Recent zbMATH articles in MSC 35Khttps://zbmath.org/atom/cc/35K2024-02-28T19:32:02.718555ZWerkzeugPartial differential equations. I: Basic theoryhttps://zbmath.org/1527.350022024-02-28T19:32:02.718555Z"Taylor, Michael E."https://zbmath.org/authors/?q=ai:taylor.michael-eugenePublisher's description: The first of three volumes on partial differential equations, this one introduces basic examples arising in continuum mechanics, electromagnetism, complex analysis and other areas, and develops a number of tools for their solution, in particular Fourier analysis, distribution theory, and Sobolev spaces. These tools are then applied to the treatment of basic problems in linear PDE, including the Laplace equation, heat equation, and wave equation, as well as more general elliptic, parabolic, and hyperbolic equations. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.
The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. In includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.
Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.
Review of first edition: ``These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.''
(Peter Lax, SIAM review, June 1998)
See the reviews of Vol. I--III of the first edition in [Zbl 0869.35002; Zbl 0869.35003; Zbl 0869.35004]. See the reviews of Vol. I--III of the second edition in [Zbl 1206.35002; Zbl 1206.35003; Zbl 1206.35004].Partial differential equations. III: Nonlinear equationshttps://zbmath.org/1527.350042024-02-28T19:32:02.718555Z"Taylor, Michael E."https://zbmath.org/authors/?q=ai:taylor.michael-eugenePublisher's description: The third of three volumes on partial differential equations, this is devoted to nonlinear PDE. It treats a number of equations of classical continuum mechanics, including relativistic versions, as well as various equations arising in differential geometry, such as in the study of minimal surfaces, isometric imbedding, conformal deformation, harmonic maps, and prescribed Gauss curvature. In addition, some nonlinear diffusion problems are studied. It also introduces such analytical tools as the theory of \(L^p\) Sobolev spaces, Holder spaces, Hardy spaces, and Morrey spaces, and also a development of Calderon-Zygmund theory and paradifferential operator calculus. The book is targeted at graduate students in mathematics and at professional mathematicians with an interest in partial differential equations, mathematical physics, differential geometry, harmonic analysis, and complex analysis.
The third edition further expands the material by incorporating new theorems and applications throughout the book, and by deepening connections and relating concepts across chapters. It includes new sections on rigid body motion, on probabilistic results related to random walks, on aspects of operator theory related to quantum mechanics, on overdetermined systems, and on the Euler equation for incompressible fluids. The appendices have also been updated with additional results, ranging from weak convergence of measures to the curvature of Kahler manifolds.
Michael E. Taylor is a Professor of Mathematics at the University of North Carolina, Chapel Hill, NC.
Review of first edition: ``These volumes will be read by several generations of readers eager to learn the modern theory of partial differential equations of mathematical physics and the analysis in which this theory is rooted.''
(Peter Lax, SIAM review, June 1998)
See the reviews of Vol. I--III of the first edition in [Zbl 0869.35002; Zbl 0869.35003; Zbl 0869.35004]. See the reviews of Vol. I--III of the second edition in [Zbl 1206.35002; Zbl 1206.35003; Zbl 1206.35004]. For Vol. I and II of the third edition see [Zbl 1527.35002; Zbl 1527.35003].The fundamental solution and blow-up problem of an anisotropic parabolic equationhttps://zbmath.org/1527.350082024-02-28T19:32:02.718555Z"Zhan, Huashui"https://zbmath.org/authors/?q=ai:zhan.huashuiSummary: This paper is devoted to the study of anisotropic parabolic equation related to the \(p_i\)-Laplacian with a source term \(f(u)\). If \(f(u)=0\), then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of \(u\) in the source term, the initial energy \(E(0)\) is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.The solution of two-dimensional coupled Burgers' equation by \(G\)-double Laplace transformhttps://zbmath.org/1527.350112024-02-28T19:32:02.718555Z"Alhefthi, Reem K."https://zbmath.org/authors/?q=ai:alhefthi.reem-k"Eltayeb, Hassan"https://zbmath.org/authors/?q=ai:eltayeb.hassan(no abstract)Refined decay estimate and analyticity of solutions to the linear heat equation in homogeneous Besov spaceshttps://zbmath.org/1527.350162024-02-28T19:32:02.718555Z"Ozawa, Tohru"https://zbmath.org/authors/?q=ai:ozawa.tohru"Takeuchi, Taiki"https://zbmath.org/authors/?q=ai:takeuchi.taikiSummary: The heat semigroup \(\{T(t)\}_{t \geq 0}\) defined on homogeneous Besov spaces \(\dot{B}_{p, q}^s(\mathbb{R}^n)\) is considered. We show the decay estimate of \(T(t)f\in\dot{B}_{p, 1}^{s+\sigma}(\mathbb{R}^n)\) for \(f\in\dot{B}_{p, \infty}^s(\mathbb{R}^n)\) with an explicit bound depending only on the regularity index \(\sigma > 0\) and space dimension \(n\). It may be regarded as a refined result compared with that of the second author
[Partial Differ. Equ. Appl. Math. 4, Article ID 100174, 12 p. (2021; \url{doi:10.1016/j.padiff.2021.100174})].
As a result of the refined decay estimate, we also improve a lower bound estimate of the radius of convergence of the Taylor expansion of \(T(\cdot)f\) in space and time. To refine the previous results, we show explicit pointwise estimates of higher order derivatives of the power function \(|\xi|^\sigma\) for \(\sigma\in\mathbb{R}\). In addition, we also refine the \(L^1\)-estimate of the derivatives of the heat kernel.Harnack's inequality for porous medium equation with singular absorption termhttps://zbmath.org/1527.350182024-02-28T19:32:02.718555Z"Zozulia, Yevhen"https://zbmath.org/authors/?q=ai:zozulia.yevhen-sIn this technical paper, the author considers equations of porous media with measurable coefficients and a lower order singular term (a potential). The aim is to demonstrate Harnack estimates for this type of equations. This result is proved thanks to the use of two theorems. First, the author proves a Keller-Osserman-type a priori estimate and then a De Giorgi-type lemma by adapting the Kilpeläinen-Malý iteration procedure onto the Keller-Osserman-type estimates demonstrated previously.
Reviewer: Vincenzo Vespri (Firenze)An \(L^1\)-theory for a nonlinear temporal periodic problem involving \(p(x)\)-growth structure with a strong dependence on gradientshttps://zbmath.org/1527.350242024-02-28T19:32:02.718555Z"Charkaoui, Abderrahim"https://zbmath.org/authors/?q=ai:charkaoui.abderrahim"Alaa, Nour Eddine"https://zbmath.org/authors/?q=ai:eddine-alaa.nourSummary: We investigate the existence of a time-periodic solution to a nonlinear evolution equation involving \(p(x)\)-growth conditions with irregular data. We tackle our problem in a suitable functional setting by considering the so-called variable exponent Lebesgue and Sobolev spaces. By assuming that the data belongs only to \(L^1\), we prove the existence of a renormalized time-periodic solution to the studied model.A perturbative approach to the parabolic optimal transport problemhttps://zbmath.org/1527.350282024-02-28T19:32:02.718555Z"Abedin, Farhan"https://zbmath.org/authors/?q=ai:abedin.farhan"Kitagawa, Jun"https://zbmath.org/authors/?q=ai:kitagawa.junSummary: Fix a pair of smooth source and target densities \(\rho\) and \(\rho^\ast\) of equal mass, supported on bounded domains \(\Omega, \Omega^\ast \subset \mathbb{R}^n\). Also fix a cost function \(c_0 \in C^{4,\alpha} (\overline{\Omega} \times \overline{\Omega^\ast})\) satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume \(\Omega\) and \(\Omega^\ast\) are uniformly \(c_0\)- and \(c_0^\ast\)-convex with respect to each other. We consider a parabolic version of the optimal transport problem between \((\Omega,\rho)\) and \((\Omega^\ast,\rho^\ast)\) when the cost function \(c\) is a sufficiently small \(C^4\) perturbation of \(c_0\), and where the size of the perturbation depends on the given data. Our main result establishes global-in-time existence of a solution \(u \in C^2_xC^1_t(\overline{\Omega} \times [0, \infty))\) of this parabolic problem, and convergence of \(u(\cdot,t)\) as \(t \to \infty\) to a Kantorovich potential for the optimal transport map between \((\Omega,\rho)\) and \((\Omega^\ast,\rho^\ast)\) with cost function \(c\). This is the first convergence result for the parabolic optimal transport problem when the cost function \(c\) fails to satisfy the weak Ma-Trudinger-Wang condition by a quantifiable amount.Limits of solutions to the semilinear plate equation with small parameterhttps://zbmath.org/1527.350292024-02-28T19:32:02.718555Z"Perjan, Andrei"https://zbmath.org/authors/?q=ai:perjan.andrei-v"Rusu, Galina"https://zbmath.org/authors/?q=ai:rusu.galinaSummary: We study the existence of the limits of solutions to the semilinear plate equation with boundary Dirichlet condition with a small parameter coefficient of the second order derivative in time. We establish the convergence of solutions to the perturbed problem and their derivatives in spacial variables to the corresponding solutions to the unperturbed problem as the small parameter tends to zero.Unique solvability and zero diffusion limits of global large solution for a nonlinear hyperbolic system with damping and diffusionhttps://zbmath.org/1527.350302024-02-28T19:32:02.718555Z"Yang, Andrew"https://zbmath.org/authors/?q=ai:yang.andrew"Zhou, Wenshu"https://zbmath.org/authors/?q=ai:zhou.wenshuSummary: We consider the Dirichlet-Neumann problem and the spatially periodic Cauchy problem for a nonlinear hyperbolic system with damping and diffusion introduced in [\textit{D. Y. Hsieh}, J. Math. Phys. 28, 1589--1597 (1987; Zbl 0657.35116)] for the study of chaos. Firstly, the existence and uniqueness of global solutions with large initial data is established. Then the zero diffusion limits are justified. Moreover, the \(L^2\) convergence rate in terms of the diffusion coefficient is obtained. Based on a new observation of the structure of the system, two equalities are found to show the existence of global large solutions of the system.Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundaryhttps://zbmath.org/1527.350352024-02-28T19:32:02.718555Z"Nandakumaran, Akambadath Keerthiyil"https://zbmath.org/authors/?q=ai:nandakumaran.akambadath-keerthiyil"Sankar, Kasinathan"https://zbmath.org/authors/?q=ai:sankar.kasinathan(no abstract)Positive steady-state solutions for a water-vegetation model with the infiltration feedback effecthttps://zbmath.org/1527.350372024-02-28T19:32:02.718555Z"Guo, Gaihui"https://zbmath.org/authors/?q=ai:guo.gaihui"Zhao, Shihan"https://zbmath.org/authors/?q=ai:zhao.shihan"Wang, Jingjing"https://zbmath.org/authors/?q=ai:wang.jingjing"Gao, Yuanxiao"https://zbmath.org/authors/?q=ai:gao.yuanxiaoSummary: In this paper, a water-vegetation model with the infiltration feedback effect is considered. Firstly, through the linear stability analysis, we get the parameter area where Turing instability can occur. Next, by maximum principle, a priori estimates for positive steady-state solutions are obtained and sufficient conditions for the nonexistence of nonconstant positive steady-state solution are given. Moreover, the steady-state bifurcations at both simple and double eigenvalues are analyzed separately. We establish the global structure of the bifurcation from simple eigenvalues and get the sufficient condition to determine the bifurcation direction. For the case of double eigenvalues, the techniques of space decomposition and the implicit function theorem are used. Finally, we verify and supplement the theoretical analysis results with numerical simulations.Spectral instability of rolls in the 2-dimensional generalized Swift-Hohenberg equationhttps://zbmath.org/1527.350442024-02-28T19:32:02.718555Z"Chae, Myeongju"https://zbmath.org/authors/?q=ai:chae.myeongju"Jung, Soyeun"https://zbmath.org/authors/?q=ai:jung.soyeunSummary: The aim of this paper is to investigate the spectral instability of roll waves bifurcating from an equilibrium in the \(2\)-dimensional generalized Swift-Hohenberg equation. We characterize unstable Bloch wave vectors to prove that the rolls are spectrally unstable in the whole parameter region where the rolls exist, while they are Eckhaus stable in \(1\) dimension [\textit{S. Jung}, Bull. Korean Math. Soc. 60, No. 1, 257--279 (2023; Zbl 1514.35032)]. As compared to [\textit{A. Mielke}, Commun. Math. Phys. 189, No. 3, 829--853 (1997; Zbl 0897.76037)], showing that the stability of the rolls in the \(2\)-dimensional Swift-Hohenberg equation without a quadratic nonlinearity is determined by Eckhaus and zigzag curves, our result says that the quadratic nonlinearity of the equation is the cause of such instability of the rolls.Quantitative analysis of pattern formation in a multistable model of glycolysis with diffusionhttps://zbmath.org/1527.350512024-02-28T19:32:02.718555Z"Bashkirtseva, Irina"https://zbmath.org/authors/?q=ai:bashkirtseva.irina-adolfovna"Pankratov, Alexander"https://zbmath.org/authors/?q=ai:pankratov.alexandr"Ryashko, Lev"https://zbmath.org/authors/?q=ai:ryashko.lev-borisovichSummary: Motivated by the problem of uncertainty quantification in self-organization, we study a spatially extended Sel'kov-Strogatz model of glycolysis. A variety of coexisting patterns induced by the Turing instability is studied in the parametric zones where the original local model without diffusion exhibits stable equilibria or self-oscillations. A phenomenon of the suppression of homogeneous self-oscillations by diffusion with formation of stationary non-homogeneous patterns-attractors is revealed. To quantify the uncertainty in the number and modality of patterns-attractors and to perform an advanced parametric analysis, we use the spectral coefficients technique and Shannon entropy.Organization of spatially localized structures near a codimension-three cusp-Turing bifurcationhttps://zbmath.org/1527.350522024-02-28T19:32:02.718555Z"Parra-Rivas, Pedro"https://zbmath.org/authors/?q=ai:parra-rivas.pedro"Champneys, Alan R."https://zbmath.org/authors/?q=ai:champneys.alan-r"Al Saadi, Fahad"https://zbmath.org/authors/?q=ai:al-saadi.fahad"Gomila, Damia"https://zbmath.org/authors/?q=ai:gomila.damia"Knobloch, Edgar"https://zbmath.org/authors/?q=ai:knobloch.edgarSummary: A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology, and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth-order scalar system. The problem contains three small parameters: two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed which involves transitions between regions of localized periodic patterns generated by standard homoclinic snaking, and regions of stationary domains of one homogeneous solution embedded in the other organized in a collapsed snaking structure. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally.Well-posedness and asymptotic behavior for the fractional Keller-Segel system in critical Besov-Herz-type spaceshttps://zbmath.org/1527.350542024-02-28T19:32:02.718555Z"Azevedo, Joelma"https://zbmath.org/authors/?q=ai:azevedo.joelma"Bezerra, Mario"https://zbmath.org/authors/?q=ai:bezerra.mario"Cuevas, Claudio"https://zbmath.org/authors/?q=ai:cuevas.claudio"Soto, Herme"https://zbmath.org/authors/?q=ai:soto.herme(no abstract)Global and local asymptotic stability of an epidemic reaction-diffusion model with a nonlinear incidencehttps://zbmath.org/1527.350632024-02-28T19:32:02.718555Z"Djebara, Lamia"https://zbmath.org/authors/?q=ai:djebara.lamia"Douaifia, Redouane"https://zbmath.org/authors/?q=ai:douaifia.redouane"Abdelmalek, Salem"https://zbmath.org/authors/?q=ai:abdelmalek.salem"Bendoukha, Samir"https://zbmath.org/authors/?q=ai:bendoukha.samir(no abstract)Large time behavior of signed fractional porous media equations on bounded domainshttps://zbmath.org/1527.350652024-02-28T19:32:02.718555Z"Franzina, Giovanni"https://zbmath.org/authors/?q=ai:franzina.giovanni"Volzone, Bruno"https://zbmath.org/authors/?q=ai:volzone.brunoSummary: Following the methodology of \textit{L. Brasco} and \textit{B. Volzone} [Adv. Math. 394, Article ID 108029, 57 p. (2022; Zbl 1480.35029)], we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane-Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.Global existence and asymptotic behavior of a predator-prey chemotaxis system with inter-species interaction coefficientshttps://zbmath.org/1527.350672024-02-28T19:32:02.718555Z"Gnanasekaran, S."https://zbmath.org/authors/?q=ai:gnanasekaran.s"Nithyadevi, N."https://zbmath.org/authors/?q=ai:nithyadevi.nagarajan"Udhayashankar, C."https://zbmath.org/authors/?q=ai:udhayashankar.cSummary: A fully parabolic predator-prey chemotaxis system with inter-species interaction coefficient
