Global existence for quasilinear wave equations close to Schwarzschild

TL;DR

This paper proves the global existence of solutions to a quasilinear wave equation near Schwarzschild spacetime, using local energy estimates for metrics with slow decay, under small initial data conditions.

Contribution

It introduces a new local energy estimate for linear waves on metrics close to Schwarzschild, enabling the proof of global solutions for the quasilinear wave equation.

Findings

Global existence for small initial data

Local energy decay estimates established

Metrics with slow decay are handled effectively

Abstract

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.