Infinite-time blow-up for the 3-dimensional energy-critical heat equation

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2 Citations (Scopus)

Abstract

We construct globally defined in time, unbounded positive solutions to the energy-critical heat equation in dimension 3 ut = Δ u + u5 in R3 × (0, ∞), u(x, 0) = u0(x) in R3. For each γ > 1 we find initial data (not necessarily radially symmetric) with lim |x| → ∞ |x| γ u0 (x) > 0 such that as t → ∞ ||u (·, t) || ∞ ~ tγ-1/2 if 1 < γ < 2, ||u (·, t) || ∞ ~ √t if γ > 2, ||u (·, t) || ∞ ~ √t(Int) -1 if γ = 2. Furthermore we show that this infinite-time blow-up is codimensional-1 stable. The existence of such solutions was conjectured by Fila and King.

Original languageEnglish
Pages (from-to)215-274
Number of pages60
JournalAnalysis and PDE
Volume13
Issue number1
DOIs
Publication statusPublished - 6 Jan 2020

Keywords

  • Blow-up
  • Critical exponents
  • Nonlinear parabolic equations

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

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