Type II finite time blow-up for the energy critical heat equation in R4

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We consider the Cauchy problem for the energy critical heat equation ut = ∆u + u3 in R4 × (0, T), (1) (u(·, 0) = u0 in R4. We find that for given points q1, q2, . . ., qk and any sufficiently small T > 0 there is an initial condition u0 such that the solution u(x, t) of (1) blows up at exactly those k points with a type II rate, namely larger than (T − t) 2 . In 1 fact ku(·, t)k∞ ∼ (T − t) 1 log2(T − t). 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)3327-3335
Number of pages9
JournalDiscrete and Continuous Dynamical Systems- Series A
Issue number6
Publication statusPublished - 30 Jun 2020


  • Construction of blow-up solution
  • Energy critical heat equation
  • Inner–outer gluing method
  • Nondegeneracy
  • Type II finite time blow-up

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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