Recent zbMATH articles in MSC 35K51https://zbmath.org/atom/cc/35K512021-06-15T18:09:00+00:00WerkzeugThe effect of nonlocal reaction in an epidemic model with nonlocal diffusion and free boundaries.https://zbmath.org/1460.354002021-06-15T18:09:00+00:00"Zhao, Meng"https://zbmath.org/authors/?q=ai:zhao.meng"Li, Wantong"https://zbmath.org/authors/?q=ai:li.wan-tong"Du, Yihong"https://zbmath.org/authors/?q=ai:du.yihongSummary: In this paper, we examine an epidemic model which is described by a system of two equations with nonlocal diffusion on the equation for the infectious agents \(u\), while no dispersal is assumed in the other equation for the infective humans \(v\). The underlying spatial region \([g(t),h(t)]\) (i.e., the infected region) is assumed to change with time, governed by a set of free boundary conditions. In the recent work [\textit{M. Zhao} et al., J. Differ. Equations 269, No. 4, 3347--3386 (2020; Zbl 1442.35486)], such a model was considered where the growth rate of \(u\) due to the contribution from \(v\) is given by \(cv\) for some positive constant \(c\). Here this term is replaced by a nonlocal reaction function of \(v\) in the form \(c\int_{g(t)}^{h(t)}K(x-y)v(t,y)dy\) with a suitable kernel function \(K\), to represent the nonlocal effect of \(v\) on the growth of \(u\). We first show that this problem has a unique solution for all \(t>0\), and then we show that its longtime behaviour is determined by a spreading-vanishing dichotomy, which indicates that the long-time dynamics of the model is not vastly altered by this change of the term \(cv\). We also obtain sharp criteria for spreading and vanishing, which reveal that changes do occur in these criteria from the earlier model in [loc. cit] where the term \(cv\) was used; in particular, small nonlocal dispersal rate of \(u\) alone no longer guarantees successful spreading of the disease as in the model of [loc. cit.].Null controllability of a system of degenerate nonlinear coupled equations derived from population dynamics.https://zbmath.org/1460.352112021-06-15T18:09:00+00:00"Birba, Mamadou"https://zbmath.org/authors/?q=ai:birba.mamadou"Traoré, Oumar"https://zbmath.org/authors/?q=ai:traore.oumarSummary: In this paper, we study the null controllability property of a nonlinear coupled model with degenerate diffusion term. Firstly, we establish a Carleman type inequality for the adjoint system of an intermediate model. From this inequality, we derive our observability inequality. Next, by a fixed point argument, we prove the null controllability result with an internal control acting on a small subset of the domain.
For the entire collection see [Zbl 1458.00035].Improvement of conditions for boundedness in a fully parabolic chemotaxis system with nonlinear signal production.https://zbmath.org/1460.351972021-06-15T18:09:00+00:00"Pan, Xu"https://zbmath.org/authors/?q=ai:pan.xu"Wang, Liangchen"https://zbmath.org/authors/?q=ai:wang.liangchenSummary: This paper deals with the chemotaxis system with nonlinear signal secretion
\[
\begin{cases}
u_t=\nabla\cdot(D(u)\nabla u-S(u)\nabla v), & x\in\Omega,\quad t>0,\\ v_t=\Delta v-v+g(u), & x\in \Omega,\quad t>0,
\end{cases}
\]
under homogeneous Neumann boundary conditions in a bounded domain \(\Omega\subset\mathbb{R}^n\) \((n\ge 2)\). The diffusion function \(D(s)\in C^2([0,\infty))\) and the chemotactic sensitivity function \(S(s)\in C^2([0,\infty))\) are given by \(D(s)\geq C_d(1+s)^{-\alpha}\) and \(0<S(s)\leq C_ss(1+s)^{\beta-1}\) for all \(s\geq 0\) with \(C_d,C_s>0\) and \(\alpha,\beta\in\mathbb{R}\). The nonlinear signal secretion function \(g(s)\in C^1([0,\infty))\) is supposed to satisfy \(g(s)\leq C_g s^{\gamma}\) for all \(s\geq 0\) with \(C_g,\gamma>0\). Global boundedness of solution is established under the specific conditions:
\[
0<\gamma\leq 1\text{ and }\alpha+\beta<\min\left\lbrace 1+\frac{1}{n},1+\frac{2}{n}-\gamma \right\rbrace.
