Higher Order Linear Stability and Instability of Reissner-Nordstr\"om's Cauchy Horizon

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner-Nordstr"om (asymptotically flat, de Sitter or anti-de Sitter) spacetime, whose restrictions to the event horizon have compact support. We provide criteria, in terms of surface gravities, for the waves to remain in $C^l$, $l\ge 1$, up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in $C^1$ and $H^1$.