Boulenger, Thomas and Himmelsbach, Dominik and Lenzmann, Enno.
(2016)
* Blowup for fractional NLS.*
Journal of Functional Analysis, 271 (9).
pp. 2569-2603.

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Official URL: http://edoc.unibas.ch/53285/

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## Abstract

We consider fractional NLS with focusing power-type nonlinearity

i∂tu=(−Δ)su−|u|2σu,(t,x)∈R×RN,

where 1/2<s<11/2<s<1 and 0<σ<∞0<σ<∞ for s⩾N/2s⩾N/2 and 0<σ⩽2s/(N−2s)0<σ⩽2s/(N−2s) for s<N/2s<N/2. We prove a general criterion for blowup of radial solutions in RNRN with N⩾2N⩾2 for L2L2-supercritical and L2L2-critical powers σ⩾2s/Nσ⩾2s/N. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain Ω⊂RNΩ⊂RN in any dimension N⩾1N⩾1 and subject to exterior Dirichlet conditions. In this setting, we prove a general blowup result without imposing any symmetry assumption on u(t,x)u(t,x).

For the blowup proof in RNRN, we derive a localized virial estimate for fractional NLS in RNRN, which uses Balakrishnan's formula for the fractional Laplacian (−Δ)s(−Δ)s from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.

i∂tu=(−Δ)su−|u|2σu,(t,x)∈R×RN,

where 1/2<s<11/2<s<1 and 0<σ<∞0<σ<∞ for s⩾N/2s⩾N/2 and 0<σ⩽2s/(N−2s)0<σ⩽2s/(N−2s) for s<N/2s<N/2. We prove a general criterion for blowup of radial solutions in RNRN with N⩾2N⩾2 for L2L2-supercritical and L2L2-critical powers σ⩾2s/Nσ⩾2s/N. In addition, we study the case of fractional NLS posed on a bounded star-shaped domain Ω⊂RNΩ⊂RN in any dimension N⩾1N⩾1 and subject to exterior Dirichlet conditions. In this setting, we prove a general blowup result without imposing any symmetry assumption on u(t,x)u(t,x).

For the blowup proof in RNRN, we derive a localized virial estimate for fractional NLS in RNRN, which uses Balakrishnan's formula for the fractional Laplacian (−Δ)s(−Δ)s from semigroup theory. In the setting of bounded domains, we use a Pohozaev-type estimate for the fractional Laplacian to prove blowup.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann) |
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UniBasel Contributors: | Lenzmann, Enno and Himmelsbach, Dominik |

Item Type: | Article, refereed |

Article Subtype: | Research Article |

Publisher: | Elsevier |

ISSN: | 0022-1236 |

e-ISSN: | 1096-0783 |

Note: | Publication type according to Uni Basel Research Database: Journal article |

Identification Number: | |

Last Modified: | 30 Oct 2017 09:10 |

Deposited On: | 30 Oct 2017 09:10 |

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