Hörmander proved global existence of solutions for sufficiently small initial data for scalar wave equations in (1+4)−dimensions of the form ✷u = Q(u, u ′ , u ′′ ) where Q vanishes to second order and (∂ 2 u 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∂αu = 1 2 ∂αu 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.

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