Cauchy problem for semilinear wave equation with time-dependent metrics

TL;DR

This paper proves the existence of weak solutions for a semilinear wave equation with time-dependent metrics, showing long-term solutions under specific conditions on the metric and initial data.

Contribution

It establishes the existence of weak solutions for a class of semilinear wave equations with time-dependent, non-trapping metrics, including long-time existence results for small initial data.

Findings

Existence of weak solutions for the semilinear wave equation.

Long time existence of solutions under non-trapping, time-periodic metrics.

Solutions exist for small initial data in the specified setting.

Abstract

We establish the existence of weak solutions $u$ of the semilinear wave equation $\partial_t^2 u-\textrm{div}_x(a(t,x)\nabla_xu)=f_k(u)$ 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 $\vert f_k(u)\vert\leq C\vert u\vert^k$. For some non-trapping time-periodic perturbations $a(t,x)$, we obtain the long time existence of solution for small initial data.