On the derivative nonlinear Schrödinger equation with weakly dissipative structure

Abstract

We consider the initial value problem for cubic derivative nonlinear Schrödinger equation in one space dimension. Under a suitable weakly dissipative condition on the nonlinearity, we show that the small data solution has a logarithmic time decay in \(L^2\).

This is a preview of subscription content, access via your institution.

References

  1. 1.

    N. Hayashi, C.  Li and P. I. Naumkin, Time decay for nonlinear dissipative Schrödinger equations in optical fields, Adv. Math. Phys., 2016, Article ID 3702738.

  2. 2.

    N. Hayashi and P. I. Naumkin, Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), no.2, 369–389.

    MathSciNet  Article  Google Scholar 

  3. 3.

    N. Hayashi and P. I. Naumkin, Asymptotics of small solutions to nonlinear Schrödinger equations with cubic nonlinearities, Int. J. Pure Appl. Math., 3 (2002), no.3, 255–273.

    MathSciNet  MATH  Google Scholar 

  4. 4.

    N. Hayashi and P. I. Naumkin, Large time behavior for the cubic nonlinear Schrödinger equation, Canad. J. Math., 54 (2002), no.5, 1065–1085.

    MathSciNet  Article  Google Scholar 

  5. 5.

    N. Hayashi and P. I. Naumkin, On the asymptotics for cubic nonlinear Schrödinger equations, Complex Var. Theory Appl., 49 (2004), no.5, 339–373.

    MathSciNet  MATH  Google Scholar 

  6. 6.

    N. Hayashi and P. I. Naumkin, Asymptotics of odd solutions for cubic nonlinear Schrödinger equations, J. Differential Equations, 246 (2009), no.4, 1703–1722.

    MathSciNet  Article  Google Scholar 

  7. 7.

    N. Hayashi and P. I. Naumkin, Global existence for the cubic nonlinear Schrödinger equation in lower order Sobolev spaces, Differential Integral Equations, 24 (2011), no.9–10, 801–828.

    MathSciNet  MATH  Google Scholar 

  8. 8.

    N. Hayashi and P. I. Naumkin, Logarithmic time decay for the cubic nonlinear Schrödinger equations, Int. Math. Res. Not. IMRN, 2015, no.14, 5604–5643.

    Article  Google Scholar 

  9. 9.

    N. Hayashi, P. I. Naumkin and H. Sunagawa, On the Schrödinger equation with dissipative nonlinearities of derivative type, SIAM J. Math. Anal., 40 (2008), no.1, 278–291.

    MathSciNet  Article  Google Scholar 

  10. 10.

    G. Hoshino, Asymptotic behavior for solutions to the dissipative nonlinear Schrödinger equations with the fractional Sobolev space, J. Math. Phys., 60 (2019), no.11, 111504, 11 pp.

  11. 11.

    G. Jin, Y. Jin and C. Li, The initial value problem for nonlinear Schrödinger equations with a dissipative nonlinearity in one space dimension, J. Evol. Equ., 16 (2016), no.4, 983–995.

    MathSciNet  Article  Google Scholar 

  12. 12.

    S. Katayama, C. Li and H. Sunagawa, A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D, Differential Integral Equations 27 (2014), no.3–4, 301–312.

    MathSciNet  MATH  Google Scholar 

  13. 13.

    S. Katayama, A. Matsumura and H. Sunagawa, Energy decay for systems of semilinear wave equations with dissipative structure in two space dimensions, NoDEA Nonlinear Differential Equations Appl., 22 (2015), no.4, 601–628.

    MathSciNet  Article  Google Scholar 

  14. 14.

    J. Kato and F. Pusateri, A new proof of long-range scattering for critical nonlinear Schrödinger equations, Differential Integral Equations 24 (2011), no.9–10, 923–940.

    MathSciNet  MATH  Google Scholar 

  15. 15.

    D. Kim, A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension, Asymptot. Anal., 98 (2016), no.1–2, 79–90.

    MathSciNet  Article  Google Scholar 

  16. 16.

