November  2020, 19(11): 5033-5057. doi: 10.3934/cpaa.2020221

Non-linear bi-harmonic Choquard equations

Department of Mathematics, College of Science and Arts in Uglat Asugour, Qassim University, Buraydah, Kingdom of Saudi Arabia

Received  January 2020 Revised  April 2020 Published  November 2020 Early access  July 2020

This note studies the fourth-order Choquard equation
$ i\dot u+\Delta^2 u\pm (I_\alpha *|u|^p)|u|^{p-2}u = 0 . $
In the mass super-critical and energy sub-critical regimes, a sharp threshold of global well-psedness and scattering versus finite time blow-up dichotomy is obtained.
Citation: Tarek Saanouni. Non-linear bi-harmonic Choquard equations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5033-5057. doi: 10.3934/cpaa.2020221
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'AveniaG. 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'AveniaG. 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

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (369)
  • HTML views (129)
  • Cited by (0)

Other articles
by authors

[Back to Top]