\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On the Cauchy problem for the Schrödinger-Hartree equation

Abstract Related Papers Cited by
  • In this paper, we undertake a comprehensive study for the Schrödinger-Hartree equation \begin{equation*} iu_t +\Delta u+ \lambda (I_\alpha \ast |u|^{p})|u|^{p-2}u=0, \end{equation*} where $I_\alpha$ is the Riesz potential. Firstly, we address questions related to local and global well-posedness, finite time blow-up. Secondly, we derive the best constant of a Gagliardo-Nirenberg type inequality. Thirdly, the mass concentration is established for all the blow-up solutions in the $L^2$-critical case. Finally, the dynamics of the blow-up solutions with critical mass is in detail investigated in terms of the ground state.
    Mathematics Subject Classification: 35Q51, 35Q55.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    P. d'Avenia and M. Squassina, Soliton dynamics for the Schrödinger-Newton system, Math. Models Methods Appl. Sci., 24 (2014), 553-572.doi: 10.1142/S0218202513500590.

    [2]

    C. Bonanno, P. d'Avenia, M. Ghimenti and M. Squassina, Soliton dynamics for the generalized Choquard equation, J. Math. Anal. Appl., 417 (2014), 180-199.doi: 10.1016/j.jmaa.2014.02.063.

    [3]

    D. Cao and Y. Su, Minimal blow-up solutions of mass-critical inhomogeneous Hartree equation, Journal of Mathematical Physics, 54 (2013), 121511, 25pp.doi: 10.1016/j.jde.2003.12.002.

    [4]

    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.

    [5]

    J. Chen and B. Guo, Strong instability of standing waves for a nonlocal Schrödinger equation, Physica D: Nonlinear Phenomena, 227 (2007), 142-148.doi: 10.1016/j.physd.2007.01.004.

    [6]

    J. Frölich, T.-P. Tsai and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys., 225 (2002), 223-274.doi: 10.1007/s002200100579.

    [7]

    J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z., 170 (1980), 109-136.doi: 10.1007/BF01214768.

    [8]

    H. Genev and G. Venkov, Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 903-923.doi: 10.3934/dcdss.2012.5.903.

    [9]

    P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.doi: 10.1051/cocv:1998107.

    [10]

    R. T. Glassey, On the blowing up of solution to the Cauchy problem for nonlinear Schrödinger operators, J. Math. Phys., 18 (1977), 1794-1797.doi: 10.1063/1.523491.

    [11]

    T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Theor., 46 (1987), 113-129.doi: 10.1016/j.jde.2003.12.002.

    [12]

    T. Kato, On nonlinear Schrödinger equations, II.$H^s$-solutions and unconditional wellposedness, J. d'Analyse. Math., 67 (1995), 281-306.doi: 10.1007/BF02787794.

    [13]

    T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, International Mathematics Research Notices, 46 (2005), 2815-2828.doi: 10.1016/j.jde.2003.12.002.

    [14]

    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.

    [15]

    X. G. Li, J. Zhang, S. Y. Lai and Y. H. Wu, The sharp threshold and limiting profile of blow-up solutions for a Davey-Stewartson system, J. Diff. Eqns., 250 (2011), 2197-2226.doi: 10.1016/j.jde.2010.10.022.

    [16]

    E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976), 93-105.doi: 10.1016/j.jde.2003.12.002.

    [17]

    E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math., 57 (1976), 93-105.doi: 10.1016/j.jde.2003.12.002.

    [18]

    E. Lieb, Analysis, 2nd ed., Graduate Studies in Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 2001.doi: 10.1090/gsm/014.

    [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]

    P.-L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1 and 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.doi: 10.1016/j.jde.2003.12.002.

    [21]

    F. Merle, Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equation with critical power, Duke Math. J., 69 (1993), 427-454.doi: 10.1215/S0012-7094-93-06919-0.

    [22]

    C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal., 253 (2007), 605-627.doi: 10.1016/j.jfa.2007.09.008.

    [23]

    C. Miao, G. Xu and L. Zhao, On the blow-up phenomenon for the mass-critical focusing Hartree equation in $\mathbbR^4$, Colloq. Math., 119 (2010), 23-50.doi: 10.4064/cm119-1-2.

    [24]

    V. Moroz and J. V. Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Funct. Anal., 265 (2013), 153-184.doi: 10.1016/j.jfa.2013.04.007.

    [25]

    R. Penrose, Quantum computation, entanglement and state reduction, Phil. Trans. R. Soc., 356 (1998), 1927-1939.doi: 10.1098/rsta.1998.0256.

    [26]

    M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.doi: 10.1016/j.jde.2003.12.002.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(364) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return