### 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) ^{-α} for α > 34. The blow-up profile around each point is of bubbling type, in the form of sharply scaled Aubin–Talenti bubbles.

Original language | English |
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Pages (from-to) | 1027-1042 |

Number of pages | 16 |

Journal | Acta Mathematica Sinica, English Series |

Volume | 35 |

Issue number | 6 |

Early online date | 20 May 2019 |

DOIs | |

Publication status | Published - 1 Jun 2019 |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Acta Mathematica Sinica, English Series*,

*35*(6), 1027-1042. https://doi.org/10.1007/s10114-019-8341-5