# American Institute of Mathematical Sciences

July  2014, 13(4): 1525-1539. doi: 10.3934/cpaa.2014.13.1525

## 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

Received  July 2013 Revised  December 2013 Published  February 2014

We consider a semi-discrete in time relaxation scheme to discretize a damped forced nonlinear Klein-Gordon Schrödinger system. This provides us with a discrete infinite-dimensional dynamical system. We prove the existence of a finite dimensional global attractor for this dynamical system.
Citation: Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] C. Besse, A relaxation scheme for nonlinear Schrödinger equations},, \emph{SIAM J. Num. Anal.}, 42 (2004), 934. doi: 10.1137/S0036142901396521. Google Scholar [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. Google Scholar [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. Google Scholar [9] E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation,, \emph{Differential Integral Equations}, 23 (2010), 237. Google Scholar [10] J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations,, \emph{J. Math. Pures et Appli.}, 66 (1987), 273. Google Scholar [11] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99. doi: 10.1080/00036819608840420. Google Scholar [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. Google Scholar [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. Google Scholar [14] J. Hale, Asymptotic behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988). Google Scholar [15] A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications, College de France Seminar, (1985). Google Scholar [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. Google Scholar [17] O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Applied Mathematical Sciences vol. 139, (1999). Google Scholar [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [6] C. Besse, A relaxation scheme for nonlinear Schrödinger equations},, \emph{SIAM J. Num. Anal.}, 42 (2004), 934. doi: 10.1137/S0036142901396521. Google Scholar [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. Google Scholar [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. Google Scholar [9] E. Ezzoug, O. Goubet and E. Zahrouni, Semi-discrete weakly damped nonlinear 2-D Schroinger equation,, \emph{Differential Integral Equations}, 23 (2010), 237. Google Scholar [10] J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations,, \emph{J. Math. Pures et Appli.}, 66 (1987), 273. Google Scholar [11] O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations,, \emph{Applicable Anal.}, 60 (1996), 99. doi: 10.1080/00036819608840420. Google Scholar [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. Google Scholar [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. Google Scholar [14] J. Hale, Asymptotic behavior of Dissipative Systems,, Math. surveys and Monographs, 25 (1988). Google Scholar [15] A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications, College de France Seminar, (1985). Google Scholar [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. Google Scholar [17] O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations,, Cambridge University Press, (1991). doi: 10.1017/CBO9780511569418. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [25] C. Sulem and P.-L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave collapse,, Applied Mathematical Sciences vol. 139, (1999). Google Scholar [26] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1997). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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