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Large time behavior of solutions to the critical dissipative nonlinear Schrödinger equation with large data

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Abstract

We consider the Cauchy problem for the dissipative nonlinear Schrödinger equation with a critical cubic nonlinearity in one space dimension. We show the global uniform bound of dissipative solutions in the Gevrey class and its \(L^2\)-decay order without any restriction of the size of smooth initial data.

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References

  1. Ahn, J., Kim, J., Seo, I., On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations, Discrete Contin. Dyn. Syst. 40 (2020), 423–439.

    Article  MathSciNet  Google Scholar 

  2. Baillon, J., Cazenave, T., Figueira, M., Équation de Schrödinger nonlinéaire, C. r. Acad. Sci. Paris 284 (1977), 869–872.

    MathSciNet  MATH  Google Scholar 

  3. Cazenave, T., Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 323pp 2003.

  4. Cazenave, T., Han, Z., Asymptotic behavior for a Schrödinger equation with nonlinear subcritical dissipation, Descrete Contin. Dyn. Syst. 40 (2020), 4801–4819.

    Article  Google Scholar 

  5. Cazenave, T., Han, Z., Naumkin, I., Asymptotic behavior for a dissipative nonlinear Schrödinger equation, Nonlinear Anal. 205 (2021), 112243, 37pp.

  6. Cazenave, T., Naumkin, I., Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Commun. Contemp. Math. 19 (2017), 1650038.

    Article  MathSciNet  Google Scholar 

  7. Cazenave, T., Naumkin, I., Modified scattering for the critical nonlinear Schrödinger equation, J. Funct. Anal. 274 (2018), 402–432.

    Article  MathSciNet  Google Scholar 

  8. Cazenave, T., Weissler, F., The Cauchy problem for the nonlinear Schrödinger equation in\(H^1\), Manuscripta Math. 61 (1988), 477–494.

    Article  MathSciNet  Google Scholar 

  9. Ginible, J., Velo, G., On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1–31.

  10. Hayashi, N., Naumkin, P., Asymptotics for large time of solutions to nonlinear Schrödinger and Hartree equations, Amer. J. Math. 120 (1998), 369–389.

    Article  MathSciNet  Google Scholar 

  11. Hayashi, N., Li, C., Naumkin, P., Time decay for nonlinear dissipative Schrödinger equations in optical fields, Adv. Math. Phys. (2016), Art. ID 3702738, 7.

  12. Hoshino, G., Global well-posedness and analytic smoothing effect for the dissipative nonlinear Schrödinger equations, J. Dynam. Differential Equations 31 (2019), 2339–2351.

    Article  MathSciNet  Google Scholar 

  13. Hoshino, G, Asymptotic behavior for solutions to the dissipative nonlinear Schrödinger equations with the fractional Sobolev space, J. Math. Phys. 60 (2019), 111504, 11.

  14. Kato, T., On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Theor. 46 (1987), 113-129.

    MathSciNet  MATH  Google Scholar 

  15. Katayama, S., Li, C., Sunagawa, H., A remark on decay rates of solutions for a system of quadratic nonlinear Schrödinger equations in 2D, Differential Integral Equations 27 (2014), 301–312.

    MathSciNet  MATH  Google Scholar 

  16. Keel, M., Tao, T., Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), 955–980.

    Article  MathSciNet  Google Scholar 

  17. Kim, D., A note on decay rates of solutions to a system of cubic nonlinear Schrödinger equations in one space dimension, Asymptot. Anal. 98 (2016), 79–90.

    MathSciNet  MATH  Google Scholar 

  18. Kim, D., Sunagawa, H., Remarks on decay of small solutions to systems of Klein-Gordon equations with dissipative nonlinearities, Nonlinaer. Anal. 97 (2014), 94–105.

    Article  MathSciNet  Google Scholar 

  19. Kita, N., Shimomura, A., Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity, J. Differential Equations 242 (2007), 192–210.

