Convergence to equilibrium for delay-diffusion equations with small delay
type: Item
instance of: scholarly article
title: Convergence to equilibrium for delay-diffusion equations with small delay
zbMATH Open document ID: 0798.35150
DOI: 10.1007/BF01063736
author: Gero Fiesecke
published in: Journal of Dynamics and Differential Equations
publication date: 1993-05-16T00:00:00Z
review text: The nonlinear reaction-diffusion equation with delay \[ \begin{aligned} u_ t- \Delta u & = f(u(t),\;u(t-\tau)),\quad\text{in }\Omega\times \mathbb{R}^ +,\\ u & = u_ 0\quad\text{in }\Omega\times [-\tau,0],\\ u & = 0\quad\text{or }\partial u/\partial\nu= 0\quad\text{on }\partial\Omega\times \mathbb{R}^ +\end{aligned} \] is considered, where \(f\) is Lipschitz and satisfies certain one-sided growth conditions. First the equation is shown to have a global unique solution. The main result shows that all bounded solutions (defined on \(C^ 0([-\tau,0]; L^ 2(\Omega))\)) tend to the equilibrium solutions, i.e. the solutions of the elliptic boundary value problem \[ -\Delta u= f(u,u)\quad\text{in }\Omega,\quad u=0\quad\text{or }\partial u/\partial\nu= 0\quad\text{on }\partial\Omega. \]
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 168054
zbMATH Keywords: nonlinear reaction-diffusion equation
zbMATH Keywords: global unique solution
zbMATH Keywords: equilibrium solutions
reviewed by: S. P. Banks
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01063736
OpenAlex ID: W2071122164
description: scientific article; zbMATH DE number 168054