Sharp energy decay estimates for the wave equation with a local degenerate dissipation

Abstract

We derive a fast decay estimate for the wave equation with a local degenerate dissipation of the type $a\left(x\right)u_t$ in a bounded domain $\Omega$. The dissipative coefficient $a\left(x\right)$ is a nonnegative function only on a neighborhood of some part of the boundary $\partial \Omega$ and may vanish somewhere in $\overline{\Omega }$. The results obtained extend and improve on earlier results of Nakao as well as those of Tcheugoué Tébou. The method of proof is based on multipliers technique and some modified interpolation and difference inequalities.