# American Institute of Mathematical Sciences

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

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##### 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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