\[
\begin{cases}
\qquad u_{1 t} = d_1 \Delta u_1 - \chi \nabla \cdot (u_1 \nabla v_1) + u_1 (\sigma_1 - a_1 u_1 + e_1 u_2), \qquad & x \in \Omega, t > 0, \\
\qquad u_{2 t} = d_2 \Delta u_2 + \xi \nabla \cdot (u_2 \nabla v_2) + u_2 (\sigma_2 - a_2 u_2 - e_2 u_1), & x \in \Omega, t > 0, \\
\qquad v_{1_t} = d_3 \Delta v_1 + \alpha_1 u_2 - \beta_1 v_1, & x \in \Omega, t > 0, \\
\qquad v_{2_t} = d_4 \Delta v_2 + \alpha_2 u_1 - \beta_2 v_2, & x \in \Omega, t > 0,
\end{cases}
\]
under the homogeneous Neumann boundary conditions in an open, bounded domain \(\Omega \subset \mathbb{R}^n\) with smooth boundary \(\partial \Omega\) is examined. The parameters are all positive constants and the initial data \((u_{10}, u_{20}, v_{1_0}, v_{2_0})\) are non negative. With some supplementary conditions imposed on the parameters, it is proved that the above system has a unique globally bounded classical solution for \(n \geq 2\). Moreover, the convergence of the solution is asserted by constructing a suitable Lyapunov functional. If \(e_2, \chi^2\) and \(\xi^2\) are sufficiently small, then the solution of the above system converges to a unique positive equilibrium. If \(e_2\) is sufficiently large and \(\chi^2\) is sufficiently small, then the solution converges to the semi-trivial equilibrium point. Remarkably, the convergence rate is exponential when \(e_2 \neq \frac{\sigma_2 a_1}{\sigma_1}\) and algebraic if \(e_2 = \frac{\sigma_2 a_1}{\sigma_1}\). Finally, the numerical examples validate the outcomes of asymptotic behavior. The results demonstrate the predominant behavior of the parameters \(a_1\) and \(a_2\) in the existence and stability.Finite-parameter feedback stabilization of original Burgers' equations and Burgers' equation with nonlocal nonlinearitieshttps://zbmath.org/1527.350682024-02-28T19:32:02.718555Z"Gumus, Serap"https://zbmath.org/authors/?q=ai:gumus.serap"Kalantarov, Varga"https://zbmath.org/authors/?q=ai:kalantarov.varga-k(no abstract)Bubble decomposition for the harmonic map heat flow in the equivariant casehttps://zbmath.org/1527.350702024-02-28T19:32:02.718555Z"Jendrej, Jacek"https://zbmath.org/authors/?q=ai:jendrej.jacek"Lawrie, Andrew"https://zbmath.org/authors/?q=ai:lawrie.andrew-g-wSummary: We consider the harmonic map heat flow for maps \(\mathbb{R}^2\rightarrow\mathbb{S}^2\), under equivariant symmetry. It is known that solutions to the initial value problem can exhibit bubbling along a sequence of times -- the solution decouples into a superposition of harmonic maps concentrating at different scales and a body map that accounts for the rest of the energy. We prove that this bubble decomposition is unique and occurs continuously in time. The main new ingredient in the proof is the notion of a collision interval from our work [``Soliton resolution for energy-critical wave wave maps in the equivariant case'', J. Am. Math. Soc. (to appear)].Asymptotic limits of viscous Cahn-Hilliard equation with homogeneous Dirichlet boundary conditionhttps://zbmath.org/1527.350712024-02-28T19:32:02.718555Z"Kagawa, Keiichiro"https://zbmath.org/authors/?q=ai:kagawa.keiichiro"Ôtani, Mitsuharu"https://zbmath.org/authors/?q=ai:otani.mitsuharuSummary: In this paper, we are concerned with the asymptotic behavior of solutions for the viscous Cahn-Hilliard equations when parameters involved in the equations tend to zero. Namely, it is shown that solutions of the viscous Cahn-Hilliard equations converge to solutions of the Allen-Cahn equation, the Cahn-Hilliard equation, and the viscous diffusion equation, when suitable parameters tend to zero. Compared with the existing results, a much wider range of chemical potentials can be treated within our setting.Large time behavior of solutions to a 3D Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturationhttps://zbmath.org/1527.350732024-02-28T19:32:02.718555Z"Ke, Yuan-yuan"https://zbmath.org/authors/?q=ai:ke.yuanyuan"Zheng, Jia-Shan"https://zbmath.org/authors/?q=ai:zheng.jiashanSummary: In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed,
\[
\begin{cases}
n_t +u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c), \quad x\in \Omega, t>0, \\
c_t +u\cdot \nabla c=\Delta c-c+n, \quad x\in\Omega, \quad t>0, \\
u_t +\nabla P=\Delta u+n\nabla \phi, \quad x\in\Omega, \quad t>0, \\
\nabla \cdot u = 0, \quad x\in\Omega, t>0
\end{cases}
\tag{KSS}
\]
subject to the boundary conditions \((\nabla n-nS(x, n, c) \nabla c) \cdot \nu = \nabla c \cdot \nu = 0\) and \(u = 0\), and suitably regular initial data \((n_0 (x), c_0 (x), u_0 (x))\), where \(\Omega \subset \mathbb{R}^3\) is a bounded domain with smooth boundary \(\partial \Omega\). Here \(S\) is a chemotactic sensitivity satisfying \(\vert S(x, n, c)\vert \leq C_S (1+n)^{-\alpha}\) with some \(C_S >0\) and \(\alpha >0\). The greatest contribution of this paper is to consider the large time behavior of solutions for the system (KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system (KSS) decays to \((\frac{1}{|\Omega|} \int_{\Omega}n_0, \frac{1}{|\Omega|} \int_{\Omega}n_0, 0)\) exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient \(C_S\) of chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.Stability of constant equilibria in a Keller-Segel system with gradient dependent chemotactic sensitivityhttps://zbmath.org/1527.350742024-02-28T19:32:02.718555Z"Kohatsu, S."https://zbmath.org/authors/?q=ai:kohatsu.s"Yokota, T."https://zbmath.org/authors/?q=ai:yokota.tomomi|yokota.tatsuya|yokota.takeru|yokota.takumi|yokota.terufumi|yokota.takashi|yokota.teruoSummary: This paper deals with the Keller-Segel system with gradient dependent chemotactic sensitivity,
\[\begin{cases}
u_1 = \Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v), & x \in \Omega, t > 0, \\
v_t = \Delta v - v + u, & x \in \Omega, t > 0,
\end{cases}\]
where \(\Omega \subset \mathbb{R}^n\) (\(n \in \mathbb{N}\)) is a bounded domain with smooth boundary, and \(\chi > 0\), \(p \in (1, \infty)\) are constants. The purpose of this paper is to establish stability of constant equilibria under some smallness conditions for the initial data.Long term spatial homogeneity for a chemotaxis model with local sensing and consumptionhttps://zbmath.org/1527.350752024-02-28T19:32:02.718555Z"Laurençot, Philippe"https://zbmath.org/authors/?q=ai:laurencot.philippeSummary: Global weak solutions to a chemotaxis model with local sensing and consumption are shown to converge to spatially homogeneous steady states in the large time limit, when the motility is assumed to be positive and \(C^1\)-smooth on \([0,\infty)\). The result is valid in arbitrary space dimension \(n \geq 1\) and extends a previous result which only deals with space dimensions \(n \in \{1,2,3 \}\).Global boundedness and asymptotic behavior in a fully parabolic attraction-repulsion chemotaxis model with logistic sourcehttps://zbmath.org/1527.350772024-02-28T19:32:02.718555Z"Liu, Chao"https://zbmath.org/authors/?q=ai:liu.chao.2|liu.chao"Liu, Bin"https://zbmath.org/authors/?q=ai:liu.bin.1|liu.bin.5|liu.bin.14|liu.bin.2|liu.bin.19|liu.bin.3|liu.bin.11|liu.bin|liu.bin.4|liu.bin.12|liu.bin.7|liu.bin.13|liu.bin.6Summary: In this paper, we consider a fully parabolic attraction-repulsion chemotaxis model with logistic source. First of all, we obtain an explicit formula \(\mu_0\) for the logistic damping rate \(\mu\) such that the model has no blow-up when \(\mu>\mu_0\). In addition, the asymptotic behavior of the solutions is studied. Our results partially generalize and improve some results in the literature, and partially results are new.A note on global stability of a degenerate diffusion avian influenza model with seasonality and spatial heterogeneityhttps://zbmath.org/1527.350782024-02-28T19:32:02.718555Z"Li, Wenjie"https://zbmath.org/authors/?q=ai:li.wenjie"Guan, Yajuan"https://zbmath.org/authors/?q=ai:guan.yajuan"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jinde"Xu, Fei"https://zbmath.org/authors/?q=ai:xu.fei|xu.fei.1|xu.fei.2|xu.fei.4|xu.fei.3Summary: This article establishes the global stability of the disease-free equilibrium in a degenerate diffusion system, which involves a degenerate diffusion system incorporating environmental transmission and spatial heterogeneity. This framework is designed to elucidate the transmission dynamics of avian influenza virus among avian, poultry, and human populations. We focus specifically on scenarios where the basic reproductive number equals 1. Our findings are an extension of the research conducted by \textit{T. Zheng} et al. [Nonlinear Anal., Real World Appl. 67, Article ID 103567, 30 p. (2022; Zbl 1492.35387)].Convergence of solutions to a convective Cahn-Hilliard-type equation of the sixth order in case of small deposition rateshttps://zbmath.org/1527.350812024-02-28T19:32:02.718555Z"Rybka, Piotr"https://zbmath.org/authors/?q=ai:rybka.piotr"Wheeler, Glen"https://zbmath.org/authors/?q=ai:wheeler.glen-eSummary: We show stabilization of solutions to the sixth-order convective Cahn-Hilliard equation. The problem has the structure of a gradient flow perturbed by a quadratic destabilizing term with coefficient \(\delta > 0\). Through application of an abstract result by Carvalho, Langa, and Robinson we show that for small \(\delta\) the equation has the structure of gradient flow in a weak sense. On the way we prove a kind of Liouville theorem for eternal solutions to parabolic problems. Finally, the desired stabilization follows from a powerful theorem due to Hale and Raugel.On stability and regularity for semilinear anomalous diffusion equations perturbed by weak-valued nonlinearitieshttps://zbmath.org/1527.350832024-02-28T19:32:02.718555Z"Van Dac, Nguyen"https://zbmath.org/authors/?q=ai:van-dac.nguyen"Dinh, Ke Tran"https://zbmath.org/authors/?q=ai:dinh.ke-tran"Thuy, Lam Tran Phuong"https://zbmath.org/authors/?q=ai:thuy.lam-tran-phuongSummary: Various diffusion processes in media with memory are depicted by anomalous diffusion equations. We are concerned with a class of anomalous diffusion equations with the nonlinearity taking values in Hilbert scales of negative order. The fundamental questions on global solvability, stability and regularity of solutions to the Cauchy problem governed by the mentioned equations are taken into account. We obtain some solvability results under different assumptions on the regularity of the nonlinear function. When the nonlinearity becomes less singular, the asymptotic stability of solutions is proved. Provided that the kernel function is 2-regular and sectorial, we show the Hölder continuity of the obtained solutions. Our approach is based on the resolvent theory, fixed point argument and embeddings of fractional Sobolev spaces. The applicability of our results is presented for some anomalous diffusion problems, where the nonlinear function is of polynomial or convection type.Uniform attractor of impulse-perturbed reaction-diffusion systemhttps://zbmath.org/1527.350862024-02-28T19:32:02.718555Z"Kapustyan, Oleksiy"https://zbmath.org/authors/?q=ai:kapustyan.oleksiy-v"Kapustian, Olena"https://zbmath.org/authors/?q=ai:kapustian.olena-a"Korol, Ihor"https://zbmath.org/authors/?q=ai:korol.ihor"Rubino, Bruno"https://zbmath.org/authors/?q=ai:rubino.brunoSummary: The paper deals with the existence and invariance of uniform attractors of impulse-perturbed reaction-diffusion system. Moments of impulsive perturbation are not fixed and are determined by the getting of trajectory into a given subset \(M\) of the phase space. At the moment of getting, the phase variable undergoes a fixed increment. It is proved that such a problem generates impulsive dynamical system that has the uniform attractor \(\Theta\), and the set \(\Theta\setminus M\) is invariant with respect to the impulsive semiflow.Global attractors of generic reaction diffusion equations under Lipschitz perturbationshttps://zbmath.org/1527.350882024-02-28T19:32:02.718555Z"Lee, Jihoon"https://zbmath.org/authors/?q=ai:lee.jihoon.1"Ngocthach Nguyen"https://zbmath.org/authors/?q=ai:ngocthach-nguyen."Pires, Leonardo"https://zbmath.org/authors/?q=ai:pires.leonardoSummary: In this paper we prove that generically the dynamical system induced by a reaction diffusion equation is Gromov-Hausdorff stable on its global attractor under Lipschitz perturbations of the domain and equation.Robust exponential attractors for the Cahn-Hilliard-Oono-Navier-Stokes systemhttps://zbmath.org/1527.350902024-02-28T19:32:02.718555Z"Nimi, Aymard Christbert"https://zbmath.org/authors/?q=ai:nimi.aymard-christbert"Reval Langa, Franck Davhys"https://zbmath.org/authors/?q=ai:langa.franck-davhys-reval"Bissouesse, Aurdeli Juves Primpha"https://zbmath.org/authors/?q=ai:bissouesse.aurdeli-juves-primpha"Moukoko, Daniel"https://zbmath.org/authors/?q=ai:moukoko.daniel"Batchi, Macaire"https://zbmath.org/authors/?q=ai:batchi.macaireSummary: We focus in this article on a Cahn-Hilliard-Oono-Navier-Stokes system, which is a model consisting of a Navier-Stokes equation governing the fluid velocity coupled to the Cahn-Hilliard-Oono equation. We start our study by establishing the a priori estimates, the existence and uniqueness of the solution, and then the existence of a continuous dissipative