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December  2015, 4(4): 431-445. doi: 10.3934/eect.2015.4.431

On the Cauchy problem for the Schrödinger-Hartree equation

Binhua Feng 1, and  Xiangxia Yuan 1,

1. 

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China, China

Received  August 2015 Revised  October 2015 Published  November 2015

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.
Keywords: minimal blow-up solutions, blow-up rate., mass concentration, Schrödinger-Hartree equation, well-posedness.
Mathematics Subject Classification: 35Q51, 35Q5.
Citation: Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431
References:
[1]

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

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[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

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

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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.  doi: 10.1007/s002200100579.  Google Scholar

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

[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.  doi: 10.3934/dcdss.2012.5.903.  Google Scholar

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

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

[13]

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

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

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

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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.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

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

[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.  doi: 10.1016/j.jfa.2007.09.008.  Google Scholar

[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.  doi: 10.4064/cm119-1-2.  Google Scholar

[24]

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

[25]

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

[26]

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

show all references

References:
[1]

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

[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.  doi: 10.1016/j.jmaa.2014.02.063.  Google Scholar

[3]

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

[4]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[5]

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

[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.  doi: 10.1007/s002200100579.  Google Scholar

[7]

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

[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.  doi: 10.3934/dcdss.2012.5.903.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.  doi: 10.1137/110846312.  Google Scholar

[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.  doi: 10.1016/j.jde.2010.10.022.  Google Scholar

[16]

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

[17]

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

[18]

E. Lieb, Analysis,, 2nd ed., (2001).  doi: 10.1090/gsm/014.  Google Scholar

[19]

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

[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.  doi: 10.1016/j.jde.2003.12.002.  Google Scholar

[21]

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

[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.  doi: 10.1016/j.jfa.2007.09.008.  Google Scholar

[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.  doi: 10.4064/cm119-1-2.  Google Scholar

[24]

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

[25]

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

[26]

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

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