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On fractional nonlinear Schrödinger equation with combined power-type nonlinearities
1. | Laboratoire Paul Painlevé UMR 8524, Université de Lille CNRS, 59655 Villeneuve d'Ascq Cedex, France |
2. | Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam |
3. | Department of Mathematics, Northwest Normal University, Lanzhou, China |
$ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $ |
$ \frac{d}{2d-1} \leq s <1 $ |
$ 0 < \alpha_1 <\alpha_2 < \frac{4s}{d-2s} $ |
$ H^s $ |
$ \dot{H}^{s_c}\cap \dot{H}^s $ |
$ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $ |
$ H^s $ |
$ L^2 $ |
$ L^2 $ |
$ \dot{H}^{s_c} \cap \dot{H}^s $ |
$ L^2 $ |
$ L^2 $ |
$ L^2 $ |
$ L^2 $ |
References:
[1] |
J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976. |
[2] |
S. Bhattarai,
On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229.
doi: 10.1016/j.jde.2017.04.034. |
[3] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.
doi: 10.1016/j.jfa.2016.08.011. |
[4] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lectures Notes in Mathematics 10, New York, AMS, 2003.
doi: 10.1090/cln/010. |
[6] |
X. Cheng, C. Miao and L. Zhao,
Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934.
doi: 10.1016/j.jde.2016.04.031. |
[7] |
Y. Cho,
Short-range scattering of Hartree type fractional NLS, J. Differential Equations, 262 (2017), 116-144.
doi: 10.1016/j.jde.2016.09.025. |
[8] |
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa,
On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224.
doi: 10.1619/fesi.56.193. |
[9] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Profile decompositions and blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29.
doi: 10.1016/j.na.2013.03.002. |
[10] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479.
doi: 10.1017/S030821051300142X. |
[11] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[12] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Profile decompositions of fractional Schrödinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037.
doi: 10.1016/j.jde.2014.01.030. |
[13] |
Y. Cho, G. Hwang and T. Ozawa,
On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differential Equations, 23 (2018), 161-192.
|
[14] |
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. |
[15] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[16] |
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. |
[17] |
M. Christ and I. 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. |
[18] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
doi: 10.1002/cpa.20029. |
[19] |
V. D. Dinh,
Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525.
|
[20] |
V. D. Dinh,
On blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Commun. Pure Appl. Anal., 18 (2019), 689-708.
|
[21] |
V. D. Dinh, A study on blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation, J. Math. Phys., 59 (2018), 071506, 25pp.
doi: 10.1063/1.5027713. |
[22] |
B. Feng,
On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220.
doi: 10.1007/s00028-017-0397-z. |
[23] |
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. |
[24] |
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. |
[25] |
B. Feng and H. Zhang,
Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507.
doi: 10.1016/j.camwa.2017.12.025. |
[26] |
R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ \mathbb R$, Acta Math.. |
[27] |
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. |
[28] |
J. Fröhlich, G. Jonsson and E. Lenzmann,
Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30.
doi: 10.1007/s00220-007-0272-9. |
[29] |
J. Ginibre and G. Velo,
The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.
doi: 10.1007/BF01168155. |
[30] |
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. |
[31] |
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. |
[32] |
Q. Guo and S. Zhu,
Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differential Equations, 264 (2018), 2802-2832.
doi: 10.1016/j.jde.2017.11.001. |
[33] |
Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. |
[34] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.
doi: 10.3934/cpaa.2015.14.2265. |
[35] |
Y. Ke,
Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861.
doi: 10.1016/j.jmaa.2011.09.039. |
[36] |
K. Kirkpatrick, E. Lenzmann and G. Staffilani,
On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591.
doi: 10.1007/s00220-012-1621-x. |
[37] |
N. Laskin,
Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304.
doi: 10.1016/S0375-9601(00)00201-2. |
[38] |
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[39] |
F. Merle and P. Raphaël,
Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978.
doi: 10.1353/ajm.0.0012. |
[40] |
C. Miao, G. Xu and L. Zhao,
The dynamics of the 3D radial NLS with combined terms, Comm. Math. Phys., 318 (2013), 767-808.
doi: 10.1007/s00220-013-1677-2. |
[41] |
C. Miao, T. Zhao and J. Zheng, On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations, 56 (2017), Art. 179, 39 pp.
doi: 10.1007/s00526-017-1264-z. |
[42] |
C. Peng and Q. Shi, Stability of standing waves for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp.
doi: 10.1063/1.5021689. |
[43] |
C. Sun, H. Wang, X. Yao and J. Zheng,
Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228.
