  We consider the problem of small data global existence for a class of
semilinear wave equations with null condition on a Lorentzian background
$(\mathbb{R}^{3+1}, g)$ with a \textbf{time dependent metric $g$} coinciding
with Minkowski metric outside the cylinder $\{(t, x)| |x|\leq R\}$. We show
that the small data global existence result can be reduced to two integrated
local energy estimates and demonstrate these estimates in the particular case
when $g$ is merely $C^1$ close to the Minkowski metric. One of the novel
aspects of this work is that it applies to equations on backgrounds which do
not settle to any particular stationary metric.
