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October  2021, 20(10): 3637-3654. 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 and Applied Analysis, 2021, 20 (10) : 3637-3654. doi: 10.3934/cpaa.2021124
References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[21]

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

[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.

[23]

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

[24]

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

[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.

[26]

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

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[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.

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