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

Abstract

Hörmander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $\square u=Q(u,u^{\prime },u^{\prime \prime })$ 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.

21 pages