  In this article we study the quasilinear wave equation $\Box_{g(u, t, x)} u =
0$ where the metric $g(u, t, x)$ is close to the Schwarzschild metric. Under
suitable assumptions of the metric coefficients, and assuming that the initial
data for $u$ is small enough, we prove global existence of the solution. The
main technical result of the paper is a local energy estimate for the linear
wave equation on metrics with slow decay to the Schwarzschild metric.
