Existence and continuity of the attractor for a singularly perturbed hyperbolic equation
type: Item
instance of: scholarly article
title: Existence and continuity of the attractor for a singularly perturbed hyperbolic equation
zbMATH Open document ID: 0846.35019
DOI: 10.1007/BF02218726
author: Ingo Witt
published in: Journal of Dynamics and Differential Equations
publication date: 1996-03-06T00:00:00Z
review text: The author considers the semilinear damped wave equation of the form \[ \varepsilon u_{tt}+ \gamma u_t+ b(u_t)= \Delta u- f(u)+ g(x), \] where \(u= u(x, t)\), \(x\in \Omega\subset \mathbb{R}^N\) with \(\Omega\) a bounded domain. The damping function \(b\) is nondecreasing. The existence of a global compact attractor in the usual energy space \((u, u_t)\in H^1_0\times L^2\) is proved under some growth restrictions concerning the function \(f\). In particular, \(f\) is supposed to be globally Lipschitz which seems to be rather restrictive. If, in addition, \(b\) is globally Lipschitz, the upper semicontinuity of the attractors with respect to the parameter \(\varepsilon\) is shown.
reviewed by: Eduard Feireisl
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L10
zbMATH DE Number: 851870
zbMATH Keywords: semilinear damped wave equation
zbMATH Keywords: global compact attractor
zbMATH Keywords: upper semicontinuity of the attractors
Wikidata QID: Q115392699
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 851870