doi: 10.3934/dcds.2018091. |
[44] |
T. Tao, M. Visan and X. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805. |
[45] |
H. Triebel, Theory of Function Spaces, Basel Birkhäuser, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[46] |
G. X. Xu and J. W. Yang,
Long time dynamics of the 3D radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.), 32 (2016), 521-540.
doi: 10.1007/s10114-016-5401-y. |
[47] |
J. Zhang and S. Zhu,
Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030.
doi: 10.1007/s10884-015-9477-3. |
[48] |
S. Zhu,
On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531.
doi: 10.1016/j.jde.2016.04.007. |
[49] |
S. Zhu,
Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021.
doi: 10.1007/s00028-016-0363-1. |
show all references
References:
[1] |
J. Bergh and J. Löfstöm, Interpolation Spaces-An Introduction, Springer-Verlag, Berlin, 1976. |
[2] |
S. Bhattarai,
On fractional Schrödinger systems of Choquard type, J. Differential Equations, 263 (2017), 3197-3229.
doi: 10.1016/j.jde.2017.04.034. |
[3] |
T. Boulenger, D. Himmelsbach and E. Lenzmann,
Blowup for fractional NLS, J. Funct. Anal., 271 (2016), 2569-2603.
doi: 10.1016/j.jfa.2016.08.011. |
[4] |
T. Cazenave and F. B. Weissler,
The Cauchy problem for the critical nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.
doi: 10.1016/0362-546X(90)90023-A. |
[5] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lectures Notes in Mathematics 10, New York, AMS, 2003.
doi: 10.1090/cln/010. |
[6] |
X. Cheng, C. Miao and L. Zhao,
Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case, J. Differential Equations, 261 (2016), 2881-2934.
doi: 10.1016/j.jde.2016.04.031. |
[7] |
Y. Cho,
Short-range scattering of Hartree type fractional NLS, J. Differential Equations, 262 (2017), 116-144.
doi: 10.1016/j.jde.2016.09.025. |
[8] |
Y. Cho, H. Hajaiej, G. Hwang and T. Ozawa,
On the Cauchy problem of fractional Schrödinger equations with Hartree type nonlimearity, Funkcial. Ekvac., 56 (2013), 193-224.
doi: 10.1619/fesi.56.193. |
[9] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Profile decompositions and blow-up phenomena of mass critical fractional Schrödinger equations, Nonlinear Anal., 86 (2013), 12-29.
doi: 10.1016/j.na.2013.03.002. |
[10] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
On finite time blow-up for the mass-critical Hartree equations, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 467-479.
doi: 10.1017/S030821051300142X. |
[11] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Well-posedness and ill-posedness for the cubic fractional Schrödinger equations, Discrete Contin. Dyn. Syst., 35 (2015), 2863-2880.
doi: 10.3934/dcds.2015.35.2863. |
[12] |
Y. Cho, G. Hwang, S. Kwon and S. Lee,
Profile decompositions of fractional Schrödinger equations with angularly regular data, J. Differential Equations, 256 (2014), 3011-3037.
doi: 10.1016/j.jde.2014.01.030. |
[13] |
Y. Cho, G. Hwang and T. Ozawa,
On the focusing energy-critical fractional nonlinear Schrödinger equations, Adv. Differential Equations, 23 (2018), 161-192.
|
[14] |
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. |
[15] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[16] |
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. |
[17] |
M. Christ and I. 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. |
[18] |
J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao,
Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $ \mathbb R^3$, Comm. Pure Appl. Math., 57 (2004), 987-1014.
doi: 10.1002/cpa.20029. |
[19] |
V. D. Dinh,
Well-posedness of nonlinear fractional Schrödinger and wave equations in Sobolev spaces, Int. J. Appl. Math., 31 (2018), 483-525.
|
[20] |
V. D. Dinh,
On blowup solutions to the focusing mass-critical nonlinear fractional Schrödinger equation, Commun. Pure Appl. Anal., 18 (2019), 689-708.
|
[21] |
V. D. Dinh, A study on blowup solutions to the focusing $L^2$-supercritical nonlinear fractional Schrödinger equation, J. Math. Phys., 59 (2018), 071506, 25pp.
doi: 10.1063/1.5027713. |
[22] |
B. Feng,
On the blow-up solutions for the nonlinear Schrödinger equation with combined power-type nonlinearities, J. Evol. Equ., 18 (2018), 203-220.
doi: 10.1007/s00028-017-0397-z. |
[23] |
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. |
[24] |
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. |
[25] |
B. Feng and H. Zhang,
Stability of standing waves for the fractional Schrödinger-Choquard equation, Comput. Math. Appl., 75 (2018), 2499-2507.
