegative function. Second, by using the refined Sobolev inequality with Coulomb norm, the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative ground state solution for the above problem.Number of synchronized and segregated solutions for linear coupled elliptic systems.https://zbmath.org/1460.351282021-06-15T18:09:00+00:00"Jin, Ke"https://zbmath.org/authors/?q=ai:jin.ke"Shen, Zifei"https://zbmath.org/authors/?q=ai:shen.zifeiSummary: In this paper, we study the linear coupled elliptic equations:
\begin{align*}
& -\varepsilon^2\-\Delta u+u=u^3+\lambda v & \text{ in } & \Omega,\\
& -\varepsilon^2\Delta v+v=v^3+\lambda u & \text{ in } & \Omega,\\
& u>0,v>0 & \text{ in } & \Omega,\tag{\(P_\varepsilon\)}\\
& \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0 & \text{ on } & \partial\Omega,
\end{align*}
where \(\Omega\) is a smooth bounded domain in \(R^3\), \(\varepsilon\) is a small parameter and \(\lambda\) is a coupling constant. Due to Lyapunov-Schmidt reduction method, we prove that \((P_\varepsilon)\) has at least \(O\left(\frac{1}{\varepsilon^3|\ln_\varepsilon|^3}\right)\) synchronized and \(O\left(\frac{1}{\varepsilon^6|\ln_\varepsilon|^6}\right)\) segregated solutions for \(\varepsilon\) sufficiently small and some \(\lambda\). Moreover, for each \(m\in(0,3)\) there exist synchronized and segregated solutions for \((P_\varepsilon)\) with energies of order \(\varepsilon^{3-m}\).An extension of the Beale-Kato-Majda criterion for the 3D Navier-Stokes equation with hereditary viscosity.https://zbmath.org/1460.352622021-06-15T18:09:00+00:00"Mohan, M. T."https://zbmath.org/authors/?q=ai:mohan.manil-tSummary: In this work, we consider the three dimensional Navier-Stokes equations on the whole space with a hereditary viscous term which depends on the past history. We study a blow-up criterion of smooth solutions to such systems. The existence and uniqueness of smooth solution is proved via a frequency truncation method. We also give the example of Maxwell's fluid flow equations, which is a linear viscoelastic fluid flow model.Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density.https://zbmath.org/1460.353022021-06-15T18:09:00+00:00"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinOn the solvability of some systems of integro-differential equations with anomalous diffusion in two dimensions.https://zbmath.org/1460.350872021-06-15T18:09:00+00:00"Vougalter, Vitali"https://zbmath.org/authors/?q=ai:vougalter.vitaliSummary: The article deals with the existence of solutions of a system of integro-differential equations in the case of anomalous diffusion with the negative Laplacian in a fractional power in two dimensions. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used.Spectral method for constructing an integral basis for a Jacobian partial differential system.https://zbmath.org/1460.350092021-06-15T18:09:00+00:00"Gorbuzov, V. N."https://zbmath.org/authors/?q=ai:gorbuzov.viktor-nikolaevich"Pronevich, A. F."https://zbmath.org/authors/?q=ai:pronevich.a-fSummary: The spectral method for building the basic first integrals of Jacobian homogeneous linear partial differential systems is elaborated.Generalized solutions to models of compressible viscous fluids.https://zbmath.org/1460.352742021-06-15T18:09:00+00:00"Abbatiello, Anna"https://zbmath.org/authors/?q=ai:abbatiello.anna"Feireisl, Eduard"https://zbmath.org/authors/?q=ai:feireisl.eduard"Novotný, Antoní"https://zbmath.org/authors/?q=ai:novotny.antoninSummary: We propose a new approach to models of general compressible viscous fluids based on the concept of dissipative solutions. These are weak solutions satisfying the underlying equations modulo a defect measure. A dissipative solution coincides with the strong solution as long as the latter exists (weak-strong uniqueness) and they