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 the problem is well-posed, b) for initial data above this threshold function , there exists no solution, c) for the singular initial datum there is non-uniqueness. The function 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 .
Cite this article
Norisuke Ioku, Bernhard Ruf, Elide Terraneo, Non-uniqueness for a critical heat equation in two dimensions with singular data. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 7, pp. 2027–2051DOI 10.1016/J.ANIHPC.2019.07.004