  Two aspects of the widely accepted heuristic picture of the final state of
gravitational collapse are the so-called Price law tails, describing the
asymptotics of the exterior region of the black hole that forms, and
Israel-Poisson's mass inflation scenario, describing the internal structure of
the black hole. (The latter scenario, if valid, would indicate that the maximal
development of initial data is extendible as a C^0 metric, putting into
question the validity of Penrose's strong cosmic censorship conjecture.) In
this talk, I shall discuss a series of rigorous results proving both Price's
law and the mass inflation scenario in an appropriate spherically symmetric
setting. The proof of Price's law is joint work with I. Rodnianski.
