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 (∂u 2Q)(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∂αu2 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.
ASJC Scopus subject areas
- Applied Mathematics