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

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

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.

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 - A, 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. BoulengerD. 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. ChenC. 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. ChoG. HwangS. 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. FrankE. 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. KenigG. 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. KriegerE. 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. BoulengerD. 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. ChenC. 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. ChoG. HwangS. 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. FrankE. 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. KenigG. 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. KriegerE. 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

[1]

Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127

[2]

Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003

[3]

Alp Eden, Elİf Kuz. Almost cubic nonlinear Schrödinger equation: Existence, uniqueness and scattering. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1803-1823. doi: 10.3934/cpaa.2009.8.1803

[4]

Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104

[5]

Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107

[6]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395

[7]

Zhengping Wang, Huan-Song Zhou. Radial sign-changing solution for fractional Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 499-508. doi: 10.3934/dcds.2016.36.499

[8]

Van Duong Dinh, Binhua Feng. On fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4565-4612. doi: 10.3934/dcds.2019188

[9]

Georgios Fotopoulos, Markus Harju, Valery Serov. Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D. Inverse Problems & Imaging, 2013, 7 (1) : 183-197. doi: 10.3934/ipi.2013.7.183

[10]

Satoshi Masaki. A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1481-1531. doi: 10.3934/cpaa.2015.14.1481

[11]

Yongsheng Jiang, Huan-Song Zhou. A sharp decay estimate for nonlinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1723-1730. doi: 10.3934/cpaa.2010.9.1723

[12]

Benjamin Dodson. Global well-posedness and scattering for the defocusing, cubic nonlinear Schrödinger equation when $n = 3$ via a linear-nonlinear decomposition. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1905-1926. doi: 10.3934/dcds.2013.33.1905

[13]

Abdelwahab Bensouilah, Van Duong Dinh, Mohamed Majdoub. Scattering in the weighted $ L^2 $-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2735-2755. doi: 10.3934/cpaa.2019122

[14]

Weiming Liu, Lu Gan. Multi-bump positive solutions of a fractional nonlinear Schrödinger equation in $\mathbb{R}^N$. Communications on Pure & Applied Analysis, 2016, 15 (2) : 413-428. doi: 10.3934/cpaa.2016.15.413

[15]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085

[16]

César E. Torres Ledesma. Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well. Communications on Pure & Applied Analysis, 2016, 15 (2) : 535-547. doi: 10.3934/cpaa.2016.15.535

[17]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

[18]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034

[19]

Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323

[20]

Boris P. Belinskiy, Peter Caithamer. Energy estimate for the wave equation driven by a fractional Gaussian noise. Conference Publications, 2007, 2007 (Special) : 92-101. doi: 10.3934/proc.2007.2007.92

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (110)
  • HTML views (260)
  • Cited by (0)

Other articles
by authors

[Back to Top]