Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schrödinger equation

$$\begin{aligned} i\partial _t u + \varDelta u - |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1, \end{aligned}$$

with \(b, \alpha >0\). We first study the decaying property of global solutions for the equation when \(0<\alpha <\alpha ^\star \) where \(\alpha ^\star = \frac{4-2b}{d-2}\) for \(d\ge 3\). The proof makes use of an argument of Visciglia (Math Res Lett 16(5):919–926, 2009). We next use this decay to show the energy scattering for the equation in the case \(\alpha _\star<\alpha <\alpha ^\star \), where \(\alpha _\star = \frac{4-2b}{d}\).

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

References

  1. 1.

    L. Bergé, Soliton stability versus collapse, Phys. Rev. E 62, No. 3, https://doi.org/10.1103/PhysRevE.62.R3071, R3071–R3074 (2000).

    MathSciNet  Article  Google Scholar 

  2. 2.

    T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics 10, Courant Institute of Mathematical Sciences, AMS, 2003.

  3. 3.

    J. Q. Chen, On a class of nonlinear inhomogeneous Schrödinger equation, J. Appl. Math. Comput. 32, https://doi.org/10.1007/s12190-009-0246-5, 237–253 (2010).

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    J. Colliander, M. Grillakis, N. Tzirakis, Tensor products and correlation estimates with applications to nonlinear Schrödinger equations, Comm. Pur Appl. Math. 62, No. 7, https://doi.org/10.1002/cpa.20278, 920–968 (2009).

    Article  MATH  Google Scholar 

  5. 5.

    V. Combet, F. Genoud, Classification of minimal mass blow-up solutions for an \(L^2\) critical inhomogeneous NLS, J. Evol. Equ. 16, No. 2, https://doi.org/10.1007/s00028-015-0309-z, 483–500 (2016).

  6. 6.

    V. D. Dinh, Scattering theory in a weighted \(L^2\) space for a class of the defocusing inhomegeneous nonlinear Schrödinger equation, preprint arXiv:1710.01392, 2017.

  7. 7.

    V. D. Dinh, Blowup of \(H^1\) solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation, Nonlinear Anal. 174, https://doi.org/10.1016/j.na.2018.04.024, 169–188 (2018).

  8. 8.

    L. G. Farah, Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ. 16, No. 1, https://doi.org/10.1007/s00028-01500298-y, 193–208 (2016).

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    L. G. Farah, C. M. Guzman, Scattering for the radial 3D cubic focusing inhomogeneous nonlinear Schrödinger equation, J. Differential Equations 262, No. 8, https://doi.org/10.1016/j.jde.2017.01.013, 4175–4231 (2017).

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    L. G. Farah, C. M. Guzman, Scattering for the radial focusing INLS equation in higher dimensions, preprint arXiv:1703.10988, 2017.

  11. 11.

    G. Fibich, X. P. Wang, Stability of solitary waves for nonlinear Schrödinger equations with inhomogeneous nonlinearities, Physica D 175, https://doi.org/10.1016/S0167-2789(02)00626-7, 96–108 (2003).

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    F. Genoud, C. A. Stuart, Schrödinger equations with a spatially decaying nonlinearity: existence and stability of standing waves, Discrete Contin. Dyn. Syst. 21, No. 1, https://doi.org/10.3934/dcds.2008.21.137, 137–286 (2008).

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    F. Genoud, An inhomogeneous, \(L^2\) -critical, nonlinear Schrödinger equation, Z. Anal. Anwend. 31, No. 3, https://doi.org/10.4171/ZAA/1460, 283–290 (2012).

  14. 14.

    T. S. Gill, Optical guiding of laser beam in nonuniform plasma, Pramana J. Phys. 55, https://doi.org/10.1007/s12043-000-0051-z, 842–845 (2000).

    Google Scholar 

  15. 15.

    J. Ginibre, G. Velo, Scattering theory in the energy space for a class of nonlinear Schödinger equations, J. Math. Pures Appl. 64, 363–401 (1985).

    MathSciNet  MATH  Google Scholar 

  16. 16.

    C. M. Guzmán, On well posedness for the inhomogneous nonlinear Schrödinger equation, Nonlinear Anal. 37, https://doi.org/10.1016/j.nonrwa.2017.02.018, 249–286 (2017).

    Article  MATH  Google Scholar 

  17. 17.

    C. S. Liu, V. K. Tripathi, Laser guiding in an axially nonuniform plasma channel, Phy. Phasmas 1, https://doi.org/10.1063/1.870501, 3100–3103 (1994).

    Article  Google Scholar 

  18. 18.

    Y. Liu, X. P. Wang, K. Wang, Instability of standing waves of the Schrödinger equations with inhomogeneous nonlinearity, Trans. Amer. Math. Soc. 358, No. 5, https://doi.org/10.1090/S0002-9947-05-03763-3, 2105–2122 (2006).

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    F. Merle, Nonexistence of minimal blow-up solutions of equations \(iu_t =-\Delta u - k(x) |u|^{\frac{4}{d}} u\) in \({\mathbb{R}}^N\), Ann. Inst. H. Poincaré Phys. Théor. 64, No. 1, 35–85 (1996).

  20. 20.

    K. Nakanishi, Energy scattering for nonlinear Klein-Gordon and Schrödinger equations in spatial dimensions 1 and 2, J. Funct. Anal. 169, 201–225 (1999).

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    P. Raphaël, J. Szeftel, Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS, J. Amer. Math. Soc. 24, No. 2, https://doi.org/10.1090/S0894-0347-2010-00688-1, 471–546 (2011).

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS Regional Conference Series in Mathematics 106, AMS, 2006.

  23. 23.

    T. Tao, M. Visan, X. Zhang, The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations 32, https://doi.org/10.1080/0360530070158880, 1281–1343 (2007).

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    I. Towers, B. A. Malomed, Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity, J. Opt. Soc. Amer. B Opt. Phys. 19, No. 3, https://doi.org/10.1364/JOSAA.19.000537, 537–543 (2002).

    MathSciNet  Article  Google Scholar 

  25. 25.

    N. Visciglia, On the decay of solutions to a class of defocusing NLS, Math. Res. Lett. 16, No. 5, 919–926 (2009).

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    S. Zhu, Blow-up solutions for the inhomogeneous Schrödinger equation with \(L^2\) supercritical nonlinearity, J. Math. Anal. Appl. 409, https://doi.org/10.1016/j.jmaa.2013.07.029, 760–776 (2014).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Van Duong Dinh.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dinh, V.D. Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation. J. Evol. Equ. 19, 411–434 (2019). https://doi.org/10.1007/s00028-019-00481-0

Download citation

Keywords

  • Inhomogeneous nonlinear Schrödinger equation
  • Scattering theory
  • Virial inequality
  • Decaying solution

Mathematics Subject Classification

  • 35G20
  • 35G25
  • 35Q55