  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.
