Asymptotic behavior in time of small solutions to nonlinear wave equations in an exterior domain

We study the initial boundary value problem for the nonlinear wave equations [GRAPHICS] where (D) over bar = (partial derivative(t), partial derivative(r)), F:R-6 --> R and epsilon(0) is sufficiently small. We assume that F(w) = F(w(1),..., w(6)) = F(u, partial derivative(tau)u, partial derivative(t)u, partial derivative(tau)(2)u, partial derivative(r)partial derivative(t)u, partial derivative(t)(2)u), is a polynomial with respect to arguments satisfying one of the following conditions [GRAPHICS] where lambda(aj) is an element of R for j = 1, ..., 6. In this paper we show global existence and asymptotic behavior in time of solutions of (*).