Rough initial data and the strength of the blue-shift instability on cosmological black holes with $\Lambda > 0$

We consider the wave equation on Reissner-Nordstr\"om-de Sitter and more generally Kerr-Newman-de Sitter black hole spacetimes with $\Lambda>0$. The strength of the blue-shift instability associated to the Cauchy horizon of these spacetimes has been the subject of much discussion, since-in contrast to the $\Lambda=0$ case-the competition with the decay associated to the region between the event and cosmological horizons is delicate. Of particular interest is the question as to whether generic, admissible initial data posed on a Cauchy surface lead to solutions whose local energy blows up at the Cauchy horizon, for this statement holds in the $\Lambda = 0$ case and would correspond precisely to the blow up required by Christodoulou's formulation of strong cosmic censorship. Some recent heuristic work suggests that the answer is in general negative for solutions arising from sufficiently smooth data, such that for all such data, the arising solutions have finite local energy at the Cauchy horizon. In this short note, we shall show in contrast that, by slightly relaxing the smoothness assumption on initial data, we are able to prove the analogue of the Christodoulou statement in the affirmative, i.e. we show that for generic data in our allowed class, the local energy blow-up statement indeed holds at the Cauchy horizon, for all subextremal black hole parameter ranges. We present two distinct proofs. Our slightly enlarged class of initial data is still sufficiently regular to ensure both stability and decay properties in the region between the event and cosmological horizons as well as the boundedness and continuous extendibility beyond the Cauchy horizon. This suggests thus that it is finally this class which may provide the correct setting to formulate the genericity condition in strong cosmic censorship.