semigroup generated by the system. Moreover, we prove the existence of the exponential attractors and, thus, of finite-dimensional global attractors. Finally, we construct a robust family of exponential attractors as a nonnegative parameter \(\alpha\) goes to zero.Asymptotic behavior of an Allen-Cahn type equation with temperaturehttps://zbmath.org/1527.350912024-02-28T19:32:02.718555Z"Ntsokongo, Armel Judice"https://zbmath.org/authors/?q=ai:ntsokongo.armel-judiceSummary: In this paper, we are interested in the study of the asymptotic behavior of an Allen-Cahn type equation with temperature and endowed Dirichlet boundary conditions. Such a model has, in particular, applications in chemistry. In particular, we obtain well-posedness results and the existence of the finite-dimensional global attractor as well as the existence of exponentials attractors by means of classical arguments.Galerkin stability of pullback attractors for nonautonomous nonlocal equationshttps://zbmath.org/1527.350922024-02-28T19:32:02.718555Z"Wang, Shulin"https://zbmath.org/authors/?q=ai:wang.shulin"Li, Yangrong"https://zbmath.org/authors/?q=ai:li.yangrongSummary: We study Galerkin approximation of pullback attractors for a nonautonomous nonlocal parabolic equation. We show that the \(n\)th Galerkin approximation system has a pullback attractor in the \(n \)-dimensional subspace of the Lebesgue space. We then prove that the sequence of Galerkin pullback attractors is uniformly backward bounded and that the sequence of Galerkin solutions converges uniformly on any bounded set. Using these results, we establish upper semi-convergence of the sequence of Galerkin pullback attractors towards the pullback attractors of the original infinite-dimensional system.Modulation theory for the flat blow-up solutions of nonlinear heat equationhttps://zbmath.org/1527.350962024-02-28T19:32:02.718555Z"Duong, Giao Ky"https://zbmath.org/authors/?q=ai:duong.giao-ky"Nouaili, Nejla"https://zbmath.org/authors/?q=ai:nouaili.nejla"Zaag, Hatem"https://zbmath.org/authors/?q=ai:zaag.hatemSummary: In this paper, we revisit the proof of the existence of a solution to the semilinear heat equation in one space dimension with a \textit{flat} blow-up profile, already proved by Bricmont and Kupainen together with Herrero and Velázquez. Though our approach relies on the well-celebrated method, based on the reduction of the problem to a finite-dimensional one, then the use of a topological ``shooting method'' to solve the latter, the novelty of our approach lays in the use of a modulation technique to control the projection of the zero eigenmode arising in the problem. Up to our knowledge, this is the first time where modulation is used with this kind of profiles. We do hope that this simplifies the argument.Fujita exponents for an inhomogeneous parabolic equation with variable coefficientshttps://zbmath.org/1527.350982024-02-28T19:32:02.718555Z"Sun, Xizheng"https://zbmath.org/authors/?q=ai:sun.xizheng"Liu, Bingchen"https://zbmath.org/authors/?q=ai:liu.bingchen"Li, Fengjie"https://zbmath.org/authors/?q=ai:li.fengjie(no abstract)The Caginalp phase field systems with logarithmic nonlinear termshttps://zbmath.org/1527.350992024-02-28T19:32:02.718555Z"Cherfils, Laurence"https://zbmath.org/authors/?q=ai:cherfils.laurence"Miranville, Alain"https://zbmath.org/authors/?q=ai:miranville.alain-mSummary: Our aim in this paper is to study the existence and uniqueness of solutions to several phase field systems with logarithmic nonlinear terms. These systems were either proposed by Gunduz Cagninalp or are variants of these models, based on other laws than the usual Fourier law for heat conduction. In particular, an essential step is the separation of the order parameter from the pure phases.The second-order gradient estimates for the \(V\)-heat kernel and its applicationshttps://zbmath.org/1527.351022024-02-28T19:32:02.718555Z"Su, Jiarong"https://zbmath.org/authors/?q=ai:su.jiarongSummary: In this paper, we derive the second-order gradient estimates for the \(V\)-heat kernel on complete Riemannian manifolds with Bakry-Emery Ricci curvature bounded from below. Applying these estimates, we proved that the \(f\)-Riesz transform on a complete manifold with nonnegative \(N\)-Bakry-Emery Ricci curvature is of weak type (1,1).Pulsating fronts of spatially periodic bistable reaction-diffusion equations around an obstaclehttps://zbmath.org/1527.351032024-02-28T19:32:02.718555Z"Jia, Fu-Jie"https://zbmath.org/authors/?q=ai:jia.fu-jie"Sheng, Wei-Jie"https://zbmath.org/authors/?q=ai:sheng.weijie"Wang, Zhi-Cheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.2Summary: In this paper, we study a spatially periodic bistable-type reaction-diffusion equation in so-called exterior domains \(\Omega =\mathbb{R}^N \backslash K\), where \(K\subset\mathbb{R}^N\) is a compact set and denotes an obstacle. For any direction \(e\in\mathbb{S}^{N-1}\), if the spatially periodic bistable reaction-diffusion equation in \(\mathbb{R}^N\) admits a moving pulsating front (i.e., the wave speed is nonzero), we first prove the existence and uniqueness of entire solution in the exterior domain \(\Omega\), which is emanated from the moving pulsating front. Assuming further that the propagation of the entire solution is complete (i.e., convergence to 1), we prove that the entire solution is a transition front connecting 0 and 1 and is trapped between two translates of the moving pulsating front as time goes to \(+\infty\). In particular, applying a Liouville-type result, we prove that the entire solution can eventually recover to the same moving pulsating front after crossing the obstacle \(K\) by providing some appropriate hypotheses.Phragmén-Lindelöf alternative results in time-dependent double-diffusive Darcy plane flowhttps://zbmath.org/1527.351052024-02-28T19:32:02.718555Z"Li, Yuanfei"https://zbmath.org/authors/?q=ai:li.yuanfei"Chen, Xuejiao"https://zbmath.org/authors/?q=ai:chen.xuejiao(no abstract)Higher Hölder regularity for nonlocal parabolic equations with irregular kernelshttps://zbmath.org/1527.351082024-02-28T19:32:02.718555Z"Byun, Sun-Sig"https://zbmath.org/authors/?q=ai:byun.sun-sig"Kim, Hyojin"https://zbmath.org/authors/?q=ai:kim.hyojin"Kim, Kyeongbae"https://zbmath.org/authors/?q=ai:kim.kyeongbaeIn this paper, the authors prove Hölder regularity for the parabolic equation
\begin{align*}
\partial_t u - \mathrm{pv}\int\limits_{\mathbb{R}^n}\Phi(u(x,t)-u(y,t))\frac{A(x,y,t)}{|x-y|^{n+2s}}\,dy=f,
\end{align*}
where \(\Phi:\mathbb{R}\to\mathbb{R}\) satisfies \(\Phi(0)=0\) and standard growth conditions
\begin{align*}
(\Phi(\xi)-\Phi(\xi'))(\xi-\xi')\gtrsim |\xi-\xi'|^2,\\
|\Phi(\xi)-\Phi(\xi')|\lesssim|\xi-\xi'|,
\end{align*}
\(A:\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) is measurable function bounded from above and below, which is ``locally close enough to being translation invariant'', and \(f\) satisfies a suitable integrability condition. This condition allows for discontinuous kernels.
For \(\Phi(t)=t\) and \(A=1\), the equation is the fractional heat equation.
The condition on \(A\) is a parabolic version of the condition presented for nonlocal elliptic equations in [\textit{S. Nowak}, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)].
The proofs use a perturbation argument similar to [\textit{S. Nowak}, Calc. Var. Partial Differ. Equ. 60, No. 1, Paper No. 24, 37 p. (2021; Zbl 1509.35087)] and the method of iterated discrete differentiation from [\textit{L. Brasco} et al., J. Evol. Equ. 21, No. 4, 4319--4381 (2021; Zbl 1486.35084)].
Additionally, the authors prove existence and local boundedness for this equation without the extra constraint on \(A\).
Reviewer: Vivek Tewary (Sri City)Regularity of stable solutions to reaction-diffusion elliptic equationshttps://zbmath.org/1527.351092024-02-28T19:32:02.718555Z"Cabré, Xavier"https://zbmath.org/authors/?q=ai:cabre.xavierSummary: The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970s. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This solves an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue question in differential geometry: the regularity of stable minimal surfaces.
For the entire collection see [Zbl 1519.00033].\(C^{1, \alpha}\)-regularity for functions in solution classes and its application to parabolic normalized \(p\)-Laplace equationshttps://zbmath.org/1527.351102024-02-28T19:32:02.718555Z"Lee, Se-Chan"https://zbmath.org/authors/?q=ai:lee.se-chan"Yun, Hyungsung"https://zbmath.org/authors/?q=ai:yun.hyungsungSummary: We establish the global \(C^{1, \alpha}\)-regularity for functions in solution classes, whenever ellipticity constants are sufficiently close. As an application, we derive the global regularity result concerning the parabolic normalized \(p\)-Laplace equations, provided that \(p\) is close enough to 2. Our analysis relies on the compactness argument with the iteration procedure.Besov regularity of inhomogeneous parabolic PDEshttps://zbmath.org/1527.351142024-02-28T19:32:02.718555Z"Schneider, Cornelia"https://zbmath.org/authors/?q=ai:schneider.cornelia"Szemenyei, Flóra Orsolya"https://zbmath.org/authors/?q=ai:szemenyei.flora-orsolyaSummary: We study the regularity of solutions of parabolic partial differential equations with inhomogeneous boundary conditions on polyhedral domains \(D\subset \mathbb{R}^3\) in the specific scale \(B^{\alpha}_{\tau ,\tau}\), \(\frac{1}{\tau}=\frac{\alpha}{3}+\frac{1}{p}\) of Besov spaces. The regularity of the solution in this scale determines the order of approximation that can be achieved by adaptive numerical schemes. We show that for all cases under consideration the Besov regularity is high enough to justify the use of adaptive algorithms. Our results are in good agreement with [\textit{S. Dahlke} and \textit{C. Schneider}, Anal. Appl., Singap. 17, No. 2, 235--291 (2019; Zbl 1420.35055)], where parabolic equations with homogeneous boundary conditions were investigated.Eventual smoothness and asymptotic stabilization in a two-dimensional logarithmic chemotaxis-Navier-Stokes system with nutrient-supported proliferation and signal consumptionhttps://zbmath.org/1527.351152024-02-28T19:32:02.718555Z"Wang, Yifu"https://zbmath.org/authors/?q=ai:wang.yifu"Liu, Ji"https://zbmath.org/authors/?q=ai:liu.jiSummary: In this paper, we study the influence of nutrient-dependent cell proliferation on large time behavior of the solutions to an associated initial-boundary problem of
\[
\begin{cases}
n_t + u \cdot \nabla n = \Delta n - \nabla \cdot (\frac{n}{c} \nabla c) + nc, & x \in \Omega, t > 0, \\
c_t + u \cdot \nabla c = \Delta c - n c, & x \in \Omega, t > 0, \\
u_t + (u \cdot \nabla) u = \Delta u + \nabla P + n \nabla \Phi, & x \in \Omega, t > 0,
\end{cases}
\]
subject to no-flux/no-flux/Dirichlet boundary conditions in a smoothly bounded domain \(\Omega \subset \mathbb{R}^2\), where \(+ n c\) in the first equation denotes cell reproduction in dependence on nutrients. It is proved that the initial-boundary problem is globally solvable in a generalized sense for any appropriately regular initial data, and that the generalized solutions thereof emanating from suitably small initial data will become smooth from a finite waiting time and exponentially approach \((\frac{1}{|\Omega|} \int_{\Omega}(n_0 + c_0), 0, 0)\) as \(t \to \infty\).
As compared to a precedent result obtained without cell proliferation, the complexity caused by \(+ nc\) in the first equation requires some conditions on the initial data \(c_0\) rather only on \(n_0\) in order to derive the large time behavior of the solutions thereof, which seems comprehensible because of the necessity for offsetting the possibly increasing trend of cell by \(+ nc\).Traveling waves for discrete reaction-diffusion equations in the general monostable casehttps://zbmath.org/1527.351182024-02-28T19:32:02.718555Z"Al Haj, M."https://zbmath.org/authors/?q=ai:haj.m-al|al-haj.mohammad"Monneau, R."https://zbmath.org/authors/?q=ai:monneau.regisSummary: We consider general fully nonlinear discrete reaction-diffusion equations \(u_t = F [u]\), described by some function \(F\). In the positively monostable case, we study monotone traveling waves of velocity \(c\), connecting the unstable state 0 to a stable state 1. Under Lipschitz regularity of \(F\), we show that there is a minimal velocity \(c_F^+\) such that there is a branch of traveling waves with velocities \(c \geq c_F^+\) and no traveling waves for \(c < c_F^+\). We also show that the map \(F \mapsto c_F^+\) is not continuous for the \(L^\infty\) norm on \(F\). Assuming more regularity of \(F\) close to the unstable state 0, we show that \(c_F^+ \geq c_F^\ast\), where the velocity \(c_F^\ast\) can be computed from the linearization of the equation around the unstable state 0. In addition, we show that the inequality can be strict for certain nonlinearities \(F\). On the contrary, under a KPP condition on \(F\), we show the equality \(c_F^+ = c_F^\ast\). Finally, we provide an example in which \(c_F^+\) is negative.Stability and uniqueness of generalized traveling front for non-autonomous Fisher-KPP equations with nonlocal diffusionhttps://zbmath.org/1527.351192024-02-28T19:32:02.718555Z"Cao, Qian"https://zbmath.org/authors/?q=ai:cao.qian"Bao, Xiongxiong"https://zbmath.org/authors/?q=ai:bao.xiongxiongSummary: This paper is concerned with the stability and uniqueness of generalized traveling front of non-autonomous Fisher-KPP equations with nonlocal diffusion. In such non-autonomous nonlocal dispersal equations, both the diffusion kernel and the reaction term generally depend on time. The existence of generalized traveling wave fronts of such equations has been studied in [\textit{A. Ducrot} and \textit{Z. Jin}, Ann. Mat. Pura Appl. (4) 201, No. 4, 1607--1638 (2022; Zbl 1495.35067)]. By using comparison principle and part metric, we show the generalized traveling front is asymptotically stable under well-fitted perturbation and it is also unique.Speed-up of traveling waves by negative chemotaxishttps://zbmath.org/1527.351202024-02-28T19:32:02.718555Z"Griette, Quentin"https://zbmath.org/authors/?q=ai:griette.quentin"Henderson, Christopher"https://zbmath.org/authors/?q=ai:henderson.christopher"Turanova, Olga"https://zbmath.org/authors/?q=ai:turanova.olgaThis very interesting paper is concerned with various asymptotic regimes of a Keller-Segel elliptic-parabolic system with negative (i.e. repulsive) chemotaxis and Fisher-KPP (i.e. logistic) reaction. Roughly speaking, it is shown that, at least as far as traveling waves are concerned, the problem converges to a porous medium type equation when the length scale chemotaxis parameter goes to zero, and to a hyperbolic system when the chemotaxis strength goes to infinity. Both limits commute in some way. In both cases, some loss of ellipticity occurs and must be overcome by new regularity estimates, exponential decay properties and so on. Finally, some upper and lower bounds on the traveling wave speeds follow, providing insight on how the negative chemotaxis may enhance spreading velocity.
