Cauchy problem for semilinear wave equation with time-dependent metrics

摘要

We establish the existence of a weak solution u of the semilinear wave equation ?t2u-div_χ(a(t,x)?_χu)=f _k(u) where a(t,x) is equal to 1 outside a compact set with respect to x and a non-linear term fk 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.