Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in \mathbb{R}^2

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March 2015, 14(2): 695-716. doi: 10.3934/cpaa.2015.14.695

Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$

Salah Missaoui ^1, and Ezzeddine Zahrouni ^2,

1.	Unité de recherche, Multifractals et Ondelettes, FSM, University of Monsatir, 5000 Monastir, Tunisia
2.	Unité de recherche : Multifractals et Ondelettes, FSM, University of Monsatir, Tunisia

Received April 2014 Revised October 2014 Published December 2014

Abstract
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we prove the existence of a global attractor to dissipative Klein-Gordon-Schrödinger (KGS) system with cubic nonlinearities in $H^1({\mathbb R}^2)\times H^1({\mathbb R}^2)\times L^2({\mathbb R}^2)$ and more particularly that this attractor is in fact a compact set of $H^2({\mathbb R}^2)\times H^2({\mathbb R}^2)\times H^1({\mathbb R}^2)$.

Keywords: Klein-Gordon-Schrödinger equation, global attractor, asymptotic compactness..

Mathematics Subject Classification: Primary: 37L30, 37L50; Secondary: 35 A0.

Citation: Salah Missaoui, Ezzeddine Zahrouni. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$. Communications on Pure and Applied Analysis, 2015, 14 (2) : 695-716. doi: 10.3934/cpaa.2015.14.695

References:

[1]	M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Diff. and Integ. Equat., 16 (2003), 573-581.
[2]	A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires, Ann. Inst. H. Poincaré Anal. Non Linéaires, 1 (1984), 453-478.
[3]	P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065.
[4]	T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matématicos, 26, 1996.
[5]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, II, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.
[6]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, III, Math. Japon., 24 (1979), 307-321.
[7]	B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbbR^3$, J. Diff. Equ., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.
[8]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbbR^2$, Ad. in Diff. Equa., 3 (1998), 337-360.
[9]	N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japon, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.
[10]	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.
[11]	C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbbR^{2+1}$, J. Diff. Equations, 227 (2006), 365-405. doi: 10.1016/j.jde.2005.10.012.
[12]	Y. Meyer, Ondelettes et opérateurs I: Ondelettes, Ed. Hermann, 1990.
[13]	J. Y. Park and J. A. Kim, Maximal attractors for the Klein-Gordon-Schrödinger equation in unbounded domain, Acta Appl. Math., 108 (2009), 197-213. doi: 10.1007/s10440-008-9309-0.
[14]	T. Runst and W. Sickel, Sobolev spaces of Fractional order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis And Applications 3. Walter de Gruyter Berlin-New York, 1996. doi: 10.1515/9783110812411.
[15]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[16]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer Applied Mathematics Sciences-Second edition, 1997. doi: 10.1007/978-1-4612-0645-3.
[17]	X. Wang, An energy equation for weakly damped driven nonlinear Schrödinger equations, Physica D, 88D (1995), 165-177.

show all references

References:

[1]	M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Diff. and Integ. Equat., 16 (2003), 573-581.
[2]	A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires, Ann. Inst. H. Poincaré Anal. Non Linéaires, 1 (1984), 453-478.
[3]	P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling, SIAM J. Math. Anal., 21 (1990), 1190-1212. doi: 10.1137/0521065.
[4]	T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matématicos, 26, 1996.
[5]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, II, J. Math. Anal. Appl., 66 (1978), 358-378. doi: 10.1016/0022-247X(78)90239-1.
[6]	I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, III, Math. Japon., 24 (1979), 307-321.
[7]	B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbbR^3$, J. Diff. Equ., 136 (1997), 356-377. doi: 10.1006/jdeq.1996.3242.
[8]	O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbbR^2$, Ad. in Diff. Equa., 3 (1998), 337-360.
[9]	N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations, J. Math. Soc. Japon, 39 (1987), 489-497. doi: 10.2969/jmsj/03930489.
[10]	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.
[11]	C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbbR^{2+1}$, J. Diff. Equations, 227 (2006), 365-405. doi: 10.1016/j.jde.2005.10.012.
[12]	Y. Meyer, Ondelettes et opérateurs I: Ondelettes, Ed. Hermann, 1990.
[13]	J. Y. Park and J. A. Kim, Maximal attractors for the Klein-Gordon-Schrödinger equation in unbounded domain, Acta Appl. Math., 108 (2009), 197-213. doi: 10.1007/s10440-008-9309-0.
[14]	T. Runst and W. Sickel, Sobolev spaces of Fractional order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter Series in Nonlinear Analysis And Applications 3. Walter de Gruyter Berlin-New York, 1996. doi: 10.1515/9783110812411.
[15]	E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[16]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer Applied Mathematics Sciences-Second edition, 1997. doi: 10.1007/978-1-4612-0645-3.
[17]	X. Wang, An energy equation for weakly damped driven nonlinear Schrödinger equations, Physica D, 88D (1995), 165-177.

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