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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 and 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), 553-572. 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.

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), 553-572. 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.

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