Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation

Abstract

We study the qualitative properties of the unique global viscosity solution of the superquadratic diffusive Hamilton-Jacobi equation with (generalized) homogeneous Dirichlet conditions. We are interested in the phenomena of gradient blow-up (GBU), loss of boundary conditions (LBC), recovery of boundary conditions and eventual regularization, and in their mutual connections.In any space dimension, we establish the sharp minimal rate of GBU. Only partial results were previously known except in one space dimension. We also obtain the corresponding minimal regularization rate.In one space dimension, under suitable conditions on the initial data, we give a quite detailed description of the behavior of solutions for all t>0. In particular, we show that nonminimal GBU solutions immediately lose the boundary conditions after the blow-up time and are immediately regularized after recovering the boundary data. Moreover, both GBU and regularization occur with the minimal rates, while loss and recovery of boundary data occur with linear rate. We describe further the intermediate singular life of those solutions in the time interval between GBU and regularization.We also study minimal GBU solutions, for which GBU occurs without LBC: those solutions are immediately regularized, but their blow-up and regularization rates are more singular.Most of our one-dimensional results crucially depend on zero-number arguments, which do not seem to have been used so far in the context of viscosity solutions of Hamilton-Jacobi equations.