Abstract and Applied Analysis

Volume 2013 (2013), Article ID 809476, 9 pages

http://dx.doi.org/10.1155/2013/809476

Research Article

Pullback Exponential Attractors for Nonautonomous Klein-Gordon-Schrödinger Equations on Infinite Lattices

Chunqiu Li,¹ Min Zhao,² and Caidi Zhao¹

¹College of Mathematics and Information Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

²College of Life and Environmental Science, Wenzhou University, Wenzhou, Zhejiang 325035, China

Received 8 April 2013; Accepted 13 June 2013

Academic Editor: Alberto Parmeggiani

Copyright © 2013 Chunqiu Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper proves the existence of the pullback exponential attractor for the process associated to the nonautonomous Klein-Gordon-Schrödinger equations on infinite lattices.

1. Introduction

Lattice dynamical systems (LDSs for short), including coupled ordinary differential equations (ODEs), coupled map lattices, and cellular automata [1], are spatiotemporal systems with discretization in some variables. In some cases, LDSs arise as the spatial discretization of partial differential equations (PDEs) on unbounded or bounded domains. LDSs occur in a wide variety of applications, ranging from image processing and pattern recognition [2–4] to electrical engineering [5], chemical reaction theory [6, 7], laser systems [8], material science [9], biology [10], and so forth.

Nowadays, LDSs have drawn much attention from mathematicians and physicists [1]. Various properties of solutions for LDSs have been widely studied. For example, the stochastic LDSs were investigated in [11, 12]. The global and uniform attractors of LDSs were examined in [13–19]. The exponential and uniform exponential attractors of LDSs were investigated by [20–24].

At the same time, the asymptotic theory of LDSs has been widely used on many concrete lattice equations from mathematical physics. For example, lattice reaction-diffusion equations [25], discrete nonlinear Schrödinger equations [26], lattice FitzHugh-Nagumo systems [27], lattice Klein-Gordon-Schrödinger (KGS) equations [28], and lattice three component reversible Gray-Scott equations [29].

Very recently, Zhou and Han [30] presented some sufficient conditions for the existence of the pullback exponential attractor for the continuous process on Banach spaces and weighted spaces of infinite sequences. Also, they applied their results to study the existence of pullback exponential attractors for first-order nonautonomous differential equations and partly dissipative differential equations on infinite lattices with time-dependent coupled coefficients and time-dependent external terms in weighted spaces.

In this paper, we will use the abstract theory of [30] to study the pullback exponential behavior of solutions for the following nonautonomous lattice systems:

with initial conditions:

where is a linear operator defined as

, and are the sets of complex and real numbers, respectively, is the set of integer numbers, is the unit of the imaginary numbers, and , and are positive constants.

Equations (1)-(2) can be regarded as a discrete analogue of the following nonautonomous KGS equations on :

Equations (5) describe the interaction of a scalar nucleon interacting with a neutral scalar meson through Yukawa coupling [31], where and represent the complex scalar nucleon field and the real meson field, respectively, and the complex-valued function and the real-valued function are the time-dependent external sources. There are many works concerning the Cauchy problem and the initial boundary value problem of the continuous model of KGS equations or its related version, see [32–36] and references therein.

We want to mention that the lattice KGS equations have been studied by [28, 37]. In [37], the authors first presented some sufficient conditions for the existence of the uniform exponential attractor for a family of continuous processes on separable Hilbert spaces and the space of infinite sequences. Then, they studied the existence of uniform exponential attractors for the dissipative nonautonomous KGS lattice system and the Zakharov lattice system driven by quasi-periodic external forces. In [28], the authors first proved the existence of compact kernel sections and gave an upper bound of the Kolmogorov -entropy for these kernel sections. Also they verified the upper semicontinuity of the kernel sections. Articles in [28, 37] use the same transformation of the variable .

The aim of the present paper is to use the abstract result of [30] to prove the existence of pullback exponential attractors for the LDSs (1)–(3). When verifying the discrete squeezing property (see Lemma 5(II)) of the generated process, we encounter the difficulty coming from the nonlinear terms and in the coupled lattice equations. To overcome this difficulty, we make a proper transformation of the variable and use the technique of cutoff functions. We want to remark that the idea concerning the transformation of the variable originates from articles in [28, 37], but our transformation is other than that of [28, 37].

The rest of the paper is organized as follows. In Section 2, we first introduce some spaces and operators. Then, we recall some results on the existence, uniqueness, and some estimates of solutions. Section 3 is devoted to proving the existence of the pullback exponential attractor for the process associated to the lattice KGS equations.

2. Preliminaries

We first introduce the mathematical setting of our problem. Set

For brevity, we use to denote or , and equip with the inner product and norm as

where denotes the conjugate of . For any two elements , we define a bilinear form on by

where is the constant in (2) and is a linear operator defined as

We also define a linear operator from to via

In fact, is the adjoint operator of and one can check that

Clearly, the bilinear form defined by (8) is also an inner product in . Since

the norm induced by is equivalent to the norm . Write

then , , and are all Hilbert spaces. Set

For any two elements , , the inner product and norm of are defined as

where stands for the conjugate of .

