-
Previous Article
Quasilinear nonlocal elliptic problems with variable singular exponent
- CPAA Home
- This Issue
- Next Article
Non-linear bi-harmonic Choquard equations
Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia |
$ i\dot u+\Delta^2 u\pm (I_\alpha *|u|^p)|u|^{p-2}u = 0 . $ |
References:
[1] |
T. Boulenger and E. Lenzmann,
Blow-up for bi-harmonic NLS, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544.
doi: 10.24033/asens.2326. |
[2] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[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] |
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. |
[5] |
T. Duyckaerts and S. Roudenko,
Going beyond the threshold: scattering and blow-up in the focusing NLS equation, Commun. Math. Phys., 334 (2015), 1573-1615.
doi: 10.1007/s00220-014-2202-y. |
[6] |
A. Elgart and B. Schlein,
Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[7] |
B. Feng and X. Yuan,
On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.
doi: 10.3934/eect.2015.4.431. |
[8] |
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. |
[9] |
E. P. Gross and E. Meeron, Physics of many-particle systems, Vol. 1, Gordon Breach, New York, (1966), 231–406. |
[10] |
C. D. Guevara,
Global behavior of finite energy solutions to the d-Dimensional focusing non-linear Schrödinger equation, Appl. Math. Res. eXpress., 2 (2014), 177-243.
doi: 10.1002/cta.2381. |
[11] |
Q. Guo,
Scattering for the focusing $L^2$-supercritical and $\dot H^2$-subcritical bi-harmonic NLS equations, Commun. Partial Differ. Equ., 41 (2016), 185-207.
doi: 10.1080/03605302.2015.1116556. |
[12] |
V. I. Karpman,
Stabilization of soliton instabilities by higher-order dispersion: fourth-order non-linear Schrödinger equation, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1016/0375-9601(95)00752-0. |
[13] |
V. I. Karpman and A. G. Shagalov,
Stability of soliton described by non-linear Schrödinger type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[14] |
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. |
[15] |
S. Le Coz,
A note on Berestycki-Cazenave classical instability result for non-linear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463.
doi: 10.1515/ans-2008-0302. |
[16] |
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. |
[17] |
M. Lewin and N. Rougerie,
Derivation of Pekar's polarons from a microscopic model of quantum crystal, SIAM J. Math. Anal., 45 (2013), 1267-1301.
doi: 10.1137/110846312. |
[18] |
E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. |
[19] |
P. L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
V. Moroz and J. V. Schaftingen,
Groundstates of non-linear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[21] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 13 (1955), 116-162.
|
[22] |
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. |
[23] |
T. Saanouni, Scattering threshold for the focusing Choquard equation, Nonlinear Differ. Equ. Appl., 26, (2019), Art. 41.
doi: 10.1007/s00030-019-0587-1. |
[24] |
R. J. Taggart,
Inhomogeneous Strichartz estimates, Forum Math., 22 (2010), 825-853.
doi: 10.1515/FORUM.2010.044. |
show all references
References:
[1] |
T. Boulenger and E. Lenzmann,
Blow-up for bi-harmonic NLS, Ann. Sci. Éc. Norm. Supér., 50 (2017), 503-544.
doi: 10.24033/asens.2326. |
[2] |
Y. Cho and T. Ozawa,
Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.
doi: 10.1142/S0219199709003399. |
[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] |
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. |
[5] |
T. Duyckaerts and S. Roudenko,
Going beyond the threshold: scattering and blow-up in the focusing NLS equation, Commun. Math. Phys., 334 (2015), 1573-1615.
doi: 10.1007/s00220-014-2202-y. |
[6] |
A. Elgart and B. Schlein,
Mean field dynamics of boson stars, Commun. Pure Appl. Math., 60 (2007), 500-545.
doi: 10.1002/cpa.20134. |
[7] |
B. Feng and X. Yuan,
On the Cauchy problem for the Schrödinger-Hartree equation, Evol. Equ. Control Theory, 4 (2015), 431-445.
doi: 10.3934/eect.2015.4.431. |
[8] |
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. |
[9] |
E. P. Gross and E. Meeron, Physics of many-particle systems, Vol. 1, Gordon Breach, New York, (1966), 231–406. |
[10] |
C. D. Guevara,
Global behavior of finite energy solutions to the d-Dimensional focusing non-linear Schrödinger equation, Appl. Math. Res. eXpress., 2 (2014), 177-243.
doi: 10.1002/cta.2381. |
[11] |
Q. Guo,
Scattering for the focusing $L^2$-supercritical and $\dot H^2$-subcritical bi-harmonic NLS equations, Commun. Partial Differ. Equ., 41 (2016), 185-207.
