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doi: 10.3934/cpaa.2021189
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Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system

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 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) $.

Citation: Salah Missaoui. Regularity of the attractor for a coupled Klein-Gordon-Schrödinger system in $ \mathbb{R}^3 $ nonlinear KGS system. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2021189
References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[10]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

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