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April  2018, 38(4): 2207-2228. doi: 10.3934/dcds.2018091

Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data

Chenmin Sun 1, , Hua Wang 2,, , Xiaohua Yao 2, and  Jiqiang Zheng 3,

1. 

Université Côte d'Azur, LJAD, 06100, France
2. 

Department of Mathematics and Hubei Province Key Laboratory of Mathematical Physics, Central China Normal University, Wuhan 430079, China
3. 

Université Côte d'Azur, LJAD, 06100, France

* Corresponding author: Hua Wang

Received  June 2017 Revised  October 2017 Published  January 2018

Fund Project: The first and last authors are financed by ERC project SCAPDE, the second author is supported by NSF grant 11101172, 11371158 and 11571131, and the third author is supported by NSF grant 11371158 and 11771165.

The aim of this paper is to adapt the strategy in [8] [ See, B. Dodson, J. Murphy, a new proof of scattering below the ground state for the 3D radial focusing cubic NLS, arXiv:1611.04195 ] to prove the scattering of radial solutions below sharp threshold for certain focusing fractional NLS. The main ingredient is to apply the fractional virial identity proved in [3] [ See, T. Boulenger, D. Himmelsbach, E. Lenzmann, Blow up for fractional NLS, J. Func. Anal, 271(2016), 2569-2603 ] to exclude the concentration of mass near the origin.

Keywords: Fractional Schrödinger equation, scattering, Morawetz estimate.
Mathematics Subject Classification: Primary: 35P25, 35Q55; Secondary: 47J35.
Citation: Chenmin Sun, Hua Wang, Xiaohua Yao, Jiqiang Zheng. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2207-2228. doi: 10.3934/dcds.2018091
References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific. J. Math., 10 (1960), 419-437.  doi: 10.2140/pjm.1960.10.419.  Google Scholar

[3]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blow up for fractional NLS, J. Functional Analysis, 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.  Google Scholar

[4]

W. Chen, C. Miao and X. Yao, Dispersive estimates with geometry of finite type, Communications in Partial Differential Equations, 37 (2012), 479-510.  doi: 10.1080/03605302.2011.641053.  Google Scholar

[5]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.  doi: 10.1142/S0219199709003399.  Google Scholar

[6]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems, 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.  Google Scholar

[7]

V. D. Dinh, On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces, preprint, arXiv: 1701.00852. Google Scholar

[8]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867, arXiv: 1611.04195. doi: 10.1090/proc/13678.  Google Scholar

[9]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacian in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[10]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[11]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, J. Anal. Math., 124 (2014), 1-38.  doi: 10.1007/s11854-014-0025-6.  Google Scholar

[12]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrodinger equation in the radial case, preprint, arXiv: 1310.6816. Google Scholar

[13]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Communications on Pure and Applied Analysis, 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[14]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. App. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[15]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[16]

J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $ L^2$-critical half-wave equations, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.  Google Scholar

[17]

N. Laskin, Fractional Schrödinger equation Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[18]

T. Ogawa and Y. Tsutsumi, Blow-up of $ H^1$ solution for the nonlinear Schrödinger equation, J. Differ. Eqns., 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[19]

E. M. Stein, Harmonic Analysis: Real-Variable Theory, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 2000.  Google Scholar

[20]

T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial. Differ.Equ., 1 (2004), 1-47.  doi: 10.4310/DPDE.2004.v1.n1.a1.  Google Scholar

show all references

References:
[1]

H. Bahouri, J. Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften, 343, Springer-Verlag, Berlin, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific. J. Math., 10 (1960), 419-437.  doi: 10.2140/pjm.1960.10.419.  Google Scholar

[3]

T. Boulenger, D. Himmelsbach and E. Lenzmann, Blow up for fractional NLS, J. Functional Analysis, 271 (2016), 2569-2603.  doi: 10.1016/j.jfa.2016.08.011.  Google Scholar

[4]

W. Chen, C. Miao and X. Yao, Dispersive estimates with geometry of finite type, Communications in Partial Differential Equations, 37 (2012), 479-510.  doi: 10.1080/03605302.2011.641053.  Google Scholar

[5]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.  doi: 10.1142/S0219199709003399.  Google Scholar

[6]

Y. Cho, G. Hwang, S. Kwon and S. Lee, Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete and Continuous Dynamical Systems, 35 (2015), 2863-2880.  doi: 10.3934/dcds.2015.35.2863.  Google Scholar

[7]

V. D. Dinh, On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces, preprint, arXiv: 1701.00852. Google Scholar

[8]

B. Dodson and J. Murphy, A new proof of scattering below the ground state for the 3D radial focusing NLS, Proc. Amer. Math. Soc., 145 (2017), 4859-4867, arXiv: 1611.04195. doi: 10.1090/proc/13678.  Google Scholar

[9]

R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacian in $ \mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9.  Google Scholar

[10]

R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591.  Google Scholar

[11]

Z. Guo and Y. Wang, Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, J. Anal. Math., 124 (2014), 1-38.  doi: 10.1007/s11854-014-0025-6.  Google Scholar

[12]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrodinger equation in the radial case, preprint, arXiv: 1310.6816. Google Scholar

[13]

Y. Hong and Y. Sire, On fractional Schrödinger equations in Sobolev spaces, Communications on Pure and Applied Analysis, 14 (2015), 2265-2282.  doi: 10.3934/cpaa.2015.14.2265.  Google Scholar

[14]

C. E. Kenig, G. Ponce and L. Vega, Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure. App. Math., 46 (1993), 527-620.  doi: 10.1002/cpa.3160460405.  Google Scholar

[15]

C. E. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[16]

J. Krieger, E. Lenzmann and P. Raphaël, Nondispersive solutions to the $ L^2$-critical half-wave equations, Arch. Ration. Mech. Anal., 209 (2013), 61-129.  doi: 10.1007/s00205-013-0620-1.  Google Scholar

[17]

N. Laskin, Fractional Schrödinger equation Phys. Rev. E, 66 (2002), 056108, 7 pp. doi: 10.1103/PhysRevE.66.056108.  Google Scholar

[18]

T. Ogawa and Y. Tsutsumi, Blow-up of $ H^1$ solution for the nonlinear Schrödinger equation, J. Differ. Eqns., 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B.  Google Scholar

[19]

E. M. Stein, Harmonic Analysis: Real-Variable Theory, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, New Jersey, 2000.  Google Scholar

[20]

T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, Dyn. Partial. Differ.Equ., 1 (2004), 1-47.  doi: 10.4310/DPDE.2004.v1.n1.a1.  Google Scholar

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