Cauchy problem for semilinear wave equation with time-dependent metrics

We establish the existence of a weak solution uu of the semilinear wave equation ∂t2u−divx(a(t,x)∇xu)=fk(u) where a(t,x)a(t,x) is equal to 11 outside a compact set with respect to xx and a non-linear term fkfk which satisfies |fk(u)|≤C|u|k|fk(u)|≤C|u|k. For some non-trapping time-periodic perturbations a(t,x)a(t,x), we obtain the long time existence of a solution from little initial data.