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

Non-uniqueness for a critical heat equation in two dimensions with singular data / N. Ioku, B. Ruf, E. Terraneo. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - 36:7(2019 Dec), pp. 2027-2051. [10.1016/j.anihpc.2019.07.004]

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

B. Ruf
;
E. Terraneo
2019

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 u˜ the problem is well-posed, b) for initial data above this threshold function u˜, there exists no solution, c) for the singular initial datum u˜ there is non-uniqueness. The function 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 u˜.
Non-existence; Non-uniqueness; Nonlinear heat equation; Singular initial data
Settore MAT/05 - Analisi Matematica
2-ago-2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/686518
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