Geometry driven type II higher dimensional blow-up for the critical heat equation

Research output: Contribution to journalArticle

Abstract

We consider the problem vt=Δv+|v|p−1vin Ω×(0,T),v=0on ∂Ω×(0,T),v>0in Ω×(0,T). In a domain Ω⊂Rd, d≥7 enjoying special symmetries, we find the first example of a solution with type II blow-up for a power p less than the Joseph-Lundgren exponent [Formula presented] No type II radial blow-up is present for p<pJL(d). We take [Formula presented], the Sobolev critical exponent in one dimension less. The solution blows up on circle contained in a negatively curved part of the boundary in the form of a sharply scaled Aubin-Talenti bubble, approaching its energy density a Dirac measure for the curve. This is a completely new phenomenon for a diffusion setting.

Original languageEnglish
Article number108788
JournalJournal of Functional Analysis
Volume280
Issue number1
Early online date28 Sep 2020
DOIs
Publication statusE-pub ahead of print - 28 Sep 2020

Keywords

  • Parabolic equation
  • Type II blow up phenomena

ASJC Scopus subject areas

  • Analysis

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