Global existence for systems of quasilinear wave equations in (1+4)-dimensions

Hormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $Box u = Q(u, u, u)$ where $Q$ vanishes to second order and $(partial_u^2 Q)(0,0,0)=0$. Without the latter condition, only almost global existence may be guaranteed. The first author and Sogge considered the analog exterior to a star-shaped obstacle. Both results relied on writing the lowest order terms $upartial_alpha u = frac{1}{2}partial_alpha u^2$ and as such do not immediately generalize to systems. The current study remedies such and extends both results to the case of multiple speed systems.