Non-uniqueness for a critical heat equation in two dimensions with singular data

Nonlinear heat equations in two dimensions with singular initial data are studied. 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 article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\widetilde u$ the problem is well-posed, b) for initial data above this threshold function $\widetilde u$, there exists no solution, c) for the singular initial datum $\widetilde u$ there is non-uniqueness. The function $\widetilde u$ 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 $\widetilde u$.