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

On the Cauchy problem for the Schrödinger-Hartree equation

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.
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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[18]

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

[19]

P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063. 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. 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. 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. 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. 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. doi: 10.1016/j.jfa.2013.04.007.

[25]

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

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[11]

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

[18]

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

[19]

P.-L. Lions, The Choquard equation and related questions,, Nonlinear Anal., 4 (1980), 1063. 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. 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. 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. 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. 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. doi: 10.1016/j.jfa.2013.04.007.

[25]

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

[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.

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