Long-time Existence for Systems of Quasilinear Wave Equations

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 (∂u2F)(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 rp-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. © The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2023.