### 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) |
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Pages (from-to) | 113-187 |

Number of pages | 75 |

Journal | Analysis and PDE |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2019 |

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### 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