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 f 1i. u j | j ∈ I iq 1 i∈Z n and (f 2i (u j | j ∈ I iq 2)) i∈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].

Upper semicontinuity of pullback attractors for non-autonomous lattice systems under singular perturbations

“…T be the solution of ( 14) with initial data ϕ 1) (τ )||, ||u (1) (τ )||, ||u (2) (τ )||, ||u (2) (τ )||)e −p(t−τ ) ,…”

“…For the attractors of first order and second order lattice systems (1)- (2), in the autonomous case (i.e. g m (t) ≡ g m is a constant), [2,8,15,16,18] studied the existence of global attractors and their upper semicontinuity of finite dimensional approximation, [12] investigated the upper semicontinuity of the global attractors for singular perturbated second order autonomous lattice system with respect to → 0 + ; in the nonautonomous case, [1,9,11,13] studied the existence of the pullback and uniform attractors for first order nonautonomous lattice systems and second order nonautonomous lattice systems with = 1.…”

“…Inspired by [1,9,12,11,13], in this paper, we consider the upper semicontinuity of the pullback attractor of the singular perturbated second order nonautonomous lattice system (2) with respect to . First, under certain conditions, by using the method of [9], we can get that system (1) has a pullback attractor {A(t) ⊂ 2 } t∈R for the solution u(t) and system (2) has a pullback attractor {A (t) ⊂ 2 × 2 } t∈R for the solution (u(t),u(t)), respectively.…”

“…and their inner product and norm are defined as follows, respectively: for ψ (j) = (u (j) , v (j) ) T ∈ E λ, , E λ and E, j = 1, 2, (ψ (1) , ψ (2) ) E λ, = (u (1) , u (2) ) λ, + (v (1) , v (2) ),…”

Attractors for first order lattice systems with almost periodic nonlinear part

“…Recently, The existence of global attractors, uniform attractors, pullback attractors, and random attractors for different types of autonomous, non-autonomous, 1242 AHMED Y. ABDALLAH and stochastic LDSs in standard and weighted spaces have been carefully investigated [1,2,3,4,5,6,9,12,13,25,26,31,34,36,37,38,39,40,41,43,44]. For first order LDSs, the existence of global attractors for autonomous systems [9,42,43] and the existence of uniform global attractors for non-autonomous systems [4,36] have been studied.…”