Global Existence of Small Solutions to Quadratic Nonlinear Wave Equations in an Exterior Domain

We consider the initial boundary value problem for the nonlinear wave equation [formula] where □ = ∂2t − ΔB = {x : |x| = [formula] > R}, ∂B = {x : |x| = R}, u0, u1 are real valued functions ϵ0 is a sufficiently small positive constant. In this paper it is shown that small solutions to (*) exist globally in time when n = 4. Our method in this paper is applicable to the more general nonlinear wave equations such that □u = F(∂tu, ∂t ∂u, ∂2tu), where F is a quadratic nonlinearity in (∂tu, part;t ∂u, ∂2tu), ∂ = (∂1, ∂2, ..., ∂n) and n ≥ 4.