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

We consider the initial-boundary value problem for the standard quasi-linear wave equation: u(tt) - div{sigma(/delu/(2))delU} + a(x)u(t) = 0 in Omega x [0, infinity) u(x,0) = u(0)(x) and u(t)(x,0) = u(1)(x) and u/(partial derivativeOmega) = 0 where Omega is an exterior domain in R-N, sigma(v) is a function like sigma(v) = 1/root1 + v and a(x) is a nonnegative function. Under two types of hypotheses on a(x) we prove existence theorems of global small amplitude solutions. We note that a(x)u(t) is required to be effective only in localized area and no geometrical condition is imposed on the boundary partial derivativeOmega.