Energy decay rates for solutions of the wave equation with linear damping in exterior domain

Moez  Daoulatli

doi:10.3934/eect.2016.5.37

Article Contents

2016, Volume 5, Issue 1: 37-59. Doi: 10.3934/eect.2016.5.37

This issue Previous Article Well productivity index for compressible fluids and gases Next Article Elliptic and parabolic equations with fractional diffusion and dynamic boundary conditions

Energy decay rates for solutions of the wave equation with linear damping in exterior domain

Moez Daoulatli^1,

1.
University of Dammam, King Faisal Road, 31952, Dammam

Received: September 30, 2014

Revised: December 31, 2015

Published: February 2016

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Abstract

In this paper we study the behavior of the energy and the $L^{2}$ norm of solutions of the wave equation with localized linear damping in exterior domain. Let $u$ be a solution of the wave system with initial data $\left( u_{0},u_{1}\right) $. We assume that the damper is positive at infinity then under the Geometric Control Condition of Bardos et al [5] (1992), we prove that:
1. If $(u_{0},u_{1}) $ belong to $H_{0}^{1}( \Omega) \times L^{2}( \Omega ) ,$ then the total energy $ E_{u}(t) \leq C_{0}(1+t) ^{-1}I_{0}$ and $\Vert u(t) \Vert _{L^{2}}^{2}\leq C_{0}I_{0},$ where \begin{equation*} I_{0}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}. \end{equation*} 2. If the initial data $\left( u_{0},u_{1}\right) $ belong to $ H_{0}^{1}\left( \Omega \right) \times L^{2}\left( \Omega \right) $ and verifies \begin{equation*} \left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}<+\infty , \end{equation*} then the total energy $E_{u}\left( t\right) \leq C_{2}\left( 1+t\right) ^{-2}I_{1}$ and $\left\Vert u\left( t\right) \right\Vert _{L^{2}}^{2} \leq C_{2} \left( 1+t\right) ^{-1}I_{1},$ where \begin{equation*} I_{1}=\left\Vert u_{0}\right\Vert _{H^{1}}^{2}+\left\Vert u_{1}\right\Vert _{L^{2}}^{2}+\left\Vert d\left( \cdot \right) \left( u_{1}+au_{0}\right) \right\Vert _{L^{2}}^{2} \end{equation*} and \begin{equation*} d\left( x\right) =\left\{ \begin{array}{lc} \left\vert x\right\vert & d\geq 3, \\ \left\vert x\right\vert \ln \left( B\left\vert x\right\vert \right) & d=2, \end{array} \right. . \end{equation*} with $B$ $\underset{x\in \Omega }{\inf } \left\vert x\right\vert \geq 2$.

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Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 35B35.

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