### Abstract

We consider the energy supercritical wave maps from R^{d} into the d-sphere S^{d} with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂_{t} ^{2}u=∂_{r} ^{2}u+[Formula presented]∂_{r}u−[Formula presented]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([Formula presented]), where Q is the stationary solution of the equation and the speed is given by the quantized rates λ(t)∼c_{u}(T−t)^{[Formula presented]},ℓ∈N^{⁎},ℓ>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 [49] for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

Original language | English (US) |
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Pages (from-to) | 2968-3047 |

Number of pages | 80 |

Journal | Journal of Differential Equations |

Volume | 265 |

Issue number | 7 |

DOIs | |

State | Published - Oct 5 2018 |

### Keywords

- Blowup profile
- Blowup solution
- Stability
- Wave maps

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

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## Cite this

*Journal of Differential Equations*,

*265*(7), 2968-3047. https://doi.org/10.1016/j.jde.2018.04.058