Global existence for semilinear parabolic systems on one-dimensional bounded domains
type: Item
instance of: scholarly article
title: Global existence for semilinear parabolic systems on one-dimensional bounded domains
zbMATH Open document ID: 0737.35034
published in: Rocky Mountain Journal of Mathematics
publication date: 1992-06-25T00:00:00Z
review text: The author is considering the following semilinear parabolic system \[ u_ t=Du_{xx}+Cu_ x+f(u), 0<x<1, \quad u(t,0)=u(t,1)=0, \quad u(0,x)=u_ 0(x), \leqno (1.1) \] where the unknown function \(u\) is vector valued and the nonlinear term \(f\) is a smooth map from \(\mathbb{R}^ m\) into \(\mathbb{R}^ m\) and \(D\) and \(C\) are matrices satisfying certain parabolicity conditions. To consider the global existence of solutions, the author assumes a kind of Lyapunov structure on the nonlinear term \(f\). That is, there exists a Lyapunov function for the following ordinary differential equation (vector field on some region of \(\mathbb{R}^ m\)), \[ dv/dt=f(v). \leqno (1.2) \] Recently, such type of methods has been employed by several people (see the references). In this paper, the author gives a generalization of his earlier work to cover the equations including a convection term. He gives conditions on the nonlinear term which ensures the global unique existence of the solution to the initial boundary value problem to (1.1).
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 14131
zbMATH Keywords: uniqueness
zbMATH Keywords: semilinear parabolic system
zbMATH Keywords: global existence
zbMATH Keywords: Lyapunov function
author: J. J. Morgan
reviewed by: Shuichi Jimbo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072963
OpenAlex ID: W2095165831
cites work: LPBounds of solutions of reaction-diffusion equations
cites work: On the existence of steady states of certain reaction-diffusion systems
cites work: Global Existence and Boundedness in Reaction-Diffusion Systems
cites work: On the global existence and asymptotic behavior of solutions of reaction- diffusion equations
cites work: Global Existence for Semilinear Parabolic Systems
DOI: 10.1216/RMJM/1181072963
description: scientific article; zbMATH DE number 14131