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Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system
1. | LAMFA, UMR CNRS 7352, Université de Picardie Jules Verne, 33 rue St Leu, 80039, Amiens Cedex |
2. | Department of Mathematics, National Technical University, Zografou Campus 157 80, Athens, Greece |
References:
[1] |
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, \emph{J. Differential Equation, 83 (1990), 85.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger Equations and dissipative dynamical systems,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 211.
|
[3] |
M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system,, \emph{Differential Integral Equations}, 16 (2003), 573.
|
[4] |
N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$,, \emph{Appl. Math. Lett.}, 12 (1999).
doi: 10.1016/S0893-9659(98)00170-0. |
[5] |
J. Ball, Global attractors for damped semilinear wave equations,, \emph{Partial differential equations and applications, 10 (2004), 31.
doi: 10.3934/dcds.2004.10.31. |
[6] |
C. Besse, A relaxation scheme for nonlinear Schrödinger equations},, \emph{SIAM J. Num. Anal.}, 42 (2004), 934.
doi: 10.1137/S0036142901396521. |
[7] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, \emph{J. of Dyn. and Diff. Equ.}, 16 (2004), 469.
doi: 10.1007/s10884-004-4289-x. |
[8] |
M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation,, \emph{J. Comput. Phys.}, 44 (1981), 277.
doi: 10.1016/0021-9991(81)90052-8. |
[9] |
E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation,, \emph{Differential Integral Equations}, 23 (2010), 237.
|
[10] |
J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations,, \emph{J. Math. Pures et Appli.}, 66 (1987), 273.
|
[11] |
O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99.
doi: 10.1080/00036819608840420. |
[12] |
O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 1429.
doi: 10.3934/cpaa.2008.7.1429. |
[13] |
B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$,, \emph{J. Diff. Eq.}, 136 (1997), 356.
doi: 10.1006/jdeq.1996.3242. |
[14] |
J. Hale, Asymptotic behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988).
|
[15] |
A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications, College de France Seminar, (1985).
|
[16] |
N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves,, \emph{Zeitschrift foangewandte Mathematik und Physik ZAMP}, (2005), 218.
doi: 10.1007/s00033-004-2095-2. |
[17] |
O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations,, Cambridge University Press, (1991).
doi: 10.1017/CBO9780511569418. |
[18] |
P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 2 (1995), 357.
doi: 10.1007/BF01261181. |
[19] |
K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, \emph{J. Diff. Eq.}, (2001), 281.
doi: 10.1006/jdeq.2000.3827. |
[20] |
A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,, Handbook of Differential Equations, ().
doi: 10.1016/S1874-5717(08)00003-0. |
[21] |
I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, \emph{Nonlinearity}, 11 (1998), 1369.
doi: 10.1088/0951-7715/11/5/012. |
[22] |
M. N. Poulou and N. M. Stavrakakis, Global Attractor for a Klein-Gordon-Schrödinger Type System in all $R$,, \emph{Nonlinear Analysis: Theory, (): 2548.
doi: 10.1016/j.na.2010.12.009. |
[23] |
M. N. Poulou and N. B. Zographopoulos, Global Attractor for a degenerate Klein - Gordon - Schrödinger Type System,, submitted., (). Google Scholar |
[24] |
G. Raugel, Global attractors in partial differential equations,, \emph{Handbook of dynamical systems}, (2002), 885.
doi: 10.1016/S1874-575X(02)80038-8. |
[25] |
C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Applied Mathematical Sciences vol. 139, (1999).
|
[26] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).
|
[27] |
B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation,, \emph{J. of Math. Phy.}, 40 (1999).
doi: 10.1063/1.532875. |
[28] |
X. Wang, An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications to their attractors,, \emph{Physica D}, 88 (1995), 167.
doi: 10.1016/0167-2789(95)00196-B. |
[29] |
Y. Yan, Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation,, \emph{Nonlinear Anal.}, 20 (1993), 1417.
doi: 10.1016/0362-546X(93)90168-R. |
show all references
References:
[1] |
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains,, \emph{J. Differential Equation, 83 (1990), 85.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
M. Abounouh, H. Al Moatassime, J-P. Chehab, S. Dumont and O. Goubet, Discrete Schrödinger Equations and dissipative dynamical systems,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 211.
