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On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators
Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730070, China |
2. | School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741000, China |
$ i\partial_{t}u = (-\Delta)^{s}u-|x|^{-b}|u|^{2\sigma}u, \, \, \, (t, x)\in \mathbb{R} \times \mathbb{R}^{N}, $ |
$ \frac{1}{2}<s<1, $ |
$ N\geq2 $ |
$ \frac{2s-b}{N}\leq \sigma<\frac{2s-b}{N-2s} $ |
$ H^{s}(\mathbb{R}^{N}) $ |
$ \mathbb{R}^{N} $ |
$ (-\Delta)^{s} $ |
$ \mathbb{R}^{N} $ |
$ L^{2} $ |
$ L^{2} $ |
$ H^{s} $ |
References:
[1] |
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. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[3] |
J. Q. Chen and B. L. Guo,
Sharp constant of improved Gagliardo-Nirenberg inequality and its application, Annali di Matematica, 190 (2011), 341-354.
doi: 10.1007/s10231-010-0152-3. |
[4] |
J. Q. Chen and B. L. Guo,
Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367.
doi: 10.3934/dcdsb.2007.8.357. |
[5] |
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. |
[6] |
Y. Cho and T. Ozawa,
Short-range scattering of Hartree type fractional NLS Ⅱ, Nonlinear Anal., 157 (2017), 62-75.
doi: 10.1016/j.na.2017.03.005. |
[7] |
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. |
[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, H. Hajaiej, G. Hwang and T. Ozawa,
On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.
doi: 10.3934/cpaa.2014.13.1267. |
[10] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[11] |
L. G. Farah,
Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.
doi: 10.1007/s00028-015-0298-y. |
[12] |
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. |
[13] |
B. H. Feng and H. Z. Zhang, Ground states for the fractional Schrödinger equation, Electron. J. Differ. Eq., 127 (2013), 11pp. |
[14] |
B. H. 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. |
[15] |
B. H. Feng and H. Z. 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. |
[16] |
F. Genoud,
An Inhomogeneous L2-critical nonlinear Schrödinger equations, Z.anal.anwend., 31 (2012), 283-290.
doi: 10.4171/ZAA/1460. |
[17] |
B. L. Guo and Z. H. Huo,
Global well-posedness for the fractional nonlinear Schrödinger equations, Comm.Partial Differential Equations, 36 (2010), 247-255.
doi: 10.1080/03605302.2010.503769. |
[18] |
Z. H. Guo and Y. Z. Wang,
Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, Journal d'Analyse Mathématique, 124 (2014), 1-38.
doi: 10.1007/s11854-014-0025-6. |
[19] |
J. Holmer and S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, AMRX.Appl.Math.Res.Express, (2007), Art. ID abm004, 31pp. |
[20] |
J. Holmer and S. Roudenko,
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm.Math.Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[21] |
C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364, 26pp.
doi: 10.1098/rspa.2014.0364. |
[22] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the L2-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[23] |
N. Laskin,
Fractional quantum mechanics and Lèvy path integrals, Physics Letter A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[24] |
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[25] |
F. Merle, Nonxistence of minimal blow up solutions of equation $i\partial_{t}u = -\Delta u-k(x)|u|^\frac{4}{N}u$ in $\mathbb{R}^{N}$, Ann.Inst.Henri Poincaré, 64 (1996), 33–85. |
[26] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[27] |
C. M. Peng and Q. H. Shi, Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp.
doi: 10.1063/1.5021689. |
[28] |
P. Raphaël and S. Jermeie,
Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass-critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546.
doi: 10.1090/S0894-0347-2010-00688-1. |
[29] |
T. Saanouni, Remark on the inhomogeneous fractional nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 081503, 14 pp.
doi: 10.1063/1.4960045. |
[30] |
Q. H. Shi and S. Wang,
Nonrelativistic approximation in the energy space for KGS system, J. Math. Anal. Appl., 462 (2018), 1242-1253.
doi: 10.1016/j.jmaa.2018.02.039. |
[31] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
doi: 10.1007/BF01208265. |
[32] |
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. |
[33] |
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. |
[34] |
S. H. 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] |
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. |
[2] |
T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics vol. 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/cln/010. |
[3] |
J. Q. Chen and B. L. Guo,
Sharp constant of improved Gagliardo-Nirenberg inequality and its application, Annali di Matematica, 190 (2011), 341-354.
