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 □ = ∂²_t − 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, ∂²_tu), where F is a quadratic nonlinearity in (∂_tu, part;_t ∂u, ∂²_tu), ∂ = (∂₁, ∂₂, … , ∂_n) and n ≥ 4.