Type II Blow-up in the 5-dimensional Energy Critical Heat Equation

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Abstract

We consider the Cauchy problem for the energy critical heat equation {ut=Δu+|u|4n−2uinRn×(0,T)u(⋅,0)=u0inRn in dimension n = 5. More precisely we find that for given points q 1 ,q 2 ,..,q k and any sufficiently small T > 0 there is an initial condition u 0 such that the solution u(x,t) of (0.1) blows-up at exactly those k points with rates type II, namely with absolute size ~(T-t) for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

Original languageEnglish
Pages (from-to)1027-1042
Number of pages16
JournalActa Mathematica Sinica, English Series
Volume35
Issue number6
Early online date20 May 2019
DOIs
Publication statusPublished - 1 Jun 2019

Keywords

  • 35B40
  • 35K58
  • bubbling phenomena
  • critical parabolic equations
  • Singularity formation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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