Quasilinear wave equations on Schwarzschild–de Sitter

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.

Keywords:

• black holes
• wave equation
• Schwarzschild de Sitter
• quasilinear wave
• stability

Acknowledgments

I would like to thank my supervisor Mihalis Dafermos, for his continuous support, for suggesting that the results of [31] may also treat the nonlinear stability problems of the present paper and for carefully reading previous versions of this paper. The author would also like to thank Christoph Kehle for valuable discussions and useful comments.