Global existence and blow-up for a nonlinear reaction-diffusion system
type: Item
instance of: scholarly article
title: Global existence and blow-up for a nonlinear reaction-diffusion system
zbMATH Open document ID: 0884.35068
author: Hongwei Chen
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-04-02T00:00:00Z
review text: The author considers the system \[ u_t-\Delta u=u^{m_1} v^{n_1}, \quad v_t-\Delta v=u^{m_2} v^{n_2} \] for \(x\in\Omega\), \(t>0\), where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with a smooth boundary, \(u\) and \(v\) have given nonnegative values initially and are zero when \(x\in \partial\Omega\), \(t>0\). Solutions here are classical solutions. The constants \(m_1\), \(n_1\), \(m_2\), \(n_2\) satisfy \(m_1\), \(n_2\geq 0\), and \(n_1,m_2>0\). The point of the paper is the proof of: i) If \(m_1\leq 1\), \(n_2\leq\) and \(m_2n_1 \leq(1-m) (1-n_2)\), then all nonnegative solutions are global. ii) If \(m_1>1\) or \(n_2>1\) or \(m_2n_1> (1-m_1) (1-n_2)\), then there are both global solutions and solutions which blow up in finite time, depending on the magnitude of the initial values.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1053997
zbMATH Keywords: blow-up
zbMATH Keywords: classical solutions
reviewed by: Ronald B. Guenther
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5522
OpenAlex ID: W2090519619
cites work: Mathematical problems from combustion theory
cites work: On nonlinear reaction-diffusion systems
cites work: The Role of Critical Exponents in Blowup Theorems
cites work: Boundedness and blow up for a semilinear reaction-diffusion system
cites work: Global existence and blow-up for a class of semilinear parabolic systems: A Cauchy Problem
cites work: Q3793087
cites work: A semilinear parabolic system in a bounded domain
cites work: Nonlinear elliptic boundary-value problems in unbounded domains
cites work: Q5562267
cites work: REMARKS ON BLOW-UP AND NONEXISTENCE THEOREMS FOR NONLINEAR EVOLUTION EQUATIONS
cites work: Single point blow-up for a semilinear initial value problem
cites work: On global solution of nonlinear hyperbolic equations
cites work: Global solutions of reaction-diffusion systems
DOI: 10.1006/JMAA.1997.5522
Recommended article: Global existence and blow-up of solutions to a nonlocal reaction-diffusion system
Recommended article: Q3371737
Recommended article: Q4655767
Recommended article: Global existence and blow-up for a class of nonlinear reaction diffusion problems
Recommended article: Global existence and blowup for a reaction-diffusion system with nonlocal boundary condition
Recommended article: Q5466106
Recommended article: Global existence and finite time blow up for a reaction-diffusion system
Recommended article: Global existence and global non-existence of solutions to a reaction-diffusion system
Recommended article: Global existence and blow-up problems for reaction diffusion model with multiple nonlinearities
Recommended article: Q4792470
description: scientific article; zbMATH DE number 1053997