Global existence for quasilinear wave equations close to Schwarzschild

Abstract

In this article we study the quasilinear wave equation ${\square }_{g(u,t,x)}u=0$ 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.

40 pages