Motivated by the Strong Cosmic Censorship Conjecture for asymptotically Anti-de Sitter (AdS) spacetimes, we initiate the study of massive scalar waves satisfying □gψ-μψ=0 on the interior of AdS black holes. We prescribe initial data on a spacelike hypersurface of a Reissner–Nordström–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.