Long-time existence for systems of quasilinear wave equations

Abstract

We consider quasilinear wave equations in $(1+3)$-dimensions where the nonlinearity $F(u,u^{\prime },u")$ 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.

27 pages