$L^p$ 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 where is an exterior domain in , is a dissipative term which is effective only near the 'critical part' of the boundary. We first give some estimates for the linear equation by combining the results of the local energy decay and 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 is odd dimensional domain . When our result is applied if . We note that no geometrical condition on the boundary is imposed. Received April 13, 2000 / Revised July 6, 2000 / Published online February 5, 2001