Decay of solutions of the wave equation with localized nonlinear damping and trapped rays
type: Item
instance of: scholarly article
title: Decay of solutions of the wave equation with localized nonlinear damping and trapped rays
zbMATH Open document ID: 1237.35020
author: K.-D. Phung
published in: Mathematical Control and Related Fields
publication date: 2012-01-18T00:00:00Z
review text: The author proves some decay estimates of the energy of the wave equation governed by localized nonlinear dissipations in a bounded domain \(\Omega\) in which trapped rays may occur. The equation is of the form \(\partial_t^2 u-\Delta u+a g(\partial_t u)=0\) with \(a\geq 0\) and homogeneous Dirichlet boundary condition \(u=0\). The result applies to many choices for the function \(g\), e.g. \(g(s)=|s|^{p-1}s\) or \(g(s)=s/\sqrt{1+s^2}\). The approach is based on a comparison with the linear damped wave equation and an interpolation argument. The result extends to the nonlinear damped wave equation the well-known optimal logarithmic decay rate for the linear damped wave equation with regular initial data.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35L71
Mathematics Subject Classification ID: 35B35
zbMATH DE Number: 5998568
zbMATH Keywords: wave equation
zbMATH Keywords: decay estimates
zbMATH Keywords: localized nonlinear damping
zbMATH Keywords: trapped rays
zbMATH Keywords: homogeneous Dirichlet boundary condition
zbMATH Keywords: interpolation argument
reviewed by: Chengbo Wang
MaRDI profile type: Publication
full work available at URL: mcrf.2011.1.251
OpenAlex ID: W2333401114
DOI: 10.3934/MCRF.2011.1.251
description: scientific article; zbMATH DE number 5998568