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

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

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

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

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     author = {Ioku, Norisuke and Ruf, Bernhard and Terraneo, Elide},
     title = {Non-uniqueness for a critical heat equation in two dimensions with singular data},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {2027--2051},
     year = {2019},
     publisher = {Elsevier},
     volume = {36},
     number = {7},
     doi = {10.1016/j.anihpc.2019.07.004},
     mrnumber = {4020532},
     zbl = {1427.35096},
     language = {en},
     url = {https://www.numdam.org/articles/10.1016/j.anihpc.2019.07.004/}
}

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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

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