solve the problem in the classical sense as soon as they are smooth (compatibility). We consider general models of compressible viscous fluids with non-linear viscosity tensor and non-homogeneous boundary conditions, for which the problem of existence of global-in-time weak/strong solutions is largely open.Global Sobolev persistence for the fractional Boussinesq equations with zero diffusivity.https://zbmath.org/1460.352902021-06-15T18:09:00+00:00"Kukavica, Igor"https://zbmath.org/authors/?q=ai:kukavica.igor"Wang, Weinan"https://zbmath.org/authors/?q=ai:wang.weinanSummary: We address the persistence of regularity for the 2D \(\alpha\)-fractional Boussinesq equations with positive viscosity and zero diffusivity in general Sobolev spaces, i.e., for \((u_0,\rho_0)\in W^{s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)\), where \(s>1\) and \(q\in(2,\infty)\). We prove that the solution \((u(t),\rho(t))\) exists and belongs to \(W^{s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)\) for all positive time \(t\) for \(q>2\), where \(\alpha\in (1,2)\) is arbitrary.Ground state radial sign-changing solutions for a gauged nonlinear Schrödinger equation involving critical growth.https://zbmath.org/1460.350062021-06-15T18:09:00+00:00"Kang, Jincai"https://zbmath.org/authors/?q=ai:kang.jincai"Tang, Chunlei"https://zbmath.org/authors/?q=ai:tang.chunleiSummary: We investigate the following gauged nonlinear Schrödinger equation \[
\begin{cases}
-\Delta u+\omega u+\lambda\bigg(\dfrac{h_u^2(|x|)}{|x|^2}+ \int_{|x|}^{+\infty}\dfrac{h_u(s)}{s}u^2(s)ds\bigg)u=f(u)\text{ in }\mathbb{R}^2,\\ u\in H_r^1(\mathbb{R}^2),
\end{cases}
\]
where \(\omega,\lambda>0\) and \(h_u(s)=\frac{1}{2}\int_0^sru^2(r)dr\). When \(f\) has exponential critical growth, by using the constrained minimization method and Trudinger-Moser inequality, it is proved that the equation has a ground state radial sign-changing solution \(u_{\lambda}\) which changes sign exactly once. Moreover, the asymptotic behavior of \(u_{\lambda}\) as \(\lambda\rightarrow 0\) is analyzed.On the Cauchy problem of 3D nonhomogeneous incompressible nematic liquid crystal flows with vacuum.https://zbmath.org/1460.352932021-06-15T18:09:00+00:00"Liu, Yang"https://zbmath.org/authors/?q=ai:liu.yang.6|liu.yang.17|liu.yang.21|liu.yang.3|liu.yang.1|liu.yang.8|liu.yang.9|liu.yang.16|liu.yang.13|liu.yang.20|liu.yang.23|liu.yang.11|liu.yang.15|liu.yang|liu.yang.10|liu.yang.12|liu.yang.2|liu.yang.18|liu.yang.19|liu.yang.5|liu.yang.4|liu.yang.22|liu.yang.14"Zhong, Xin"https://zbmath.org/authors/?q=ai:zhong.xinSummary: This paper deals with the Cauchy problem of three-dimensional (3D) nonhomogeneous incompressible nematic liquid crystal flows. The global well-posedness of strong solutions with large velocity is established provided that \(\|\rho_0\|_{L^\infty}+\|\nabla d_0\|_{L^3}\) is suitably small. In particular, the initial density may contain vacuum states and even have compact support. Furthermore, the large time behavior of the solution is also obtained.Global entropy solutions to the compressible Euler equations in the isentropic nozzle flow.https://zbmath.org/1460.350682021-06-15T18:09:00+00:00"Tsuge, Naoki"https://zbmath.org/authors/?q=ai:tsuge.naokiThe auhtor studies the one-dimensional non-steady isentropic compressible Euler flow in a nozzle. The nozzle is infinitely long and described by an \(x\)-dependent cross-section function in the equations. Equations are written in terms of density and momentum. The gas is barotropic, so that pressure is a power function of density with the adiabatic exponent is from [1,5/3]. The Cauchy problem for arbitrarily large initial data is studied. The aim is to establish the global existence of the entropy solution with sonic state.
For the entire collection see [Zbl 1453.35003].
