Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system

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

Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system

Olivier Goubet ^1, and Marilena N. Poulou ^2,

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

Abstract
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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.

Keywords: relaxation scheme., fractal dimension, global attractor, Klein-Gordon Schrödinger system.

Mathematics Subject Classification: Primary: 35Q55, 37L3.

Citation: Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure and 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, J. Differential Equation, 83 (1990), 85-108. 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, Comm. in Pure and Applied Analysis, 7 (2008), 211-227.
[3]	M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system, Differential Integral Equations, 16 (2003), 573-581.
[4]	N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $\mathbb{R}^{N}$, Appl. Math. Lett., 12 (1999), 458. doi: 10.1016/S0893-9659(98)00170-0.
[5]	J. Ball, Global attractors for damped semilinear wave equations, Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[6]	C. Besse, A relaxation scheme for nonlinear Schrödinger equations}, SIAM J. Num. Anal., 42 (2004), 934-952. doi: 10.1137/S0036142901396521.
[7]	I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equ., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.
[8]	M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288. 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, Differential Integral Equations, 23 (2010), 237-252.
[10]	J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations, J. Math. Pures et Appli., 66 (1987), 273-319.
[11]	O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420.
[12]	O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Comm. in Pure and Applied Analysis, 7 (2008), 1429-1442. doi: 10.3934/cpaa.2008.7.1429.
[13]	B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$, J. Diff. Eq., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.
[14]	J. Hale, Asymptotic behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence.
[15]	A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications College de France Seminar, vol 7, J. L. Lions and H. Brezis, Pitmann, London, 1985.
[16]	N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves, Zeitschrift foangewandte Mathematik und Physik ZAMP, (2005) Volume 56, Issue 2 , pp 218-238. doi: 10.1007/s00033-004-2095-2.
[17]	O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[18]	P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369. doi: 10.1007/BF01261181.
[19]	K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Diff. Eq., 170 (2001), 281-316. 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, Evolutionary Partial Differential Equations, C. M. Dafermos and M. Pokorny eds., Elsevier, Amsterdam, to appear. doi: 10.1016/S1874-5717(08)00003-0.
[21]	I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. 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 $\mathbb{R}^{N}$, Nonlinear Analysis: Theory, Methods and Applications, Volume 74, Issue 7, Pages 2548-2562. 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.
[24]	G. Raugel, Global attractors in partial differential equations, Handbook of dynamical systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. 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, Springer, 1999.
[26]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997.
[27]	B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation, J. of Math. Phy., 40 (1999), 2445. 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, Physica D, 88 (1995), 167-175. 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, Nonlinear Anal., 20 (1993), 1417-1452. 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, J. Differential Equation, 83 (1990), 85-108. 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, Comm. in Pure and Applied Analysis, 7 (2008), 211-227.
[3]	M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrodinger system, Differential Integral Equations, 16 (2003), 573-581.
[4]	N. Akroune, Regularity of the attractor for a weakly damped Schrodinger equation on $\mathbb{R}^{N}$, Appl. Math. Lett., 12 (1999), 458. doi: 10.1016/S0893-9659(98)00170-0.
[5]	J. Ball, Global attractors for damped semilinear wave equations, Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31.
[6]	C. Besse, A relaxation scheme for nonlinear Schrödinger equations}, SIAM J. Num. Anal., 42 (2004), 934-952. doi: 10.1137/S0036142901396521.
[7]	I. Chueshov and I. Lasiecka, Attractors for second-order evolution equations with a nonlinear damping, J. of Dyn. and Diff. Equ., 16 (2004), 469-512. doi: 10.1007/s10884-004-4289-x.
[8]	M. Delfour, M. Fortin and G. Payre, Finite-difference solutions of a nonlinear Schrödinger equation, J. Comput. Phys., 44 (1981), 277-288. 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, Differential Integral Equations, 23 (2010), 237-252.
[10]	J. M. Ghidaglia and R. Temam, Attractors for damped hyperbolic equations, J. Math. Pures et Appli., 66 (1987), 273-319.
[11]	O. Goubet, Regularity of the attractor for the weakly damped nonlinear Schrödinger equations, Applicable Anal., 60 (1996), 99-119. doi: 10.1080/00036819608840420.
[12]	O. Goubet and E. Zahrouni, On a time discretization of a weakly damped forced nonlinear Schrödinger equation, Comm. in Pure and Applied Analysis, 7 (2008), 1429-1442. doi: 10.3934/cpaa.2008.7.1429.
[13]	B. Guo and Y. Li, Attractor for dissipative Klein-Gordon-Schrodinger equation in $R^3$, J. Diff. Eq., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.
[14]	J. Hale, Asymptotic behavior of Dissipative Systems, Math. surveys and Monographs, 25 (1988), AMS, Providence.
[15]	A. Haraux, Two Remarks on Dissipative Hyperbolic Problems in nonlinear Partial Differential Equations and Their Applications College de France Seminar, vol 7, J. L. Lions and H. Brezis, Pitmann, London, 1985.
[16]	N. Karachalios, M. Stavrakakis and P. Xanthopoulos, Parametric exponential energy decay for dissipative electron-ion plasma waves, Zeitschrift foangewandte Mathematik und Physik ZAMP, (2005) Volume 56, Issue 2 , pp 218-238. doi: 10.1007/s00033-004-2095-2.
[17]	O. Ladyzhenskaya, Attractors of Semigroups and Evolution Equations, Cambridge University Press, Cambridge, 1991. doi: 10.1017/CBO9780511569418.
[18]	P. Laurençot, Long-time behaviour for weakly damped driven nonlinear Schroinger equations in $R^N$, $N\leq 3$, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 357-369. doi: 10.1007/BF01261181.
[19]	K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Diff. Eq., 170 (2001), 281-316. 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, Evolutionary Partial Differential Equations, C. M. Dafermos and M. Pokorny eds., Elsevier, Amsterdam, to appear. doi: 10.1016/S1874-5717(08)00003-0.
[21]	I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. 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 $\mathbb{R}^{N}$, Nonlinear Analysis: Theory, Methods and Applications, Volume 74, Issue 7, Pages 2548-2562. 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.
[24]	G. Raugel, Global attractors in partial differential equations, Handbook of dynamical systems, Vol. 2, 885-982, North-Holland, Amsterdam, 2002. 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, Springer, 1999.
[26]	R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, Second Edition, 1997.
[27]	B. Wang and H. Lange, Attractors for the Klein-Gordon-Schrödinger equation, J. of Math. Phy., 40 (1999), 2445. 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, Physica D, 88 (1995), 167-175. 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, Nonlinear Anal., 20 (1993), 1417-1452. doi: 10.1016/0362-546X(93)90168-R.

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