Cauchy problem for semilinear wave equation with time-dependent metrics

Abstract

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