For convenience, we will express (1)–(3) as an abstract Cauchy problem of first-order ODE with respect to time in . To this end, we put , , , , , , , and , . Then, we rewrite (1)–(3) in a vector form as

Set

, ,

Then, (16)–(18) can be written as

Let be the set of continuous and bounded functions from into , then for each , we have . Write

We next recall some results of solutions to (21)-(22).

Lemma 1 (see [28]). Let , . Then, for any initial data , there exists a unique solution of (21)-(22), such that . Moreover, the mapping

generates a continuous process on , where .

Lemma 2 (see [28]). Let , . Then, the solution of (21)-(22) corresponding to initial data satisfies

where , and are positive constants independent of and .

Lemma 3 (see [28]). Let , . Then, the process corresponding to (21)-(22) possesses a uniformly bounded absorbing set , such that for any bounded set of , there exists a time yielding

where is a closed ball centered at with radius .

Lemma 3 shows that there exists a time , such that

Lemma 4 (see [28]). Let with and with , respectively. Then, for any , there exist and , such that when , the solution of (21)-(22) with satisfies

where .

3. Existence of the Pullback Exponential Attractors

In this section, we prove the existence of the pullback exponential attractor for the process defined by (24). Write

then is a -dimensional subspace of . Define a bounded projection by

Lemma 5. (I) For any , there exists some (independent of ), such that for every and any ,

(II) There exist two positive constants and , and a -dimensional orthogonal projection for some , such that for every and ,

Proof. (I) For any , let

be two solutions of (21)-(22) with initial conditions , respectively. Set

From (21)-(22), we get

Taking the real part of the inner product of (35) with in , we obtain

Since is a bounded linear operator, is a locally Lipschitz continuous operator (see Lemma 2.2 in [28]), and is a bounded set in , we see that there exist two positive constants and , such that

Combining (36) and (37), we get

where . Applying Gronwall inequality to (38) on with , we obtain

Thus,

where does not depend on .

(II) Define a smooth function , such that

Set

where is a positive integer that will be specified later. From (16), we see that

Taking the imaginary part of the inner product of (43) with in , we get

Now, we have

where locates between and . According to Lemma 4, we know that there exist and , such that when and , we obtain

which implies that when and

Then, taking (44)–(47) into account, we obtain for every and that

From (17) and (19), we obtain

Taking the inner product of (49) with in , we obtain

It is clear that . Then, (50) can be rewritten as

Also, we have

By some computations, we get

Here locates between and . Inserting (53) into (52), we get

According to Lemma 4, we can see that there exist and , such that when and ,

It follows from (51) and (54)-(55) that when and , we have

Combining (48) and (56), when , we obtain

Since , we get for any that

Thus,

We then conclude from (57) and (59) that when and ,

Choosing , we obtain

Applying Gronwall inequality to (61) from to with , we get for every and that

By (39), when and we have

Thus, it follows from (62)-(63) that for any and ,

Pick two sufficient large numbers and to satisfy

Then, from (64), we have for that

that is,

where . The proof is complete.

Now, we can state the main result of this paper.

Theorem 6. Let the conditions of Lemma 4 hold. Then, the process associated with (21)-(22) possesses a pullback exponential attractor , satisfying

(1)(compactness and finiteness of dimension) for each , is a compact set of , and the fractal dimension is finite and uniformly bounded in ; that is,

(2) (positive invariant property) for all ;(3)(pullback exponential attractivity) there exist an exponent and two positive-valued functions , such that for any bounded set ,

where is the Hausdorff semidistance between two subsets of .

Proof. Using Lemmas 2.3 and 3.1 and Theorem 2 of [30], we obtain the result.

Remark 7. The spectrum of Lyapunov exponents is the most precise tool for identification of the character of motion of a dynamical system [38]. There are some works on the estimation of the dominant Lyapunov exponent of nonsmooth systems by means of synchronization method, one can refer to the articles of Stefański et al. [38–40]. In [38], Stefański and Kapitaniak presented a method to estimate the value of largest Lyapunov exponent both for discrete dynamical systems of known difference equations and also for discrete maps reconstructed from the time evolution of the given system. Following this clue, we can ask naturally the problem that whether the method presented in [38] could be applied to estimate Lyapunov exponents for the trajectories on the pullback attractor . It is an interesting and challenging issue for us to investigate.

Acknowledgments

The authors warmly thank the anonymous referee for useful comments and bringing them the issue on Lyapunov exponents for the trajectories in the pullback attractor into their attention. This paper is supported by National NSFC (no. 11271290), the National Key Basic Research Program of China (973 Program with Grant no. 2012CB426510), and NSF of Wenzhou University (2008YYLQ01).

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