### 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 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

**Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent.** / Seki, Yukihiro.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 275, no. 12, pp. 3380-3456. https://doi.org/10.1016/j.jfa.2018.05.008

}

TY - JOUR

T1 - Type II blow-up mechanisms in a semilinear heat equation with critical Joseph–Lundgren exponent

AU - Seki, Yukihiro

PY - 2018/12/15

Y1 - 2018/12/15

N2 - We are concerned with blow-up mechanisms in a semilinear heat equation ut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>pJL, where pJL stands for the Joseph–Lundgren exponent: pJL={+∞,N≤10,1+[Formula presented],N≥11. On the other hand, it is also known that, if pJL, 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=pJL,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.

AB - We are concerned with blow-up mechanisms in a semilinear heat equation ut=Δu+|u|p−1u,x∈RN,t>0, where p>1 is a constant. It is well known that type II blow-up does occur if N≥11 and p>pJL, where pJL stands for the Joseph–Lundgren exponent: pJL={+∞,N≤10,1+[Formula presented],N≥11. On the other hand, it is also known that, if pJL, 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=pJL,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.

UR - http://www.scopus.com/inward/record.url?scp=85048471748&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85048471748&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2018.05.008

DO - 10.1016/j.jfa.2018.05.008

M3 - Article

AN - SCOPUS:85048471748

VL - 275

SP - 3380

EP - 3456

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 12

ER -