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

Non-linear bi-harmonic Choquard equations

Tarek Saanouni

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.
Keywords: Fourth-order Choquard equation, Gagliardo-Nirenberg inequality, global existence, scattering, blow-up.
Mathematics Subject Classification: 35Q55.
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'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.

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