Energy decay rate for a quasi-linear wave equation with localized strong dissipation

Abstract

The purpose of this paper is to derive a sharp energy decay estimate for a quasi-linear wave equation with localized strong dissipation of the type $-\nabla \cdot \left(a\left(x\right)\nabla u_t\right)$ in a domain $\Omega$ of ${\mathbb{R}}^N$, where $a\left(x\right)$ is a nonnegative function supported only on a part of the boundary $\partial \Omega$. We note that the index of algebraic decay depends on dimension $N$ and no geometrical condition is imposed on the boundary $\partial \Omega$.