Uniform boundedness and continuity at the Cauchy horizon for linear waves on Reissner-Nordstr\"om-AdS black holes

Understand

Motivated by the Strong Cosmic Censorship Conjecture for asymptotically AdS spacetimes, we initiate the study of massive scalar waves satisfying $\Box_g \psi - \mu \psi =0$ on the interior of Anti-de Sitter (AdS) black holes.

• We prescribe initial data on a spacelike hypersurface of a Reissner--Nordstr\"om--AdS black hole and impose Dirichlet (reflecting) boundary conditions at infinity.
• It was known previously that such waves only decay at a sharp logarithmic rate (in contrast to a polynomial rate as in the asymptotically flat regime) in the black hole exterior.
• In view of this slow decay, the question of uniform boundedness in the black hole interior and continuity at the Cauchy horizon has remained up to now open.