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October  2021, 20(10): 3621-3638. doi: 10.3934/cpaa.2021124

Energy scattering for the focusing fractional generalized Hartree equation

Tarek Saanouni 1,2,,

1. 

Departement of Mathematics, College of Sciences and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia
2. 

University of Tunis El Manar, Faculty of Science of Tunis, LR03ES04 partial differential Equations and applications, 2092 Tunis, Tunisia

* Corresponding author

Received  February 2021 Revised  June 2021 Published  October 2021 Early access  July 2021

This note studies the asymptotics of radial global solutions to the non-linear fractional Schrödinger equation
$ i\dot u-(-\Delta)^s u+|u|^{p-2}(I_\alpha *|u|^p)u = 0. $
Indeed, using a new method due to Dodson-Murphy [10], one proves that, in the inter-critical regime, under the ground state threshold, the radial global solutions scatter in the energy space.
Keywords: Fractional Hartree equation, global existence, scattering.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B35.
Citation: Tarek Saanouni. Energy scattering for the focusing fractional generalized Hartree equation. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3621-3638. doi: 10.3934/cpaa.2021124
References:
[1]

R. Adams, Sobolev Spaces, Academic, New York, 1975.  Google Scholar

[2]

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

[3]

M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[4]

Y. Cho, G. Hwang and T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differ. Equ., 23, (2018), 161–192.  Google Scholar

[5]

Y. Cho, G. Hwang and Y-S. Shim, Energy concentration of the focusing energy-critical fNLS, J. Math. Anal. Appl., 437 (2016), 310-329.  doi: 10.1016/j.jmaa.2015.12.060.  Google Scholar

[6]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020.  doi: 10.1512/iumj.2013.62.4970.  Google Scholar

[7]

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

[8]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128.  doi: 10.3934/cpaa.2011.10.1121.  Google Scholar

[9]

P. D'avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Model. Meth. Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.  Google Scholar

[10]

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

[11]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[12]

B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804.  doi: 10.3934/cpaa.2018085.  Google Scholar

[13]

B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364.  doi: 10.1016/j.jmaa.2017.11.060.  Google Scholar

[14]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.  doi: 10.1007/BF01214768.  Google Scholar

[15]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 15 (2018), 265-282.  doi: 10.4310/dpde.2018.v15.n4.a2.  Google Scholar

[16]

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 equations, J. Anal. Math., 124 (2014), 1-38.  doi: 10.1007/s11854-014-0025-6.  Google Scholar

[17]

J. Holmer and S. Roudenko, A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation, Commun. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.  Google Scholar

[18]

C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201, (2008), 147–212. doi: 10.1007/s11511-008-0031-6.  Google Scholar

[19]

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

[20]

N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A., 268 (2000), 298-304.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[21]

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

[22]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.  Google Scholar

[23]

E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.2307/3621022.  Google Scholar

[24]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[25]

C. Peng and D. Zhao, Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation, Discr. Contin. Dyn. Systems-B, 24 (2019), 3335-3356.  doi: 10.3934/dcdsb.2018323.  Google Scholar

[26]

R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256.  Google Scholar

[27]

T. Saanouni, Strong instability of standing waves for the fractional Choquard equation, J. Math. Phys., 59 (2018), 081509. doi: 10.1063/1.5043473.  Google Scholar

[28]

T. Saanouni, A note on the fractional Schrödinger equation of Choquard type, J. Math. Anal. Appl., 470 (2019), 1004-1029.  doi: 10.1016/j.jmaa.2018.10.045.  Google Scholar

[29]

T. Saanouni, Potential well theory for the focusing fractional Choquard equation, J. Math. Phys., 61 (2020), 061502. doi: 10.1063/5.0002234.  Google Scholar

[30]

Z. Shen, F. Gao and M. Yang, Ground states for nonlinear fractional Choquard equations with general non-linearities, Math. Meth. App. Sci., 39 (2016), 4082-4098.  doi: 10.1002/mma.3849.  Google Scholar

[31]

C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discr. Cont. Dyn. Syst., 38 (2018), 2207-2228.  doi: 10.3934/dcds.2018091.  Google Scholar

[32]

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

[33]

S. Zhu, Existence of Stable Standing Waves for the Fractional Schrödinger Equations with Combined Non-linearities, J. Evol. Equ., 17 (2017), 1003-1021.  doi: 10.1007/s00028-016-0363-1.  Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces, Academic, New York, 1975.  Google Scholar

[2]

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

[3]

M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[4]

Y. Cho, G. Hwang and T. Ozawa, On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differ. Equ., 23, (2018), 161–192.  Google Scholar

