Stabilization of the wave equations with potential and indefinite damping
type: Item
instance of: scholarly article
title: Stabilization of the wave equations with potential and indefinite damping
zbMATH Open document ID: 1004.35023
DOI: 10.1016/S0022-247X(02)00062-8
author: Kangsheng Liu
author: Bopeng Rao
author: Xu Zhang
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-09-08T00:00:00Z
review text: In this paper the authors study the energy \({\mathcal E}(t)\) of the wave equation \[ \begin{cases} y_{tt}-\Delta y+\bigl[a(x) +b(x)\bigr]y_t +c(x)y=0, \quad & \text{in }(0,\infty) \times\Omega,\\ y=0,\quad &\text{on }(0,\infty)\times \Gamma,\\ y(0)=y_0,\;y_t(0)= y_1,\quad &\text{in }\Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain with \(C^2\) boundary \(\Gamma\), \(a(\cdot) \geq 0\), \(b(\cdot)\leq 0\) and \(c(\cdot)\geq 0\). It is assumed that the damping \(a(\cdot)+ b(\cdot)>0\) in a subdomain \(G\) of \(\Omega\), which is a neighborhood of the ``star-shaped'' part of \(\Gamma\), and \(|b |_{L^\infty (\Omega)}\) is small compared to \(|a|_{L^\infty (\Omega)}\). Using a global Carleman-type inequality and a classical perturbation result, an explicit exponential decay rate of \({\mathcal E}(t)\) is obtained. Also, in the same context, an upper bound estimate of the negative damping \(b(\cdot)\) is given.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 1797507
zbMATH Keywords: energy decay rate
zbMATH Keywords: global Carleman-type inequality
zbMATH Keywords: exponential decay rate
zbMATH Keywords: upper bound estimate of the negative damping
reviewed by: Rodica Luca
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1797507