On global smooth solutions to the initial-boundary value problem for quasilinear wave equations in exterior domains

Abstract

We consider the initial-boundary value problem for the standard quasilinear wave equation:

$u_{\mathrm{tt}}-\text{div}\left\{\sigma \right(\left|{\nabla }_u{|}^2\right){\nabla }_u\}+a(x)u_t=0$ in $\Omega \times [0,\infty )$

$u(x,0)=u_0\left(x\right)$ and $u_t(x,0)=u_1\left(x\right)$ and $u{|}_{\partial \Omega }=0$

where $\Omega$ is an exterior domain in $R^N,\sigma \left(v\right)$ is a function like $\sigma \left(v\right)=1/\sqrt{1+v}$ and $a\left(x\right)$ is a nonnegative function. Under two types of hypotheses on $a\left(x\right)$ we prove existence theorems of global small amplitude solutions. We note that $a\left(x\right)u_t$ is required to be effective only in localized area and no geometrical condition is imposed on the boundary $\partial \Omega$.