Reviewer: Thomas Giletti (Clermont-Ferrand)Stability of monostable traveling waves in diffusive three-species competition systemshttps://zbmath.org/1527.351212024-02-28T19:32:02.718555Z"Guo, Jong-Shenq"https://zbmath.org/authors/?q=ai:guo.jong-shenq"Guo, Karen"https://zbmath.org/authors/?q=ai:guo.karen"Shimojo, Masahiko"https://zbmath.org/authors/?q=ai:shimojo.masahikoSummary: In this note, we derive the stability of various monostable traveling waves in two different classes of three-species competition systems. This includes cases of three weak competitors and two-weak-one-strong competitors.Propagation phenomena of a vector-host disease modelhttps://zbmath.org/1527.351222024-02-28T19:32:02.718555Z"Lin, Guo"https://zbmath.org/authors/?q=ai:lin.guo"Wang, Xinjian"https://zbmath.org/authors/?q=ai:wang.xinjian"Zhao, Xiao-Qiang"https://zbmath.org/authors/?q=ai:zhao.xiaoqiang|zhao.xiao-qiangSummary: This paper is devoted to the study of spreading properties and traveling wave solutions for a vector-host disease system, which models the invasion of vectors and hosts to a new habitat. Combining the uniform persistence idea from dynamical systems with the properties of the corresponding entire solutions, we investigate the propagation phenomena in two different cases: (1) fast susceptible vector; (2) slow susceptible vector when the disease spreads. It turns out that in the former case, the susceptible vector may spread faster than the infected vector and host under appropriate conditions, which leads to multi-front spreading with different speeds; while in the latter case, the infected vector and host always catch up with the susceptible vector, and they spread at the same speed. We further obtain the existence and nonexistence of traveling wave solutions connecting zero to the endemic equilibrium. We also conduct numerical simulations to illustrate our analytic results.A time-periodic competition model with nonlocal dispersal and bistable nonlinearity: propagation dynamics and stabilityhttps://zbmath.org/1527.351232024-02-28T19:32:02.718555Z"Ma, Manjun"https://zbmath.org/authors/?q=ai:ma.manjun"Meng, Wentao"https://zbmath.org/authors/?q=ai:meng.wentao"Ou, Chunhua"https://zbmath.org/authors/?q=ai:ou.chunhuaSummary: This paper is concerned with traveling waves to a time-periodic bistable Lotka-Volterra competition system with nonlocal dispersal. We first establish the existence, uniqueness and stability of traveling wave solutions for this system. By utilizing comparison principle and the stability property, the relationship among the bistable wave speed, the asymptotic propagation speeds of the associated monotone subsystems and the speed of upper/lower solutions is obtained. Explicit sufficient conditions for positive and negative bistable wave speeds are derived. Our explicit results are derived by constructing particular and novel upper/lower solutions with specific asymptotical behaviors, which can be seen as case studies applicable to further investigations and improvements. Finally, the theoretical results are corroborated under weak conditions by direct simulations of the underlying time-periodic system with nonlocal dispersal. The combined impact of competition, dispersal and seasonality on the invasion direction has shed new light on the modelings and analysis of population competition and species invasion in heterogeneous media.Speed determinacy of the traveling waves for a three species time-periodic Lotka-Volterra competition systemhttps://zbmath.org/1527.351252024-02-28T19:32:02.718555Z"Wu, Qiong"https://zbmath.org/authors/?q=ai:wu.qiong"Pan, Chaohong"https://zbmath.org/authors/?q=ai:pan.chaohong"Wang, Hongyong"https://zbmath.org/authors/?q=ai:wang.hongyong(no abstract)The global generalized solution of the chemotaxis-Navier-Stokes system with logistic sourcehttps://zbmath.org/1527.351382024-02-28T19:32:02.718555Z"Ding, Dandan"https://zbmath.org/authors/?q=ai:ding.dandan"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong|tan.zhong.1"Wu, Zhonger"https://zbmath.org/authors/?q=ai:wu.zhongerSummary: In this paper, we consider the initial boundary value problem of the chemotaxis-Navier-Stokes system with low regularity, and we show that the system has a global generalized solution, which was first introduced by \textit{M. Winkler} [SIAM J. Math. Anal. 47, No. 4, 3092--3115 (2015; Zbl 1330.35202)].Global weak solution to a generic reaction-diffusion nonlinear parabolic systemhttps://zbmath.org/1527.351392024-02-28T19:32:02.718555Z"Hana, Matallah"https://zbmath.org/authors/?q=ai:hana.matallah"Messaoud, Maouni"https://zbmath.org/authors/?q=ai:messaoud.maouni"Hakim, Lakhal"https://zbmath.org/authors/?q=ai:hakim.lakhal(no abstract)On the dynamics of aeolian sand rippleshttps://zbmath.org/1527.351432024-02-28T19:32:02.718555Z"Coclite, Giuseppe Maria"https://zbmath.org/authors/?q=ai:coclite.giuseppe-maria"di Ruvo, Lorenzo"https://zbmath.org/authors/?q=ai:di-ruvo.lorenzoSummary: The dynamics of aeolian sand ripples is described by a 1D non-linear evolutive fourth order equation. In this paper, we prove the well-posedness of the classical solutions of the Cauchy problem, associated with this equation.Heat equations and wavelets on Mumford curves and their finite quotientshttps://zbmath.org/1527.351542024-02-28T19:32:02.718555Z"Bradley, Patrick Erik"https://zbmath.org/authors/?q=ai:bradley.patrick-erikSummary: A class of heat operators over non-archimedean local fields acting on \(L_2\)-function spaces on holed discs in the local field are developed and seen as being operators previously introduced by Zúñiga-Galindo, and if the underlying trees are regular, they are associated here with certain finite Kronecker product graphs. \(L_2\)-spaces and integral operators invariant under the action of a finite group acting on a holed disc are studied, and then applied to Mumford curves. It is found that the spectral gap in families of Mumford curves can become arbitrarily small.Hidden dissipation and convexity for Kimura equationshttps://zbmath.org/1527.351552024-02-28T19:32:02.718555Z"Casteras, Jean-Baptiste"https://zbmath.org/authors/?q=ai:casteras.jean-baptiste"Monsaingeon, Léonard"https://zbmath.org/authors/?q=ai:monsaingeon.leonardSummary: In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein-Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic process to nonfixation in order to get rid of singularities on the boundaries, and then studying the conditioned \(Q\)-process from a more traditional and variational point of view. In doing so we complete the work initiated in [\textit{F. A. C. C. Chalub} et al., Acta Appl. Math. 171, Paper No. 24, 51 p. (2021; Zbl 1473.35590)], where the gradient flow was identified only formally. The approach is based on the \textit{energy dissipation inequality} and \textit{evolution variational inequality} notions of metric gradient flows. Building up on some convexity of the driving entropy functional, we obtain new contraction estimates and quantitative long-time convergence towards the stationary distribution.Uniqueness and stability analysis to a system of nonlocal partial differential equations related to an epidemic modelhttps://zbmath.org/1527.351562024-02-28T19:32:02.718555Z"Wu, Wenbing"https://zbmath.org/authors/?q=ai:wu.wenbing(no abstract)Study of weak solutions to a nonlinear fourth-order parabolic equation with boundary degeneracyhttps://zbmath.org/1527.351572024-02-28T19:32:02.718555Z"Liang, Bo"https://zbmath.org/authors/?q=ai:liang.bo"Wang, Ying"https://zbmath.org/authors/?q=ai:wang.ying.35"Qu, Chengyuan"https://zbmath.org/authors/?q=ai:qu.chengyuan(no abstract)Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domainhttps://zbmath.org/1527.351582024-02-28T19:32:02.718555Z"Guo, Hongpeng"https://zbmath.org/authors/?q=ai:guo.hongpeng"Guo, Zhiming"https://zbmath.org/authors/?q=ai:guo.zhiming"Li, Yijie"https://zbmath.org/authors/?q=ai:li.yijieSummary: This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.Propagation dynamics of nonlocal dispersal competition systems in time-periodic shifting habitatshttps://zbmath.org/1527.351592024-02-28T19:32:02.718555Z"Qiao, Shao-Xia"https://zbmath.org/authors/?q=ai:qiao.shao-xia"Li, Wan-Tong"https://zbmath.org/authors/?q=ai:li.wan-tong"Wang, Jia-Bing"https://zbmath.org/authors/?q=ai:wang.jiabingSummary: This paper is concerned with the propagation dynamics of the time periodic Lotka-Volterra competition systems with nonlocal dispersal in a shifting habitat. We first obtain three types of time-periodic forced waves connecting the extinction state to the co-existence state, itself and the semi-trivial state, which describe the conversion from the state of two aboriginal co-existent competing species, two invading alien competitors, and a saturated aboriginal competitor with another invading alien competitor to the extinction state, respectively. This provides a comprehensive explanation of the point-wise extinction dynamics of these two competing species under such a time-periodic worsening habitat. Then, we establish the spreading properties of the associated Cauchy problem depending on the range of the shifting speed. More specifically, we give a complete description on the threshold values for the extinction as well as persistence (by moving with asymptotic speed). Our results reveal the possibility that a competitively weaker species with a much faster spreading speed can drive a competitively stronger species with a slower spreading speed to extinction. The discussion in this paper applies to both cases of weak competition and strong-weak competition. In particular, we need to point out that some combined effects of nonlocal dispersal, two-variable coupling and time-periodic shifting heterogeneity in this system pose extra difficulties in mathematical treatment, which are dealt with by introducing new approaches.Finite-time synchronization of intermittently controlled reaction-diffusion systems with delays: a weighted LKF methodhttps://zbmath.org/1527.351602024-02-28T19:32:02.718555Z"Tang, Rongqiang"https://zbmath.org/authors/?q=ai:tang.rongqiang"Yuan, Shuang"https://zbmath.org/authors/?q=ai:yuan.shuang"Yang, Xinsong"https://zbmath.org/authors/?q=ai:yang.xinsong"Shi, Peng"https://zbmath.org/authors/?q=ai:shi.peng"Xiang, Zhengrong"https://zbmath.org/authors/?q=ai:xiang.zhengrongSummary: Considering the fact that existing methodologies for finite-time control are difficult to simultaneously overcome the difficulties induced by the effects of reaction-diffusion and time delay when intermittent control is confronted, this paper explores a novel Lyapunov-Krasovskii functional (LKF) method to investigate the finite-time synchronization of delayed reaction-diffusion systems. By designing a simple intermittent control and a weighted LKF, a general finite-time stability criterion is established first. Then, sufficient conditions for the finite-time synchronization of the interested system are given, where the weight factor of the LKF has a heavy influence on the settling time. Several important corollaries are also given to specify the usefulness and generality of the weighted LKF method and the finite-time stability criterion. Finally, a numerical example is provided to verify the new findings, and an image encryption algorithm is presented to validate the useful application of theoretical results.Propagation dynamics in a heterogeneous reaction-diffusion system under a shifting environmenthttps://zbmath.org/1527.351612024-02-28T19:32:02.718555Z"Wu, Chufen"https://zbmath.org/authors/?q=ai:wu.chufen"Xu, Zhaoquan"https://zbmath.org/authors/?q=ai:xu.zhaoquanThe object of this paper is a class of cooperative reaction-diffusion systems of the logistic/KPP type, under a shifting heterogeneity depending monotonically on a moving variable. Forced waves (i.e. stationary solutions in the moving frame of the heterogeneity) are shown to exist uniquely and to be globally stable.
Reviewer: Thomas Giletti (Clermont-Ferrand)Optimal control of a parabolic equation with a nonlocal nonlinearityhttps://zbmath.org/1527.351622024-02-28T19:32:02.718555Z"Kenne, Cyrille"https://zbmath.org/authors/?q=ai:kenne.cyrille"Djomegne, Landry"https://zbmath.org/authors/?q=ai:djomegne.landry"Mophou, Gisèle"https://zbmath.org/authors/?q=ai:mophou.gisele-massengoSummary: This paper proposes an optimal control problem for a parabolic equation with a nonlocal nonlinearity. The system is described by a parabolic equation involving a nonlinear term that depends on the solution and its integral over the domain. We prove the existence and uniqueness of the solution to the system and the boundedness of the solution. Regularity results for the control-to-state operator, the cost functional and the adjoint state are also proved. We show the existence of optimal solutions and derive first-order necessary optimality conditions. In addition, second-order necessary and sufficient conditions for optimality are established.Some further progress for boundedness of solutions to a quasilinear higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusionhttps://zbmath.org/1527.351632024-02-28T19:32:02.718555Z"Zheng, Jiashan"https://zbmath.org/authors/?q=ai:zheng.jiashan"Xie, Jianing"https://zbmath.org/authors/?q=ai:xie.jianingSummary: This study focuses on the \(N\)-dimensional chemotaxis-haptotaxis model with nonlinear diffusion that was initially proposed by \textit{M. A. J. Chaplain} and \textit{G. Lolas} (see [Netw. Heterog. Media 1, No. 3, 399--439 (2006; Zbl 1108.92023)]) to describe the interactions between cancer cells, the matrix-degrading enzyme, and the host tissue during cancer cell invasion. Accordingly, we consider the diffusion coefficient \(D(u)\) of cancer cells to be a nonlinear function satisfying \(D(u)\geq C_D u^{m-1}\) for all \(u > 0\) with some \(C_D >0\) and \(m>0\). Relying on a \textbf{new energy inequality} and \textbf{iteration argument}, this paper proves that under the mild condition
\[
m>\dfrac{2N(N+1)[\max_{2\leq s\leq N+2}\lambda_{0, s}^{\frac{1}{s}}(\chi +\xi \|w_0 \|_{L^{\infty}(\Omega)})-\mu]_+}{(N+2) [(N+1)\max_{2\leq s\leq N+2}\lambda_{0, s}^{\frac{1}{s}} (\chi +\xi\|w_0 \|_{L^{\infty}(\Omega)})-N\mu]_+},
\]
and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem has at least one globally bounded classical solution when \(D(0) > 0 \) (the case of nondegenerate diffusion), while if \(D(0)\geq 0 \) (the case of possibly degenerate diffusion), the existence of bounded weak solutions for the system is shown, where the positive parameters \(\xi, \chi\), and \(\mu > 0\) measure the chemotactic and haptotactic sensitivities and proliferation rate of the cells, respectively.Doubly nonlinear equations for the 1-Laplacianhttps://zbmath.org/1527.351642024-02-28T19:32:02.718555Z"Mazón, J. M."https://zbmath.org/authors/?q=ai:mazon-ruiz.jose-m"Molino, A."https://zbmath.org/authors/?q=ai:molino.alexis"Toledo, J."https://zbmath.org/authors/?q=ai:toledo.jefferson-m|toledo.jesus|toledo.juan-a|toledo.j-julian|toledo.julian|toledo.jonay|toledo.jonathanSummary: This paper is concerned with the Neumann problem for a class of doubly nonlinear equations for the 1-Laplacian,
\[
\frac{\partial v}{\partial t} - \Delta_1 u \ni 0 \text{ in } (0, \infty) \times \Omega, \quad v\in \gamma (u),
\]
and initial data in \(L^1 (\Omega)\), where \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^N\) and \(\gamma\) is a maximal monotone graph in \(\mathbb{R}\times\mathbb{R}\). We prove that, under certain assumptions on the graph \(\gamma\), there is existence and uniqueness of solutions. Moreover, we proof that these solutions coincide with the ones of the Neumann problem for the total variational flow. We show that such assumptions are necessary.Boundedness in a three-dimensional chemotaxis-Stokes system involving a subcritical sensitivity and indirect signal productionhttps://zbmath.org/1527.351652024-02-28T19:32:02.718555Z"Ren, Guoqiang"https://zbmath.org/authors/?q=ai:ren.guoqiang"Liu, Bin"https://zbmath.org/authors/?q=ai:liu.bin.6Summary: This paper is concerned with the Keller-Segel-Stokes system
\[
\begin{cases}
n_t + \mathbf{u} \cdot \nabla n =\nabla \cdot (D(n)n) - \nabla \cdot (S(n)\nabla v), \\
v_t + \mathbf{u} \cdot \nabla v = \Delta v-v+w, \\
w_t +\mathbf{u} \cdot \nabla w = \Delta w-w+n, \\
\mathbf{u}_t = \Delta \mathbf{u} + \nabla P + n\nabla \phi, \quad \nabla \cdot u=0,
\end{cases}
\eqno{(*)}
\]
under no-flux/no-flux/no-flux/Dirichlet boundary conditions in smoothly bounded domains \(\Omega \subset \mathbb{R}^3\), with given suitably regular functions \(D, S\) and \(\phi\). Under the assumption that there exist \(m^0 \in \mathbb{R}\), \(m \geq m_0\), \(k_D > 0\) and \(K_D > 0\) such that
\[
k_Ds^{m_0-1}\leq D(s)\leq K_Ds^{m-1} \text{ for all } s>1,
\]
and that \(S(0)=0\) as well as
\[
\frac{|S(s)|}{D(s)}\leq K_0s^\alpha \text{ for all } s>1
\]
with \(K_0 > 0\), it is shown that for all suitably regular initial data an associated initial-boundary value problem \((*)\) possesses a globally defined bounded classical solution provided \(\alpha <\frac{8}{9}\). We underline that the same results were established for the corresponding system with direct signal production in a well-known result for \(\alpha < \frac{2}{3}\) in
[\textit{X. Cao}, Z. Angew. Math. Phys. 71, No. 2, Paper No. 61, 21 p. (2020; Zbl 1439.35234)] and [\textit{M. Winkler}, Appl. Math. Lett. 112, Article ID 106785, 7 p. (2021; Zbl 1453.35115)].
