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

Dedicated to Professor Atsushi Yoshikawa on his sixtieth birthday

Abstract

We consider the initial-boundary value problem for the standard quasi-linear wave equation: u_tt-div{σ(|∇_u|²)∇_u}+a(x)u_t=0 in Ω×[0, ∞) u(x, 0)=u₀(x) and u_t(x, 0)=u₁(x) and u|_∂Ω=0 where Ω is an exterior domain in R^N, σ(v) is a function like σ(v)=1/√{1+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 ∂Ω.