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