Philippe Souplet

Université Sorbonne Paris Nord

Diffusive Hamilton–Jacobi equations and their singularities

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: single-point GBU, time rate, 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 at multiple times.

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

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.