Blow-up problems for a semilinear heat equation with large diffusion

Kazuhiro Ishige, Hiroki Yagisita

Research output: Contribution to journalArticlepeer-review

27 Citations (Scopus)

Abstract

We consider the blow-up problem of a semilinear heat equation, ut = DΔu + up in Ω × (0, TD), ∂u/∂ν(x, t) = 0 on ∂Ω × (0,TD), u(x, 0) = φ(x) ≥ 0 in Ω, where Ω is a bounded smooth domain in RN, TD > 0, D > 0, and p > 1. We study the blow-up time, the location of the blow-up set, and the blow-up profile of the blow-up solution for sufficiently large D. In particular, we prove that, for almost all initial data φ, if D is sufficiently large, then the solution blows-up only near the maximum points of the orthogonal projection of the initial data φ from L2 (Ω) onto the second Neumann eigenspace.

Original languageEnglish
Pages (from-to)114-128
Number of pages15
JournalJournal of Differential Equations
Volume212
Issue number1
DOIs
Publication statusPublished - 2005 May 1

Keywords

  • Blow-up profile
  • Blow-up set
  • Blow-up time
  • Nonlinear diffusion equation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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