Thursday, July 22, 2021 — 4:00 pm — aula M1.7, edificio Matematica, and live streaming

Non-uniqueness for a critical heat equation in two dimensions


In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this talk we consider a specific square-exponential nonlinearity in two dimensions . We prove that for initial data strictly below a certain singular threshold function $\widetilde u$ the problem is well-posed, for initial data above this threshold function $\widetilde u$, there exists no solution and for the singular initial datum $\widetilde u$ there is non-uniqueness: $\widetilde u$ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum $\widetilde u$. This is a joint work with Norisuke Ioku and Bernhard Ruf.