Energy decay for the wave equation in exterior domains with some half-linear dissipation

We study the decay estimates of the energy for the wave equation in an exterior domain with a localized dissipation. The dissipative term consists of the following two parts: The first part may be nonlinear and localized in a suitable bounded area, while the second part is linear in the outside of a big ball. So we may call such a dissipation ``half-linear" dissipation. We note that no geometrical condition is imposed on the boundary. As an application of the decay estimates we prove some global existence theorems for the wave equation with a nonlinear source term.