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

We study the conformal wave equation ${\mathrm{\square }}_g\psi +\frac{2}{l^2}\psi =0$ 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.