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

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  • 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 $B$ $\underset{x\in \Omega }{\inf } \left\vert x\right\vert \geq 2$.
    Mathematics Subject Classification: Primary: 35L05, 35B40; Secondary: 35B35.


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