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 and 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), 241-276.

[2]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.

[3]

T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.

[4]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.

[5]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in "Nonlinear Semigroups, Partial Differential Equations and Attractors" (Washington, DC, 1987), Lecture Notes in Math., 1394, Springer, Berlin, (1989), 18-29.

[6]

G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 2004, 31 pp.

[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), 957-983. doi: 10.1016/j.matpur.2004.09.016.

[8]

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

[9]

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

[10]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 2005, 2815-2828. doi: 10.1155/IMRN.2005.2815.

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[13]

M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.

[14]

M. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363.

[15]

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

[16]

E. Lieb, The stability of matter and quantum electrodynamics, Milan J. Math., 71 (2003), 199-217. doi: 10.1007/s00032-003-0020-3.

[17]

E. Lieb and M. Loss, "Analysis," Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.

[18]

P.-L. Lions, Some remarks on Hartree equation, Nonlinear Anal., 5 (1981), 1245-1256. doi: 10.1016/0362-546X(81)90016-X.

[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), 109-145.

[20]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672.

[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]

F. Merle, Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two, Comm. Pure Appl. Math., 49 (1996), 765-794. doi: 10.1002/(SICI)1097-0312(199608)49:8<765::AID-CPA1>3.0.CO;2-6.

[23]

E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," With the assistance of Timothy S. Murphy, Princeton Math. Ser., 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.

[24]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

[25]

Y. Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724. doi: 10.1016/0362-546X(90)90088-X.

[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), 15-28.

[27]

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

[28]

M. Weinstein, The nonlinear Schrödinger equation-singularity formation, stability and dispersion, in "The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems" (Boulder, CO, 1987), Contemporary Math., 99, Amer. Math. Soc., Providence, RI, (1989), 213-232.

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), 241-276.

[2]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.

[3]

T. Cazenave, "Semilinear Schrödinger Equations," Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence, RI, 2003.

[4]

T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561. doi: 10.1007/BF01403504.

[5]

T. Cazenave and F. Weissler, Some remarks on the nonlinear Schrödinger equation in the critical case, in "Nonlinear Semigroups, Partial Differential Equations and Attractors" (Washington, DC, 1987), Lecture Notes in Math., 1394, Springer, Berlin, (1989), 18-29.

[6]

G. M. Coclite and V. Georgiev, Solitary waves for Maxwell-Schrödinger equations, Electron. J. Differential Equations, 2004, 31 pp.

[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), 957-983. doi: 10.1016/j.matpur.2004.09.016.

[8]

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

[9]

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

[10]

T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., 2005, 2815-2828. doi: 10.1155/IMRN.2005.2815.

[11]

T. Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., 46 (1987), 113-129.

[12]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.

[13]

M. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\mathbbR^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.

[14]

M. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363.

[15]

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

[16]

E. Lieb, The stability of matter and quantum electrodynamics, Milan J. Math., 71 (2003), 199-217. doi: 10.1007/s00032-003-0020-3.

[17]

E. Lieb and M. Loss, "Analysis," Graduate Studies in Mathematics, 14, American Mathematical Society, Providence, RI, 1997.

[18]

P.-L. Lions, Some remarks on Hartree equation, Nonlinear Anal., 5 (1981), 1245-1256. doi: 10.1016/0362-546X(81)90016-X.

[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), 109-145.

[20]

P.-L. Lions, Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109 (1987), 33-97. doi: 10.1007/BF01205672.

[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]

F. Merle, Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two, Comm. Pure Appl. Math., 49 (1996), 765-794. doi: 10.1002/(SICI)1097-0312(199608)49:8<765::AID-CPA1>3.0.CO;2-6.

[23]

E. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals," With the assistance of Timothy S. Murphy, Princeton Math. Ser., 43, Monographs in Harmonic Analysis, III, Princeton University Press, Princeton, NJ, 1993.

[24]

W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55 (1977), 149-162. doi: 10.1007/BF01626517.

[25]

Y. Tsutsumi, Rate of $L^2$ concentration of blow-up solutions for the nonlinear Schrödinger equation with critical power, Nonlinear Anal., 15 (1990), 719-724. doi: 10.1016/0362-546X(90)90088-X.

[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), 15-28.

[27]

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

[28]

M. Weinstein, The nonlinear Schrödinger equation-singularity formation, stability and dispersion, in "The Connection Between Infinite-Dimensional and Finite-Dimensional Dynamical Systems" (Boulder, CO, 1987), Contemporary Math., 99, Amer. Math. Soc., Providence, RI, (1989), 213-232.

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