Our result rigorously confirms that the indirect signal production mechanism genuinely contributes to the global solvability of the three-dimensional Keller-Segel-Stokes system.A limiting regularity result for some parabolic problems with data in Zygmund spaceshttps://zbmath.org/1527.351662024-02-28T19:32:02.718555Z"Kbiri Alaoui, Mohammed"https://zbmath.org/authors/?q=ai:kbiri-alaoui.mohammed"Aharouch, Lahsen"https://zbmath.org/authors/?q=ai:aharouch.lahsen"Di Fazio, Giuseppe"https://zbmath.org/authors/?q=ai:di-fazio.giuseppe"Altalhan, Ali"https://zbmath.org/authors/?q=ai:altalhan.aliSummary: The aim of this paper is to study the model problem:
\[
\left.
\begin{cases}
\frac{ \partial u}{ \partial t} - \Delta_p u = f & \text{in } Q \\ u = 0 & \text{on } \partial \Omega \times (0, T) \\ u (x, 0) = u_0 & \text{in } \Omega.
\end{cases}
\right.
\]
The main purpose of this work is to prove the existence and a limiting regularity result for the solution \(u\) of the above problem having right-hand side \(f \in L^\beta(0, T; L \operatorname{Log}^\alpha L(\Omega))\). In particular, we will consider the cases:
\begin{itemize}
\item[(i)] If \(p \geq 2, \beta = 1\), and \(\alpha > \frac{ N - 1}{ N} \), then \(u \in L^{p - 1}(0, T, W_0^{1, \overline{q}}(\Omega)) \left(\overline{q} = \frac{ N (p - 1)}{ N - 1}\right)\).
\item[(ii)] If \(2 - \frac{ 1}{ N} < p < N, \beta = N^\prime \), and \(\alpha = \frac{ N - 1}{ N} \), then \(u \in L^{\overline{q}}(0, T, W_0^{1, \overline{q}}(\Omega))\).
\item[(iii)] If \(p = N, \beta = N^\prime \), and \(\alpha = \frac{ N - 1}{ N} \), then \(u \in L^N(0, T, W_0^{1, N}(\Omega))\).
\item[(iv)] If \(p \geq 2, f \in L^1(0, T; L^m \operatorname{Log}^\alpha L(\Omega))\), and \(\alpha > \frac{ N - m}{ N} \), then \(u \in L^{p - 1}(0, T, W_0^{1, m^\ast (p - 1)}(\Omega))\).
\end{itemize}
{{\copyright} 2022 John Wiley \& Sons, Ltd.}Critical exponents for the \(p\)-Laplace heat equations with combined nonlinearitieshttps://zbmath.org/1527.351672024-02-28T19:32:02.718555Z"Torebek, Berikbol T."https://zbmath.org/authors/?q=ai:torebek.berikbol-tillabayulySummary: This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by \textit{M. Jleli} et al. [Proc. Am. Math. Soc. 148, No. 6, 2579--2593 (2020; Zbl 1439.35199)]. Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli et al. [loc. cit.] for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.Spectral and soliton structures for the four-component Kaup-Newell type negative flow equationhttps://zbmath.org/1527.351952024-02-28T19:32:02.718555Z"Yan, Feiying"https://zbmath.org/authors/?q=ai:yan.feiying"Geng, Xianguo"https://zbmath.org/authors/?q=ai:geng.xianguo"Li, Ruomeng"https://zbmath.org/authors/?q=ai:li.ruomengSummary: Based on the spectral analysis of Lax pairs and the inverse scattering method, we construct a matrix Riemann-Hilbert problem of the four-component Kaup-Newell type negative flow equation associated with a \(4\times 4\) matrix spectral problem and investigate the evolution of scattering data. Then \(N\)-soliton formulas of the four-component Kaup-Newell type negative flow equation are obtained by solving the irregular Riemann-Hilbert problem in the reflectionless case. As an application, the one-soliton solutions and the two-soliton solutions of the four-component Kaup-Newell type negative flow equation are given explicitly. In addition, the interaction dynamics of the soliton solutions are analyzed and illustrated.Asymptotic stability of explicit infinite energy blowup solutions of the 3D incompressible Navier-Stokes equationshttps://zbmath.org/1527.352072024-02-28T19:32:02.718555Z"Han, Fangyu"https://zbmath.org/authors/?q=ai:han.fangyu"Tan, Zhong"https://zbmath.org/authors/?q=ai:tan.zhong|tan.zhong.1Summary: In this paper, we study the dynamical stability of a family of explicit blowup solutions of the three-dimensional (3D) incompressible Navier-Stokes (NS) equations with smooth initial values, which is constructed in [\textit{B.-L. Guo} et al., Chin. Phys. Lett. 25, No. 6, 2115--2117 (2008; \url{doi:10.1088/0256-307X/25/6/052})]. This family of solutions has finite energy in any bounded domain of \(\mathbb{R}^3\), but unbounded energy in \(\mathbb{R}^3\). Based on similarity coordinates, energy estimates and the Nash-Moser-Hörmander iteration scheme, we show that these solutions are asymptotically stable in the backward light-cone of the singularity. Furthermore, the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing data. Finally, the result also shows that in the absence of physical boundaries, the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.Simple two-layer dispersive models in the Hamiltonian reduction formalismhttps://zbmath.org/1527.352312024-02-28T19:32:02.718555Z"Camassa, R."https://zbmath.org/authors/?q=ai:camassa.roberto"Falqui, G."https://zbmath.org/authors/?q=ai:falqui.gregorio"Ortenzi, G."https://zbmath.org/authors/?q=ai:ortenzi.giovanni"Pedroni, M."https://zbmath.org/authors/?q=ai:pedroni.marco"Vu Ho, T. T."https://zbmath.org/authors/?q=ai:ho.t-t-vuSummary: A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of \textit{T. B. Benjamin} [J. Fluid Mech. 165, 445--474 (1986; Zbl 0595.76023)] for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac's theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids. To assess the performance of the asymptotic models, special solutions are studied and compared with those of the parent equations.Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaceshttps://zbmath.org/1527.352722024-02-28T19:32:02.718555Z"Cuvillier, Ophélie"https://zbmath.org/authors/?q=ai:cuvillier.ophelie"Fanelli, Francesco"https://zbmath.org/authors/?q=ai:fanelli.francesco"Salguero, Elena"https://zbmath.org/authors/?q=ai:salguero.elenaSummary: In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain \(\mathbb{T}^d\), for space dimensions \(d=2,3\). We admit the average turbulent kinetic energy \(k\) to vanish in part of the domain, \textit{i.e.} we consider the case \(k \geq 0\); in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces \(H^s\), for any \(s>1+d/2\). We expect this regularity to be optimal, due to the degeneracy of the system when \(k \approx 0\). We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations.A simple solution formula for the Stokes equations in the half spacehttps://zbmath.org/1527.352832024-02-28T19:32:02.718555Z"Hirata, Daisuke"https://zbmath.org/authors/?q=ai:hirata.daisukeSummary: This note studies the Stokes equations in the half space \(\mathbb{R}_+^d\) with the non-slip boundary condition. We present an explicit solution formula by using the hybrid Fourier-Fourier sine transform, which is simpler than already known ones.On the Neumann problem for the nonstationary Stokes system in angles and coneshttps://zbmath.org/1527.352932024-02-28T19:32:02.718555Z"Kozlov, Vladimir"https://zbmath.org/authors/?q=ai:kozlov.vladimir-a"Rossmann, Jürgen"https://zbmath.org/authors/?q=ai:rossmann.jurgenSummary: The authors consider the Neumann problem for the nonstationary Stokes system in a two-dimensional angle or a three-dimensional cone. They obtain existence and uniqueness results for solutions in weighted Sobolev spaces and prove a regularity assertion for the solutions.
{{\copyright} 2023 The Authors. Mathematische Nachrichten published by Wiley-VCH GmbH.}Global large solutions to the Navier-Stokes-Nernst-Planck-Poisson equations in Fourier-Besov spaceshttps://zbmath.org/1527.353062024-02-28T19:32:02.718555Z"Xiao, Weiliang"https://zbmath.org/authors/?q=ai:xiao.weiliang"Kang, Wenyu"https://zbmath.org/authors/?q=ai:kang.wenyu(no abstract)BV entropy solutions of two-dimensional nonstationary Prandtl boundary layer systemhttps://zbmath.org/1527.353112024-02-28T19:32:02.718555Z"Zhan, Huashui"https://zbmath.org/authors/?q=ai:zhan.huashuiSummary: A new kind of BV entropy solution matching up with the degenerate parabolic equation arising from the two-dimensional Prandtl boundary layer system is introduced. If there are some restrictions on the coefficients of the system, then by means of Kruzkov's bi-variables method, the stability of entropy solutions is proved independent of the boundary value condition. For some domains which are regular in a special sense, by the inverse transformation of the Crocco transformation, the two-dimensional Prandtl boundary layer systems are well-posedness.On properties of aggregated regularized systems of equations for a homogeneous multicomponent gas mixturehttps://zbmath.org/1527.353162024-02-28T19:32:02.718555Z"Zlotnik, Alexander"https://zbmath.org/authors/?q=ai:zlotnik.alexander-a"Fedchenko, Anna"https://zbmath.org/authors/?q=ai:fedchenko.anna(no abstract)Standing waves for Schrödinger equations with Kato-Rellich potentialshttps://zbmath.org/1527.353702024-02-28T19:32:02.718555Z"Ćwiszewski, Aleksander"https://zbmath.org/authors/?q=ai:cwiszewski.aleksander"Kokocki, Piotr"https://zbmath.org/authors/?q=ai:kokocki.piotrSummary: We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman-Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski. The obtained results extend those by \textit{M. Prizzi} [Fundam. Math. 176, No. 3, 261--275 (2003; Zbl 1022.37013)] for the non-resonant case, resolve the existence problem at resonance and complete those from \textit{A. Ćwiszewski} and \textit{W. Kryszewski} [Calc. Var. Partial Differ. Equ. 58, No. 1, Paper No. 13, 23 p. (2019; Zbl 1404.35127)] where the bifurcation of stationary solutions from infinity was studied.On the safe storage of Bagassehttps://zbmath.org/1527.354182024-02-28T19:32:02.718555Z"Mitchell, S. L."https://zbmath.org/authors/?q=ai:hedetniemi.sandra-mitchell|mitchell.sarah-l"Myers, T. G."https://zbmath.org/authors/?q=ai:myers.timothy-g|myers.tim-gSummary: In this paper, we investigate the thermal evolution in a one-dimensional bagasse stockpile. The mathematical model involves four unknowns: the temperature, oxygen content, liquid water content and water vapour content. We first nondimensionalize the model to identify dominant terms and so simplify the system. We then calculate solutions for the approximate and full system. It is shown that under certain conditions spontaneous combustion will occur. Most importantly, we show that spontaneous combustion can be avoided by sequential building. To be specific, in a situation where, say, a 4.7 m stockpile can spontaneously combust, we could construct a 3 m pile and then some days later add another 1.7 m to produce a stable 4.7 m pile.On numerical approximations of fractional and nonlocal mean field gameshttps://zbmath.org/1527.354282024-02-28T19:32:02.718555Z"Chowdhury, Indranil"https://zbmath.org/authors/?q=ai:chowdhury.indranil"Ersland, Olav"https://zbmath.org/authors/?q=ai:ersland.olav"Jakobsen, Espen R."https://zbmath.org/authors/?q=ai:jakobsen.espen-robstadSummary: We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.Monotone methods in counterparty risk models with nonlinear Black-Scholes-type equationshttps://zbmath.org/1527.354292024-02-28T19:32:02.718555Z"Alziary, Bénédicte"https://zbmath.org/authors/?q=ai:alziary.benedicte"Takáč, Peter"https://zbmath.org/authors/?q=ai:takac.peterSummary: A nonlinear Black-Scholes-type equation is studied within \textbf{counterparty risk models}. The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black-Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.On a rumor propagation model with spatial heterogeneityhttps://zbmath.org/1527.354302024-02-28T19:32:02.718555Z"Chen, Mengxin"https://zbmath.org/authors/?q=ai:chen.mengxin"Srivastava, Hari Mohan"https://zbmath.org/authors/?q=ai:srivastava.hari-mohanSummary: Rumors or wrong information are always spread in different spatial locations by virtue of various available media. Therefore, the propagation ability of rumors or wrong information should be different in different geographical locations. In this paper, we report the dynamical behaviors of a diffusive SI (susceptible-infected) type rumor propagation model in a spatially heterogeneous environment. In view of the fact that rumor-refuting is a common phenomenon in the real world, we introduce this concern in the rumor propagation model. We first derive the properties of uniform boundedness and permanence of the rumor propagation model. These results indicate that the rumor propagation model has at least one rumor-spreading steady state. Thereafter, the asymptotic profiles of the rumor-spreading steady state are reported. We thus find that such rumor-spreading steady state exists if one of the migration rates of the rumor-infected individuals or the rumor-susceptible individuals tends to zero and infinity, respectively. Our theoretical results reveal that this rumor propagation model can admit wealthy dynamical profiles in a spatially heterogeneous environment. Some numerical results are also presented in order to check the theoretical conclusions.Adaptation in a heterogeneous environment. II: To be three or not to behttps://zbmath.org/1527.354322024-02-28T19:32:02.718555Z"Alfaro, Matthieu"https://zbmath.org/authors/?q=ai:alfaro.matthieu"Hamel, François"https://zbmath.org/authors/?q=ai:hamel.francois"Patout, Florian"https://zbmath.org/authors/?q=ai:patout.florian"Roques, Lionel"https://zbmath.org/authors/?q=ai:roques.lionel-jSummary: We propose a model to describe the adaptation of a phenotypically structured population in a \(H\)-patch environment connected by migration, with each patch associated with a different phenotypic optimum, and we perform a rigorous mathematical analysis of this model. We show that the large-time behaviour of the solution (persistence or extinction) depends on the sign of a principal eigenvalue, \( \lambda_H\), and we study the dependency of \(\lambda_H\) with respect to \(H\). This analysis sheds new light on the effect of increasing the number of patches on the persistence of a population, which has implications in agroecology and for understanding zoonoses; in such cases we consider a pathogenic population and the patches correspond to different host species. The occurrence of a \textit{springboard} effect, where the addition of a patch contributes to persistence, or on the contrary the emergence of a detrimental effect by increasing the number of patches on the persistence, depends in a rather complex way on the respective positions in the phenotypic space of the optimal phenotypes associated with each patch. From a mathematical point of view, an important part of the difficulty in dealing with \(H\ge 3\), compared to \(H=1\) or \(H=2\), comes from the lack of symmetry. Our results, which are based on a fixed point theorem, comparison principles, integral estimates, variational arguments, rearrangement techniques, and numerical simulations, provide a better understanding of these dependencies. In particular, we propose a precise characterisation of the situations where the addition of a third patch increases or decreases the chances of persistence, compared to a situation with only two patches.Traveling waves and free boundaries arising in tumor angiogenesishttps://zbmath.org/1527.354392024-02-28T19:32:02.718555Z"Fasano, Antonio"https://zbmath.org/authors/?q=ai:fasano.antonio"Sinisgalli, Carmela"https://zbmath.org/authors/?q=ai:sinisgalli.carmelaSummary: In [\textit{A. Gandolfi} et al., J. Theor. Biol. 512, Article ID 110526, 15 p. (2021; \url{doi:10.1016/j.jtbi.2020.110526})], a mathematical model was developed for tumor invasion with vessel cooption, including a fine analysis of angiogenesis driven by chemotaxis. Numerical solutions in spherical symmetry revealed that a traveling wave sets in. Results were in agreement with experimental data. In the present paper, we introduce some nontrivial changes in the model and we further analyze the structure of the solutions as well as their dependence on some critical biological parameters, emphasizing for instance which are the most active zones where angiogenesis takes place. Moreover, we propose an alternative model characterized by the presence of a free boundary (playing the role of the invasion front), showing that the new formulation (which is advantageous from the computational point of view) matches the results of the previous model for some biologically significant range of the parameters.Tumor growth with nutrients: regularity and stabilityhttps://zbmath.org/1527.354412024-02-28T19:32:02.718555Z"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: In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative \(L^1\)-contraction estimates and establish the \(C^{1,\alpha}\)-boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient \(n_0\) either lies entirely above or entirely below the critical value \(n_0=1\), we are able to give a complete characterization of the long-time behavior of the system. When \(n_0\) is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells.Global bounded solution of a forager-exploiter model with logistic sources and different taxis mechanismshttps://zbmath.org/1527.354422024-02-28T19:32:02.718555Z"Liu, Changfeng"https://zbmath.org/authors/?q=ai:liu.changfeng"Guo, Shangjiang"https://zbmath.org/authors/?q=ai:guo.shangjiangThe authors consider a parabolic system of three equations, resembling models of chemotactic type together with logistic type growth terms, which is proposed to model interactions of foragers and exploiters (or producers-scroungers). Under some structure conditions, the existence of global-in-time bounded solutions is shown.
