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 ? Δu+a(x)u _t =0 in Ω × [0, ∞) with the boundary conditionu/?Ω, wherea(x) is a nonnegative function on $bar Omega $ satisfying $$a(x) > a.e. x in omega andsmallint _omega frac{1}{{a(x)^P }}dx< infty for some 0< p< 1$$ for an open set $omega subset bar Omega $ including a part of ?Ω 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.