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

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} .