Non-linear instability of the Kerr Cauchy horizon near $i_+$

We consider solutions of the Einstein vacuum equations which arise from smooth initial data on a hypersurface slightly inside a dynamical black hole settling down to a subextremal Kerr black hole, and satisfying a precise non-linear Price's law-type estimate (which we expect to hold generically). We prove that the corresponding maximal globally hyperbolic development admits a non-trivial piece of future null boundary - the Cauchy horizon - emanating from timelike infinity $i_+$, which exhibits a kind of curvature blow-up, and across which the spacetime metric is Lipschitz-inextendible. Our results thus imply a Lipschitz version of Strong Cosmic Censorship for Kerr spacetimes near timelike infinity under this Price's law-type assumption. The analysis relies on the proof of the $C^0$ stability of the Kerr Cauchy horizon by Dafermos and Luk, on the non-integrable formalism of Giorgi-Klainerman-Szeftel and principal temporal gauge of Klainerman and Szeftel used in the proof of the exterior stability of slowly rotating Kerr black holes, on the linearized analysis for the Teukolsky equation inside subextremal Kerr black holes by the author (following an earlier paper on the scalar wave equation by Ma and Zhang), and on Sbierski's criterion for Lipschitz inextendibility. More precisely, we proceed by decomposing the black hole interior into different regions equipped with appropriate gauges, allowing for a proof of stability estimates and a thorough analysis of the non-linear analog of the Teukolsky equation, from which we infer our instability results.