Resumen: We consider the diffusive Hamilton-Jacobi equation $u_t-Delta u=|
abla u|^p$ with homogeneous Dirichlet boundary conditions, which plays an important role in stochastic optimal control theory and in certain models of surface growth (KPZ). Despite its simplicity, in the superquadratic case p>2 it displays a variety of interesting and surprising behaviors and we will discuss two classes of phenomena:

- Gradient blow-up (GBU): localization of singularities on the boundary, single-point GBU, time rate of GBU, space and time-space profiles, Liouville type theorems and their applications;
- Continuation after GBU as a global viscosity solution: GBU with or without loss of boundary conditions (LBC), recovery of boundary conditions with or without regularization, GBU and LBC at multiple times.

In particular, in one space dimension, we will present the recently obtained, complete classification of all GBU and recovery rates.

This talk is based on a series of joint works in collaboration with A. Attouchi, R. Filippucci, Y. Li, N. Mizoguchi, A. Porretta, P. Pucci, Q. Zhang.