The Klein-Gordon equation on the toric adS-Schwarzschild black hole

TL;DR

This paper studies the behavior of solutions to the Klein-Gordon equation on a toric anti-de Sitter Schwarzschild black hole, establishing decay estimates and demonstrating the necessity of derivative loss in energy estimates.

Contribution

It introduces a non-degenerate energy controlling the $H^1$ norm, proves decay and integrated decay, and shows derivative loss is unavoidable via Gaussian beam constructions.

Findings

Established decay and integrated decay of energy.

Defined a non-degenerate energy controlling the $H^1$ norm.

Proved the necessity of derivative loss in energy estimates.

Abstract

We consider the Klein-Gordon equation on the exterior of the toric anti de-Sitter Schwarzschild black hole with Dirichlet, Neumann and Robin boundary conditions at $\mathcal{I}$. We define a non-degenerate energy for the equation which controls the renormalised $H^1$ norm of the field. We then establish both decay and integrated decay of this energy through vector field methods. Finally we demonstrate the necessity of `losing a derivative' in the integrated energy estimate through the construction of a Gaussian beam staying in the exterior of the event horizon for arbitrary long co-ordinate time.