On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation

Abstract

We consider the dynamic elasticity equations with a locally distributed damping in a bounded domain. The local dissipation of the form $a(x)y_{t}$ allows coefficients $a$ that lie in some $L^r(\Omega)$, with $(r>2)$. Using multiplier techniques, interpolation inequalities, and a judicious application of the Hölder inequality, we prove sharp energy decay estimates for all $r>Max(2,N)$, where $N$ denotes the space dimension. All space dimensions are considered; the results obtained generalize and improve earlier works where $r$ is required to satisfy $r \! \geq \! {3N+\sqrt{9N^2-16N}\over4}$, for $N\geq3$.