doi: 10.1080/03605302.2015.1116556. |
[12] |
V. I. Karpman,
Stabilization of soliton instabilities by higher-order dispersion: fourth-order non-linear Schrödinger equation, Phys. Rev. E, 53 (1996), 1336-1339.
doi: 10.1016/0375-9601(95)00752-0. |
[13] |
V. I. Karpman and A. G. Shagalov,
Stability of soliton described by non-linear Schrödinger type equations with higher-order dispersion, Phys. D, 144 (2000), 194-210.
doi: 10.1016/S0167-2789(00)00078-6. |
[14] |
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. |
[15] |
S. Le Coz,
A note on Berestycki-Cazenave classical instability result for non-linear Schrödinger equations, Adv. Nonlinear Stud., 8 (2008), 455-463.
doi: 10.1515/ans-2008-0302. |
[16] |
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. |
[17] |
M. Lewin and N. Rougerie,
Derivation of Pekar's polarons from a microscopic model of quantum crystal, SIAM J. Math. Anal., 45 (2013), 1267-1301.
doi: 10.1137/110846312. |
[18] |
E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001. |
[19] |
P. L. Lions,
The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.
doi: 10.1016/0362-546X(80)90016-4. |
[20] |
V. Moroz and J. V. Schaftingen,
Groundstates of non-linear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.
doi: 10.1016/j.jfa.2013.04.007. |
[21] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 13 (1955), 116-162.
|
[22] |
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. |
[23] |
T. Saanouni, Scattering threshold for the focusing Choquard equation, Nonlinear Differ. Equ. Appl., 26, (2019), Art. 41.
doi: 10.1007/s00030-019-0587-1. |
[24] |
R. J. Taggart,
Inhomogeneous Strichartz estimates, Forum Math., 22 (2010), 825-853.
doi: 10.1515/FORUM.2010.044. |
[1] |
Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang. Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4321-4335. doi: 10.3934/dcdss.2021108 |
[2] |
Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170 |
[3] |
Long Wei, Zhijun Qiao, Yang Wang, Shouming Zhou. Conserved quantities, global existence and blow-up for a generalized CH equation. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1733-1748. doi: 10.3934/dcds.2017072 |
[4] |
Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
[5] |
Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 |
[6] |
Akmel Dé Godefroy. Existence, decay and blow-up for solutions to the sixth-order generalized Boussinesq equation. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 117-137. doi: 10.3934/dcds.2015.35.117 |
[7] |
Ronghua Jiang, Jun Zhou. Blow-up and global existence of solutions to a parabolic equation associated with the fraction p-Laplacian. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1205-1226. doi: 10.3934/cpaa.2019058 |
[8] |
Shuyin Wu, Joachim Escher, Zhaoyang Yin. Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation. Discrete and Continuous Dynamical Systems - B, 2009, 12 (3) : 633-645. doi: 10.3934/dcdsb.2009.12.633 |
[9] |
Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 |
[10] |
Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169 |
[11] |
Nadjat Doudi, Salah Boulaaras, Nadia Mezouar, Rashid Jan. Global existence, general decay and blow-up for a nonlinear wave equation with logarithmic source term and fractional boundary dissipation. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022106 |
[12] |
Jinmyong An, Roesong Jang, Jinmyong Kim. Global existence and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022111 |
[13] |
Guangyu Xu, Jun Zhou. Global existence and blow-up of solutions to a singular Non-Newton polytropic filtration equation with critical and supercritical initial energy. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1805-1820. doi: 10.3934/cpaa.2018086 |
[14] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations and Control Theory, 2022, 11 (2) : 515-536. doi: 10.3934/eect.2021011 |
[15] |
Marco Donatelli, Luca Vilasi. Existence of multiple solutions for a fourth-order problem with variable exponent. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2471-2481. doi: 10.3934/dcdsb.2021141 |
[16] |
Shiming Li, Yongsheng Li, Wei Yan. A global existence and blow-up threshold for Davey-Stewartson equations in $\mathbb{R}^3$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1899-1912. doi: 10.3934/dcdss.2016077 |
[17] |
Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 |
[18] |
Bin Li. On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks. Kinetic and Related Models, 2019, 12 (5) : 1131-1162. doi: 10.3934/krm.2019043 |
[19] |
Lili Du, Chunlai Mu, Zhaoyin Xiang. Global existence and blow-up to a reaction-diffusion system with nonlinear memory. Communications on Pure and Applied Analysis, 2005, 4 (4) : 721-733. doi: 10.3934/cpaa.2005.4.721 |
[20] |
Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]