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

Norisuke Ioku, Bernhard Ruf, Elide Terraneo

Research output: Contribution to journalArticle

1 Citation (Scopus)

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˜.

Original languageEnglish
Pages (from-to)2027-2051
Number of pages25
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume36
Issue number7
DOIs
Publication statusPublished - 2019 Nov 1

Keywords

  • Non-existence
  • Non-uniqueness
  • Nonlinear heat equation
  • Singular initial data

ASJC Scopus subject areas

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

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