Global existence of solutions to multiple speed systems of quasilinear   wave equations in exterior domains
  In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time $L^2$ estimates. 
Jason Metcalfe, Makoto Nakamura, and Christopher D. Sogge