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

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


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.



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.


  1. 1.
    J. Bellazzini, V. Georgiev, N. Visciglia, Long time dynamics for semirelativistic NLS and half wave in arbitrary dimension. Math. Annalen. 371(1–2), 707–740 (2018)MathSciNetCrossRefGoogle Scholar
  2. 2.
    J.P. Borgna, D.F. Rial, Existence of ground states for a one-dimensional relativistic Schödinger equation. J. Math. Phys. 53, 062301 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A.P. Calderón, A. Zygmund, On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)MathSciNetCrossRefGoogle Scholar
  4. 4.
    T. Cazenave, Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10 (American Mathematical Society/New York University/Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, 2003)Google Scholar
  5. 5.
    T. Cazenave, F.B. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in Nonlinear Semigroups, Partial Differential Equations and Attractors, ed. by T.L. Gill, W.W. Zachary (Washington, DC, 1987). Lecture Notes in Mathematics, vol. 1394 (Springer, Berlin, 1989), pp. 18–29Google Scholar
  6. 6.
    T. Cazenave, F.B. Weissler, The Cauchy problem for the critical nonlinear Schrödinger equation in H s. Nonlinear analysis. Theory, methods & applications. Int. Multidiscip. J. Ser. A Theory Methods 14, 807–836 (1990)Google Scholar
  7. 7.
    Y. Cho, T. Ozawa, Sobolev inequalities with symmetry. Commun. Contemp. Math. 11, 355–365 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Forcella, K. Fujiwara, V. Georgiev, T. Ozawa, Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential. Discrete Contin. Dyn. Syst. 39, 2661–2678 (2019). arXiv:1804.02524Google Scholar
  10. 10.
    K. Fujiwara, Remark on local solvability of the Cauchy problem for semirelativistic equations. J. Math. Anal. Appl. 432, 744–748 (2015)MathSciNetCrossRefGoogle Scholar
  11. 11.
    K. Fujiwara, V. Georgiev, T. Ozawa, Blow-up for self-interacting fractional Ginzburg-Landau equation. Dyn. Partial Differ. Equ. 15, 175–182 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    K. Fujiwara, V. Georgiev, T. Ozawa, On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases. arXiv: 1611.09674 (2016)Google Scholar
  13. 13.
    J. Ginibre, T. Ozawa, G. Velo, On the existence of the wave operators for a class of nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Phys. Théor. 60, 211–239 (1994)MathSciNetzbMATHGoogle Scholar
  14. 14.
    J. Ginibre, G. Velo, Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 50–68 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    L. Grafakos, S. Oh, The Kato-Ponce Inequality. Comm. Partial Differ. Equ. 39, 1128–1157 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Ikeda, T. Inui, Some non-existence results for the semilinear Schrödinger equation without gauge invariance. J. Math. Anal. Appl. 425, 758–773 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Ikeda, Y. Wakasugi, Small-data blow-up of L 2-solution for the nonlinear Schrödinger equation without gauge invariance. Differ. Integral Equ. 26, 11–12 (2013)zbMATHGoogle Scholar
  18. 18.
    T. Inui, Some nonexistence results for a semirelativistic Schrödinger equation with nongauge power type nonlinearity. Proc. Am. Math. Soc. 144, 2901–2909 (2016)CrossRefGoogle Scholar
  19. 19.
    T. Kato, G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations. Comm. Pure Appl. Math. 41, 891–907 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices. Duke Math. J. 71, 1–21 (1993)MathSciNetCrossRefGoogle Scholar
  21. 21.
    S. Klainerman, M. Machedon, Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993)MathSciNetCrossRefGoogle Scholar
  22. 22.
    A. Kufner, B. Opic, Hardy-Type Inequalities. Pitman Research Notes in Mathematics Series (Longman Scientific & Technical, Harlow, 1990)Google Scholar
  23. 23.
    N. Laskin, Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A. 268, 298–305 (2000)MathSciNetCrossRefGoogle Scholar
  24. 24.
    E. Lenzmann, A. Schikorra, Sharp commutator estimates via harmonic extensions. arXiv: 1609.08547Google Scholar
  25. 25.
    D. Li, On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoamericana. (in press). arXiv:1609.01780v2. It appeared on arXiv in 2016 and revised in 2018Google Scholar
  26. 26.
    M. Nakamura, T. Ozawa, The Cauchy problem for nonlinear Klein-Gordon equations in the Sobolev spaces. Publ. Res. Inst. Math. Sci. 37, 255–293 (2001)MathSciNetCrossRefGoogle Scholar
  27. 27.
    T. Ozawa, Remarks on proofs of conservation laws for nonlinear Schrödinger equations. Calc. Var. Partial Differ. Equ. 25, 403–408 (2006)CrossRefGoogle Scholar
  28. 28.
    W. Sickel, L. Skrzypczak, Radial subspaces of Besov and Lizorkin-Triebel classes: extended Strauss lemma and compactness of embeddings. J. Fourier Anal. Appl. 6, 639–662 (2000)MathSciNetCrossRefGoogle Scholar

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© 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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