# The Fuchsian approach to global existence for hyperbolic equations

## TL;DR

This paper develops a Fuchsian approach to prove global existence and decay for symmetric hyperbolic equations with singularities, applying it to wave and Euler equations in various spacetime models.

## Contribution

It introduces a new Fuchsian method for analyzing hyperbolic equations with singularities and applies it to several important physical models.

## Key findings

- Established global existence for symmetric hyperbolic equations with Fuchsian singularities.
- Derived decay estimates for solutions approaching the singularity.
- Extended the theory to wave equations on Minkowski and Schwarzschild spacetimes, and Euler equations on Kasner spacetimes.

## Abstract

We analyze the Cauchy problem for symmetric hyperbolic equations with a time singularity of Fuchsian type and establish a global existence theory along with decay estimates for evolutions towards the singular time under a small initial data assumption. We then apply this theory to semilinear wave equations near spatial infinity on Minkowski and Schwarzschild spacetimes, and to the relativistic Euler equations with Gowdy symmetry on Kasner spacetimes.