Some asymptotic profiles for the viscous Burgers equation with infinite mass

This is a summary of the papers [6], [7] done in collaboration with Tej-Eddine Ghoul and Nader Masmoudi and [2] done in collaboration with Nicola de Nitti. We are interested in the long time behavior of solutions of the viscous Burgers equation for a class of initial data with infinite mass. We show that, up to a rescaling, they converge to a limit profile that has a discontinuity. We investigate this profile and explain how to understand this discontinuity point.