### Abstract

We consider u (x,t) a solution of ∂_{t}u = Δu + |u|^{p-1}u which blows up at some time T > 0, where u : ℝ^{N} × [0,T) → ℝ, p > 1 and (N-2)p < N + 2. Define S ⊂ ℝ^{N} to be the blow-up set of u, that is, the set of all blow-up points. Under suitable non-degeneracy conditions, we show that if S contains an (N-ℓ-dimensional continuum for some ℓϵ{1,...,N-1}, then S is in fact a ^{2} manifold. The crucial step is to make a refined study of the asymptotic behavior of u near blow-up. In order to make such a refined study, we have to abandon the explicit profile function as a first-order approximation and take a non-explicit function as a first-order description of the singular behavior. This way we escape logarithmic scales of the variable (T-t) and reach significant small terms in the polynomial order (T-t)^{μ} for some μ > 0. Knowing the refined asymptotic behavior yields geometric constraints of the blow-up set, leading to more regularity on S.

Original language | English (US) |
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Pages (from-to) | 31-54 |

Number of pages | 24 |

Journal | Advanced Nonlinear Studies |

Volume | 17 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2017 |

### Keywords

- Blow-Up Profile
- Blow-Up Set
- Blow-Up Solution
- Regularity
- Semilinear Heat Equation

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematics(all)

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## Cite this

*Advanced Nonlinear Studies*,

*17*(1), 31-54. https://doi.org/10.1515/ans-2016-6005