  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.
