Long-time existence for systems of quasilinear wave equations

TL;DR

This paper proves almost global existence for coupled systems of quasilinear wave equations in three spatial dimensions, extending previous results for scalar equations by employing advanced energy estimates and decay techniques.

Contribution

It extends almost global existence results from scalar to coupled systems of quasilinear wave equations with solution-dependent nonlinearities.

Findings

Established almost global existence for coupled systems.

Utilized a variant of $r^p$-weighted local energy estimates with ghost weights.

Applied space-time Klainerman-Sobolev estimates for decay.

Abstract

We consider quasilinear wave equations in $(1+3)$-dimensions where the nonlinearity $F(u,u',u")$ is permitted to depend on the solution rather than just its derivatives. For scalar equations, if $(\partial_u^2 F)(0,0,0)=0$, almost global existence was established by Lindblad. We seek to show a related almost global existence result for coupled systems of such equations. To do so, we will rely upon a variant of the $r^p$-weighted local energy estimate of Dafermos and Rodnianski that includes a ghost weight akin to those used by Alinhac. The decay that is needed to close the argument comes from space-time Klainerman-Sobolev type estimates from the work of Metcalfe, Tataru, and Tohaneanu.