Quasilinear wave equations on Schwarzschild-de Sitter

TL;DR

This paper presents a new, elementary method to prove global existence and exponential decay of solutions to quasilinear wave equations on Schwarzschild-de Sitter backgrounds, using local energy estimates and iteration.

Contribution

It introduces a simple, local energy estimate-based approach for establishing global existence and decay in black hole spacetimes, improving understanding of wave behavior.

Findings

Solutions exist globally for small initial data.

Solutions decay exponentially over time.

The method is based on local energy estimates and iteration.

Abstract

We give an elementary new argument for global existence and exponential decay of solutions of quasilinear wave equations on Schwarzschild-de Sitter black hole backgrounds, for appropriately small initial data. The core of the argument is entirely local, based on time translation invariant energy estimates in spacetime slabs of fixed time length. Global existence then follows simply by iterating this local result in consecutive spacetime slabs. We infer that an appropriate future energy flux decays exponentially with respect to the energy flux of the initial data.