  We consider a system of quasilinear wave equations on the product space
$\mathbb{R}^{1+3}\times \mathbb{S}^1$, which we want to see as a toy model for
Einstein equations with additional compact dimensions. We show global existence
for small and regular initial data with polynomial decay at infinity. The
method combines energy estimates on hyperboloids inside the light cone and
weighted energy estimates outside the light cone.
