Attractors for first order lattice systems with almost periodic nonlinear part

Ahmed Y. Abdallah

doi:10.3934/dcdsb.2019218

Article Contents

2020, Volume 25, Issue 4: 1241-1255. Doi: 10.3934/dcdsb.2019218

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Attractors for first order lattice systems with almost periodic nonlinear part

Ahmed Y. Abdallah

Department of Mathematics, The University of Jordan, Amman 11942, Jordan

Received: January 31, 2019

Revised: May 31, 2019

Early access: September 2019

Published: April 2020

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Abstract

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) $.

Keywords:

Non-autonomous lattice dynamical system,
uniform absorbing set,
uniform global attractor,
almost periodic symbol.

Mathematics Subject Classification: Primary: 37L30; Secondary: 37L60.

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