Higher order linear stability and instability of Reissner–Nordström’s Cauchy horizon

We consider smooth solutions of the wave equation, on a fixed black hole region of a subextremal Reissner–Nordstrom (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 $$C^l$$ , $$l\geqslant 1$$ , up to and including the Cauchy horizon. We also provide sufficient conditions for the blow up of solutions in $$C^1$$ and $$H^1$$ .