Boundedness and decay for the conformal wave equation in Schwarzschild-AdS under dissipative boundary conditions

TL;DR

This paper proves polynomial decay of solutions to the conformal wave equation in Schwarzschild-AdS spacetime with dissipative boundary conditions, showing improved decay rates over Dirichlet conditions and robustness against trapping effects.

Contribution

It establishes boundedness and polynomial decay rates for the conformal wave equation under dissipative boundary conditions in Schwarzschild-AdS, extending previous results.

Findings

Polynomial decay rate of (1+v)^{-n} for the energy of solutions.

Decay unaffected by trapping at the photon sphere.

Contrast with inverse logarithmic decay under Dirichlet boundary conditions.

Abstract

We study the conformal wave equation $\square_g \psi + \frac{2}{l^2} \psi = 0$ on 4-dimensional Schwarzschild--Anti de Sitter spacetimes under dissipative boundary conditions. We prove boundedness and decay of the non-degenerate energy of $\psi$ at an arbitrary polynomial rate of $(1+v)^{-n}$ provided that we control the (up to) $n$-times $T$-commuted energy. This contrasts with the inverse logarithmic decay obtained under Dirichlet boundary conditions and is in line with the result obtained in the pure Anti-de Sitter case under dissipative boundary conditions. In particular, the decay is not affected by the additional trapping at the photon sphere.