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 quasilinear 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.