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H\"ormander proved global existence of solutions for sufficiently smallinitial data for scalar wave equations in $(1+4)-$dimensions of the form $\Boxu = Q(u, u', u'')$ where $Q$ vanishes to second order and $(\partial_u^2Q)(0,0,0)=0$. Without the latter condition, only almost global existence may beguaranteed. The first author and Sogge considered the analog exterior to astar-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 notimmediately generalize to systems. The current study remedies such and extendsboth results to the case of multiple speed systems.

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