A priori bounds and stability in the retarded nonlinear wave equation
type: Item
instance of: scholarly article
title: A priori bounds and stability in the retarded nonlinear wave equation
zbMATH Open document ID: 0755.35007
DOI: 10.1016/0022-247X(92)90310-A
author: Bo Zhang
author: T. A. Burton
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: The object of the authors' work is to give sufficient conditions to ensure that the zero solution of certain generalizations of \(u_{tt}=(a(x)g(u_ x(t-h(t),x)))_ x+(f(u_ x))_{xt}\) is asymptotically stable and to study limit sets. Here, \(0\leq h(t)\leq h\) say, \(a(x)>0\), \(xg(x)>0\) if \(x\neq 0\), \(f'(x)>0\) and \(g'(x)>0\). They impose the boundary conditions \(u(t,0)=u(t,1)=0\). Among other conditions, it is crucial to have \[ \int^ 1_ 0(a(x)g(u_ x))_ xu_ t dx=- \int^ 1_ 0(a(x)g(u_ x))u_{tx} dx. \]
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35L05
zbMATH DE Number: 62827
zbMATH Keywords: sufficient conditions for asymptotic stability
reviewed by: Anthony D. Osborne
MaRDI profile type: Publication
cites work: Functional differential equations and Jensen's inequality
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cites work: Q5539892
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cites work: Geometric theory of semilinear parabolic equations
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cites work: A model for one-dimensional, nonlinear viscoelasticity
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cites work: Existence and Asymptotic Behavior for a Strongly Damped Nonlinear Wave Equation
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cites work: Q4724978
cites work: Q5334517
cites work: Q3223357
full work available at URL: 0022-247x(92)90310-a
OpenAlex ID: W2044096483
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description: scientific article; zbMATH DE number 62827