doi: 10.1007/s10231-010-0152-3. |
[4] |
J. Q. Chen and B. L. Guo,
Sharp global existence and blowing up results for inhomogeneous Schrödinger equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 357-367.
doi: 10.3934/dcdsb.2007.8.357. |
[5] |
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. |
[6] |
Y. Cho and T. Ozawa,
Short-range scattering of Hartree type fractional NLS Ⅱ, Nonlinear Anal., 157 (2017), 62-75.
doi: 10.1016/j.na.2017.03.005. |
[7] |
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. |
[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, H. Hajaiej, G. Hwang and T. Ozawa,
On the orbital stability of fractional Schrödinger equations, Commun. Pure Appl. Anal., 13 (2014), 1267-1282.
doi: 10.3934/cpaa.2014.13.1267. |
[10] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[11] |
L. G. Farah,
Global well-posedness and blow-up on the energy space for the inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ., 16 (2016), 193-208.
doi: 10.1007/s00028-015-0298-y. |
[12] |
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. |
[13] |
B. H. Feng and H. Z. Zhang, Ground states for the fractional Schrödinger equation, Electron. J. Differ. Eq., 127 (2013), 11pp. |
[14] |
B. H. 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. |
[15] |
B. H. Feng and H. Z. 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. |
[16] |
F. Genoud,
An Inhomogeneous L2-critical nonlinear Schrödinger equations, Z.anal.anwend., 31 (2012), 283-290.
doi: 10.4171/ZAA/1460. |
[17] |
B. L. Guo and Z. H. Huo,
Global well-posedness for the fractional nonlinear Schrödinger equations, Comm.Partial Differential Equations, 36 (2010), 247-255.
doi: 10.1080/03605302.2010.503769. |
[18] |
Z. H. Guo and Y. Z. Wang,
Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equation, Journal d'Analyse Mathématique, 124 (2014), 1-38.
doi: 10.1007/s11854-014-0025-6. |
[19] |
J. Holmer and S. Roudenko, On blow-up solutions to the 3D cubic nonlinear Schrödinger equation, AMRX.Appl.Math.Res.Express, (2007), Art. ID abm004, 31pp. |
[20] |
J. Holmer and S. Roudenko,
A sharp condition for scattering of the radial 3D cubic nonlinear Schrödinger equation, Comm.Math.Phys., 282 (2008), 435-467.
doi: 10.1007/s00220-008-0529-y. |
[21] |
C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364, 26pp.
doi: 10.1098/rspa.2014.0364. |
[22] |
J. Krieger, E. Lenzmann and P. Raphaël,
Nondispersive solutions to the L2-critical half-wave equation, Arch. Ration. Mech. Anal., 209 (2013), 61-129.
doi: 10.1007/s00205-013-0620-1. |
[23] |
N. Laskin,
Fractional quantum mechanics and Lèvy path integrals, Physics Letter A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[24] |
N. Laskin, Fractional Schrödinger equations, Physics Review E, 66 (2002), 056108, 7pp.
doi: 10.1103/PhysRevE.66.056108. |
[25] |
F. Merle, Nonxistence of minimal blow up solutions of equation $i\partial_{t}u = -\Delta u-k(x)|u|^\frac{4}{N}u$ in $\mathbb{R}^{N}$, Ann.Inst.Henri Poincaré, 64 (1996), 33–85. |
[26] |
T. Ogawa and Y. Tsutsumi,
Blow-up of $H^1$ solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.
doi: 10.1016/0022-0396(91)90052-B. |
[27] |
C. M. Peng and Q. H. Shi, Stability of standing wave for the fractional nonlinear Schrödinger equation, J. Math. Phys., 59 (2018), 011508, 11pp.
doi: 10.1063/1.5021689. |
[28] |
P. Raphaël and S. Jermeie,
Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass-critical NLS, J. Amer. Math. Soc., 24 (2011), 471-546.
doi: 10.1090/S0894-0347-2010-00688-1. |
[29] |
T. Saanouni, Remark on the inhomogeneous fractional nonlinear Schrödinger equations, J. Math. Phys., 57 (2016), 081503, 14 pp.
doi: 10.1063/1.4960045. |
[30] |
Q. H. Shi and S. Wang,
Nonrelativistic approximation in the energy space for KGS system, J. Math. Anal. Appl., 462 (2018), 1242-1253.
doi: 10.1016/j.jmaa.2018.02.039. |
[31] |
M. I. Weinstein,
Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
doi: 10.1007/BF01208265. |
[32] |
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. |
[33] |
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. |
[34] |
S. H. 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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