Non-uniqueness for a critical heat equation in two dimensions with singular data |
| |
Authors: | Norisuke Ioku Bernhard Ruf Elide Terraneo |
| |
Affiliation: | 1. Mathematical Institute, Tohoku University, Aramaki 6-3, Sendai 980-8578, Japan;2. Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via C. Saldini 50, Milano 20133, Italy |
| |
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 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 . |
| |
Keywords: | Corresponding author. Nonlinear heat equation Singular initial data Non-uniqueness Non-existence |
本文献已被 ScienceDirect 等数据库收录! |
|