Second order non-autonomous lattice systems and their uniform attractors

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


1.
Introduction. Lattice dynamical systems (LDSs) have attracted much attention in the literature. They arise naturally in a wide variety of applications, for instance, in propagation of nerve pulses in myelinated axons, electrical engineering, pattern recognition, image processing, chemical reaction theory, etc. In each case, they have their own form, but in some cases, they appear as spatial discretizations of continuous partial differential equations [9,13].
In this work we study the existence of the uniform global attractor for the following second order non-autonomous LDS: ..
u j | j ∈ I iq1 +f 2i (u j | j ∈ I iq2 ) = g i (t) , i ∈ Z n , t > τ, τ ∈ R, (1) with initial data u i (τ ) = u 0i,τ , where Z n is the product of n integer sets, A 1 , A 2 are linear operators, f 1i , f 2i are nonlinear functions, and g i is an external term, to be determined later. Considering the nonlinear functions f 1i and f 2i of (1), one can see that such a non-autonomous LDS covers a wide range of second order autonomous and non-autonomous second order LDSs. In fact for particular choices of the nonlinear part, we get the autonomous and non-autonomous second order systems given by [1,25,26].
Since LDSs are infinite systems of ordinary differential equations or of difference equations, indexed by points in a lattice such as the unbounded n-dimensional integer lattice Z n , proving the asymptotic compactness of the solution operators is a crucial step towards proving the existence of attractors for the system. Such compactness property in autonomous systems is obtained by the uniform estimates on the tails of solutions with a bounded set of initial data when ∞. In the present case, the LDS (1)-(2) is non-autonomous with almost periodic symbols and therefore it defines a family of processes, instead of a semigroup in the autonomous case. In such a case, the estimates on the tails of the solutions must be uniform with respect to initial data from a bounded set as well as all translations of the time symbol g ∈ H (g 0 ) in the system. This paper is organized as follows. In section 2, the non-autonomous LDS (6)-(7) with almost periodic symbols is introduced, where abstract assumptions on the linear and nonlinear parts of the system and some introductory results are presented. In section 3, the non-autonomous LDS (6)- (7) is written in the abstract form (40)-(41) in a suitable extended phase space, where the well-posedness of the system (40)-(41) is established, a family of processes associated with this system is defined, and the existence of a uniform absorbing set for this family of processes is verified. In section 4, the uniform estimates on the tails of solutions with respect to initial data from the uniform absorbing set and the time symbol are introduced, where such estimates are needed to obtain the asymptotic compactness of the solution semigroup, then by the semigroup theory, the uniform global attractor is presented.