Reviewer: Ilya A. Chernov (Petrozavodsk)Remarks on the derivation of finite energy weak solutions to the QHD system.https://zbmath.org/1460.352752021-06-15T18:09:00+00:00"Antonelli, Paolo"https://zbmath.org/authors/?q=ai:antonelli.paoloSummary: In this note we give an alternative proof of existence of finite energy weak solutions to the quantum hydrodynamics (QHD) system. The main novelty in our approach is that no regularization procedure or approximation is needed, as it is only based on the integral formulation of NLS equation and the a priori bounds given by the Strichartz estimates. The main advantage of this proof is that it can be applied to a wider class of QHD systems.Normalized solutions for 3-coupled nonlinear Schrödinger equations.https://zbmath.org/1460.350932021-06-15T18:09:00+00:00"Liu, Chuangye"https://zbmath.org/authors/?q=ai:liu.chuangye"Tian, Rushun"https://zbmath.org/authors/?q=ai:tian.rushunSummary: In this paper, we study the existence of \(L^2\)-normalized solutions for the following 3-coupled nonlinear Schrödinger equations in \([H_r^1(\mathbb{R}^N)]^3\),
\[
\begin{cases}
-\Delta u_i=\lambda_iu_i+\mu_i|u_i|^{p_i-2}u_i+\beta r_i|u_i|^{r_i-2}\big(\sum\limits_{j\neq i}|u_j|^{r_j}\big)u_i,\\
|u_i|_2^2=a_i, \quad i,j=1,2,3,
\end{cases}
\]
where \(\mu_i,\beta\) and \(a_i\) are given positive constants, \(\lambda_i\) appear as unknown parameters, and \(H_r^1(\mathbb{R}^N)\) denotes the radial subspace of Hilbert space \(H^1(\mathbb{R}^N)\). For \(p_i, r_i\) satisfying \(L^2\)-subcritical or \(L^2\)-supercritical conditions, we obtain positive solutions of this system using variational methods and perturbation methods.Existence and concentration of nodal solutions for a subcritical \textit{p\&q} equation.https://zbmath.org/1460.351752021-06-15T18:09:00+00:00"Costa, Gustavo S."https://zbmath.org/authors/?q=ai:costa.gustavo-s"Figueiredo, Giovany M."https://zbmath.org/authors/?q=ai:figueiredo.giovany-malcherSummary: In this paper we prove existence and concentration results for a family of nodal solutions for a some quasilinear equation with subcritical growth, whose prototype is
\[
-\Delta_pu-\Delta_qu+V( x)(|u|^{p-2}u+|u|^{q-2}u)=f(u)\text{ in }\mathbb{R}^N. \]
Each nodal solution changes sign exactly once in \(\mathbb{R}^N\) and has an exponential decay at infinity. Here we use variational methods and \textit{M. A. del Pino} and \textit{P. L. Felmer}'s technique [Calc. Var. Partial Differ. Equ. 4, No. 2, 121--137 (1996; Zbl 0844.35032)] in order to overcome the lack of compactness.On a class of ultradifferentiable functions.https://zbmath.org/1460.460282021-06-15T18:09:00+00:00"Pilipović, Stevan"https://zbmath.org/authors/?q=ai:pilipovic.stevan-r"Teofanov, Nenad"https://zbmath.org/authors/?q=ai:teofanov.nenad"Tomić, Filip"https://zbmath.org/authors/?q=ai:tomic.filipSummary: We introduce a class of ultradifferentiable functions which contains Gevrey functions and study its basic properties. In particular, we investigate the continuity properties of certain (ultra)differentiable operators. Finally, we discuss microlocal properties in appropriate dual spaces.Infinitely many periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients.https://zbmath.org/1460.351122021-06-15T18:09:00+00:00"Wei, Hui"https://zbmath.org/authors/?q=ai:wei.huiSummary: We consider the periodic solutions for a semilinear Euler-Bernoulli beam equation with variable coefficients, which is used to describe the infinitesimal undamped transverse vibration of a thin straight elastic beam in a plane. The presence of variable coefficients leads to the destruction of spectral separability, which implies a loss of compactness on the