### 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) |
---|---|

Pages (from-to) | 31-54 |

Number of pages | 24 |

Journal | Advanced Nonlinear Studies |

Volume | 17 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2 2017 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematics(all)

### Cite this

*Advanced Nonlinear Studies*,

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

**Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation.** / Ghoul, Tej-eddine; Nguyen, Van Tien; Zaag, Hatem.

Research output: Contribution to journal › Article

*Advanced Nonlinear Studies*, vol. 17, no. 1, pp. 31-54. https://doi.org/10.1515/ans-2016-6005

}

TY - JOUR

T1 - Refined regularity of the blow-up set linked to refined asymptotic behavior for the semilinear heat equation

AU - Ghoul, Tej-eddine

AU - Nguyen, Van Tien

AU - Zaag, Hatem

PY - 2017/1/2

Y1 - 2017/1/2

N2 - We consider u (x,t) a solution of ∂tu = Δu + |u|p-1u which blows up at some time T > 0, where u : ℝN × [0,T) → ℝ, p > 1 and (N-2)p < N + 2. Define S ⊂ ℝN 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.

AB - We consider u (x,t) a solution of ∂tu = Δu + |u|p-1u which blows up at some time T > 0, where u : ℝN × [0,T) → ℝ, p > 1 and (N-2)p < N + 2. Define S ⊂ ℝN 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.

KW - Blow-Up Profile

KW - Blow-Up Set

KW - Blow-Up Solution

KW - Regularity

KW - Semilinear Heat Equation

UR - http://www.scopus.com/inward/record.url?scp=85011688987&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85011688987&partnerID=8YFLogxK

U2 - 10.1515/ans-2016-6005

DO - 10.1515/ans-2016-6005

M3 - Article

AN - SCOPUS:85011688987

VL - 17

SP - 31

EP - 54

JO - Advanced Nonlinear Studies

JF - Advanced Nonlinear Studies

SN - 1536-1365

IS - 1

ER -