    N. Kita, Existence of blowing-up solutions to some Schrödinger equations including nonlinear amplification with small initial data, preprint, 2019.

  17. 17.

    N. Kita and C. Li, Decay estimate of solutions to dissipative nonlinear Schrödinger equations, preprint, 2017.

  18. 18.

    N. Kita and Y. Nakamura, Decay estimate and asymptotic behavior of small solutions to Schrödinger equations with subcritical dissipative nonlinearity, Adv. Stud. Pure Math., 81 (2019), 121–138.

    Article  Google Scholar 

  19. 19.

    N. Kita and A. Shimomura, Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan, 61 (2009), no.1, 39–64.

    MathSciNet  Article  Google Scholar 

  20. 20.

    C. Li, Y. Nishii, Y. Sagawa and H. Sunagawa, Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension, to appear in Funkcialaj Ekvacioj [arXiv:1905.07123].

  21. 21.

    C. Li, Y. Nishii, Y. Sagawa and H. Sunagawa, Large time asymptotics for a cubic nonlinear Schrödinger system in one space dimension, II, to appear in Tokyo J. Math. [arXiv:2001.10682].

  22. 22.

    C. Li and H. Sunagawa, On Schrödinger systems with cubic dissipative nonlinearities of derivative type, Nonlinearity, 29 (2016), no.5, 1537–1563; Corrigendum, ibid., no.12, C1–C2.

  23. 23.

    C. Li and H. Sunagawa, Remarks on derivative nonlinear Schrödinger systems with multiple masses, Adv. Stud. Pure Math., 81 (2019), 173–195.

    Article  Google Scholar 

  24. 24.

    J. Murphy and F. Pusateri, Almost global existence for cubic nonlinear Schrödinger equations in one space dimension, Discrete Contin. Dyn. Syst., 37 (2017), no.4, 2077–2102.

    MathSciNet  Article  Google Scholar 

  25. 25.

    P. I. Naumkin, The dissipative property of a cubic non-linear Schrödinger equation, Izv. Math., 79 (2015), no.2, 346–374.

    MathSciNet  Article  Google Scholar 

  26. 26.

    Y. Nishii, Non-decay of the energy for a system of semilinear wave equations, in preparation.

  27. 27.

    Y. Nishii and H. Sunagawa, On Agemi-type structural conditions for a system of semilinear wave equations, to appear in J. Hyperbolic Differential Equations [arXiv:1904.09083].

  28. 28.

    Y. Nishii, H. Sunagawa and H. Terashita, Energy decay for small solutions to semilinear wave equations with weakly dissipative structure, to appear in J. Math. Soc. Japan [arXiv:2002.09639].

  29. 29.

    Y. Sagawa and H. Sunagawa, The lifespan of small solutions to cubic derivative nonlinear Schrödinger equations in one space dimension, Discrete Contin. Dyn. Syst., 36 (2016), no.10, 5743–5761; Corrigendum, ibid., 40 (2020), no.7, 4577–4578.

  30. 30.

    A. Shimomura, Asymptotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Comm. Partial Differential Equations, 31 (2006), no.7–9, 1407–1423.

    MathSciNet  Article  Google Scholar 

  31. 31.

    H. Sunagawa, Lower bounds of the lifespan of small data solutions to the nonlinear Schrödinger equations, Osaka J. Math. 43 (2006), no.4, 771–789.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hideaki Sunagawa.

Additional information

Dedicated to Professor Akitaka Matsumura on the occasion of his seventieth birthday

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is partly supported by Osaka City University Advanced Mathematical Institute (MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849). The work of H. S. is supported by Grant-in-Aid for Scientific Research (C) (No. 17K05322), JSPS.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, C., Nishii, Y., Sagawa, Y. et al. On the derivative nonlinear Schrödinger equation with weakly dissipative structure. J. Evol. Equ. (2020). https://doi.org/10.1007/s00028-020-00634-6

Download citation

Keywords

  • Cubic derivative nonlinear Schrödinger equation
  • Large time behavior
  • Weakly dissipative structure

Mathematics Subject Classification

  • 35Q55
  • 35B40