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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 & 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.  Google Scholar

[2]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.  doi: 10.1142/S0219199709003399.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. P. Gross and E. Meeron, Physics of many-particle systems, Vol. 1, Gordon Breach, New York, (1966), 231–406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[19]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[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.  Google Scholar

[21]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 13 (1955), 116-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. J. Taggart, Inhomogeneous Strichartz estimates, Forum Math., 22 (2010), 825-853.  doi: 10.1515/FORUM.2010.044.  Google Scholar

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.  Google Scholar

[2]

Y. Cho and T. Ozawa, Sobolev inequalities with symmetry, Commun. Contemp. Math., 11 (2009), 355-365.  doi: 10.1142/S0219199709003399.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

E. P. Gross and E. Meeron, Physics of many-particle systems, Vol. 1, Gordon Breach, New York, (1966), 231–406.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[19]

P. L. Lions, The Choquard equation and related questions, Nonlinear Anal., 4 (1980), 1063-1072.  doi: 10.1016/0362-546X(80)90016-4.  Google Scholar

[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.  Google Scholar

[21]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 13 (1955), 116-162.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. J. Taggart, Inhomogeneous Strichartz estimates, Forum Math., 22 (2010), 825-853.  doi: 10.1515/FORUM.2010.044.  Google Scholar

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