  H\"ormander 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
$u\partial_\alpha u = \frac{1}{2}\partial_\alpha u^2$ 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.
