Attractors for first order lattice systems with almost periodic nonlinear part

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 $ f\left( u,t\right) $ presents the main difficultly of this work. To overcome this difficulty we introduce a suitable Banach space $ W $ of functions satisfying (3)-(7) with norm (8) such that $ f_{0}\left( \cdot ,t\right) $ is an almost periodic function of $ t $ with values in $ W $ and $ \left( g,f\right) \in \mathcal{H}\left( \left( g_{0},f_{0}\right) \right) $.