|
[3] |
M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system,, \emph{Differential Integral Equations}, 16 (2003), 573.
|
[4] |
N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $R$,, \emph{Appl. Math. Lett.}, 12 (1999).
doi: 10.1016/S0893-9659(98)00170-0. |
[5] |
J. Ball, Global attractors for damped semilinear wave equations,, \emph{Partial differential equations and applications, 10 (2004), 31.
doi: 10.3934/dcds.2004.10.31. |
[6] |
C. Besse, A relaxation scheme for nonlinear Schrödinger equations},, \emph{SIAM J. Num. Anal.}, 42 (2004), 934.
doi: 10.1137/S0036142901396521. |
[7] |
I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping,, \emph{J. of Dyn. and Diff. Equ.}, 16 (2004), 469.
doi: 10.1007/s10884-004-4289-x. |
[8] |
M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation,, \emph{J. Comput. Phys.}, 44 (1981), 277.
doi: 10.1016/0021-9991(81)90052-8. |
[9] |
E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation,, \emph{Differential Integral Equations}, 23 (2010), 237.
|
[10] |
J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations,, \emph{J. Math. Pures et Appli.}, 66 (1987), 273.
|
[11] |
O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99.
doi: 10.1080/00036819608840420. |
[12] |
O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation,, \emph{Comm. in Pure and Applied Analysis}, 7 (2008), 1429.
doi: 10.3934/cpaa.2008.7.1429. |
[13] |
B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$,, \emph{J. Diff. Eq.}, 136 (1997), 356.
doi: 10.1006/jdeq.1996.3242. |
[14] |
J. Hale, Asymptotic behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988).
|
[15] |
A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications, College de France Seminar, (1985).
|
[16] |
N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves,, \emph{Zeitschrift foangewandte Mathematik und Physik ZAMP}, (2005), 218.
doi: 10.1007/s00033-004-2095-2. |
[17] |
O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations,, Cambridge University Press, (1991).
doi: 10.1017/CBO9780511569418. |
[18] |
P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 2 (1995), 357.
doi: 10.1007/BF01261181. |
[19] |
K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, \emph{J. Diff. Eq.}, (2001), 281.
doi: 10.1006/jdeq.2000.3827. |
[20] |
A. Miranville and S. Zelik, Attractors for Dissipative Partial Differential Equations in Bounded and Unbounded Domains,, Handbook of Differential Equations, ().
doi: 10.1016/S1874-5717(08)00003-0. |
[21] |
I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations,, \emph{Nonlinearity}, 11 (1998), 1369.
doi: 10.1088/0951-7715/11/5/012. |
[22] |
M. N. Poulou and N. M. Stavrakakis, Global Attractor for a Klein-Gordon-Schrödinger Type System in all $R$,, \emph{Nonlinear Analysis: Theory, (): 2548.
doi: 10.1016/j.na.2010.12.009. |
[23] |
M. N. Poulou and N. B. Zographopoulos, Global Attractor for a degenerate Klein - Gordon - Schrödinger Type System,, submitted., (). Google Scholar |
[24] |
G. Raugel, Global attractors in partial differential equations,, \emph{Handbook of dynamical systems}, (2002), 885.
doi: 10.1016/S1874-575X(02)80038-8. |
[25] |
C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Applied Mathematical Sciences vol. 139, (1999).
|
[26] |
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997).
|
[27] |
B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation,, \emph{J. of Math. Phy.}, 40 (1999).
doi: 10.1063/1.532875. |
[28] |
X. Wang, An energy equation for the weakly damped driven nonlinear Schrodinger equations and its applications to their attractors,, \emph{Physica D}, 88 (1995), 167.
doi: 10.1016/0167-2789(95)00196-B. |
[29] |
Y. Yan, Attractors and dimensions for discretizations of a weakly damped Schrödinger equations and a sine-Gordon equation,, \emph{Nonlinear Anal.}, 20 (1993), 1417.
doi: 10.1016/0362-546X(93)90168-R. |
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