Cauchy problem for semilinear wave equation with time-dependent metrics

We establish the existence of weak solutions $u$ of the semilinear wave equation ${\mathrm{\partial }}_t^{2}u-{\text{div}}_x(a(t,x){\mathrm{\nabla }}_{x}u)=f_k(u)$ 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 $\mathrm{\mid }{f}_k(u)\mathrm{\mid }\le C\mathrm{\mid }u{\mathrm{\mid }}^k$. For some non-trapping time-periodic perturbations $a(t,x)$, we obtain the long time existence of solution for small initial data.