Decay of solutions of the wave equation with a local degenerate dissipation

We derive a precise decay estimate of the solutions to the initial-boundary value problem for the wave equation with a dissipation: u(tt) - Delta u + a(x)u(t) = 0 in Omega x [0,infinity) with the boundary condition u\(partial derivative Omega) = 0, where a(x) is a nonnegative function on <(Omega)over bar> satisfying a(x) > 0 a.e. x is an element of omega and integral(omega) 1/a(x)(p) dx < infinity for some 0 < p < 1 for an open set omega subset of <(Omega)over bar> including apart of partial derivative Omega with a specific property. The result is applied to prove a global existence and decay of smooth solutions for a semilinear wave equation with such a weak dissipation.