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

Abstract

We consider the initial-boundary value problem for the semilinear wave equation \[u_{tt}-\Delta u + a(x)u_t = f(u) \mbox{ in } \Omega \times [0,\infty),\] \[u(x,0)=u_0(x), u_t(x,0)=u_1(x) \mbox{ and } u|_{\partial\Omega}=0,\] where $\Omega$ 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 $\Omega$ is odd dimensional domain . When $N=3 \mbox{ and } f=|u|^\alpha u $ our result is applied if $\alpha > 2\sqrt{3}-1$ . We note that no geometrical condition on the boundary $\partial \Omega$ is imposed.