Global existence for Dirichlet-wave equations with quadratic   nonlinearties in high dimensions
  We prove global existence of solutions to quasilinear wave equations with quadratic nonlinearities exterior to nontrapping obstacles in spatial dimensions four and higher. This generalizes a result of Shibata and Tsutsumi in spatial dimensions greater than or equal to six. The technique of proof would allow for more complicated geometries provided that an appropriate local energy decay exists for the associated linear wave equation. 
Jason Metcalfe and Christopher D. Sogge