Bounded Energy Waves on the Black Hole Interior of Reissner-Nordstrom-de Sitter

Motivated by the strong cosmic censorship conjecture, in the presence of a cosmological constant, we consider solutions of the scalar wave equation on fixed subextremal Reissner-Nordstrom-de Sitter backgrounds , without imposing symmetry assumptions on . We provide a sufficient condition, in terms of surface gravities and a parameter for an exponential decaying Price Law, for a local energy of the waves to remain bounded up to the Cauchy horizon. The energy we consider controls, in particular, regular transverse derivatives at the Cauchy horizon; this allows us to extend the solutions with bounded energy, to the Cauchy horizon, as functions in . Our results correspond to another manifestation of the potential breakdown of strong cosmic censorship in the positive cosmological constant setting.