\]
The purpose of this work is to remove the upper bound of the diffusion condition assumed in [\textit{X. Tao} et al., J. Math. Anal. Appl. 474, No. 1, 733--747 (2019; Zbl 07056518)], and we also give the necessary constraint \(\alpha+\beta<1+\frac{1}{n}\), which is ignored in [loc. cit., Theorem 1.1].Passivity and synchronization of coupled reaction-diffusion complex-valued memristive neural networks.https://zbmath.org/1460.351942021-06-15T18:09:00+00:00"Huang, Yanli"https://zbmath.org/authors/?q=ai:huang.yanli"Hou, Jie"https://zbmath.org/authors/?q=ai:hou.jie"Yang, Erfu"https://zbmath.org/authors/?q=ai:yang.erfuSummary: This paper considers two types of coupled reaction-diffusion complex-valued memristive neural networks (CRDCVMNNs). The nodes of the first type CRDCVMNN are coupled through their state and the second one is coupled by spatial diffusion coupling term. For the former, some novel criteria for the passivity and synchronization are derived by constructing an appropriate controller and utilizing some inequality techniques as well as Lyapunov functional method. For the latter, we establish some sufficient conditions which guarantee that this type of CRDCVMNNs can realize passivity and synchronization. Finally, the effectiveness and correctness of the acquired theoretical results are verified by two numerical examples.On cross-diffusion systems for two populations subject to a common congestion effect.https://zbmath.org/1460.351952021-06-15T18:09:00+00:00"Laborde, Maxime"https://zbmath.org/authors/?q=ai:laborde.maximeLet \(\Omega\) be a convex and relatively compact open subset of \(\mathbb{R}^n\) with smooth boundary and \(|\Omega|>2\) and consider two potentials \(V_i\in W^{1,\infty}(\Omega)\), \(i=1,2\). Existence of global weak solutions is proved for the system \begin{align*} \partial_t \rho_1 - \mathrm{div}\left( \nabla\rho_1 + \rho_1 \left( \nabla V_1 + \nabla p \right) \right)=0\text{ in }(0,\infty)\times \Omega\,, \\ \partial_t \rho_2 - \mathrm{div}\left( \nabla\rho_2 + \rho_2 \left( \nabla V_2 + \nabla p \right) \right)=0\text{ in }(0,\infty)\times \Omega\,, \end{align*} supplemented with constraints \[ p\ge 0\,, \qquad \rho_1+\rho_2 \le 1\,, \] along with no flux boundary conditions and initial conditions \((\rho_{1,0},\rho_{2,0})\in \mathcal{P}^{ac}(\Omega;\mathbb{R}^2)\) (that is, probability measures on \(\Omega\) which are absolutely continuous with respect to the Lebesgue measure) satisfying \(\rho_{1,0},\rho_{2,0}\le 1\) a.e. in \(\Omega\). In addition, when \(m\ge 1\), existence of global weak solutions is also established for the system \begin{align*} \partial_t \rho_1 - \mathrm{div}\left( \nabla\rho_1 + \rho_1 \left( \nabla V_1 + \frac{m}{m-1} \nabla [(\rho_1+\rho_2)^{m-1}] \right) \right)=0\text{ in }(0,\infty)\times \Omega\,, \\ \partial_t \rho_2 - \mathrm{div}\left( \nabla\rho_2 + \rho_2 \left( \nabla V_2 + \frac{m}{m-1} \nabla [(\rho_1+\rho_2)^{m-1}] \right) \right)=0\text{ in }(0,\infty)\times \Omega\,,\end{align*} supplemented with no flux boundary conditions and initial conditions \((\rho_{1,0},\rho_{2,0})\in \mathcal{P}^{ac}(\Omega;\mathbb{R}^2)\) (when \(m=1\), \(m(\rho_1+\rho_2)^{m-1}/(m-1)\) has to be replaced by \(\ln{(\rho_1+\rho_2)}\) as usual). In both cases, the proof relies on the underlying gradient flow structure with respect to the \(2\)-Wasserstein distance and makes use of the flow interchange technique to obtain the compactness estimates needed for the convergence of the variational scheme. The last section is devoted to the particular case \(\nabla V_1=\nabla V_2\), for which \(\rho=\rho_1+\rho_2\) solves a closed equation. Additional regularity on \(\rho\), including \(L^\infty\)-estimates, are derived. Numerical simulations are also provided.
