Energy decay for the wave equation with boundary and localized dissipations in exterior domains

We study a decay property of solutions for the wave equation with a localized dissipation and a boundary dissipation in an exterior domain Ω with the boundary ∂Ω, = Γ₀ ∪ Γ₁, Γ₀ ∩ Γ₁ = Ø. We impose the homogeneous Dirichlet condition on Γ₀ and a dissipative Neumann condition on Γ₁. Further, we assume that a localized dissipation a(x)u_t is effective near infinity and in a neighborhood of a certain part of the boundary Γ₀. Under these assumptions we derive an energy decay like E(t) ≤ C(1 + t)^-1 and some related estimates.