Second order non-autonomous lattice systems and their uniform attractors

Abstract

The existence of the uniform global attractor for a second order non-autonomous lattice dynamical system (LDS) with almost periodic symbols has been carefully studied. Considering the nonlinear operators $ \left( f_{1i}\left( \overset{.}{u}_{j}\mid j\in I_{iq_{1}}\right) \right) _{i\in \mathbb{Z} ^{n}} $ and $ \left( f_{2i}\left( u_{j}\mid j\in I_{iq_{2}}\right) \right) _{i\in \mathbb{Z} ^{n}} $ of this LDS, up to our knowledge it is the first time to investigate the existence of uniform global attractors for such second order LDSs. In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear parts, cf. [3, 24]. Moreover, the LDS under consideration covers a wide range of second order LDSs. In fact, for specific choices of the nonlinear functions $ f_{1i} $ and $ f_{2i} $ we get the autonomous and non-autonomous second order systems given by [1, 25, 26].