Nonlinear abstract wave equations with strong damping
type: Item
instance of: scholarly article
title: Nonlinear abstract wave equations with strong damping
zbMATH Open document ID: 0918.35094
author: Yuhua Lin
author: Naoki Tanaka
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-08-19T00:00:00Z
review text: The authors consider an initial-value problem for the equation \[ \ddot u- Au- B(u)= 0\tag{1} \] in a general Banach space \(X\). Here \(A\) is a closed linear operator in \(X\) satisfying the Hille-Yosida condition with the exception of the density of the domain \(D(A)\) of \(A\), and \(B\) is a nonlinear operator defined on \(D(A)\). The purpose of this paper is to study equation (1) under weaker conditions than those previously considered in literature. The authors prove the loal existence theorem and give a sufficient condition for the global existence. The authors' abstract results are applied to two examples, namely the quasilinear hyperbolic equation with linear viscosity, \[ \ddot u- \sigma(u_x)u_x- \mu\dot u_{xx}= 0,\quad \mu>0, \] with \(\sigma\in C^2(\mathbb{R})\), and a hyperbolic equation with strong damping: \[ \ddot u+|\sqrt A u|^{2\alpha}Au+ \mu A\dot u= 0,\quad \alpha\geq 1, \] where \(A\) is a nonnegative selfadjoint operator in a Hilbert space.
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 47H20
zbMATH DE Number: 1218636
zbMATH Keywords: nondense domain
zbMATH Keywords: strong damping
zbMATH Keywords: Hille-Yosida condition
zbMATH Keywords: quasilinear hyperbolic equation with linear viscosity
reviewed by: I. V. Barashenkov
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5999
OpenAlex ID: W1981727875
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DOI: 10.1006/JMAA.1998.5999
description: scientific article; zbMATH DE number 1218636