2019/03/19 by Ioku, Norisuke, Ruf, Bernhard, Terraneo, Elide · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.08013
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.