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

2016; American Institute of Mathematical Sciences; Volume: 5; Issue: 1 Linguagem: Inglês

10.3934/eect.2016.5.37

2163-2480

Moez Daoulatli,

Advanced Mathematical Modeling in Engineering

In this paper we study the behavior of the energy and the $L^{2}$ norm ofsolutions of the wave equation with localized linear damping in exteriordomain. 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 thenunder the Geometric Control Condition of Bardos et al [5] (1992), weprove 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 $\Vertu(t) \Vert _{L^{2}}^{2}\leq C_{0}I_{0},$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) $ andverifies\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 $\left\Vert u\left( t\right) \right\Vert _{L^{2}}^{2} \leqC_{2} \left( 1+t\right) ^{-1}I_{1},$ 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 $B$ $\underset{x\in \Omega }{\inf } \left\vert x\right\vert \geq 2$.