On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

Tej-eddine Ghoul, Slim Ibrahim, Van Tien Nguyen

Research output: Contribution to journalArticle

Abstract

We consider the energy-supercritical harmonic heat flow from ℝ d into the d-sphere S[double-struck] d with d ≥ 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation ∂ t =∂ r 2 u+(d-1)/r ∂ r u-(d-1)/2r 2 sin(2u) We construct for this equation a family of C solutions which blow up in finite time via concentration of the universal profile u(r,t)Q(r/λ(t)), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t) c u (T-t) λ/γ , ℓ ∈ ℕ *, 2ℓ > γ=γ(d) ∈ (1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski (Camb. J. Math. 3:4 (2015), 439-617) for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer (Anal. PDE 7:8 (2014), 1713-1805) for the energy critical harmonic heat flow. Then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed-point theorem. Moreover, our constructed solutions are in fact .(ℓ-1)-codimension stable under perturbations of the initial data. As a consequence, the case ℓ=1 corresponds to a stable type II blowup regime.

Original languageEnglish (US)
Pages (from-to)113-187
Number of pages75
JournalAnalysis and PDE
Volume12
Issue number1
DOIs
StatePublished - Jan 1 2019

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Heat Flow
Blow-up
Harmonic
Heat transfer
Energy
Nonlinear equations
Brouwer Fixed Point Theorem
Semilinear Heat Equation
Blow-up Solution
Modulation
Energy Method
Stationary Solutions
Codimension
Nonlinear Equations
Perturbation
Symmetry
Hot Temperature

Keywords

  • Blowup
  • Differential geometry
  • Harmonic heat flow
  • Stability

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Applied Mathematics

Cite this

On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow. / Ghoul, Tej-eddine; Ibrahim, Slim; Nguyen, Van Tien.

In: Analysis and PDE, Vol. 12, No. 1, 01.01.2019, p. 113-187.

Research output: Contribution to journalArticle

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abstract = "We consider the energy-supercritical harmonic heat flow from ℝ d into the d-sphere S[double-struck] d with d ≥ 7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one-dimensional semilinear heat equation ∂ t =∂ r 2 u+(d-1)/r ∂ r u-(d-1)/2r 2 sin(2u) We construct for this equation a family of C ∞ solutions which blow up in finite time via concentration of the universal profile u(r,t)Q(r/λ(t)), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t) c u (T-t) λ/γ , ℓ ∈ ℕ *, 2ℓ > γ=γ(d) ∈ (1,2]. The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Rapha{\"e}l and Rodnianski (Camb. J. Math. 3:4 (2015), 439-617) for the energy supercritical nonlinear Schr{\"o}dinger equation and by Rapha{\"e}l and Schweyer (Anal. PDE 7:8 (2014), 1713-1805) for the energy critical harmonic heat flow. Then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed-point theorem. Moreover, our constructed solutions are in fact .(ℓ-1)-codimension stable under perturbations of the initial data. As a consequence, the case ℓ=1 corresponds to a stable type II blowup regime.",
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