October  2012, 5(5): 903-923. doi: 10.3934/dcdss.2012.5.903

Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation

1. 

Sofia University St. Kl. Ohridski, Faculty of Mathematics and Informatics, Bulgaria

2. 

Technical University of Sofia, Faculty of Applied Mathematics and Informatics, Bulgaria

Received  December 2010 Revised  June 2011 Published  January 2012

We study the Cauchy problem for the focusing time-dependent Schrödinger - Hartree equation $$i \partial_t \psi + \triangle \psi = -({|x|^{-(n-2)}}\ast |\psi|^{\alpha})|\psi|^{\alpha - 2} \psi, \quad \alpha\geq 2,$$ for space dimension $n \geq 3$. We prove the existence of solitary wave solutions and give conditions for formation of singularities in dependence of the values of $\alpha\geq 2$ and the initial data $\psi(0,x)=\psi_0(x)$.
Citation: Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
References:
[1]

R. Agemi, K. Kubota and H. Takamura, On certain integral equations related to nonlinear wave equations,, Hokkaido Math. J., 23 (1994), 241.   Google Scholar

[2]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.   Google Scholar

[3]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003).   Google Scholar

[4]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549.  doi: 10.1007/BF01403504.  Google Scholar

[5]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, in, 1394 (1989), 18.   Google Scholar

[6]

G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations,, Electron. J. Differential Equations, 2004 (2004).   Google Scholar

[7]

V. Georgiev and N. Visciglia, Solitary waves for Klein-Gordon–Maxwell system with external Coulomb potential,, J. Mathematiques Pures et Appliques (9), 84 (2005), 957.  doi: 10.1016/j.matpur.2004.09.016.  Google Scholar

[8]

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

[9]

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

[10]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, Int. Math. Res. Not., 2005 (2005), 2815.  doi: 10.1155/IMRN.2005.2815.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113.   Google Scholar

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[13]

M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243.   Google Scholar

[14]

M. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations,, Trans. Amer. Math. Soc., 333 (1992), 339.   Google Scholar

[15]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93.   Google Scholar

[16]

E. Lieb, The stability of matter and quantum electrodynamics,, Milan J. Math., 71 (2003), 199.  doi: 10.1007/s00032-003-0020-3.  Google Scholar

[17]

E. Lieb and M. Loss, "Analysis,", Graduate Studies in Mathematics, 14 (1997).   Google Scholar

[18]

P.-L. Lions, Some remarks on Hartree equation,, Nonlinear Anal., 5 (1981), 1245.  doi: 10.1016/0362-546X(81)90016-X.  Google Scholar

[19]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109.   Google Scholar

[20]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems,, Commun. Math. Phys., 109 (1987), 33.  doi: 10.1007/BF01205672.  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]

F. Merle, Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two,, Comm. Pure Appl. Math., 49 (1996), 765.  doi: 10.1002/(SICI)1097-0312(199608)49:8<765::AID-CPA1>3.0.CO;2-6.  Google Scholar

[23]

E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", With the assistance of Timothy S. Murphy, 43 (1993).   Google Scholar

[24]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[25]

Y. Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power,, Nonlinear Anal., 15 (1990), 719.  doi: 10.1016/0362-546X(90)90088-X.  Google Scholar

[26]

G. Venkov, Small data global existence and scattering for the mass-critical nonlinear Schrödinger equation with power convolution in $\R^3$,, Cubo, 11 (2009), 15.   Google Scholar

[27]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.  doi: 10.1007/BF01208265.  Google Scholar

[28]

M. Weinstein, The nonlinear Schrödinger equation-singularity formation, stability and dispersion,, in, 99 (1989), 213.   Google Scholar

show all references

References:
[1]

R. Agemi, K. Kubota and H. Takamura, On certain integral equations related to nonlinear wave equations,, Hokkaido Math. J., 23 (1994), 241.   Google Scholar

[2]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313.   Google Scholar

[3]

T. Cazenave, "Semilinear Schrödinger Equations,", Courant Lecture Notes in Mathematics, 10 (2003).   Google Scholar

[4]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations,, Comm. Math. Phys., 85 (1982), 549.  doi: 10.1007/BF01403504.  Google Scholar

[5]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case,, in, 1394 (1989), 18.   Google Scholar

[6]

G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations,, Electron. J. Differential Equations, 2004 (2004).   Google Scholar

[7]

V. Georgiev and N. Visciglia, Solitary waves for Klein-Gordon–Maxwell system with external Coulomb potential,, J. Mathematiques Pures et Appliques (9), 84 (2005), 957.  doi: 10.1016/j.matpur.2004.09.016.  Google Scholar

[8]

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

[9]

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

[10]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited,, Int. Math. Res. Not., 2005 (2005), 2815.  doi: 10.1155/IMRN.2005.2815.  Google Scholar

[11]

T. Kato, On nonlinear Schrödinger equations,, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113.   Google Scholar

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[13]

M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$,, Arch. Rational Mech. Anal., 105 (1989), 243.   Google Scholar

[14]

M. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations,, Trans. Amer. Math. Soc., 333 (1992), 339.   Google Scholar

[15]

E. Lieb, Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation,, Studies in Appl. Math., 57 (): 93.   Google Scholar

[16]

E. Lieb, The stability of matter and quantum electrodynamics,, Milan J. Math., 71 (2003), 199.  doi: 10.1007/s00032-003-0020-3.  Google Scholar

[17]

E. Lieb and M. Loss, "Analysis,", Graduate Studies in Mathematics, 14 (1997).   Google Scholar

[18]

P.-L. Lions, Some remarks on Hartree equation,, Nonlinear Anal., 5 (1981), 1245.  doi: 10.1016/0362-546X(81)90016-X.  Google Scholar

[19]

P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I.,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109.   Google Scholar

[20]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems,, Commun. Math. Phys., 109 (1987), 33.  doi: 10.1007/BF01205672.  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]

F. Merle, Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two,, Comm. Pure Appl. Math., 49 (1996), 765.  doi: 10.1002/(SICI)1097-0312(199608)49:8<765::AID-CPA1>3.0.CO;2-6.  Google Scholar

[23]

E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals,", With the assistance of Timothy S. Murphy, 43 (1993).   Google Scholar

[24]

W. A. Strauss, Existence of solitary waves in higher dimensions,, Comm. Math. Phys., 55 (1977), 149.  doi: 10.1007/BF01626517.  Google Scholar

[25]

Y. Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power,, Nonlinear Anal., 15 (1990), 719.  doi: 10.1016/0362-546X(90)90088-X.  Google Scholar

[26]

G. Venkov, Small data global existence and scattering for the mass-critical nonlinear Schrödinger equation with power convolution in $\R^3$,, Cubo, 11 (2009), 15.   Google Scholar

[27]

M. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates,, Comm. Math. Phys., 87 (): 567.  doi: 10.1007/BF01208265.  Google Scholar

[28]

M. Weinstein, The nonlinear Schrödinger equation-singularity formation, stability and dispersion,, in, 99 (1989), 213.   Google Scholar

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