### Abstract

We are concerned with blow-up mechanisms in a semilinear heat equation u_{t}=Δu+|u|^{p−1}u,x∈R^{N},t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>p_{JL}, where p_{JL} stands for the Joseph–Lundgren exponent: p_{JL}={+∞,N≤10,1+[Formula presented],N≥11. On the other hand, it is also known that, if p<p_{JL}, type II blow-up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned. It has long remained open whether or not type II blow-up occurs for the borderline case p=p_{JL},N≥11. In the present article we prove constructively the existence of type II blow-up solutions for the Joseph–Lundgren critical case, thus giving an answer to this question.

Original language | English |
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Pages (from-to) | 3380-3456 |

Number of pages | 77 |

Journal | Journal of Functional Analysis |

Volume | 275 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 15 2018 |

### All Science Journal Classification (ASJC) codes

- Analysis