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 $\mathbb{R}^2$

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

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 $H^2({\mathbb R}^2)\times H^2({\mathbb R}^2)\times H^1({\mathbb R}^2)$.
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 & 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,, \emph{Diff. and Integ. Equat.}, 16 (2003), 573. Google Scholar

[2]

A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'aires}, 1 (1984), 453. Google Scholar

[3]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling,, \emph{SIAM J. Math. Anal.}, 21 (1990), 1190. doi: 10.1137/0521065. Google Scholar

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T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, \emph{Textos de Metodos Mat\'ematicos}, 26 (1996). Google Scholar

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I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, II,, \emph{J. Math. Anal. Appl.}, 66 (1978), 358. doi: 10.1016/0022-247X(78)90239-1. Google Scholar

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I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, III,, \emph{Math. Japon.}, 24 (1979), 307. Google Scholar

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B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbbR^3$,, \emph{J. Diff. Equ.}, 136 (1997), 356. doi: 10.1006/jdeq.1996.3242. Google Scholar

[8]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbbR^2$,, \emph{Ad. in Diff. Equa.}, 3 (1998), 337. Google Scholar

[9]

N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations,, \emph{J. Math. Soc. Japon}, 39 (1987), 489. doi: 10.2969/jmsj/03930489. Google Scholar

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K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, \emph{J. Diff. Eq.}, 170 (2001), 281. doi: 10.1006/jdeq.2000.3827. Google Scholar

[11]

C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbbR^{2+1}$,, \emph{J. Diff. Equations}, 227 (2006), 365. doi: 10.1016/j.jde.2005.10.012. Google Scholar

[12]

Y. Meyer, Ondelettes et opérateurs I: Ondelettes,, Ed. Hermann, (1990). Google Scholar

[13]

J. Y. Park and J. A. Kim, Maximal attractors for the Klein-Gordon-Schrödinger equation in unbounded domain,, \emph{Acta Appl. Math.}, 108 (2009), 197. doi: 10.1007/s10440-008-9309-0. Google Scholar

[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. Google Scholar

[15]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

[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. Google Scholar

[17]

X. Wang, An energy equation for weakly damped driven nonlinear Schrödinger equations,, \emph{Physica D}, 88D (1995), 165. Google Scholar

show all references

References:
[1]

M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system,, \emph{Diff. and Integ. Equat.}, 16 (2003), 573. Google Scholar

[2]

A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'aires}, 1 (1984), 453. Google Scholar

[3]

P. Biler, Attractors for the system of Schrödinger and Klein-Gordon equations with Yukawa coupling,, \emph{SIAM J. Math. Anal.}, 21 (1990), 1190. doi: 10.1137/0521065. Google Scholar

[4]

T. Cazenave, An Introduction to Nonlinear Schrödinger Equations,, \emph{Textos de Metodos Mat\'ematicos}, 26 (1996). Google Scholar

[5]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, II,, \emph{J. Math. Anal. Appl.}, 66 (1978), 358. doi: 10.1016/0022-247X(78)90239-1. Google Scholar

[6]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, III,, \emph{Math. Japon.}, 24 (1979), 307. Google Scholar

[7]

B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbbR^3$,, \emph{J. Diff. Equ.}, 136 (1997), 356. doi: 10.1006/jdeq.1996.3242. Google Scholar

[8]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbbR^2$,, \emph{Ad. in Diff. Equa.}, 3 (1998), 337. Google Scholar

[9]

N. Hayashi and W. von Wahl, On the global strong solutions of coupled Klein-Gordon-Schrödinger equations,, \emph{J. Math. Soc. Japon}, 39 (1987), 489. doi: 10.2969/jmsj/03930489. Google Scholar

[10]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains,, \emph{J. Diff. Eq.}, 170 (2001), 281. doi: 10.1006/jdeq.2000.3827. Google Scholar

[11]

C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbbR^{2+1}$,, \emph{J. Diff. Equations}, 227 (2006), 365. doi: 10.1016/j.jde.2005.10.012. Google Scholar

[12]

Y. Meyer, Ondelettes et opérateurs I: Ondelettes,, Ed. Hermann, (1990). Google Scholar

[13]

J. Y. Park and J. A. Kim, Maximal attractors for the Klein-Gordon-Schrödinger equation in unbounded domain,, \emph{Acta Appl. Math.}, 108 (2009), 197. doi: 10.1007/s10440-008-9309-0. Google Scholar

[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. Google Scholar

[15]

E. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970). Google Scholar

[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. Google Scholar

[17]

X. Wang, An energy equation for weakly damped driven nonlinear Schrödinger equations,, \emph{Physica D}, 88D (1995), 165. Google Scholar

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