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

Abstract

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 $\tilde{u}$ the problem is well-posed, b) for initial data above this threshold function $\tilde{u}$, there exists no solution, c) for the singular initial datum $\tilde{u}$ there is non-uniqueness. The function $\tilde{u}$ 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 $\tilde{u}$.