The Palais-Smale condition for the energy of some semilinear parabolic equations
type: Item
instance of: scholarly article
title: The Palais-Smale condition for the energy of some semilinear parabolic equations
zbMATH Open document ID: 0953.35067
author: Ryo Ikehata
published in: Hiroshima Mathematical Journal
publication date: 2001-01-15T00:00:00Z
review text: The author of this interesting paper considers an initial and boundary value problem for the nonlinear parabolic equation \(u_t(t,x)-\triangle u(t,x)=|u(t,x)|^{p-1}u(t,x)\), \((t,x)\in (0,T)\times\Omega \), \(u(0,x)=u_0(x)\), \(x\in \Omega \), \(u|_{\partial \Omega }=0\), \(t\in (0,T)\), where \(1<p\leq (N+2)/(N-2)\) and \(\Omega \subset \mathbb{R}^N\) \((N\geq 3)\) is a bounded domain with smooth boundary \(\partial \Omega \). Of special interest here are the solutions which neither blowup nor decay. The main result is that all the global solutions for some semilinear parabolic equations naturally contain a Palais-Smale sequence as a subsequence and then one can apply a global compactness result due to M. Struwe. The finite-time blowup problems are discussed as well.
Mathematics Subject Classification ID: 35K60
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 1484787
zbMATH Keywords: Palais-Smale sequence
zbMATH Keywords: global compactness result
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1484787