We study the existence of the uniform global attractor for a family of infinite dimensional first order non-autonomous lattice dynamical systems of the following form:
$ \begin{equation*} \overset{.}{u}+Au+\alpha u+f\left( u,t\right) = g\left( t\right) ,\,\,\left( g,f\right) \in \mathcal{H}\left( \left( g_{0},f_{0}\right) \right) ,t>\tau ,\tau \in \mathbb{R}, \end{equation*} $
with initial data
$ \begin{equation*} u\left( \tau \right) = u_{\tau }. \end{equation*} $
The nonlinear part of the system
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