Reaction-diffusion equations with nonlinear boundary delay
type: Item
instance of: scholarly article
title: Reaction-diffusion equations with nonlinear boundary delay
zbMATH Open document ID: 0933.35105
DOI: 10.1023/A:1021929413376
author: Sergio Muniz Oliva
published in: Journal of Dynamics and Differential Equations
publication date: 2000-01-04T00:00:00Z
review text: The following problem (with a boundary condition including a small delay) is considered: \(u_t-\text{Div}(a\nabla u)+ \sum^n_{j=1} B_j(x) {\partial u\over\partial x_j}+ \lambda u=f(u)\), in \(\Omega\times \mathbb{R}^+\), \({\partial u\over \partial n_a}=g(u(t), u(t-r))\) in \(\mathbb{R}^+\), \(u=0\) on \(\partial \Omega\times [-r,0]\); \(\Omega\) is an open bounded domain in \(\mathbb{R}^n\), \(r>0\) is the delay, \(a\in C^1(\overline \Omega)\), \((a(x)>m_0>0\), \(x\in\Omega)\), \(\partial u/ \partial n_a=\langle a\nabla u,\vec n\rangle\) \((n\) is the outward normal), \(\lambda\) is a positive constant, \(B_j(x)\) is continuous in \(\overline \Omega\) \((j=1, \dots,n)\), \(f:\mathbb{R}\to\mathbb{R}\) and \(g:\mathbb{R}^2 \to\mathbb{R}\) are smooth functions. Using semigroup theory, global existence and uniqueness of solutions in a convenient fractional power space \((H^s_p(\Omega))\) is shown. It is also proved that all bounded solutions are asymptotic to the set of equilibria as \(t\to\infty\).
reviewed by: Ion Onciulescu
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35R10
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K57
zbMATH DE Number: 1333042
zbMATH Keywords: small delay
zbMATH Keywords: semigroup theory
zbMATH Keywords: fractional power space
MaRDI profile type: Publication
full work available at URL: a:1021929413376
OpenAlex ID: W2183588469
description: scientific article; zbMATH DE number 1333042