Quasilinear reaction-diffusion systems with nonlinear boundary conditions
type: Item
instance of: scholarly article
title: Quasilinear reaction-diffusion systems with nonlinear boundary conditions
zbMATH Open document ID: 0926.35059
author: Mingxin Wang
author: Shu Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-11-16T00:00:00Z
review text: This paper analyses the global solvability of the problem \[ \begin{aligned} u_t = \nabla (u^m\nabla u), \;\;v_t = \nabla (v^n\nabla v) \quad &\text{ in } \Omega\times (0,\infty), \\ \frac {\partial u} { \partial \eta} = u^\alpha v^p,\;\;\frac {\partial v}{\partial \eta} = u^qv^\beta \quad &\text{ on }\partial\Omega\times(0,\infty), \\ u(\cdot ,0)=u_0,\;\;v(\cdot,0)=v_0 \quad &\text{ in } \Omega, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with sufficiently smooth boundary and unit outward normal \(\eta\), \(m\), \(n\),\(\alpha\), and \(\beta\) are nonnegative parameters, and \(p\) and \(q\) are positive parameters. In addition, \(u_0\) and \(v_0\) are positive \(C^1\) functions which satisfy the boundary conditions. It is shown that this problem has a solution (for all time) if and only if \[ \max\{m+\alpha,n+\beta\} \leq 1, \quad pq \leq (1-m-\alpha)(1-n-\beta). \tag{*} \] Previous results were proved in case \(m=n=0\). The proof proceeds by showing that (*) implies the existence of a positive supersolution for all time and that a nonnegative subsolution exists, which becomes infinite in finite time, if (*) does not hold.
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1274768
zbMATH Keywords: finite time blow-up
zbMATH Keywords: global solutions
reviewed by: Gary M. Lieberman
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6220
OpenAlex ID: W2004619715
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DOI: 10.1006/JMAA.1998.6220
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description: scientific article; zbMATH DE number 1274768