doi: 10.1016/j.camwa.2017.12.025. |
[26] |
R. L. Frank and E. Lenzmann, Uniqueness of nonlinear ground states for fractional Laplacians in $ \mathbb R$, Acta Math.. |
[27] |
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. |
[28] |
J. Fröhlich, G. Jonsson and E. Lenzmann,
Boson stars as solitary waves, Comm. Math. Phys., 274 (2007), 1-30.
doi: 10.1007/s00220-007-0272-9. |
[29] |
J. Ginibre and G. Velo,
The global Cauchy problem for the nonlinear Klein-Gordon equation, Math. Z., 189 (1985), 487-505.
doi: 10.1007/BF01168155. |
[30] |
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. |
[31] |
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. |
[32] |
Q. Guo and S. Zhu,
Sharp threshold of blow-up and scattering for the fractional Hartree equation, J. Differential Equations, 264 (2018), 2802-2832.
doi: 10.1016/j.jde.2017.11.001. |
[33] |
Q. Guo and S. Zhu, Sharp criteria of scattering for the fractional NLS, preprint, arXiv: 1706.02549. |
[34] |
Y. Hong and Y. Sire,
On fractional Schrödinger equations in Sobolev spaces, Commun. Pure Appl. Anal., 14 (2015), 2265-2282.
doi: 10.3934/cpaa.2015.14.2265. |
[35] |
Y. Ke,
Remark on the Strichartz estimates in the radial case, J. Math. Anal. Appl., 387 (2012), 857-861.
doi: 10.1016/j.jmaa.2011.09.039. |
[36] |
K. Kirkpatrick, E. Lenzmann and G. Staffilani,
On the continuum limit for discrete NLS with long-range lattice interactions, Comm. Math. Phys., 317 (2013), 563-591.
doi: 10.1007/s00220-012-1621-x. |
[37] |
N. Laskin,
Fractional Quantum Mechanics and Lévy Path Integrals, Physics Letter A, 268 (2000), 298-304.
doi: 10.1016/S0375-9601(00)00201-2. |
[38] |
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[39] |
F. Merle and P. Raphaël,
Blow up of critical norm for some radial $L^2$ super critical nonlinear Schrödinger equations, Amer. J. Math., 130 (2008), 945-978.
doi: 10.1353/ajm.0.0012. |
[40] |
C. Miao, G. Xu and L. Zhao,
The dynamics of the 3D radial NLS with combined terms, Comm. Math. Phys., 318 (2013), 767-808.
doi: 10.1007/s00220-013-1677-2. |
[41] |
C. Miao, T. Zhao and J. Zheng, On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold, Calc. Var. Partial Differential Equations, 56 (2017), Art. 179, 39 pp.
doi: 10.1007/s00526-017-1264-z. |
[42] |
C. Peng and Q. Shi, Stability of standing waves for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp.
doi: 10.1063/1.5021689. |
[43] |
C. Sun, H. Wang, X. Yao and J. Zheng,
Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data, Discrete Contin. Dyn. Syst., 38 (2018), 2207-2228.
doi: 10.3934/dcds.2018091. |
[44] |
T. Tao, M. Visan and X. Zhang,
The nonlinear Schrödinger equation with combined power-type nonlinearities, Comm. Partial Differential Equations, 32 (2007), 1281-1343.
doi: 10.1080/03605300701588805. |
[45] |
H. Triebel, Theory of Function Spaces, Basel Birkhäuser, 1983.
doi: 10.1007/978-3-0346-0416-1. |
[46] |
G. X. Xu and J. W. Yang,
Long time dynamics of the 3D radial NLS with the combined terms, Acta Math. Sin. (Engl. Ser.), 32 (2016), 521-540.
doi: 10.1007/s10114-016-5401-y. |
[47] |
J. Zhang and S. Zhu,
Stability of standing waves for the nonlinear fractional Schrödinger equation, J. Dynam. Differential Equations, 29 (2017), 1017-1030.
doi: 10.1007/s10884-015-9477-3. |
[48] |
S. Zhu,
On the blow-up solutions for the nonlinear fractional Schrödinger equation, J. Differential Equations, 261 (2016), 1506-1531.
doi: 10.1016/j.jde.2016.04.007. |
[49] |
S. Zhu,
Existence of stable standing waves for the fractional Schrödinger equations with combined nonlinearities, J. Evol. Equ., 17 (2017), 1003-1021.
doi: 10.1007/s00028-016-0363-1. |
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