Abstract: We consider the diffusive Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ 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. We will discuss two classes of phenomena: - Gradient blow-up (GBU) on the boundary: time rate, single-point GBU, space and time-space profiles, Liouville type theorems and their applications; - Continuation after GBU as a global viscosity solution with loss and recovery of boundary conditions. In particular, we will present the complete classification of solutions in one space dimension, which describes the losses and recoveries of boundary conditions at multiple times, as well as all the possible 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.

Those interested in participating in the lecture should send an email requesting a link through the email: Este endereço de email está sendo protegido de spambots. Você precisa do JavaScript ativado para vê-lo.