Reviewer: Piotr Biler (Wrocław)Global dynamics of a reaction-diffusion brucellosis model with spatiotemporal heterogeneity and nonlocal delayhttps://zbmath.org/1527.354432024-02-28T19:32:02.718555Z"Liu, Shu-Min"https://zbmath.org/authors/?q=ai:liu.shumin"Bai, Zhenguo"https://zbmath.org/authors/?q=ai:bai.zhenguo"Sun, Gui-Quan"https://zbmath.org/authors/?q=ai:sun.guiquanSummary: The increase of animal transportation and livestock lead to repeated outbreaks of brucellosis, and its transmission process is extremely complex. According to the clinical symptoms and infectious differences of the diseased sheep, it is further divided into acute infection and chronic infection, and a reaction-diffusion SLICB (susceptible-latent-acute infected-chronic infected-brucella) model with seasonality, spatial heterogeneity and nonlocal delay is proposed to understand the transmission law of brucellosis and analyze its transmission risk. Based on the basic reproduction number \(\boldsymbol{\mathcal{R}}_0\) of the model, the final development trend of brucellosis transmission among sheep is analyzed by persistence theory. The \(\boldsymbol{\mathcal{R}}_0\) of the model is numerically calculated by the generalized power method, and simulation analysis shows that: (i) extending the latent period and increasing the random walk rate of infected sheep can effectively prevent brucellosis from developing into an endemic disease; (ii) the greater the density of acute infections, the higher the risk of brucellosis transmission, and the density of infected sheep and the time to reach the stable state will have a large deviation if only consider acute or chronic sheep; (iii) reaching the peak time will be delayed if the peak time of sheep birth is delayed. These results can provide some suggestions for the control of brucellosis.
{{\copyright} 2023 IOP Publishing Ltd \& London Mathematical Society}Solutions to the Keller-Segel system with non-integrable behavior at spatial infinityhttps://zbmath.org/1527.354482024-02-28T19:32:02.718555Z"Winkler, Michael"https://zbmath.org/authors/?q=ai:winkler.michaelThis is a study on the minimal Keller-Segel model in chemotaxis with radially symmetric nonintegrable initial data with the main stress put on the finite time blowup of solutions. Results include precise (and sometimes optimal) conditions on either singularities or tails of the initial data (compared to the steady state Chandrasekhar solution) that lead either to global-in-time existence or to blowup. Techniques combine delicate comparison principles and Kaplan type arguments.
Reviewer: Piotr Biler (Wrocław)A two-strain malaria transmission model with seasonality and incubation periodhttps://zbmath.org/1527.354492024-02-28T19:32:02.718555Z"Zhou, Rong"https://zbmath.org/authors/?q=ai:zhou.rong"Wu, Shi-Liang"https://zbmath.org/authors/?q=ai:wu.shiliangSummary: Malaria is one of the most common mosquito-borne diseases in the world. To understand the joint effects of the vector-bias, seasonality, spatial heterogeneity multi-strain and the extrinsic incubation period of the parasite on the dynamics of malaria, we formulate a time-periodic two-strain malaria reaction-diffusion model with delay and nonlocal terms. We then consider threshold conditions that determine whether malaria will spread. More specifically, the basic reproduction number \(\mathcal{R}_i\) for single strain-\(i\) and invasion number \(\hat{\mathcal{R}}_i\) for each strain-\(i\) (\(i = 1,2\)) are derived. Our results imply that if \(\max\{\mathcal{R}_1, \mathcal{R}_2\} < 1\), then the disease-free periodic solution is globally attractive; if \(\mathcal{R}_i > 1 > \mathcal{R}_j\) (\(i,j = 1,2\), \(i \neq j\)), then competitive exclusion, where the \(j\)th strain dies out and the \(i\)th strain persists; if \(\min\{\hat{\mathcal{R}}_1, \hat{\mathcal{R}}_2, \mathcal{R}_1, \mathcal{R}_2\} > 1\), then the disease persists. Our numerical simulations show that spatially heterogeneous infection can increase the basic reproduction number and the omission of the vector-bias effect will underestimate the infection risk of the disease.On null-controllability of the heat equation on infinite strips and control cost estimatehttps://zbmath.org/1527.354512024-02-28T19:32:02.718555Z"Egidi, Michela"https://zbmath.org/authors/?q=ai:egidi.michelaSummary: We consider an infinite strip \(\Omega_L = (0, 2 \pi L)^{d - 1} \times \mathbb{R}\), \(d \geq 2\), \(L > 0\), and study the control problem of the heat equation on \(\Omega_L\) with Dirichlet or Neumann boundary conditions, and control set \(\omega \subset \Omega_L\). We provide a sufficient and necessary condition for null-controllability in any positive time \(T > 0\), which is a geometric condition on the control set \(\omega\). This is referred to as ``thickness with respect to \(\Omega_L\)'' and implies that the set \(\omega\) cannot be concentrated in a particular region of \(\Omega_L\). We compare the thickness condition with a previously known necessity condition for null-controllability and give a control cost estimate which only shows dependence on the geometric parameters of \(\omega\) and the time \(T\).
{{\copyright} 2021 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH}Aronson-Bénilan and Harnack estimates for the discrete porous medium equationhttps://zbmath.org/1527.354552024-02-28T19:32:02.718555Z"Kräss, Sebastian"https://zbmath.org/authors/?q=ai:krass.sebastian"Zacher, Rico"https://zbmath.org/authors/?q=ai:zacher.ricoSummary: We consider the porous medium equation (PME) on a locally finite graph and identify suitable curvature-dimension (CD) conditions under which a discrete version of the fundamental Aronson-Bénilan estimate holds true for positive solutions of the PME on finite graphs. We also show that these estimates allow to prove Harnack inequalities which are structurally similar to the continuous case. The new CD conditions are illustrated with several concrete examples, e.g. complete and chain-like graphs.Non-existence results for \(p \)-Laplacian parabolic problems on the Heisenberg group \(\mathbb{H}^n \)https://zbmath.org/1527.354582024-02-28T19:32:02.718555Z"Goldstein, Gisèle Ruiz"https://zbmath.org/authors/?q=ai:ruiz-goldstein.gisele"Goldstein, Jerome A."https://zbmath.org/authors/?q=ai:goldstein.jerome-a"Kömbe, Ismail"https://zbmath.org/authors/?q=ai:kombe.ismailSummary: Let \(\mathbb{H}^n = \mathbb{C}^n\times \mathbb{R}\) be the \(2n+1 \)-dimensional Heisenberg group and \(\Omega\) be a bounded domain with smooth boundary \(\partial\Omega\) in \(\mathbb{H}^n \). This paper deals with the nonexistence of positive solutions to the problem
\[ \begin{cases}
u_t(z, l, t) = \mathscr{L} u+\eta u^{p-1}|\nabla_{\mathbb{H}^n}u|^s+V(x)u^{p-1}+\lambda u^q \quad & \text{in } \Omega \times (0, T ), \\
u(z, l, 0) = u_0(z, l)\geq 0 \quad & \text{in } \Omega, \\
u(z, l, t) = 0 \quad & \text{on } \partial\Omega\times (0, T), \end{cases} \]
where \(\mathscr{L} u\) is the subelliptic \(p \)-Laplacian operator on the Heisenberg group \(\mathbb{H}^n\), \(p>1\), \(\eta \in \mathbb{R}\), \(s>0\), \(V\in L_{\text{loc}}^1(\Omega)\), \(\lambda\in \mathbb{R}\) and \(q>0 \). We also demonstrate several applications of our main result using concrete potentials with sharp constants derived from Hardy and Leray type inequalities.A novel robust fractional-time anisotropic diffusion for multi-frame image super-resolutionhttps://zbmath.org/1527.354642024-02-28T19:32:02.718555Z"Ben-loghfyry, Anouar"https://zbmath.org/authors/?q=ai:ben-loghfyry.anouar"Hakim, Abdelilah"https://zbmath.org/authors/?q=ai:hakim.abdelilahSummary: In this paper, we propose an image multi-frame Super Resolution (SR) method based on fractional-time Caputo derivative combined with Weickert-type diffusion process idea. We provide the existence and uniqueness results with a detailed discretization using the finite difference scheme. Our approach is based on anisotropic diffusion behavior with coherence enhancing diffusion tensor together with the fractional-time derivative to benefit from its memory effect potential and to control the smoothing process near strong edges and flat regions while avoiding tiny corners destruction. The experimental results confirm the effectiveness of fractional-time derivative and the robustness of the proposed Partial Differential Equation (PDE) compared with some competitive super-resolution methods.Terminal value problem for nonlinear parabolic and pseudo-parabolic systemshttps://zbmath.org/1527.354652024-02-28T19:32:02.718555Z"Binh, Ho Duy"https://zbmath.org/authors/?q=ai:binh.ho-duy"Tien, Nguyen van"https://zbmath.org/authors/?q=ai:van-tien.nguyen"Minh, Vo Ngoc"https://zbmath.org/authors/?q=ai:minh.vo-ngoc"Can, Nguyen Huu"https://zbmath.org/authors/?q=ai:can.nguyen-huuSummary: In this paper, we study on the backward problem for parabolic system and pseudo-parabolic system with conformable derivative. Two these problems have many applications in engineering such as image processing, geophysics, biology, etc. There are two main contributions in this paper. The first major result deals with ill-posedness and regularization for terminal value problem for parabolic system. By applying a new truncation method, we construct a regularized solution. We investigate the existence and uniqueness of regularized problem. Under some suitable conditions of the terminal data, we provide the error estimate between the mild and regularized solutions. The second contribution concerns the existence of a global solution of pseudo-parabolic system. Finally, we also obtain the convergence of the mild solution as the order of derivative approaches \(1^-\).Well-posedness results for nonlinear fractional diffusion equation with memory quantityhttps://zbmath.org/1527.354802024-02-28T19:32:02.718555Z"Tuan, Nguyen Huy"https://zbmath.org/authors/?q=ai:nguyen-huy-tuan."Nguyen, Anh Tuan"https://zbmath.org/authors/?q=ai:nguyen.anh-tuan"Debbouche, Amar"https://zbmath.org/authors/?q=ai:debbouche.amar"Antonov, Valery"https://zbmath.org/authors/?q=ai:antonov.valerySummary: We study the well-posedness for solutions of an initial-value boundary problem on a two-dimensional space with source functions associated to nonlinear fractional diffusion equations with the Riemann-Liouville derivative and nonlinearities with memory on a two-dimensional domain. In order to derive the existence and uniqueness for solutions, we mainly proceed on reasonable choices of Hilbert spaces and the Banach fixed point principle. Main results related to the Mittag-Leffler functions such as its usual lower or upper bound and the relationship with the Mainardi function are also applied. In addition, to set up the global-in-time results, \( L^p-L^q\) estimates and the smallness assumption on the initial data function are also necessary to be applied in this research. Finally, the work also considers numerical examples to illustrate the graphs of analytic solutions.Solving the backward problem for time-fractional wave equations by the quasi-reversibility regularization methodhttps://zbmath.org/1527.354822024-02-28T19:32:02.718555Z"Wen, Jin"https://zbmath.org/authors/?q=ai:wen.jin"Li, Zhi-Yuan"https://zbmath.org/authors/?q=ai:li.zhiyuan"Wang, Yong-Ping"https://zbmath.org/authors/?q=ai:wang.yongpingSummary: This paper is devoted to the backward problem of determining the initial value and initial velocity simultaneously in a time-fractional wave equation, with the aid of extra measurement data at two fixed times. Uniqueness results are obtained by using the analyticity and the asymptotics of the Mittag-Leffler functions provided that the two fixed measurement times are sufficiently close. Since this problem is ill-posed, we propose a quasi-reversibility method whose regularization parameters are given by the a priori parameter choice rule. Finally, several one- and two-dimensional numerical examples are presented to show the accuracy and efficiency of the proposed regularization method.Event-triggered boundary consensus control for multi-agent systems of fractional reaction-diffusion PDEshttps://zbmath.org/1527.354832024-02-28T19:32:02.718555Z"Zhao, Lirui"https://zbmath.org/authors/?q=ai:zhao.lirui"Wu, Huaiqin"https://zbmath.org/authors/?q=ai:wu.huaiqin"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jindeSummary: The distributed consensus is considered for multi-agent systems (MASs), which characterized by fractional reaction-diffusion partial differential equations (RDPDEs) in this paper. Based on Lyapunov technique and linear matrix inequalities (LMIs) theory, the consensus can be realized via two novel event-triggered boundary control schemes. Firstly, a novel convergence principle subject to finite time is presented for the continuously differentiable function. Secondly, the cooling fin on surface of high-speed aerospace vehicle is remodeled by fractional RDPDEs system, and the well-posedness of presented system is discussed applying the monotone iterative approach. Thirdly, according to the presented static event-triggered boundary control strategy, the consensus criterion in finite time is addressed in the form of LMIs, in addition, the settling time is calculated accurately. Applying the dynamic event-triggered control protocol, the Mittag-Leffler (M-L) consensus condition is achieved. Moreover, the Zeno behaviors are ruled out for proposed event-triggered mechanisms. Finally, the high-speed aerospace vehicle model is presented to verify the effectiveness of the control performance.The linear stability for a free boundary problem modeling multilayer tumor growth with time delayhttps://zbmath.org/1527.355042024-02-28T19:32:02.718555Z"He, Wenhua"https://zbmath.org/authors/?q=ai:he.wenhua"Xing, Ruixiang"https://zbmath.org/authors/?q=ai:xing.ruixiang"Hu, Bei"https://zbmath.org/authors/?q=ai:hu.bei(no abstract)The global existence and numerical method for the free boundary problem of ductal carcinoma in situhttps://zbmath.org/1527.355052024-02-28T19:32:02.718555Z"Liu, Dan"https://zbmath.org/authors/?q=ai:liu.dan.2"Liu, Keji"https://zbmath.org/authors/?q=ai:liu.keji"Xu, Xinyi"https://zbmath.org/authors/?q=ai:xu.xinyi"Yu, Jianfeng"https://zbmath.org/authors/?q=ai:yu.jianfeng(no abstract)Dynamics of a Leslie-Gower predator-prey model with advection and free boundarieshttps://zbmath.org/1527.355072024-02-28T19:32:02.718555Z"Zhang, Yingshu"https://zbmath.org/authors/?q=ai:zhang.yingshu"Li, Yutian"https://zbmath.org/authors/?q=ai:li.yutianSummary: This paper investigates a Leslie-Gower predator-prey model with advection and double free boundaries. Firstly, we establish the global existence and uniqueness of the solution. Subsequently, we analyze the long-time behaviors of the solution and demonstrate a spreading-vanishing dichotomy. If the initial habitat size or the expanding capacity of the boundaries is large, the invasive predator will spread throughout the entire space; otherwise, both the prey and predator will eventually vanish. When spreading occurs, two subcases arise, depending on the interaction coefficient \(c\) and the environmental support \(\delta\). If \(c\delta < 1\), the prey and predator coexist, which we refer to as the weakly hunting case; conversely, if \(c\delta \geq 1\), the predator survives and spreads throughout the space, while the prey vanishes, which we term the strongly hunting case. In the event of spreading, we also provide an estimate of the spreading speed. Finally, we introduce a front-fixing implicit-explicit finite difference method for the free boundary problem. Numerical simulations validate our theoretical findings and uncover some intriguing new phenomena, prompting further investigation into this and general free boundary problems.A dengue fever model with free boundary incorporating the time-periodicity and spatial-heterogeneityhttps://zbmath.org/1527.355082024-02-28T19:32:02.718555Z"Zhu, Min"https://zbmath.org/authors/?q=ai:zhu.min.3|zhu.min.4|zhu.min|zhu.min.2|zhu.min.1"Xu, Yong"https://zbmath.org/authors/?q=ai:xu.yong.6"Zhang, Lai"https://zbmath.org/authors/?q=ai:zhang.lai"Cao, Jinde"https://zbmath.org/authors/?q=ai:cao.jinde(no abstract)Exact sharp-fronted solutions for nonlinear diffusion on evolving domainshttps://zbmath.org/1527.355092024-02-28T19:32:02.718555Z"Johnston, Stuart T."https://zbmath.org/authors/?q=ai:johnston.stuart-t"Simpson, Matthew J."https://zbmath.org/authors/?q=ai:simpson.matthew-jSummary: Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either report numerical solutions or require an assumption of linear diffusion to determine exact solutions. Unfortunately, numerical solutions do not reveal the relationship between the model parameters and the solution features. Additionally, experimental observations typically report the presence of sharp fronts, which are not captured by linear diffusion. Here we address both limitations by presenting exact sharp-fronted solutions to a model of degenerate nonlinear diffusion on a growing domain. We obtain the solution by identifying a series of transformations that converts the model of a nonlinear diffusive process on an evolving domain to a nonlinear diffusion equation on a fixed domain, which admits known exact solutions for certain choices of diffusivity functions. We determine expressions for critical time scales and domain growth rates such that the diffusive population never reaches the domain boundaries and hence the solution remains valid.
