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On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions
On the Cauchy problem for the Schrödinger-Hartree equation
1. | Department of Mathematics, Northwest Normal University, Lanzhou 730070, China, China |
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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