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

Tej Eddine Ghoul, Van Tien Nguyen, Hatem Zaag

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)31-54
Number of pages24
JournalAdvanced Nonlinear Studies
Volume17
Issue number1
DOIs
StatePublished - 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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