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Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case

  • Luigi Forcella
  • Kazumasa FujiwaraEmail author
  • Vladimir Georgiev
  • Tohru Ozawa
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

The aim of this work is to give a complete picture concerning the asymptotic behaviour of the solutions to fractional Ginzburg-Landau equation. In previous works, we have shown global well-posedness for the past interval in the case where spatial dimension is less than or equal to 3. Moreover, we have also shown blow-up of solutions for the future interval in one dimensional case. In this work, we summarise the asymptotic behaviour in the case where spatial dimension is less than or equal to 3 by proving blow-up of solutions for a future time interval in multidimensional case. The result is obtained via ODE argument by exploiting a new weighted commutator estimate.

Notes

Acknowledgements

V. Georgiev was supported in part by INDAM, GNAMPA – Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni, by Institute of Mathematics and Informatics, Bulgarian Academy of Sciences and Top Global University Project, Waseda University and the Project PRA 2018 – 49 of University of Pisa. The authors are grateful to the referees for their helpful comments.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Luigi Forcella
    • 1
  • Kazumasa Fujiwara
    • 2
    Email author
  • Vladimir Georgiev
    • 3
    • 4
    • 5
  • Tohru Ozawa
    • 6
  1. 1.Bâtiment des MathématiquesÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland
  2. 2.Centro di Ricerca Matematica Ennio De GiorgiScuola Normale SuperiorePisaItaly
  3. 3.Department of MathematicsUniversity of PisaPisaItaly
  4. 4.Faculty of Science and EngineeringWaseda UniversityShinjuku-kuJapan
  5. 5.IMI–BASSofiaBulgaria
  6. 6.Department of Applied PhysicsWaseda UniversityShinjuku-kuJapan

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