We study the long time behaviour of the solutions for a weakly damped forced nonlinear fractional Klein-Gordon-Schrödinger system i u t + i ν u - ( - Δ ) α u + v u - | u | 2 u = f , v tt + γ v t + ( - Δ ) α v + v + v 3 - | u | 2 = g , for a given α ∈ ( 1 2 , 1 ) considered in the whole space R. We prove that this system provides an infinite dimensional dynamical system in H α ( R ) × H α ( R ) × L 2 ( R ) that possesses a global attractor A α in the same space and more particularly that this attractor is in fact a compact subset of H 2 α ( R ) × H 2 α ( R ) × H α ( R ) .
