Asymptotic behavior of the Klein-Gordon-Schrödinger lattice dynamical systems (Q2469349)

The author deals with the lattice dynamical system of a Klein-Gordon-Schrödinger type equation in a suitable Hilbert space, which can be considered as a spatial discretization of the following system of partial differential equations with spatial variable \(x\in\mathbb{R}^d\), and \(t\in\mathbb{R}_+\), \[ \psi_t+ \Delta_x\psi+ i\alpha\psi+ f(\psi,\varphi)= g,\tag{1} \] \[ \varphi_{tt}+ \beta\varphi_t- \Delta_x\varphi+ \gamma\varphi= |\psi|^\theta+ h,\tag{2} \] where \(\alpha,\beta,\gamma\in (0,+\infty)\) and \(\theta\in [1,+\infty)\). Complex-valued function \(g\) and the real-valued function \(h\) represent the external forces. Under some natural assumptions on the data of (1)--(2), the author studies the existence of the global attractor and its upper semicontinuity for the corresponding to (1)--(2) lattice dynamical systems.