Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations

We derive the total energy decay E(t) ≤ I₀(1 + t)^-1 and L² boundedness ∥u(t)∥2 ≤ CI_o for the solutions to the initial boundary value problem for the wave equation in an exterior domain Ω: u_tt - Δu + a(x)u_t = 0 in Ω × (0, ∞) with u(x, 0) = u₀(x), u_t(x, 0) = u₁(x) and u|∂Ω = 0, where I₀ = ∥u₀∥H₁ + ∥u₁∥₂ and a(x) is a nonnegative function which is positive near some part of the boundary ∂Ω and near infinity. We apply these estimates to prove the global existence of decaying solutions for semilinear wave equations with nonlinearity f(u) like |u|^αu, α > 0. We note that no geometrical condition is imposed on the boundary ∂Ω.