Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data
type: Item
instance of: scholarly article
title: Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data
zbMATH Open document ID: 1042.35006
published in: Journal of Differential Equations
publication date: 2002-07-08T00:00:00Z
review text: This paper is devoted to the asymptotic of the solutions of the following reaction-diffusion equation with nonlinear boundary conditions, that is \[ \begin{cases} u_t-d\Delta u+f(u)=0\quad &\text{in } \Omega\\ d\frac{\partial u}{\partial n}+g(u)=0\quad &\text{on } \Gamma=\partial\Omega\\ u(0,x)=u_0(x),\end{cases}\tag{1} \] where \(\Omega\subset\mathbb{R}^N\), \(N\geq 1\), is a bounded smooth set and \(d>0\). The author deals with singular initial data, that is \(u_0\in L^p(\Omega)\), \(1<p<\infty\) or \(u_0\in W^{1,p}(\Omega)\), \(1<p<N\) and special attention is given to the case that the nonlinearities grow critically. The author's main goal is to give conditions on the nonlinear terms such that (1) is globally well-posed and moreover has a global compact attractor.
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 1764712
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: nonlinear boundary conditions
zbMATH Keywords: singular initial data
zbMATH Keywords: nonlinearities
zbMATH Keywords: globally well-posed
zbMATH Keywords: global compact attractor
author: Anibal Rodriguez-Bernal
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4072
OpenAlex ID: W2049865803
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DOI: 10.1006/JDEQ.2001.4072
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description: scientific article; zbMATH DE number 1764712