Cauchy problem for semilinear wave equation with time-dependent metrics

We establish the existence of a weak solution u of the semilinear wave equation where a(t,x) is equal to 1 outside a compact set with respect to x and a non-linear term f_k which satisfies |f_k(u)|≤C|u|^k. For some non-trapping time-periodic perturbations a(t,x), we obtain the long time existence of a solution from little initial data.