  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. We answer this
question in the affirmative.
