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 ) O(|x|^{-1}) decay of solutions of the wave equation as much as the O ( | t | − 1 ) 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 ) O(1/t) decay of solutions of the inhomogeneous linear wave equation in terms of a O ( 1 / | x | ) O(1/|x|) -weighted norm on the forcing term. A weighted L 2 L^{2} space-time estimate for inhomogeneous wave equations is also important in making the spatial decay useful for the long-term existence argument.