Global existence theorem for nonlinear wave equation in exterior domain (Q1058067)

The authors consider nonlinear wave equations of the form \(\square u+F(t,x,\Lambda u)=f(t,x)\) in [0,\(\infty)\times \Omega\) subject to the following boundary and initial conditions: \(u=0\) on [0,\(\infty)\times \partial \Omega\); \(u(0,x)=\phi_ 0(x)\), \(\partial u(0,x)/\partial t=\phi_ 1(x)\) in \(\Omega\). Here \(\Omega\) is an unbounded domain in \({\mathbb{R}}^ n(n\geq 3)\) such that its boundary \(\partial \Omega\) is \(c^{\infty}\) and compact, and \(\Lambda u=(\partial_ iu,\partial_ j\partial_ ku,\quad i,j,k=0,...,n)\) with \(\partial_ i\) defined by \(\partial_ 0=\partial /\partial t\), \(\partial_ j=\partial /\partial x_ j\) \((j=1,...,n).\) The aim of the paper is to report that if the spatial dimension is not smaller than 3 and the initial data are small and smooth, then the above mentioned problem has a unique classical global solution. The results are phrased in two theorems without giving the proofs.