JournalsaihpcVol. 36, No. 7pp. 2027–2051

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

  • Norisuke Ioku

    Mathematical Institute, Tohoku University, Aramaki 6-3, Sendai 980-8578, Japan
  • Bernhard Ruf

    Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via C. Saldini 50, Milano 20133, Italy
  • Elide Terraneo

    Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, via C. Saldini 50, Milano 20133, Italy
Non-uniqueness for a critical heat equation in two dimensions with singular data cover
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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\limits^{˜} the problem is well-posed, b) for initial data above this threshold function u\limits^{˜}, there exists no solution, c) for the singular initial datum u\limits^{˜} there is non-uniqueness. The function u\limits^{˜} 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\limits^{˜}.

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. 2027–2051

DOI 10.1016/J.ANIHPC.2019.07.004