2.
Preliminaries. For a positive integer n, we consider the Hilbert space whose inner product and norm are given by: In the Hilbert space H = l 2 × l 2 , we consider the inner product and norm, For m = 1, 2, i = (i 1 , . . . , i n ) ∈ Z n , and a nonnegative integer q m , the set I iqm is defined as follows: Let g 0 : R → l 2 with g 0 (t) = (g 0i (t)) i∈Z n , be an almost periodic function. From [16], we know that such a function is bounded and uniformly continuous on R, therefore g 0 ∈ C b R, l 2 , where C b R, l 2 is the space of bounded continuous functions on R with the norm Moreover, since g 0 : R → l 2 is an almost periodic function, by Bochner's criterion [16], the set of translations Then for any g ∈ H (g 0 ), g is almost periodic and H (g) = H (g 0 ). Here H(g) is called the hull of the function g.
In the extended phase space E = H × H(g 0 ), we consider the norm: Considering the symbol space H(g 0 ) and the LDS (1)-(2), we shall study the existence of the uniform global attractor (with respect to g ∈ H(g 0 )) for the following second order non-autonomous LDS in H: where ) i∈Z n , for m = 1, 2, and g (t) = (g i (t)) i∈Z n . Up to our knowledge it is the first time to study the existence of the uniform global attractor for such a second order non-autonomous LDS with nonlinear part of the form In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear part, cf. [3,24].
Within this work, we assume that: (A0) g ∈ H (g 0 ), where g 0 : R → l 2 is an almost periodic function.
(A1) For m = 1, 2, A m : l 2 → l 2 is a positive self-adjoint bounded linear operator which can be represented in the following form such that for k = 1, · · · , n, there exists a bounded linear operator D mk : l 2 → l 2 with l m is a positive integer, d mk,l ∈ R, l = −l m , . . . , l m , not all of them are zeros, where C m is a positive constant, · O is the norm of a bounded linear operator in l 2 , and D * mk is the adjoint operator of D mk . That is, for u = (u i ) i∈Z n , v = (v i ) i∈Z n ∈ l 2 , Since D mk O ≤ C m , it is clear that (A2) For m = 1, 2, considering the nonlinear function f mi : R (2qm+1) n → R, there exist positive constants c m , r 2 and a positive integer I m such that for (u i ) i∈Z n ∈ l 2 and i ∈ Z n , f mi (u j = 0 | j ∈ I iqm ) = 0, where (A3) There exists a positive constant c 3 such that for (u i ) i∈Z n , (A4) There exists a positive continuous increasing function M 2 : R + →R + such that for (u i ) i∈Z n ∈ l 2 , R > 0, and i ∈ Z n , where f 2i,k (u j | j ∈ I iq2 ) is the derivative of f 2i (u j | j ∈ I iq2 ) with respect to k ∈ I iq2 . There exists δ 2 > 0 such that for (u i ) i∈Z n ∈ l 2 and i ∈ Z n , where f 2i is the function f 2i but u i in f 2i is replaced by r inf 2i . The positive constant δ 2 is sufficiently small such that there exists 0 < µ < 1 with where β 2 is given by Lemma 2.1. One can show that the assumptions (A2)-(A4) related to the nonlinear part of the system (6) are acceptable for the following particular choices: where h 2 is a positive integer and there exists λ * > 0 such that Remark 1. For m = 1, 2, (u i ) i∈Z n ∈ l 2 and i = (i 1 , i 2 , · · · , i n ) ∈ Z n , let us choose then LDS (6) can be regarded as a spatial discretization of the following damped non-autonomous nonlinear hyperbolic family of equations with continuous spatial variable x, In some compact domain, Chepyzhov and Vishik [10,11,12] studied the existence of a uniform global attractor for similar non-autonomous systems. Some compactness results are not available whenever we study the existence of global attractors on unbounded domains and usually the global attractors are infinite dimensional, cf. [19]. Therefore, it is important to investigate the existence of the uniform global attractor for the LDS (6) since it can be regarded as a discrete analogue of the continuous nonlinear family of equations (21) on the unbounded domain R n .
Here we present some well known results which will be frequently used within this work, for u = (u i ) i∈Z n ∈ l 2 we have i∈Z n j∈Iiq m and for u = (u i ) i∈Z n , v = (v i ) i∈Z n ∈ l 2 , the mean value theorem implies Here we show that f 1 and f 2 map l 2 into l 2 . Indeed, for (u i ) i∈Z n ∈ l 2 , considering (12) and (16), we get and from (22)- (23), we find Using (12), (17), and (25), and the fact that M 2 is a positive continuous increasing function, it follows that and from (22)- (23), and again the fact that M 2 is a positive continuous increasing function, we obtain Let Recalling (19) and using the mean value theorem for integrals, there exists c i between 0 and u i such that and following (28), we find and Using (13) and (15), we obtain From (28), it is clear that In such a case, using (14)- (15), there exists a constant β 1 = β 1 (n, The assumptions on the nonlinear operators f 1 and f 2 of the LDS (6) present the main difficulty of this work. Lemmas 2.1 and 2.2 will be helpful to overcome this difficulty. where Proof. Taking into account (33)-(35), the proof is similar to that of Lemma 2 in [1].
The following is a Gronwall-type lemma, cf. [6], which will be helpful to introduce global solutions, a uniform absorbing set, and uniform tail estimates of the solutions.
Lemma 2.2. Let X be a Banach space, Π : X → R be a given functional, and 3. Global solutions and uniform absorbing sets.
. Then the LDS (6)- (7) can be written in the abstract form: where C : H → H is the bounded linear operator: I : l 2 → l 2 is the identity operator, and F : [τ, +∞) × H → H is the nonlinear operator: .

