Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media

We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background with a time dependent metric g coinciding with the Minkowski metric outside the cylinder . We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate that these estimates work 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.