Junior Analysis Seminar – Alex Tullini (Münster)

We study the conformal wave equation on the exterior of Schwarzschild–Anti-de Sitter, subject to dissipative boundary conditions. We prove boundedness of the non-degenerate energy and an integrated local energy decay estimate with one derivative loss, from which we deduce energy decay at an arbitrarily fast inverse polynomial rate, at the cost of controlling up to n times T-commuted energies. The proof combines a Morawetz estimate for the T-commuted equation and an elliptic estimate for the spatial derivatives. A key feature of the result is that the additional trapping at the photon sphere does not produce further derivative loss compared to the pure Anti-de Sitter case. This contrasts with the inverse logarithmic decay rate obtained under Dirichlet boundary conditions and supports the expectation that Schwarzschild-AdS is asymptotically stable under dissipative boundary conditions.