Speaker
Description
Self-similar behavior is a well-studied phenomenon in extended systems. However, the consideration is often restricted to scalar problems having exact self-similar solutions (e.g. the porous medium equation) or to problems with a trivial behavior at infinity. In this talk, we study a reaction-diffusion system and other related dissipative systems on the whole real line with prescribed non-trivial limits at infinity to investigate their solutions' long-time behavior. The system under consideration has the special property that it possesses a continuum of constant solutions. By assuming that the solutions are in equilibria at infinity, we study the convergence towards so-called self-similar profiles. With this, we answer how the solutions mix the two stable asymptotic boundary values when time increases. The key idea is to rescale space and time into parabolic scaling variables and to derive energy-dissipation estimates for the relative Boltzmann entropy. In the original variables, these profiles correspond to asymptotically self-similar behavior describing the phenomenon of diffusive mixing of the different states at infinity.
