Global existence for coupled reaction-diffusion systems
type: Item
instance of: scholarly article
title: Global existence for coupled reaction-diffusion systems
zbMATH Open document ID: 0963.35077
author: Nassima Boudiba
author: Michel Pierre
published in: Journal of Mathematical Analysis and Applications
publication date: 2001-06-18T00:00:00Z
review text: The following initial-boundary value problem for a reaction-diffusion system is considered in the paper: \[ \begin{aligned} &\frac {\partial u}{\partial t}-a\Delta u=f(x,t,u,v) \quad \text{in } \{x\in\Omega,\;t\in (0,\infty)\}\\ &\frac {\partial v}{\partial t}-c\Delta u-d\Delta v=g(x,t,u,v) \quad \text{in } \{x\in\Omega,\;t\in (0,\infty)\}\\&u=v=0 \quad \text{on \;}\{x\in\partial \Omega,\;t\in (0,\infty)\}\\ &v(x,0)=v_0(x),\;u(x,0)=u_0(x), \;x\in\Omega, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\) with a regular boundary, \(a>0\), \(d>0\), \(c\in\mathbb{R}\). It is proved that the system has a unique global classical solution if \(f\) and \(g\) satisfy the following conditions: \(f\), \(g\) are measurable locally Lipschitz continuous in \(u,v\); \((\text{sign }u) f(x,t,u,v)\leq 0\) for all \(u,v\) and a.e. \(x,t\); there exists an \(\alpha> 0\) such that \((\text{sign }u) f(x,t,u,v)+\alpha(\text{sign }v) g(x,t,u,v)\leq 0\) for all \(u,v\) and a.e. \(x,t\); there exist constants \(L(r)\), \(M(r)>0\), \(\sigma\geq 1\) such that \[ |f(x,t,u,v)|+|g(x,t,u,v)|\leq L(r)|v|^{\sigma}+M(r) \] for all \(u,v\) with \(|u|\leq r\) and a.e. \(x,t\). Some examples are discussed in the article.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1539394
zbMATH Keywords: semilinear parabolic system
zbMATH Keywords: unique global classical solution
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6895
OpenAlex ID: W2067609272
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DOI: 10.1006/JMAA.2000.6895
description: scientific article; zbMATH DE number 1539394