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

Ioku, Norisuke; Ruf, Bernhard; Terraneo, Elide

doi:10.1016/j.anihpc.2019.07.004

Geodesic

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

Ioku, Norisuke ¹ ; Ruf, Bernhard ² ; Terraneo, Elide ²

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

Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 2027-2051.

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Résumé

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 $\tilde{u}$ .

MR Zbl

DOI : 10.1016/j.anihpc.2019.07.004

Keywords: Nonlinear heat equation, Singular initial data, Non-uniqueness, Non-existence

Affiliations des auteurs :

Ioku, Norisuke ¹ ; Ruf, Bernhard ² ; Terraneo, Elide ²

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     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
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Ioku, Norisuke; Ruf, Bernhard; Terraneo, Elide. Non-uniqueness for a critical heat equation in two dimensions with singular data. Annales de l'I.H.P. Analyse non linéaire, Tome 36 (2019) no. 7, pp. 2027-2051. doi : 10.1016/j.anihpc.2019.07.004. http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2019.07.004/

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