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February  2022, 21(2): 567-584. doi: 10.3934/cpaa.2021189

Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system

Salah Missaoui 1,2,

1. 

Unité de recherche, Multifractals et Ondelettes, FSM, University of Monastir, 5019 Monastir, Tunisia
2. 

IPEIK, University of Kairouan, 3100 Kairouan, Tunisia

Received  October 2020 Revised  April 2021 Published  February 2022 Early access  November 2021

The main goal of this paper is to study the asymptotic behavior of a coupled Klein-Gordon-Schrödinger system in three dimensional unbounded domain. We prove the existence of a global attractor of the systems of the nonlinear Klein-Gordon-Schrödinger (KGS) equations in $ H^1({\mathbb R}^3)\times H^1({\mathbb R}^3)\times L^2({\mathbb R}^3) $ and more particularly that this attractor is in fact a compact set of $ H^2({\mathbb R}^3)\times H^2({\mathbb R}^3)\times H^1({\mathbb R}^3) $.

Keywords: Klein-Gordon-Schrödinger equation, global attractor, asymptotic compactness.
Mathematics Subject Classification: Primary: 37L30, 37L50; Secondary: 35A01.
Citation: Salah Missaoui. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system. Communications on Pure and Applied Analysis, 2022, 21 (2) : 567-584. doi: 10.3934/cpaa.2021189
References:
[1]

M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Differ. Integral Equ., 16 (2003), 573-581. 

[2]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 10 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.

[3]

A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires, Ann. Inst. H. Poincaré Anal. Non Lináires, 1 (1984), 453-478. 

[4]

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.

[5]

M. M. Cavalcanti and V. N. Domingos Cavalcanti, Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations, Nonlinear Differ. Equ. Appl., 7 (2000), 285–307. Birkhäuser Verlag, Basel, 2000 doi: 10.1007/PL00001426.

[6]

T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matématicos, 1996.

[7]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the Amerrican Mathematical Society, 2008. doi: 10.1090/memo/0912.

[8]

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.

[9]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, Ⅲ, Math. Japon., 24 (1979), 307-321. 

[10]

J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures et Appl., 66 (1987), 273-319. 

[11]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbb{R}^2$,, Adv. Differ. Equ., 3 (1998), 337-360. 

[12]

B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Differ. Equ., 136 (1997), 356-377.  doi: 10.1006/jdeq.1996.3242.

[13]

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.

[14]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differ. Equ., 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.

[15]

Y. Meyer, Ondelettes et Opérateurs Ⅰ: Ondelettes, Ed. Hermann, 1990.

[16]

C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbb{R}^{2+1}$, J. Differ. Equ., 227 (2006), 365-405.  doi: 10.1016/j.jde.2005.10.012.

[17]

S. Missaoui and E. Zahrouni, Regularity of the Attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$, Commun. Pure Appl. Anal., 14 (2015), 695-716.  doi: 10.3934/cpaa.2015.14.695.

[18]

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.

[19]

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.

[20] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[21]

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.

[22]

X. Wang, An energy equation for weakly damped driven nonlinear Schrödinger equations, Physica D, 88D (1995), 165-177.  doi: 10.1016/0167-2789(95)00196-B.

show all references

References:
[1]

M. Abounouh, O. Goubet and A. Hakim, Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system, Differ. Integral Equ., 16 (2003), 573-581. 

[2]

B. Alouini, Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain, Commun. Pure Appl. Anal., 10 (2015), 1781-1801.  doi: 10.3934/cpaa.2015.14.1781.

[3]

A. Bachelot, Problème de Cauchy pour des systèmes hyperboliques semi-linéaires, Ann. Inst. H. Poincaré Anal. Non Lináires, 1 (1984), 453-478. 

[4]

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.

[5]

M. M. Cavalcanti and V. N. Domingos Cavalcanti, Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations, Nonlinear Differ. Equ. Appl., 7 (2000), 285–307. Birkhäuser Verlag, Basel, 2000 doi: 10.1007/PL00001426.

[6]

T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Metodos Matématicos, 1996.

[7]

I. D. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations With Nonlinear Damping, Memoirs of the Amerrican Mathematical Society, 2008. doi: 10.1090/memo/0912.

[8]

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.

[9]

I. Fukuda and M. Tsutsumi, On coupled Klein-Gordon-Schrödinger equations, Ⅲ, Math. Japon., 24 (1979), 307-321. 

[10]

J. M. Ghidaglia and R. Temam, Attractors for damped nonlinear hyperbolic equations, J. Math. Pures et Appl., 66 (1987), 273-319. 

[11]

O. Goubet, Regularity of the attractor for a weakly damped nonlinear Schrödinger equation in $\mathbb{R}^2$,, Adv. Differ. Equ., 3 (1998), 337-360. 

[12]

B. Guo and Y. Li, Attractors for Klein-Gordon-Schrödinger equations in $\mathbb{R}^3$, J. Differ. Equ., 136 (1997), 356-377.  doi: 10.1006/jdeq.1996.3242.

[13]

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.

[14]

K. Lu and B. Wang, Global attractors for the Klein-Gordon-Schrödinger equation in unbounded domains, J. Differ. Equ., 170 (2001), 281-316.  doi: 10.1006/jdeq.2000.3827.

[15]

Y. Meyer, Ondelettes et Opérateurs Ⅰ: Ondelettes, Ed. Hermann, 1990.

[16]

C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrödinger system with rough data in $\mathbb{R}^{2+1}$, J. Differ. Equ., 227 (2006), 365-405.  doi: 10.1016/j.jde.2005.10.012.

[17]

S. Missaoui and E. Zahrouni, Regularity of the Attractor for a coupled Klein-Gordon-Schrödinger system with cubic nonlinearities in $\mathbb{R}^2$, Commun. Pure Appl. Anal., 14 (2015), 695-716.  doi: 10.3934/cpaa.2015.14.695.

[18]

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.

[19]

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.

[20] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. 
[21]

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.

[22]

X. Wang, An energy equation for weakly damped driven nonlinear Schrödinger equations, Physica D, 88D (1995), 165-177.  doi: 10.1016/0167-2789(95)00196-B.

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