Stabilization of the total energy for a system of elasticity with localized dissipation

Abstract

In this paper we study the system of elastic waves in a bounded domain in ${{\mathbb R}}^3$ with a localized dissipation given by a function $q(x)$ in $L^r(\Omega )$. The dissipation is effective only on a neighborhood of part of the boundary of the domain. We prove algebraic decay rate of energy. In order to obtain this result we develop new energy identities for this system. Our result extend previous results for the system of elasticity and wave equation.