{{\copyright} 2023 The Author(s). Published by IOP Publishing Ltd}Singular limits for stochastic equationshttps://zbmath.org/1527.355122024-02-28T19:32:02.718555Z"Blömker, Dirk"https://zbmath.org/authors/?q=ai:blomker.dirk"Tölle, Jonas M."https://zbmath.org/authors/?q=ai:tolle.jonas-mSummary: We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and insufficient regularity, where the equation in the limit with noise would not be defined due to lack of regularity. We recover previously known results on vanishing small noise with increasing roughness, but our main focus is to study for fixed noise the singular limit where the leading order differential operator in the equation may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We separate the analysis of the equation from the convergence of stochastic terms and give a general framework for the main error estimates. This first reduces the result to bounds on a residual and in a second step to various bounds on the stochastic convolution. Moreover, as examples we apply our result to the singularly regularized Allen-Cahn (AC) equation with a vanishing Bilaplacian, and the Cahn-Hilliard/AC homotopy with space-time white noise in two spatial dimensions.On a non-local Sobolev-Galpern-type equation associated with random noisehttps://zbmath.org/1527.355152024-02-28T19:32:02.718555Z"Long Le Dinh"https://zbmath.org/authors/?q=ai:long-le-dinh."Duc Phuong Nguyen"https://zbmath.org/authors/?q=ai:duc-phuong-nguyen."Ragusa, Maria Alessandra"https://zbmath.org/authors/?q=ai:ragusa.maria-alessandraSummary: This paper aims to retrieve the initial value for a non-local fractional Sobolev-Galpern problem. The given data are subject to noise by the discrete random model. We show that the solution to the problem is ill-posed in the sense of Hadamard. In this paper, we applied the Fourier truncation method to construct the regularized solution. We estimate the convergence between the solution and the regularized solution. In addition, the numerical example is also proposed to assess the efficiency of the theory.Time-step heat problem on the mesh: asymptotic behavior and decay rateshttps://zbmath.org/1527.390032024-02-28T19:32:02.718555Z"Abadias, Luciano"https://zbmath.org/authors/?q=ai:abadias.luciano"González-Camus, Jorge"https://zbmath.org/authors/?q=ai:gonzalez-camus.jorge"Rueda, Silvia"https://zbmath.org/authors/?q=ai:rueda.silviaSummary: In this article, we study the asymptotic behavior and decay of the solution of the fully discrete heat problem. We show basic properties of its solutions, such as the mass conservation principle and their moments, and we compare them to the known ones for the continuous analogue problems. We present the fundamental solution, which is given in terms of spherical harmonics, and we state pointwise and \(\ell^p\) estimates for that. Such considerations allow to prove decay and large-time behavior results for the solutions of the fully discrete heat problem, giving the corresponding rates of convergence on \(\ell^p\) spaces.Differential Harnack inequalities for semilinear parabolic equations on \(\mathrm{RCD}^{\ast} (K,N)\) metric measure spaceshttps://zbmath.org/1527.530372024-02-28T19:32:02.718555Z"Lu, Zhihao"https://zbmath.org/authors/?q=ai:lu.zhihaoSummary: We present a unified method to derive differential Harnark inequalities for locally positive weak solutions to semilinear parabolic equations on \(\mathrm{RCD}^{\ast} (K,N)\) metric measure spaces, as introduced by \textit{N. Gigli} [On the differential structure of metric measure spaces and applications. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1325.53054)] and \textit{M. Erbar} et al. [invent. Math. 201, No.3, 993--1071 (2015; Zbl 1329.53059)]. As its application, on the one hand, we recover many known differential Harnack inequalities on heat equation, on the other hand, for logarithmic type equation and Yamabe type equation, we get some sharp differential Harnack inequalities on \(\mathrm{RCD}^{\ast} (0,N)\) metric measure spaces. As its corollary, we get some sharp Harnack inequalities and Liouville theorems.Ricci flow of \(W^{2, 2}\)-metrics in four dimensionshttps://zbmath.org/1527.530892024-02-28T19:32:02.718555Z"Lamm, Tobias"https://zbmath.org/authors/?q=ai:lamm.tobias"Simon, Miles"https://zbmath.org/authors/?q=ai:simon.milesThe paper studies the existence problem of Ricci-DeTurck flow starting from metrics with rough regularity in dimension four, and the instantaneous smoothing. More precisely, the authors establish the quantitative short-time existence starting from rough metrics which are bi-lipschitz and $W^{2,2}$ metrics with respect to bounded geometry background metrics. As an application, the authors define the notion of scalar curvature lower bound for this class of nonsmooth metrics.
Reviewer: Man-Chun Lee (Evanston)Transition probability estimates for subordinate random walkshttps://zbmath.org/1527.600322024-02-28T19:32:02.718555Z"Cygan, Wojciech"https://zbmath.org/authors/?q=ai:cygan.wojciech"Šebek, Stjepan"https://zbmath.org/authors/?q=ai:sebek.stjepanSummary: Let \(S_n\) be a symmetric simple random walk on the integer lattice \(\mathbb{Z}^d\). For a Bernstein function \(\phi\) we consider a random walk \(S_n^\phi\) which is subordinated to \(S_n\). Under a certain assumption on the behaviour of \(\phi\) at zero we establish global estimates for the transition probabilities of the random walk \(S_n^\phi\). The main tools that we apply are a parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk.
{{\copyright} 2021 The Authors. \textit{Mathematische Nachrichten} published by Wiley-VCH GmbH}Survey on path-dependent PDEshttps://zbmath.org/1527.600412024-02-28T19:32:02.718555Z"Peng, Shige"https://zbmath.org/authors/?q=ai:peng.shige"Song, Yongsheng"https://zbmath.org/authors/?q=ai:song.yongsheng"Wang, Falei"https://zbmath.org/authors/?q=ai:wang.faleiSummary: In this paper, the authors provide a brief introduction of the path-dependent partial differential equations (PDEs for short) in the space of continuous paths, where the path derivatives are in the Dupire (rather than Fréchet) sense. They present the connections between Wiener expectation, backward stochastic differential equations (BSDEs for short) and path-dependent PDEs. They also consider the well-posedness of path-dependent PDEs, including classical solutions, Sobolev solutions and viscosity solutions.The stochastic heat equation with multiplicative Lévy noise: existence, moments, and intermittencyhttps://zbmath.org/1527.600442024-02-28T19:32:02.718555Z"Berger, Quentin"https://zbmath.org/authors/?q=ai:berger.quentin"Chong, Carsten"https://zbmath.org/authors/?q=ai:chong.carsten"Lacoin, Hubert"https://zbmath.org/authors/?q=ai:lacoin.hubertSummary: We study the stochastic heat equation (SHE) \(\partial_t u = \frac{1}{2} \Delta u + \beta u \xi\) driven by a multiplicative Lévy noise \(\xi\) with positive jumps and coupling constant \(\beta >0\), in arbitrary dimension \(d\ge 1\). We prove the existence of solutions under an optimal condition if \(d=1\), \(2\) and a close-to-optimal condition if \(d\ge 3\). Under an assumption that is general enough to include stable noises, we further prove that the solution is unique. By establishing tight moment bounds on the multiple Lévy integrals arising in the chaos decomposition of \(u\), we further show that the solution has finite \(p\) th moments for \(p>0\) whenever the noise does. Finally, for any \(p>0\), we derive upper and lower bounds on the moment Lyapunov exponents of order \(p\) of the solution, which are asymptotically sharp in the limit as \(\beta \rightarrow 0\). One of our most striking findings is that the solution to the SHE exhibits a property called \textit{strong intermittency} (which implies moment intermittency of all orders \(p>1\) and pathwise mass concentration of the solution), for any non-trivial Lévy measure, at any disorder intensity \({\beta}>0\), in any dimension \(d\ge 1\). This behavior contrasts with that observed for the SHE on \(\mathbb{Z}^d\) and for the SHE on \(\mathbb{R}^d\) with Gaussian noise, for which intermittency does not occur in high dimensions for small \(\beta\).Stochastic heat equations with logarithmic nonlinearityhttps://zbmath.org/1527.600502024-02-28T19:32:02.718555Z"Shang, Shijie"https://zbmath.org/authors/?q=ai:shang.shijie"Zhang, Tusheng"https://zbmath.org/authors/?q=ai:zhang.tusheng-sThis paper treats stochastic heat equations with logarithmic nonlinearity
\[
du(t,x) = \varDelta u(t,x) dt + u(t,x) \log \vert u(t,x) \vert dt + \sigma ( u(t,x) ) d B_t, \quad t >0, x \in D, \tag{3}
\]
driven by a one-dimensional standard Brownian motion \(B\) on a bounded domain \(D\) \(( \subset {\mathbb R}^d )\) in the setting of a \(L^2(D)\) space, with \(u(t,x) = 0\) for \(t > 0\) and \(x \in \partial D\) (smooth boundary), and \(u(0,x) = u_0(x)\) for \(x \in D\). The coefficient \(\sigma\) : \({\mathbb R} \to {\mathbb R}\) is a deterministic continuous function. The authors establish the global existence and uniqueness of solutions to the above-mentioned equation, on the assumption that the initial value \(u_0\) is a deterministic function in \(L^2(D)\), where the logarithmic Sobolev inequality plays an important role in derivation of fundamental estimates. The result is also new even in the deterministic setting. Based on a new estimate of the difference of two logarithmic terms and a nonlinear type of Gronwall's inequality, the authors prove the uniqueness of the solutions in the \(L^2(D)\) space when the diffusion coefficient \(\sigma\) satisfies a local Lipschitz conditions. To obtain the existence of solutions, they use the Galerkin methods.
More precisely, we set \(H := L^2(D)\), \(V := H_0^1(D)\),
\[
\Vert u \Vert_V^2 = \int_D \vert \nabla u \vert^2 (x) dx,\text{ and }\log_+ z := \log ( 1 \vee z ).
\]
Furthermore, we put \(u(t) (x) := u(t,x)\), \(( u(t) \log \vert u(t) \vert ) (x)\) \(:=\) \(u(t,x) \log \vert u(t,x) \vert\), and \(\sigma( u(t) ) (x)\) \(:=\) \(\sigma ( u(t,x) )\). The process \(u\) is said to be a solution of the stochastic evolution equation associated with (1), if \(u\) is an \(H\)-valued continuous and \({\mathcal F}_t\)-adapted stochastic process such that \(u \in L^2( [0, T]; V)\) for any \(T > 0\), \(P\)-a.s. and \(u\) satisfies the equation
\[
u(t) = u_0 + \int_0^t \varDelta u(s) ds + \int_0^t u(s) \log \vert u(s) \vert ds + \int_0^t \sigma ( u(s) ) d B_s, \text{ in }V^*, \quad P-a.s.
\]
for any \(t \geq 0\) with \(u(0) = u_0 \in H\). In addition, assume that \(\sigma\) satisfies a local Lipschitz condition:
(H.1) there exist \(L_1\), \(L_2 > 0\) such that for \(x, y \in {\mathbb R}\)
\[
\vert \sigma (x) - \sigma(y) \vert \leqslant L_1 \vert x - y \vert + L_2 \vert x - y \vert ( \log_+ ( \vert x \vert \vee \vert y \vert ) )^{1/2}.
\]
Moreover, suppose that \par (H.2) \(\sigma\) is continuous, and there exist \(\theta \in [0, 1)\), and \(C_1, C_2 > 0\) such that for any \(x \in {\mathbb R}\)
\[
\vert \sigma(x) \vert \leqslant C_1 + C_2 \vert x \vert^{\theta}.
\]
(H.3) \(\sigma\) is continuous, and there exist \(C_3, C_4 > 0\) such that for any \(x \in {\mathbb R}\)
\[
\vert \sigma (x) \vert \leqslant C_3 + C_4 \vert x \vert ( \log_+ \vert x \vert )^{1/2}.
\]
The first result is about the uniqueness.
Theorem A. Suppose that (H.1) holds. Then the pathwise uniqueness holds for equation (3) in \(L^2(D)\).