[5]

Y. Cho, G. Hwang and Y-S. Shim, Energy concentration of the focusing energy-critical fNLS, J. Math. Anal. Appl., 437 (2016), 310-329.  doi: 10.1016/j.jmaa.2015.12.060.  Google Scholar

[6]

Y. Cho and S. Lee, Strichartz estimates in spherical coordinates, Indiana Univ. Math. J., 62 (2013), 991-1020.  doi: 10.1512/iumj.2013.62.4970.  Google Scholar

[7]

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

[8]

Y. Cho, T. Ozawa and S. Xia, Remarks on some dispersive estimates, Commun. Pure Appl. Anal., 10 (2011), 1121-1128.  doi: 10.3934/cpaa.2011.10.1121.  Google Scholar

[9]

P. D'avenia, G. Siciliano and M. Squassina, On fractional Choquard equations, Math. Model. Meth. Appl. Sci., 25 (2015), 1447-1476.  doi: 10.1142/S0218202515500384.  Google Scholar

[10]

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

[11]

A. Elgart and B. Schlein, Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.  doi: 10.1002/cpa.20134.  Google Scholar

[12]

B. Feng, On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities, Commun. Pure Appl. Anal., 17 (2018), 1785-1804.  doi: 10.3934/cpaa.2018085.  Google Scholar

[13]

B. Feng and H. Zhang, Stability of standing waves for the fractional Schrödinger-Hartree equation, J. Math. Anal. Appl., 460 (2018), 352-364.  doi: 10.1016/j.jmaa.2017.11.060.  Google Scholar

[14]

J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.  doi: 10.1007/BF01214768.  Google Scholar

[15]

Z. Guo, Y. Sire, Y. Wang and L. Zhao, On the energy-critical fractional Schrödinger equation in the radial case, Dyn. Partial Differ. Equ., 15 (2018), 265-282.  doi: 10.4310/dpde.2018.v15.n4.a2.  Google Scholar

[16]

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 equations, J. Anal. Math., 124 (2014), 1-38.  doi: 10.1007/s11854-014-0025-6.  Google Scholar

[17]

J. Holmer and S. Roudenko, A Sharp Condition for Scattering of the Radial 3D Cubic Nonlinear Schrödinger Equation, Commun. Math. Phys., 282 (2008), 435-467.  doi: 10.1007/s00220-008-0529-y.  Google Scholar

[18]

C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201, (2008), 147–212. doi: 10.1007/s11511-008-0031-6.  Google Scholar

[19]

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

[20]

N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A., 268 (2000), 298-304.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[21]

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

[22]

E. Lenzmann, Well-posedness for semi-relativistic Hartree equations of critical type, Math. Phys. Anal. Geom., 10 (2007), 43-64.  doi: 10.1007/s11040-007-9020-9.  Google Scholar

[23]

E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2001. doi: 10.2307/3621022.  Google Scholar

[24]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[25]

C. Peng and D. Zhao, Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation, Discr. Contin. Dyn. Systems-B, 24 (2019), 3335-3356.  doi: 10.3934/dcdsb.2018323.  Google Scholar

[26]

R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc., 356 (1998), 1927-1939.  doi: 10.1098/rsta.1998.0256.  Google Scholar

[27]

T. Saanouni, Strong instability of standing waves for the fractional Choquard equation, J. Math. Phys., 59 (2018), 081509. doi: 10.1063/1.5043473.  Google Scholar

[28]

T. Saanouni, A note on the fractional Schrödinger equation of Choquard type, J. Math. Anal. Appl., 470 (2019), 1004-1029.  doi: 10.1016/j.jmaa.2018.10.045.  Google Scholar

[29]

T. Saanouni, Potential well theory for the focusing fractional Choquard equation, J. Math. Phys., 61 (2020), 061502. doi: 10.1063/5.0002234.  Google Scholar

[30]

Z. Shen, F. Gao and M. Yang, Ground states for nonlinear fractional Choquard equations with general non-linearities, Math. Meth. App. Sci., 39 (2016), 4082-4098.  doi: 10.1002/mma.3849.  Google Scholar

[31]

C. Sun, H. Wang, X. Yao and J. Zheng, Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discr. Cont. Dyn. Syst., 38 (2018), 2207-2228.  doi: 10.3934/dcds.2018091.  Google Scholar

[32]

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

[33]

S. Zhu, Existence of Stable Standing Waves for the Fractional Schrödinger Equations with Combined Non-linearities, J. Evol. Equ., 17 (2017), 1003-1021.  doi: 10.1007/s00028-016-0363-1.  Google Scholar

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