### Abstract

Let Ω be a smooth bounded domain in ℝn, n ≥ 5. We consider the classical semilinear heat equation at the critical Sobolev exponent (Equation Presented) Let G(x, y) be the Dirichlet Green function of - A in Q and H(x, y) its regular part. Let qj e Q, j = 1,⋯, k, be points such that the matrix (Equation Presented) is positive definite. For any k ≥ 1 such points indeed exist. We prove the existence of a positive smooth solution u(x, t) which blows up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form (Equation Presented).

Original language | English |
---|---|

Pages (from-to) | 283-344 |

Number of pages | 62 |

Journal | Journal of the European Mathematical Society |

Volume | 22 |

Issue number | 1 |

Early online date | 1 Oct 2019 |

DOIs | |

Publication status | Published - 31 Jan 2020 |

### Keywords

- Critical exponent
- Green's function
- Infinite-time blow-up

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Green's function and infinite-time bubbling in the critical nonlinear heat equation.** / Cortazar, Carmen; Musso, Monica; Del Pino, Manuel.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 22, no. 1, pp. 283-344. https://doi.org/10.4171/JEMS/922

}

TY - JOUR

T1 - Green's function and infinite-time bubbling in the critical nonlinear heat equation

AU - Cortazar, Carmen

AU - Musso, Monica

AU - Del Pino, Manuel

PY - 2020/1/31

Y1 - 2020/1/31

N2 - Let Ω be a smooth bounded domain in ℝn, n ≥ 5. We consider the classical semilinear heat equation at the critical Sobolev exponent (Equation Presented) Let G(x, y) be the Dirichlet Green function of - A in Q and H(x, y) its regular part. Let qj e Q, j = 1,⋯, k, be points such that the matrix (Equation Presented) is positive definite. For any k ≥ 1 such points indeed exist. We prove the existence of a positive smooth solution u(x, t) which blows up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form (Equation Presented).

AB - Let Ω be a smooth bounded domain in ℝn, n ≥ 5. We consider the classical semilinear heat equation at the critical Sobolev exponent (Equation Presented) Let G(x, y) be the Dirichlet Green function of - A in Q and H(x, y) its regular part. Let qj e Q, j = 1,⋯, k, be points such that the matrix (Equation Presented) is positive definite. For any k ≥ 1 such points indeed exist. We prove the existence of a positive smooth solution u(x, t) which blows up by bubbling in infinite time near those points. More precisely, for large time t, u takes the approximate form (Equation Presented).

KW - Critical exponent

KW - Green's function

KW - Infinite-time blow-up

UR - http://www.scopus.com/inward/record.url?scp=85077345524&partnerID=8YFLogxK

U2 - 10.4171/JEMS/922

DO - 10.4171/JEMS/922

M3 - Article

VL - 22

SP - 283

EP - 344

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 1

ER -