### 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 language | English (US) |
---|---|

Pages (from-to) | 113-187 |

Number of pages | 75 |

Journal | Analysis and PDE |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Blowup
- Differential geometry
- Harmonic heat flow
- Stability

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Applied Mathematics

### Cite this

*Analysis and PDE*,

*12*(1), 113-187. https://doi.org/10.2140/apde.2019.12.113

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

Research output: Contribution to journal › Article

*Analysis and PDE*, vol. 12, no. 1, pp. 113-187. https://doi.org/10.2140/apde.2019.12.113

}

TY - JOUR

T1 - On the stability of type II blowup for the 1-corotational energy-supercritical harmonic heat flow

AU - Ghoul, Tej-eddine

AU - Ibrahim, Slim

AU - Nguyen, Van Tien

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

KW - Blowup

KW - Differential geometry

KW - Harmonic heat flow

KW - Stability

UR - http://www.scopus.com/inward/record.url?scp=85061727157&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061727157&partnerID=8YFLogxK

U2 - 10.2140/apde.2019.12.113

DO - 10.2140/apde.2019.12.113

M3 - Article

AN - SCOPUS:85061727157

VL - 12

SP - 113

EP - 187

JO - Analysis and PDE

JF - Analysis and PDE

SN - 2157-5045

IS - 1

ER -