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

## TL;DR

This paper extends the global existence results for small data solutions of quasilinear wave equations to systems with multiple speeds in (1+4)-dimensions, overcoming previous limitations related to scalar equations and obstacles.

## Contribution

It generalizes prior scalar wave equation results to systems with multiple speeds, including cases with obstacles, by developing new analytical techniques.

## Key findings

- Established global existence for small data in multi-speed systems
- Extended results to exterior domains with obstacles
- Overcame limitations of previous scalar-focused methods

## Abstract

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.