Pullback Exponential Attractor for Second Order Nonautonomous Lattice System
We first present some sufficient conditions for the existence of a pullback exponential attractor for continuous process on the product space of the weighted spaces of infinite sequences. Then we prove the existence and continuity of a pullback exponential attractor for second order lattice system with time-dependent coupled coefficients in the weighted space of infinite sequences. Moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.
Lattice dynamical systems (LDSs), which include coupled systems of infinite ordinary differential equations and coupled map lattices, have drawn more and more attention because these systems appear in various fields [1, 2]. In recent years, global attractors, uniform attractors, pullback attractors (or kernel sections), and random attractor for autonomous, nonautonomous, and stochastic LDSs have been studied; see [3–12]. However, these attractors sometimes attract orbits at a relatively slow speed, so that it might take an unexpected long time to be reached. Besides, it is usually difficult to estimate the attracting rate in terms of physical parameters of the model. Appropriate alternatives are exponential attractors and pullback exponential attractors which contain the global attractors and pullback attractors and attract all bounded sets exponentially [13–20]. For LDSs, [13, 16, 21] studied the existence of exponential attractors for first order, second order, and partly dissipative autonomous LDSs, respectively. Zhou and Han  presented some sufficient conditions for the existence of pullback exponential attractors for LDSs in and provided applications to first order and partly dissipative nonautonomous LDSs. As we know, there are no results on pullback exponential attractors for second order nonautonomous LDSs which have been studied extensively [4–7, 9–13, 21–23]. In this paper, we study the following second order nonautonomous LDSs: where for , , ; ; , are locally integrable in , and , are positive constants. First we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space of infinite sequences, which are direct results of . Then we prove the existence of a pullback exponential attractor for system (1) in weighted space ; moreover, we obtain the upper bound of fractal dimension and attracting rate for the attractor.
In this section, we present some sufficient conditions for the existence of a pullback exponential attractor for a continuous process on the product space of weighted space of infinite sequences.
Let be positive-valued function such that , where are positive constants, , and let which is Banach space with norm for , . Let be a product space of spaces , . Write Then is -dimensional subspaces of . Let and define a bounded projection as follows: for , ,
Consider a two-parameter continuous process on : , , satisfying the following: , for all ; , ; is continuous for .
Definition 1. A family of subsets of is called a pullback exponential attractor for the continuous process , if (i)each is a compact set of and its fractal dimension is uniformly bounded in ; that is, ;(ii)it is positively invariant; that is, for all ;(iii)there exist an exponent and two positive-valued functions , such that for any bounded set where “” is the Hausdorff semidistance between two subsets of .
As a direct consequence of Theorem 2 of , we have the following theorem.
Theorem 2. Let be a continuous process in . Assume that there exists a uniformly bounded absorbing set for : for any bounded set , there exists a constant such that for all , . For any , set . Ifthere exist and such that, for every , any , and , there exist a positive constant and a -dimensional bounded projection such that, for every and , Then possesses a pullback exponential attractor satisfying and, for any bounded set , where is the minimal number of closed balls of with radius covering the closed unit ball of centered at , and with , , and .
3. Pullback Exponential Attractor for Second Order Nonautonomous Lattice System
In this section, we study the existence of a pullback exponential attractor for the continuous processes associated with the second order nonautonomous lattice system (1). Note that system (1) with initial conditions can be written as a vector form where , , , , .
Throughout this section, let be a positive weight function from to satisfying(P0), , , for some positive constants and .
Consider the weighted Hilbert space of infinite sequences endowed with inner product and norm for , . For any , , define an inner product on by , then the norm include by is equivalent to the norm include by . Let
We make the following assumptions on , , , and , for , .(H1)There exist two positive constants and such that (H2)Let satisfy the following:(H2a) in ;(H2b)there exists a continuous positive valued function such that where is a positive constant;(H2c)there exist and such that (H3)For all , let satisfy the following:(H3a) is differentiable in and continuous in ;(H3b)for any , , ;(H3c)there exist functions and such that (H4), , where and , , denotes the space of all continuous bounded functions from into .
Letting then the system (10) is equivalent to the following evolution equation: where
Definition 3. The function is called a mild solution of the following lattice differential equations: if and
Theorem 4. Assume that (P0) and (H1)–(H4) hold. Then for any fixed and any initial data , problem (19) admits a unique mild solution with and being continuous in , and the mapping generates a continuous process on .
Proof. Let and then is continuous in and locally integrable in from into . For any bounded set with , let Then for , , , by (P0), (H1)–(H4), we have By the approximating method , it follows that problem (19) possesses a unique local mild solution . Similar to the proof of Theorem 5 below concerning the existence of a uniformly bounded absorbing set, we can prove that .
In the following part of this section, we assume that (P0) and (H1)–(H4) and hold.
Theorem 5. Assume that (P0) and (H1)–(H4) hold. There exists a uniform bounded closed absorbing ball of centered at with radius (independent of , ) such that, for any bounded subset , there exists yielding for all .
Proof. Let be a mild solution of (19), . Since the set of continuous functions is dense in , by (H2a), there exist sequences of continuous functions , , such that
Consider the following lattice differential equations: where Combining the continuity of in with the proof of Theorem 4, (36) has a unique strong solution , satisfying Taking the inner product of (29) with , we obtain By some computation, we have that Then we have that, for , Applying Gronwall’s inequality to (35) on (), we obtain It then follows that, for , for some independent of and , and then, for any , , implies the equicontinuity of . By the Arzela-Ascoli Theorem, there exists a convergent subsequence of , such that and is continuous in ; moreover, by (36), for . By the Lebesgue Convergence Theorem, we have Thus, by replacing with in (31) and letting , we have By the uniqueness of the mild solutions of (19), we have for . By replacing with and letting in (36), we have that for, , where By (H2b), there exists such that for, and , Let and then Thus, for any bounded subset of and , where Again, by (H2b), It then follows that is a uniformly bounded closed absorbing set for .
Lemma 6. Assume that (P0) and (H1)–(H4) hold. For any , there exist and such that, for , the mild solution , of system (19) with satisfies
Proof. Let be a smooth increasing function which satisfies Let be a solution of (29) with . Let ( is in (H2c)) be a suitable large integer, and set Taking the inner product of (29) with , we have And we have the following estimates: where Therefore, Applying Gronwall’s inequality to (57) on , we have that, for , By (44)-(45), for , we have as . This means that, for any , there exists such that when , By , , and (H2b), there exists such that, for , From (47), By (44), we have And, for any , there exists such that, for , Let and . It follows immediately that
Theorem 7. Assume that (P0) and (H1)–(H4) hold.(a)There exists a continuous positive value function in such that, for every , (b)There exist positive constants , , and a -dimensional orthogonal projection , such that, for every and , (c)For each fixed , .
Proof. For any , initial data , . Let , , and ; then, , , .
(a) For any , let , be two solutions of (29) with initial data , and set , and then satisfies
Taking the inner product of (69) with , we have Since , , by (36) and (66), for , , , thus , , and for . By (33), we have and, by (H2)-(H3), Thus, Applying Gronwall’s inequality to (73) on , , we have that is, where By replacing with in (75) and letting , we have
(b) For , let and , where is as in (52) and . Taking the inner product of (69) with in , we have that, for , For the terms in the right hand of (78), we have where By the continuity of and (see (H3c)), there exists such that By Lemma 6 and for , there exist , such that, for ,