Global attractor for a system of Klein–Gordon–Schrödinger type in all R

In this paper we study the long time behavior of solutions for the following system of Klein–Gordon–Schrödinger type iψt+κψxx+iαψ=ϕψ+f,iψt+κψxx+iαψ=ϕψ+f, Turn MathJax on View the MathML sourceϕtt−ϕxx+ϕ+λϕt=−Reψx+g, Turn MathJax on View the MathML sourceψ(x,0)=ψ0(x),ϕ(x,0)=ϕ0(x),ϕt(x,0)=ϕ1(x), Turn MathJax on View the MathML sourcelimx→±∞ψ(x,t)=limx→±∞ϕ(x,t)=0,t>0, Turn MathJax on where View the MathML sourcex∈R,κ>0,α>0,λ>0. First, the existence, uniqueness and continuity of the solutions on the initial data are proved. Then the asymptotic compactness of the solutions and the existence of a global compact attractor are shown.