### Abstract

We construct finite time blow-up solutions to the 3-dimensional harmonic map flow into the sphere S^{2}, ut = ∆u + |∇u|^{2}u in Ω × (0, T) u = ub on ∂Ω × (0, T) u(·, 0) = u_{0} in Ω, with u(x, t): Ω^{¯} × [0, T) → S^{2}. Here Ω is a bounded, smooth axially symmetric domain in R^{3}. We prove that for any circle Γ ⊂ Ω with the same axial symmetry, and any sufficiently small T > 0 there exist initial and boundary conditions such that u(x, t) blows-up exactly at time T and precisely on the curve Γ, in fact |∇u(·, t)|^{2} * |∇u∗|^{2} + 8πδ_{Γ} as t → T. for a regular function u∗(x), where δ_{Γ} denotes the Dirac measure supported on the curve. This the first example of a blow-up solution with a spacecodimension 2 singular set, the maximal dimension predicted in the partial regularity theory by Chen-Struwe and Cheng [5, 6].

Original language | English |
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Pages (from-to) | 6913-6943 |

Number of pages | 31 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 39 |

Issue number | 12 |

DOIs | |

Publication status | Published - 31 Dec 2019 |

### Keywords

- Blow-up
- Codimension 2 singular set
- Finite time blow-up
- Harmonic map flow
- Semilinear parabolic equation

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*S*.

^{2}*Discrete and Continuous Dynamical Systems- Series A*,

*39*(12), 6913-6943. https://doi.org/10.3934/dcds.2019237