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Parabolic problems in generalized Sobolev spaces
Energy scattering for the focusing fractional generalized Hartree equation
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 |
$ i\dot u-(-\Delta)^s u+|u|^{p-2}(I_\alpha *|u|^p)u = 0. $ |
References:
[1] | |
[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] | |
[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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