Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation
type: Item
instance of: scholarly article
title: Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation
zbMATH Open document ID: 0922.35094
author: Sang Ro Park
author: James Serrin
author: Howard A. Levine
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-04-27T00:00:00Z
review text: The global existence and global nonexistence of solutions of the Cauchy problem for wave equations with nonlinear damping \[ u_{tt}+ Q(t,x,u_t)-\Delta u= f(t,u),\quad (t,x)\in [0,\infty)\times \mathbb{R}^n \] are investigated. The authors establish various global nonexistence of weak solutions in terms of the function \(a(t,x)\) for which the inequality \(| Q(t,x,v)|\leq a(t,x)| v|^{m-1}\) holds, the right member \(f(x,u)\) and its integral for compactly supported initial data with negative or sufficiently negative initial energy, and the global continuability of local weak solutions in terms of the function \(\widetilde a(t,x)\) such that the inequality \(| Q(t,x,v)|\geq \widetilde a(t,x)| v|^{m-1}\) holds and the Fréchet derivative of \(f(t,u)\) with respect to \(u\). The results can be applied to the case \[ Q(x,t,u)= au_t| u_t|^{m-2},\quad f(x,u)=bu| u|^{p-2}. \]
reviewed by: Hiroki Tanabe
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35L15
zbMATH DE Number: 1246345
zbMATH Keywords: unbounded domain
zbMATH Keywords: compactly supported initial data
zbMATH Keywords: negative or sufficiently negative initial energy
MaRDI profile type: Publication
full work available at URL: jmaa.1998.6126
OpenAlex ID: W1984259737
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DOI: 10.1006/JMAA.1998.6126
description: scientific article; zbMATH DE number 1246345