### 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) |
---|---|

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 |

### Fingerprint

### Keywords

- Blowup profile
- Blowup solution
- Stability
- Wave maps

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Differential Equations*,

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

**Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps.** / Ghoul, Tej-eddine; Ibrahim, S.; Nguyen, V. T.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 265, no. 7, pp. 2968-3047. https://doi.org/10.1016/j.jde.2018.04.058

}

TY - JOUR

T1 - Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

AU - Ghoul, Tej-eddine

AU - Ibrahim, S.

AU - Nguyen, V. T.

PY - 2018/10/5

Y1 - 2018/10/5

N2 - We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[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)∼cu(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.

AB - We consider the energy supercritical wave maps from Rd into the d-sphere Sd with d≥7. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation ∂t 2u=∂r 2u+[Formula presented]∂ru−[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)∼cu(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.

KW - Blowup profile

KW - Blowup solution

KW - Stability

KW - Wave maps

UR - http://www.scopus.com/inward/record.url?scp=85047061519&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85047061519&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2018.04.058

DO - 10.1016/j.jde.2018.04.058

M3 - Article

VL - 265

SP - 2968

EP - 3047

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 7

ER -