Late-time tails and mass inflation for the spherically symmetric Einstein-Maxwell-scalar field system

We establish a decay result in the black hole exterior region of spherically symmetric solutions to the Einstein-Maxwell-scalar field system arising from compactly supported admissible data. Our result allows for large initial data, and it is the first decay statement for higher order derivatives of the scalar field. Solutions to this model generically develop a singularity in the black hole interior. Indeed, Luk--Oh (arxiv:1702.05715, arxiv:1702.05716) identify a generic class of initial data that produces $C^2$-future-inextendible solutions. However, they leave open the question of mass inflation: does the Hawking mass become identically infinite at the Cauchy horizon? By work of Luk--Oh--Shlapentokh-Rothman (arxiv:2201.12294), our decay result implies mass inflation for sufficiently regular solutions in the generic class considered by Luk--Oh (arxiv:1702.05715, arxiv:1702.05716). Together with the methods and results of Luk--Oh (arXiv:2404.02220), our estimates imply a late-time tails result for the scalar field. This result provides another proof of generic mass inflation, through a result of Dafermos (arXiv:arch-ive/0307013). Another application of our late-time tails result, due to Van de Moortel, is the global construction of two-ended black holes that contain null and spacelike singularities.