Nested Discontinuous Asymptotic Profiles for the Viscous Burgers’ Equation with Infinite Mass

We study the viscous Burgers’ equation with a family of initial data having infinite mass. After rescaling, the solution converges toward a bounded discontinuous profile in the long-time limit. We investigate the solution near this discontinuity point, and show that, by changing once again the scale depending on time near it, we find a new profile, that is also discontinuous. This process can be repeated any arbitrary number of times. In other words, if we think of the discontinuity in the limit profile as a boundary layer, we can change the scale in time near it to get a new profile, that will also have a boundary layer, and this repeats at infinity.