range. By translating the spectrum, we construct a suitable function space which plays a crucial role in this paper. On this basis, we establish a theorem on the existence of infinitely many periodic solutions for the nonlinearity satisfying sublinear growth.A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain.https://zbmath.org/1460.352342021-06-15T18:09:00+00:00"Fino, Ahmad Z."https://zbmath.org/authors/?q=ai:fino.ahmad-z"Chen, Wenhui"https://zbmath.org/authors/?q=ai:chen.wenhuiSummary: We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity \(|u|^p+|u_t|^q\) in an exterior domain, where \(p,q>1\). We prove global (in time) existence of small data solution with suitable higher regularity by using a weighted energy method, and assuming some conditions on powers of nonlinearity.Existence and symmetry breaking of ground state solutions for Schrödinger-Poisson systems.https://zbmath.org/1460.350882021-06-15T18:09:00+00:00"Wu, Tsung-fang"https://zbmath.org/authors/?q=ai:wu.tsungfang|wu.tsung-fangSummary: We study the Schrödinger-Poisson system:
\[
\begin{cases}
-\Delta u+u+\lambda \phi u=a (x) |u|^{p-2} u \quad &\text{in }\mathbb{R}^3, \\
-\Delta \phi =u^2 \quad &\text{in } \mathbb{R}^3, \end{cases}
\]
where parameter \(\lambda >0\), \(2<p<3\) and \(a( x)\) is a positive continuous function in \(\mathbb{R}^3\). Assuming that \(a( x) \geq \lim_{|x|\rightarrow\infty} a(x) =a_\infty >0\) and other suitable conditions, we explore the energy functional corresponding to the system which is bounded below on \(H^1(\mathbb{R}^3)\) and the existence and multiplicity of positive (ground state) solutions for \(\left[\frac{A(p)}{p} a_\infty\right]^{2/(p-2)}<\lambda \leq \left[\frac{A(p)}{p}a_1\right]^{2/(p-2)}\), where \(A(p) :=2^{(6-p)/2}(3-p)^{3-p} (p-2)^{(p-2)}\) and \(a_\infty <a_1<a_{\max} :=\sup_{x\in \mathbb{R}^3} a(x)\). More importantly, when \(a(x) =a(|x|)\) and \(a(0) =a_{\max}\), we establish the existence of non-radial ground state solutions.Periodic solutions of an age-structured epidemic model with periodic infection rate.https://zbmath.org/1460.353532021-06-15T18:09:00+00:00"Kang, Hao"https://zbmath.org/authors/?q=ai:kang.hao"Huang, Qimin"https://zbmath.org/authors/?q=ai:huang.qimin"Ruan, Shigui"https://zbmath.org/authors/?q=ai:ruan.shiguiSummary: In this paper we consider an age-structured epidemic model of the susceptible-exposed-infectious-recovered (SEIR) type. To characterize the seasonality of some infectious diseases such as measles, it is assumed that the infection rate is time periodic. After establishing the well-posedness of the initial-boundary value problem, we study existence of time periodic solutions of the model by using a fixed point theorem. Some numerical simulations are presented to illustrate the obtained results.Semilinear elliptic problems involving exponential critical growth in the half-space.https://zbmath.org/1460.351672021-06-15T18:09:00+00:00"Felix, Diego D."https://zbmath.org/authors/?q=ai:felix.diego-dias"Furtado, Marcelo F."https://zbmath.org/authors/?q=ai:furtado.marcelo-fernandes"Medeiros, Everaldo S."https://zbmath.org/authors/?q=ai:medeiros.everaldo-sSummary: We perform an weighted Sobolev space approach to prove a Trudinger-Moser type inequality in the upper half-space. As applications, we derive some existence and multiplicity results for the problem
\[
\begin{cases}
-\Delta u+h(x)|u|^{q-2}u=a(x) f(u), & \text{ in }\mathbb{R}^2_+,\\
\dfrac{\partial u}{\partial \nu}+u=0, & \text{ on }\partial\mathbb{R}^2_+, \end{cases}
\]
under some technical condition on \(a,b\) and the the exponential nonlinearity \(f\). The ideas can also be used to deal with Neumann boundary conditions.