Nonlinear nonlocal reaction-diffusion problem with local reaction
type: Item
instance of: scholarly article
title: Nonlinear nonlocal reaction-diffusion problem with local reaction
zbMATH Open document ID: 1498.37115
published in: Discrete and Continuous Dynamical Systems
publication date: 2022-02-28T00:00:00Z
review text: Let \(\Omega\) be a measure metric space. Assume \(u(x, t)\) is the density of some population at the point \(x\in\Omega\) at time \(t\), \(u_0\) is the initial distribution density of the population, \(J(x, y)\) is a positive function defined in \(\Omega\times\Omega\) that represents the fraction of the population jumping from a location \(y\) to location \(x\), per unit time, and \(f: \Omega\times\mathbb{R}\rightarrow\mathbb{R}\) is a nonlinear term which describes the local rate of production/consumption of the magnitude \(u\) at each point \(x\in\Omega\). The aim of this paper is to make a general study of the nonlinear problem \[ \begin{cases} u_{t}(x,t)= \int_{\Omega}J(x,y)u(y,t)~dy+f(x,u(x,t)) \quad\mbox{if }\quad x\in\Omega, t>0, \\ u(x,0)= u_{0}(x) \quad\mbox{if}\quad x\in\Omega. \end{cases} \tag{1} \] Firstly, the authors prove the global existence of solutions for (1) for initial \(u_0\in X\) for some classes of nonlinear terms \(f(x,u)\) and show some similarities and differences between (1) and the following local reaction-diffusion: \[ u_{t}-\Delta u=f(x,u).\tag{2} \] When \(f\) is globally Lipschitz, they prove that solutions of (1) satisfy both weak, strict or strong comparison and maximum principles, depending on the conditions for the kernel \(J\). These results are analogous to (2) and differs when \(f(x, u)\) is locally Lipschitz in the variable \(u\in \mathbb{R}\), uniformly with respect to \(x\in \Omega\), and satisfies a sign condition. Secondly, the authors give some asymptotic estimates of the solution. In particular they prove the existence of two extremal ordered equilibria of (1) in \(L^{\infty}(\Omega)\) which enclose all other equilibria. They also prove that the maximal extremal equilibria is ``stable from above'' and the minimal extremal equilibria is ``stable from below''. Thirdly, for nonnegative solutions they prove that if \(f(x,0)\) is nonnegative then there exists a minimal nonnegative equilibrium such that the solutions of (1) …
reviewed by: Mohamed Zitane
Mathematics Subject Classification ID: 37L15
Mathematics Subject Classification ID: 37L05
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45M10
Mathematics Subject Classification ID: 45M20
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 7481818
zbMATH Keywords: nonlocal diffusion
zbMATH Keywords: local reaction
zbMATH Keywords: maximum principles
zbMATH Keywords: asymptotic bounds
zbMATH Keywords: extremal solutions
zbMATH Keywords: logistic models
author: Anibal Rodriguez-Bernal
author: Silvia Sastre-Gómez
MaRDI profile type: Publication
full work available at URL: dcds.2021170
OpenAlex ID: W3212150906
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DOI: 10.3934/DCDS.2021170
description: scientific article; zbMATH DE number 7481818