We consider the initial-boundary value problem for the semilinear wave equation utt - Δu + a(x)ut = f(u) in Ω x [0, ∞), u(x, 0) = u0(x), ut(x, 0) = u1(x) and u|∂Ω = 0, where Ω is an exterior domain in RN, a(x)ut is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some LP estimates for the linear equation by combining the results of the local energy decay and LP 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.
ASJC Scopus subject areas
- 数学 (全般)