### Abstract

We consider the blow-up problem for a semilinear heat equation, (equation presented) where ε> 0, p> 1, N≥1, Ω is a domain in ℝ_{N}, and φ is a nonnegative smooth bounded function in Ω. It is known that, under suitable assumptions, if ε is sufficiently small, then the solution of (E) blows up only near the maximum points of the initial function φ (see, for example, [7]). In this paper, as a continuation of [7], we study the relationship between the location of the blow-up set and the level sets of the initial function φ. We also prove that the location of the blow-up set depends on the mean curvature of the graph of the initial function on its maximum points.

Original language | English |
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Pages (from-to) | 627-663 |

Number of pages | 37 |

Journal | Indiana University Mathematics Journal |

Volume | 61 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2012 Dec 1 |

### Keywords

- Blow-up set
- Mean curvature
- Small diffusion

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Fujishima, Y., & Ishige, K. (2012). Blow-up set for a semilinear heat equation and pointedness of the initial data.

*Indiana University Mathematics Journal*,*61*(2), 627-663. https://doi.org/10.1512/iumj.2012.61.4596