Advertisement

Scattering threshold for the focusing Choquard equation

  • Tarek SaanouniEmail author
Article
  • 87 Downloads

Abstract

It is the purpose of this note, to obtain a scattering versus finite time blow-up dichotomy for a mass super-critical and energy sub-critical Choquard equation in the energy space.

Keywords

Nonlinear Schrödinger–Choquard equation Global existence Scattering Blow-up 

Mathematics Subject Classification

35Q55 

Notes

References

  1. 1.
    Adams, R.: Sobolev Spaces. Academic, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Bonanno, C., d’Avenia, P., Ghimenti, M., Squassina, M.: Soliton dynamics for the generalized Choquard equation. J. Math. Anal. Appl. 417, 180–199 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chen, J., Guo, B.: Strong instability of standing waves for a nonlocal Schrödinger equation. Physica D Nonlinear Phenom. 227, 142–148 (2007)CrossRefGoogle Scholar
  4. 4.
    Christ, M., Weinstein, M.: Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation. J. Funct. Anal. 100, 87–109 (1991)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cho, Y., Hwang, G., Ozawa, T.: Global well-posedness of critical nonlinear Schrödinger equations below \(L^2\). Discrete Contin. Dyn. Syst. A 44, 1389–1405 (2013)zbMATHGoogle Scholar
  6. 6.
    Cho, Y., Ozawa, T.: Sobolev inequalities with symmetry. Commun. Contemp. Math. 11(3), 355–365 (2009). MR 2538202 (2010h:46039)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Duyckaerts, T., Merle, F.: Dynamic of thresholds solutions for energy-critical NLS. Geom. Funct. Anal. 18(6), 1787–1840 (2009)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duyckaerts, T., Roudenko, S.: Going beyond the threshold: scattering and blow-up in the focusing NLS equation. Commun. Math. Phys. 334, 1573–1615 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Elgart, A., Schlein, B.: Mean field dynamics of boson stars. Commun. Pure Appl. Math. 60, 500–545 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Feng, B., Yuan, X.: On the Cauchy problem for the Schrödinger–Hartree equation. Evolut. Equ. Control Theory 4(4), 431–445 (2015)CrossRefGoogle Scholar
  11. 11.
    Gaididei, Y.B., Rasmussen, K.O., Christiansen, P.L.: Nonlinear excitations in two-dimensional molecular structures with impurities. Phys. Rev. E 52, 2951–2962 (1995)CrossRefGoogle Scholar
  12. 12.
    Gao, Y., Wang, Z.: Scattering versus blow-up for the focusing \(L^2\) supercritical Hartree equation. Math. Phys. 65, 179–202 (2014)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Genev, H., Venkov, G.: Soliton and blow-up solutions to the time-dependent Schrödinger Hartree equation. Discrete Contin. Dyn. Syst. Ser. S 5, 903–923 (2012)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gross, E.P., Meeron, E.: Physics of Many-Particle Systems, vol. 1, pp. 231–406. Gordon Breach, New York (1966)Google Scholar
  15. 15.
    Guevara, C.D.: Global behavior of finite energy solutions to the d-dimensional focusing nonlinear Schrödinger equation. Appl. Math. Res. eXpress 2, 177–243 (2014)zbMATHGoogle Scholar
  16. 16.
    Guo, Q.: Scattering for the focusing \(L^2\) -supercritical and \({\dot{H}}^2\)-subcritical biharmonic NLS equations. Commun. Partial Differ. Equ. 41(2), 185–207 (2016)CrossRefGoogle Scholar
  17. 17.
    Guo, Z., Wang, Y.: Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations. J. Anal. Math. 124(1), 1–38 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Holmer, J., Roudenko, S.: A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation. Commun. Math. Phys. 282, 435–467 (2008)CrossRefGoogle Scholar
  19. 19.
    Kenig, C., Merle, F.: Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math. 201(2), 147–212 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Le Coz, S.: A note on Berestycki–Cazenave classical instability result for nonlinear Schrödinger equations. Adv. Nonlinear Stud. 8(3), 455–463 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lenzmann, E.: Well-posedness for semi-relativistic Hartree equations of critical type. Math. Phys. Anal. Geom. 10, 43–64 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lewin, M., Rougerie, N.: Derivation of Pekar’s polarons from a microscopic model of quantum crystal. SIAM J. Math. Anal. 45, 1267–1301 (2013)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lieb, E.: Analysis, Graduate Studies in Mathematics, vol. 14, 2nd edn. American Mathematical Society, Providence (2001)Google Scholar
  24. 24.
    Lions, P.-L.: Symétrie et compacité dans les espaces de Sobolev. J. Funct. Anal. 49(3), 315–334 (1982)CrossRefGoogle Scholar
  25. 25.
    Lions, P.-L.: The Choquard equation and related questions. Nonlinear Anal. 4, 1063–1072 (1980)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Miao, C., Xu, G., Zhao, L.: Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data. J. Func. Anal. 253(2), 605–627 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Miao, C., Xu, G., Zhao, L.: The cauchy problem of the Hartree equation. J. PDE 21, 22–44 (2008)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Moroz, V., Schaftingen, J.V.: Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics. J. Funct. Anal. 265, 153–184 (2013)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Payne, L.E., Sattinger, D.H.: Saddle points and instability of non-linear hyperbolic equations. Isr. J. Math. 22, 273–303 (1976)CrossRefGoogle Scholar
  30. 30.
    Tao, T., Visan, M.: Stability of energy-critical nonlinear Schrödinger equations in high dimensions. Electron. J. Differ. Equ. 2005, 1–28 (2005)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Qassim UniversityBuraidahKingdom of Saudi Arabia
  2. 2.LR03ES04 Partial Differential Equations and Applications, Faculty of Science of TunisUniversity of Tunis El ManarTunisTunisia

Personalised recommendations