L^P estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains

We consider the initial-boundary value problem for the semilinear wave equation u_tt - Δu + a(x)u_t = f(u) in Ω x [0, ∞), u(x, 0) = u₀(x), u_t(x, 0) = u₁(x) and u|_∂Ω = 0, where Ω is an exterior domain in R^N, a(x)u_t is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some L^P estimates for the linear equation by combining the results of the local energy decay and L^P estimates for the Cauchy problem in the whole space. Next, on the basis of these estimates we prove global existence of small amplitude solutions for semilinear equations when Ω is odd dimensional domain. When N = 3 and f = |u|^αu our result is applied if α > 2√3-1. We note that no geometrical condition on the boundary ∂Ω is imposed.