Reviewer: Philippe Laurençot (Toulouse)Exponential attractor for Hindmarsh-Rose equations in neurodynamics.https://zbmath.org/1460.350492021-06-15T18:09:00+00:00"Phan, Chi"https://zbmath.org/authors/?q=ai:phan.chi"You, Yuncheng"https://zbmath.org/authors/?q=ai:you.yunchengSummary: The existence of exponential attractor for the diffusive Hindmarsh-Rose equations on a three-dimensional bounded domain in the study of neurodynamics is proved through uniform estimates and a new theorem on the squeezing property of the abstract reaction-diffusion equation established in this paper. This result on the exponential attractor infers that the global attractor whose existence has been proved in [the authors and \textit{J. Su},``Global attractors for Hindmarsh-Rose equationsin neurodynamics'', Preprint, \url{arXiv:1907.13225}] for the diffusive Hindmarsh-Rose semiflow has a finite fractal dimension.Dimension estimate of attractors for complex networks of reaction-diffusion systems applied to an ecological model.https://zbmath.org/1460.350452021-06-15T18:09:00+00:00"Cantin, Guillaume"https://zbmath.org/authors/?q=ai:cantin.guillaume"Aziz-Alaoui, M. A."https://zbmath.org/authors/?q=ai:aziz-alaoui.m-aSummary: The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a complex network of competing species systems modeling an heterogeneous biological ecosystem and propose a series of numerical simulations which underpin our theoretical statements.Global synchronization of coupled reaction-diffusion neural networks with general couplings via an iterative approach.https://zbmath.org/1460.351982021-06-15T18:09:00+00:00"Tseng, Jui-Pin"https://zbmath.org/authors/?q=ai:tseng.jui-pinSummary: We establish a framework to investigate the global synchronization of coupled reaction-diffusion neural networks with time delays. The coupled networks under consideration can incorporate both the internal delays in each individual network and the transmission delays across different networks. The coupling scheme for the coupled networks is rather general, and its performance is not adversely affected by the restrictions commonly imposed by existing relevant investigations. Based on the proposed iterative approach, the problem of global synchronization is transformed into that of solving the corresponding homogeneous linear system of algebraic equations. The synchronization criterion is subsequently derived and can be verified with simple computations. Three numerical examples are presented to illustrate the effectiveness of the synchronization theory presented in this paper.Stable asymmetric spike equilibria for the Gierer-Meinhardt model with a precursor field.https://zbmath.org/1460.350142021-06-15T18:09:00+00:00"Kolokolnikov, Theodore"https://zbmath.org/authors/?q=ai:kolokolnikov.theodore"Paquin-Lefebvre, Frédéric"https://zbmath.org/authors/?q=ai:paquin-lefebvre.frederic"Ward, Michael J."https://zbmath.org/authors/?q=ai:ward.michael-jSummary: Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1D Gierer-Meinhardt (GM) model, we show that a non-constant precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric and two-spike patterns, corresponding to localized peaks in the activator of different heights. These stable, asymmetric patterns co-exist in the same parameter space as symmetric two-spike patterns. This is in contrast to a constant precursor case, for which asymmetric spike patterns are known to be unstable. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach, we analyse the spectrum of the linearization of the GM model around the two-spike steady state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis, a new class of vector-valued non-local eigenvalue problem is derived and analysed.Quasi linear parabolic PDE posed on a network with non linear Neumann boundary condition at vertices.https://zbmath.org/1460.351962021-06-15T18:09:00+00:00"Ohavi, Isaac"https://zbmath.org/authors/?q=ai:ohavi.isaacSummary: The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.Solvability of a coupled quasilinear reaction-diffusion system.https://zbmath.org/1460.351932021-06-15T18:09:00+00:00"Ambrazevičius, A."https://zbmath.org/authors/?q=ai:ambrazevicius.algirdas"Skakauskas, V."https://zbmath.org/authors/?q=ai:skakauskas.vladasSummary: The aim of this paper is to investigate the existence, uniqueness, and long-time behaviour of the classical solutions to a coupled system of four quasilinear parabolic equations. Two of them are solved in a domain and the other two are determined on the domain boundary. Such coupled systems arise in modelling of surface reactions between two reactants.Local existence and nonexistence for fractional in time weakly coupled reaction-diffusion systems.https://zbmath.org/1460.353792021-06-15T18:09:00+00:00"Suzuki, Masamitsu"https://zbmath.org/authors/?q=ai:suzuki.masamitsuSummary: We study a fractional in time weakly coupled reaction-diffusion system in a bounded domain with the Dirichlet boundary condition. The domain is imbedded in an \(N\)-dimensional space and it has \(C^2\) boundary, and fractional derivatives are meant in a generalized Caputo sense. The system can be referred to as a standard reaction-diffusion system in two components with polynomial growth. We obtain integrability conditions on the initial state functions which determine the existence/nonexistence of a local in time mild solution.Global attractors and exponential stability of partly dissipative reaction diffusion systems with exponential growth nonlinearity.https://zbmath.org/1460.350472021-06-15T18:09:00+00:00"Lee, Jihoon"https://zbmath.org/authors/?q=ai:lee.jihoon"Toi, Vu Manh"https://zbmath.org/authors/?q=ai:toi.vu-manhSummary: We study the long-time behavior of the solutions of the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with exponential growth nonlinearity. More precisely, we prove the existence of weak solutions, the regularity of the global attractor and the exponential stability of stationary solutions of the systems.