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

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 −∇⋅(a(x)∇u_t) in a domain Ω of R^N, where a(x) is a nonnegative function supported only on a part of the boundary ∂Ω. We note that the index of algebraic decay depends on dimension N and no geometrical condition is imposed on the boundary ∂Ω.