LONG-TIME EXISTENCE OF SOLUTIONS FOR DIFFERENTIAL-EQUATION-2/DIFFERENTIAL-EQUATION-T(2)U(X,T) + DIFFERENTIAL-EQUATION/DIFFERENTIAL-EQUATION-T-ALPHA(U(X,T)) = DIFFERENTIAL-EQUATION-2/DIFFERENTIAL-EQUATION-X(2)BETA(U(X,T))

For a class of initial-value problems associated with the nonlinear wave equation of the title, we establish, under relatively mild conditions on the functions alpha(.) and beta(.), the following result on almost global existence, in time, of a classical solution: if t infinity, the maximal time of existence of a C1 solution, is finite for arbitrarily small initial data, then for all initial data that are sufficiently small, t infinity is bounded from below by a function of the initial data which increases without bound as the magnitude of the initial data goes to zero.