Uniform global attractors.
Here we present uniform estimates on the tails of solutions with respect to initial data from the uniform absorbing set O and g ∈ H (g 0 ). Such estimates are needed to obtain the asymptotic compactness of the solution semigroup, then by the semigroup theory we present the uniform global attractor.

Proof. Consider a smooth increasing function
and there exists a constant B such that From Lemma 3.2, there exists T 1 = T 1 (R 0 ) > 0 such that for g ∈ H (g 0 ), τ ∈ R, Considering this solution, let w = (w i ) i∈Z n where w i = θ i 0 L u i and L be a positive integer such that where I 1 and I 2 are given by assumption (A2). In such a case considering the inner product of (6) with where and Recalling (15), (39), and (79), we find
Substituting the values of a 3 and b 3 into (104), we get i∈Z n Since g : R → l 2 is almost periodic, the set (g i (t)) i∈Z n : t ∈ R is precompact in l 2 , which implies that for given > 0, there exists a constant N 1 = N 1 (g, ) depending on g and such that But g ∈ H (g 0 ) = {g 0 (· + h) : h ∈ R} and the set H (g 0 ) is compact in C b R, l 2 , taking into account (106), it follows that for given > 0 there exists N 2 = N 2 ( ) depending only on and independent of g such that Given η > 0, taking into account (107), we shall fix L = L (η) to be sufficiently large such that It is clear that b5 a3 , where a 3 and b 5 are given by (91) and (101), respectively, depends on L (η). In such a case, for T = T (η) = max T 1 , b5 a3 and I = I (η) = 2L (η), 1844 AHMED Y. ABDALLAH AND RANIA T. WANNAN taking into account (105) and (108), we find The proof is completed.
Here we study the existence of a uniform global attractor for the family of processes {Φ g (t, τ ) : t ≥ τ, τ ∈ R} g∈H(g0) associated with the LDS (40)-(41) in H by following the semigroup theory. Along the lines of [12], we define the nonlinear semigroup {S (t)} t≥0 associated with LDS (40)-(41) acting on the extended phase space E = H × H (g 0 ) by where {S (t)} t≥0 satisfies the semigroup identities: and I E : E → E is the identity operator. is asymptotically compact, that is, if {(ϕ n , g n )} ∞ n=1 is bounded in E, and t n → ∞, then {S (t n ) (ϕ n , g n )} ∞ n=1 is precompact in E. Proof. Using Lemmas 3.2 and 4.1, the proof is similar to that of Lemma 5.4 [20]. Definition 4.4. Given g ∈ H (g 0 ), a curve t → ϕ (t) ∈ H is said to be a complete solution for the process Φ g (t, τ ), if it satisfies Φ g (t, τ ) ϕ (τ ) = ϕ (t) , ∀τ ∈ R, t ≥ τ.
The kernel of the process Φ g (t, τ ) is the collection K g of all its bounded complete solutions, that is, The kernel section of the process Φ g (t, τ ) at time s ∈ R is the set K g (s) = {ϕ (s) : ϕ (·) ∈ K g } .
Following the uniform attractor theory [12], we obtain the following proposition, where F 1 and F 2 , given below, are the projectors from E = H × H (g 0 ) onto H and H (g 0 ), respectively. Proposition 1. In E, if the semigroup {S (t)} t≥0 is continuous, point dissipative, and asymptotically compact, then it has a compact global attractor A S . Furthermore, in H, A = F 1 A S is the compact uniform attractor for the family of processes {Φ g (t, τ )} g∈H(g0) . In addition, (a) A S = g∈H(g0) K g (0) × {g} , Theorem 4.5. In H, the family of processes {Φ g (t, τ )} g∈H(g0) associated with the LDS (40)-(41) has a compact uniform attractor A with respect to g ∈ H (g 0 ) .
Following the same procedure one can prove the upper semicontinuity of the uniform global attractor A associated with the family of processes {Φ g (t, τ )} g∈H(g0) of (40)-(41). That is, the uniform global attractor A generated by the infinite-dimensional non-autonomous LDS (40)-(41) can be approached by the uniform global attractors of finite-dimensional truncated ordinary differential systems.