The authors first establish the existence of a probabilistic weak solution by showing the tightness of the approximate solutions and identifying any their limit as the solution of the stochastic heat equation. The existence of probabilistic strong solutions then follows by appealing to the Yamada-Watanabe theorem. They have two results on the existence of the solutions. The first one is obtained under the sublinear growth condition on the diffusion coefficient \(\sigma\). In this case, they also obtain a global moment estimate of the solution. The second one is established under the superlinear growth condition on \(\sigma\). However, they do not have the global moment estimate for the latter. That is to say:
Theorem B. Suppose that (H.1) and (H.2) hold. Then there exists a unique global strong solution (in the probabilistic sense) \(u\) to (3) for every initial value \(u_0 \in L^2(D)\). Moreover, for any \(T > 0\) and \(p \geq 2\), we have
\[
E[ \sup_{ t \in [0, T] } \Vert u(t) \Vert^p ]+ E[ \int_0^T \Vert u \Vert^{p-2} \Vert u \Vert_V^2 ds ] < \infty.
\]
Theorem C. Suppose that (H.1) and (H.3) hold. Then there exists a unique global strong solution (in the probabilistic sense) to (3) for any initial value \(u_0 \in L^2(D)\). \par For other related works, see, e.g., [\textit{M. Alfaro} and \textit{R. Carles}, Dyn. Partial Differ. Equ. 14, No. 4, 343--358 (2017; Zbl 1386.35129)] for superexponential growth or decay in the heat equation with a logarithmic nonlinearity, [\textit{H. Chen} et al., J. Math. Anal. Appl. 422, No. 1, 84--98 (2015; Zbl 1302.35071)] for global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity, and [\textit{H. Chen} and \textit{S. Tian}, J. Differ. Equations 258, No. 12, 4424--4442 (2015; Zbl 1370.35190)] for initial boundary value problem for a class of semilinear heat equation with logarithmic nonlinearity. On the other hand, \textit{M. Dozzi} and \textit{J. A. López-Mimbela} [Stochastic Processes Appl. 120, No. 6, 767--776 (2010; Zbl 1193.35258)] studied the equation (1) with drift \(b( u(t,x)) = ( u(t,x) )^{1 + p}\) for some \(p > 0\) and \(\sigma(z) = z\).
Reviewer: Isamu Dôku (Saitama)Errors of an implicit variable-step BDF2 method for a molecular beam epitaxial model with slope selectionhttps://zbmath.org/1527.650762024-02-28T19:32:02.718555Z"Zhao, Xuan"https://zbmath.org/authors/?q=ai:zhao.xuan"Zhang, Haifeng"https://zbmath.org/authors/?q=ai:zhang.haifeng"Sun, Hong"https://zbmath.org/authors/?q=ai:sun.hongSummary: Unconditionally stable and convergent variable-step BDF2 scheme for solving the MBE model with slope selection is derived. Discrete orthogonal convolution kernels of the variable-step BDF2 method are commonly utilized for solving the phase field models. We present new inequalities, concerning the vector forms, for the kernels especially dealing with nonlinear terms in the slope selection model. The convergence rate of the fully discrete scheme is proved to be two both in time and space in \(L^2\) norm under the setting of the variable time steps. Energy dissipation law is proved rigorously with a modified energy by adding a small term to the discrete version of the original free energy functional. Two numerical examples including an adaptive time-stepping strategy are given to verify the convergence rate and the energy dissipation law.Shannon-Taylor technique for solving one-dimensional inverse heat conduction problemhttps://zbmath.org/1527.650852024-02-28T19:32:02.718555Z"Annaby, M. H."https://zbmath.org/authors/?q=ai:annaby.mahmoud-h"Al-Abdi, I. A."https://zbmath.org/authors/?q=ai:al-abdi.i-a"Ghaleb, A. F."https://zbmath.org/authors/?q=ai:ghaleb.ahmed-f"Abou-Dina, M. S."https://zbmath.org/authors/?q=ai:abou-dina.moustafa-sSummary: The Shannon-Taylor interpolation technique was introduced by \textit{P. L. Butzer} and \textit{W. Engels} in [IEEE Trans. Inf. Theory 29, 314--318 (1983; Zbl 0513.65100)]. In this work, the sinc-function is replaced by a Taylor approximation polynomial. In this work, we implement the Shannon-Taylor approximations to solve a one-dimensional heat conduction problem. One of the major advantages of this approach is that the resulting linear system of equations of the approximation procedure has an explicit coefficient matrix. This is not the case of the classical sinc methods due to finite integrals involving \(e^{-x^2}\). We establish rigorous error estimates involving an additional Taylor's series tail. Numerical illustrations are depicted.\(L2-1_\sigma\) finite element method for time-fractional diffusion problems with discontinuous coefficientshttps://zbmath.org/1527.650962024-02-28T19:32:02.718555Z"Chen, Yanping"https://zbmath.org/authors/?q=ai:chen.yanping.2"Tan, Xuejiao"https://zbmath.org/authors/?q=ai:tan.xuejiao"Huang, Yunqing"https://zbmath.org/authors/?q=ai:huang.yunqingSummary: A time-fractional diffusion equation with an interface problem caused by discontinuous coefficients is considered. To solve it, in the temporal direction Alikhanov's \(L2-1_\sigma\) method with graded mesh is presented to deal with the weak singularity at \(t = 0,\) while in the spatial direction a finite element method with uniform mesh is employed to handle the discontinuous coefficients. Then, with the help of discrete fractional Grönwall inequality and the robustness theory of \(\alpha \rightarrow 1^-\), we show that the method has stable error bounds at \(\alpha \rightarrow 1^-\), the fully discrete schemes \(L^2(\ohm)\) norm and \(H^1(\ohm)\) semi-norm are unconditionally stable, and the optimal convergence order is \(\mathscr{O}(h^2 + N^{-\min\{r \alpha ,2\}})\) and \(\mathscr{O}(h + N^{-\min\{r \alpha ,2\}})\), respectively, where, \(h\), \(N\), \(\alpha\), \(r\) is the total number of spatial parameter, the time-fractional order coefficient, and the time grid constant. Finally, three numerical examples are provided to illustrate our theoretical results.Adaptive space-time finite element methods for parabolic optimal control problemshttps://zbmath.org/1527.650992024-02-28T19:32:02.718555Z"Langer, Ulrich"https://zbmath.org/authors/?q=ai:langer.ulrich"Schafelner, Andreas"https://zbmath.org/authors/?q=ai:schafelner.andreasSummary: We present, analyze, and test locally stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard \(L_2\)-regularization. We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space -- time cylinder.Linearized stable spectral method to analyze two-dimensional nonlinear evolutionary and reaction-diffusion modelshttps://zbmath.org/1527.651052024-02-28T19:32:02.718555Z"Hamid, Muhammad"https://zbmath.org/authors/?q=ai:hamid.muhammad"Usman, Muhammad"https://zbmath.org/authors/?q=ai:usman.muhammad"Haq, Rizwan UI"https://zbmath.org/authors/?q=ai:haq.rizwan-ui"Tian, Zhenfu"https://zbmath.org/authors/?q=ai:tian.zhenfu"Wang, Wei"https://zbmath.org/authors/?q=ai:wang.wei.8Summary: The work is devoted to the development of a new spectral method based on higher dimensional orthogonal polynomials. Firstly, the concept of traditional Chelyshkov polynomials is extended for the function of more than one variable while definitions and theorems are presented with proofs. The operational matrices of derivative have been constructed assisted by defined higher-order polynomials and used to the development of a spectral method. The method is further coupled with a Picard-iterative scheme to tackle the high nonlinearities and termed as the Picard-Chelyshkov polynomial method (PCPM). The convergence and error-bound have been analyzed through theorems and their proofs in order to prove the mathematical authenticity of the method. The PCPM is applied for some two-dimensional unsteady nonlinear fractional partial differential equations and efficient results have been attained. In addition, it is evident from the comparative analysis with existing literature that the proposed method is fair enough in terms of accuracy, efficiency, and cost to deal with the problem in higher dimensions whilst could be further modified for other classes of dynamical problems.
{{\copyright} 2020 Wiley Periodicals LLC.}Multigrid treatment of implicit continuum diffusionhttps://zbmath.org/1527.651372024-02-28T19:32:02.718555Z"Francisquez, M."https://zbmath.org/authors/?q=ai:francisquez.manaure"Zhu, B."https://zbmath.org/authors/?q=ai:zhu.ben"Rogers, B. N."https://zbmath.org/authors/?q=ai:rogers.barrett-nSummary: Implicit treatment of diffusive terms of various differential orders common in continuum mechanics modeling, such as computational fluid dynamics, is investigated with spectral and multigrid algorithms in non-periodic 2D domains. In doubly periodic time dependent problems these terms are handled efficiently by spectral methods, but in non-periodic systems solved with distributed memory parallel computing and 2D domain decomposition, this efficiency is lost for a large number of processors. We built and present here a multigrid algorithm for these types of problems that outperforms a spectral solution employing the highly optimized FFTW library. This solver is suitable for high performance computing and may be able to efficiently treat implicit diffusion of arbitrary order by introducing auxiliary equations of lower order. We test these solvers for fourth and sixth order diffusion with harmonic test functions as well as turbulent 2D magnetohydrodynamic simulations. It is also shown that an anisotropic operator without mixed-derivative terms improves model accuracy and speed, and we examine the impact that the various diffusion operators have on the energy, the enstrophy, and the qualitative aspect of a simulation.Boundedness in a two-species chemotaxis system with nonlinear resource consumptionhttps://zbmath.org/1527.920132024-02-28T19:32:02.718555Z"Ou, Houzuo"https://zbmath.org/authors/?q=ai:ou.houzuo"Wang, Liangchen"https://zbmath.org/authors/?q=ai:wang.liangchenIn this paper, the authors have conducted a study on a two-species chemotaxis system that incorporates nonlinear resource consumption. By imposing appropriate conditions on the parameter, the authors have successfully demonstrated that the solution remains both global and bounded in time for all initial data that satisfy suitable regularity conditions.
These findings are undoubtedly innovative and captivating. The derivation is presented in a highly structured and coherent manner, effectively leveraging existing concepts from the literature while appropriately acknowledging the relevant references. Furthermore, the work introduces substantial and noteworthy new ideas to the field.
Reviewer: Pan Zheng (Chongqing)Carleman estimate and null controllability for a degenerate parabolic equation with a slightly superlinear reaction termhttps://zbmath.org/1527.930222024-02-28T19:32:02.718555Z"Wang, Chunpeng"https://zbmath.org/authors/?q=ai:wang.chunpeng"Zhou, Yanan"https://zbmath.org/authors/?q=ai:zhou.yananThis paper proves the null controllability of the following degenerate parabolic equation:
\[
u_t - (x^\alpha u_x)_x + p(t,x,u) = h \chi_\omega \text{ in } (0, 1) \times (0, T),
\]
with \(0 < \alpha < \frac{1}{2}\) and \(p\) a Lipschitz continuous function with respect to \(u\) satisfying a growth condition at infinity similar to the one encountered in the case of the null-controllability of the semi-linear heat equation by the means of the distributed control \(h\). The idea of the proof is to show that a uniform Carleman estimate holds for a regularized dual equation. This Carleman estimate implies an observability inequality for the linear regularized dual equation which holds uniformly with respect to the regularization parameter. Finally, the null controllability of the non-linear degenerate equation follows by applying Kakutani fixed point theorem.
Reviewer: Nicolae Cîndea (Aubière)Spatio-temporal sampled-data control for delay reaction-diffusion systemshttps://zbmath.org/1527.932642024-02-28T19:32:02.718555Z"Wang, Yun-Zhu"https://zbmath.org/authors/?q=ai:wang.yunzhu"Wang, Zhen"https://zbmath.org/authors/?q=ai:wang.zhen.17"Wu, Kai-Ning"https://zbmath.org/authors/?q=ai:wu.kaining"Wang, Chen-Xu"https://zbmath.org/authors/?q=ai:wang.chenxu.1Summary: This article considers the exponential stabilization and \(H_\infty\) performance for delay reaction-diffusion systems (DRDSs), with spatial sampled-data controller (SSDC) and spatio-temporal sampled-data controller (STSDC). Firstly, an SSDC is designed to stabilize the DRDSs. Using Lyapunov functional and Wirtinger's inequality technique, we obtain sufficient conditions to ensure the exponential stability of DRDSs. When there exist external disturbances, \( H_\infty\) performance is considered and criteria are provided for the disturbed DRDSs, under the designed SSDC. Then, an STSDC is adopted for DRDSs. Time discrete item brings new difficulties for the analysis of the desired properties. To overcome these difficulties, a novel Lyapunov functional is constructed and Halaney's inequality is used. Moreover, the descriptor method is adopted. Under these techniques, delay-dependent conditions are obtained to guarantee the exponential stability. The \(H_\infty\) performance is also considered for the disturbed DRDSs under the designed STSDC. A Razumikhin-type method is introduced together with the Lyapunov-like functional. Sufficient conditions are presented to achieve \(H_\infty\) performance for DRDSs with STSDC. Our theoretical results show that the spatial sampling interval does affect the desired properties, the shorter the spatial sampling interval, the easier to achieve the desired properties. Moreover, the time delay also influences the exponential stability of DRDSs, and smaller time delay is beneficial to the achievement of the stability. Finally, examples are given to illustrate the validity of results.
{{\copyright} 2021 John Wiley \& Sons Ltd.}Lyapunov-based nonlinear boundary control design with predefined convergence for a class of 1D linear reaction-diffusion equationshttps://zbmath.org/1527.934002024-02-28T19:32:02.718555Z"Zekraoui, Salim"https://zbmath.org/authors/?q=ai:zekraoui.salim"Espitia, Nicolas"https://zbmath.org/authors/?q=ai:espitia.nicolas"Perruquetti, Wilfrid"https://zbmath.org/authors/?q=ai:perruquetti.wilfridSummary: In this paper, we treat the problem of Lyapunov-based nonlinear boundary stabilization of a class of one-dimensional reaction-diffusion systems with any predefined convergence (asymptotic or non-asymptotic). As an application, we focus on the non-asymptotic notions (finite-time and fixed-time) for which we give some particular explicit control designs followed by some numerical simulations. The key idea of our approach is to use a ``spatially weighted \(L^2\)-norm'' as a Lyapunov functional to design a nonlinear controller and to ensure stability with any desired convergence.Modal consensus observers for distributed parameter systemshttps://zbmath.org/1527.934092024-02-28T19:32:02.718555Z"Demetriou, Michael A."https://zbmath.org/authors/?q=ai:demetriou.michael-aSummary: This article proposes a new type of a consensus protocol for the synchronization of distributed observers in systems governed by parabolic partial differential equations. Addressing the goal of sharing useful information among distributed observers, it delves into the details governing the modal decompositions of distributed parameter systems. Assuming that two different groups of sensors are available to provide process information to the two distributed observers, the proposed modal consensus design ensures that only useful information is transmitted to the requisite modal components of each of the observers. Without any consensus protocol, the observers capture different frequency content of the spatial process in differing degrees, as it relates to the concept of modal observability. Their modal components exhibit different learning behavior toward the process state. In the extreme case, it turns out that certain modal components of the distributed observers occasionally behave as naïve observers. To ensure that, both collectively and modal componentwise, the observers agree both with the process state and with each other, a modal component consensus protocol is proposed. Such a consensus protocol is mono-directional and provides only useful information necessary to the appropriate modal component of the distributed filters that behaves as a naïve modal observer. This protocol, when abstracted and applied to different state decompositions can be viewed as mono-directional projections of information transmitted and received by the participating distributed observers. Detailed numerical studies of advection PDE in one and two spatial dimensions are included to elucidate the details of the proposed modal consensus observers.
{{\copyright} 2021 John Wiley \& Sons Ltd.}