Almost global existence for quasilinear wave equations in three space dimensions

Abstract:

We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only use the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as much as the $O(|t|^{-1})$ decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equation that gives $O(1/t)$ decay of solutions of the inhomogeneous linear wave equation in terms of a $O(1/|x|)$-weighted norm on the forcing term. A weighted $L^{2}$ space-time estimate for inhomogeneous wave equations is also important in making the spatial decay useful for the long-term existence argument.