    Article  MathSciNet  Google Scholar 

  20. Kita, N., Shimomura, A., Large time behavior of solutions to Schrödinger equations with a dissipative nonlinearity for arbitrarily large initial data, J. Math. Soc. Japan 61 (2009), 39–64.

    Article  MathSciNet  Google Scholar 

  21. Liskevich, V. A., Perelmuter, M. A., Analyticity of submarkovian semigroups, Proc. Amer. Math. Soc. 123 (1995), 1097–1104.

    MathSciNet  MATH  Google Scholar 

  22. Li, C., Nishii, Y., Sagawa, Y., Sunagawa, H., Upper and lower \(L^2\)-decay bounds for a class of derivative nonlinear Schrödinger equations, preprint.

  23. Li, C., Sunagawa, H., On Schrödinger systems with cubic dissipative nonlinearities of derivative type, Nonlinearity 29 (2016), 1537–1563.

    Article  MathSciNet  Google Scholar 

  24. Nishii, Y., Sunagawa, H., On Agemi-type structural conditions for a system of semilinear wave equations, J. Hypabolic Differ. Equ. 17 (2020), 459–473.

    Article  MathSciNet  Google Scholar 

  25. Ogawa, T., Sato, T., \(L^2\)-decay rate for the critical nonlinear Schrödinger equation with a small smooth data, Nonlinear Differ. Equ. Appl. 27 (2020), 18.

    Article  Google Scholar 

  26. Okazawa, N., Yokota, T., Global existence and smoothing effect for the complex Ginzburg-Landau equation with \(p\)-Laplacian, J. Differential Equations 182 (2002), 541–576.

    Article  MathSciNet  Google Scholar 

  27. Sato, T., \(L^2\)-decay estimate for the dissipative nonlinear Schrödinger equation in the Gevrey class, Arch. Math. 115 (2020), 575–588.

    Article  MathSciNet  Google Scholar 

  28. Sato, T., Lower bound estimate for the dissipative nonlinear Schrödinger equation, SN Partial Differ. Equ. Appl. 2 (2021), 66, 11.

  29. Stanley, R. P., Catalan addendum, http://www-math.mit.edu/rstan/ec/catadd.pdf

  30. Sunagawa, H., Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. Math. Soc. Japan 58 (2006), 379–400.

    Article  MathSciNet  Google Scholar 

  31. Shimomura, A., Asymtotic behavior of solutions for Schrödinger equations with dissipative nonlinearities, Commun. Partial Differ. Equ. 31 (2006), 1407–1423.

    Article  Google Scholar 

  32. Tesfahun, A., On the radius of spatial analyticity for cubic nonlinear Schrödinger equations, J. Differential Equations 263 (2017), 7496–7512.

    Article  MathSciNet  Google Scholar 

  33. Tesfahun, A., Remark on the persistence of spatial analyticity for cubic nonlinear Schrödinger equation on the circle, Nonlinear Differ. Equ. Appl. 26 (2019), 12, 13.

  34. Tsutsumi, Y., \(L^2\)solution for nonlinear Schrödinger equation and nonlinear groups, Funk. Ekva. 30 (1987), 115–125.

    MATH  Google Scholar 

  35. Tsutsumi, Y., Global strong solutions for nonlinear Schrödinger equations, Nonlinear Anal. T. M. A. 11 (1987), 1143–1154.

    Article  Google Scholar 

  36. Yajima, K., Existence of solutions for Schrödinger evolution equations, Commun. Math. Phys. 110 (1987), 415–426.

    Article  Google Scholar 

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Acknowledgements

The author would like to express cordial appreciation to Professor Hideo Kozono for his important suggestions. The author is deeply grateful to the referee for reviewing this paper and for valuable advice. The author is supported by JSPS grant-in-aid for Early-Career Scientists #22K13937 and partially supported by JSPS grant-in-aid for Scientific Research (S) #19H05597.

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  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Takuya Sato

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Sato, T. Large time behavior of solutions to the critical dissipative nonlinear Schrödinger equation with large data. J. Evol. Equ. 22, 59 (2022). https://doi.org/10.1007/s00028-022-00815-5

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