# The Klein-Gordon equation on the hyperboloidal anti-de Sitter Schwarzschild black hole

## TL;DR

This paper studies the behavior of solutions to the Klein-Gordon equation in a specific black hole spacetime, proving energy decay under certain conditions and instability in others, using advanced mathematical techniques.

## Contribution

It establishes energy decay and instability results for the Klein-Gordon equation on hyperboloidal anti-de Sitter Schwarzschild black holes, employing vector field methods and energy renormalization.

## Key findings

- Energy decay for solutions with specific boundary conditions.
- Existence of negative energy solutions indicating linear instability.
- Results depend on the mass squared parameter region.

## Abstract

In this paper we establish energy decay for solutions to the Klein-Gordon equation on the positive mass hyperboloidal anti-de Sitter Schwarzschild black hole, subject to Dirichlet, Neumann and Robin boundary conditions at infinity, for a range of the (negative) mass squared parameter. To do so we use vector field methods with a renormalised energy to avoid divergences that would otherwise appear in the energy integrals. For another region of the parameter space, we use the existence of negative energy solutions to demonstrate linear instability.