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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   \begin{document}$ H^1({\mathbb R}^3)\times H^1({\mathbb R}^3)\times L^2({\mathbb R}^3) $\end{document}   and more particularly that this attractor is in fact a compact set of   \begin{document}$ H^2({\mathbb R}^3)\times H^2({\mathbb R}^3)\times H^1({\mathbb R}^3) $\end{document}  .

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