Uniform Attractor for the Partly Dissipative Nonautonomous Lattice Systems

Abstract

The existence of uniform attractor in is proved for the partly dissipative nonautonomous lattice systems with a new class of external terms belonging to , which are locally asymptotic smallness and translation bounded but not translation compact in . It is also showed that the family of processes corresponding to nonautonomous lattice systems with external terms belonging to weak topological space possesses uniform attractor, which is identified with the original one. The upper semicontinuity of uniform attractor is also studied.

1. Introduction

This paper is concerned with the long-time behavior of the following non-autonomous lattice systems:

(11)

(12)

with initial conditions

(13)

where is the integer lattice; is a nonlinear function satisfying ; is a positive self-adjoint linear operator; belong to certain metric space, which will be given in the following.

Lattice dynamical systems occur in a wide variety of applications, where the spatial structure has a discrete character, for example, chemical reaction theory, electrical engineering, material science, laser, cellular neural networks with applications to image processing and pattern recognition; see [1–4]. Thus, a great interest in the study of infinite lattice systems has been raising. Lattice differential equations can be considered as a spatial or temporal discrete analogue of corresponding partial differential equations on unbounded domains. It is well known that the long-time behavior of solutions of partial differential equations on unbounded domains raises some difficulty, such as well-posedness and lack of compactness of Sobolev embeddings for obtaining existence of global attractors. Authors in [5–7] consider the autonomous partial equations on unbounded domain in weighted spaces, using the decaying of weights at infinity to get the compactness of solution semigroup. In [8–10], asymptotic compactness of the solutions is used to obtain existence of global compact attractors for autonomous system on unbounded domain. Authors in [11] consider them in locally uniform space. For non-autonomous partial differential equations on bounded domain, many studies on the existence of uniform attractor have been done, for example [12–14].

For lattice dynamical systems, standard theory of ordinary differential equations can be applied to get the well-posedness of it. "Tail ends" estimate method is usually used to get asymptotic compactness of autonomous infinite-dimensional lattice, and by this the existence of global compact attractor is obtained; see [15–17]. Authors in [18, 19] also prove that the uniform smallness of solutions of autonomous infinite lattice systems for large space and time variables is sufficient and necessary conditions for asymptotic compactness of it. Recently, "tail ends" method is extended to non-autonomous infinite lattice systems; see [20–22]. The traveling wave solutions of lattice differential equations are studied in [23–25]. In [18, 26, 27], the existence of global attractors of autonomous infinite lattice systems is obtained in weighted spaces, which do not exclude traveling wave.

In this paper, we investigate the existence of uniform attractor for non-autonomous lattice systems (1.1)–(1.3). The external term in [20] is supposed to belong to and to be almost periodic function. By Bochner-Amerio criterion, the set of this external term's translation is precompact in . Based on ideas of [28], authors in [14] introduce uniformly -limit compactness, and prove that the family of weakly continuous processes with respect to (w.r.t.) certain symbol space possesses compact uniform attractors if the process has a bounded uniform absorbing set and is uniformly -limit compact. Motivated by this, we will prove that the process corresponding to problem (1.1)–(1.3) with external terms being locally asymptotic smallness (see Definition 4.5) possesses a compact uniform attractor in , which coincides with uniform attractor of the family of processes with external terms belonging to weak closure of translation set of locally asymptotic smallness function in . We also show that locally asymptotic functions are translation bounded in , but not translation compact (tr.c.) in . Since the locally asymptotic smallness functions are not necessary to be translation compact in , compared with [20], the conditions on external terms of (1.1)–(1.3) can be relaxed in this paper.

This paper is organized as follows. In Section 2, we give some preliminaries and present our main result. In Section 3, the existence of a family of processes for (1.1)–(1.3) is obtained. We also show that the family of processes possesses a uniformly (w.r.t ) absorbing set. In Section 4, we prove the existence of uniform attractor. In Section 5, the upper semicontinuity of uniform attractor will be studied.

2. Main Result

In this section, we describe our main result. Denote by the Hilbert space defined by

(21)

with the inner product and norm given by

(22)

For , we endow with the inner and norm as. For

(23)

Denote by the space of function with values in that locally 2-power integrable in the Bochner sense, that is,

(24)

It is equipped with the local 2-power mean convergence topology. Then, is a metrizable space. Let be a space of functions from such that

(25)

Denote by the space endow with the local weak convergence topology.

For each sequence define linear operators on by

(26)

Then

(27)

For convenience, initial value problem (1.1)–(1.3) can be written as

(28)

(29)

with initial conditions

(210)

where , .

In the following, we give some assumption on nonlinear function , and :

(211)

There exists a positive-value continuous function such that

(212)

There exist positive constants such that

(213)

Let the external term belong to , it follows from the standard theory of ordinary differential equations that there exists a unique local solution for problem (2.8)–(2.10) if (H ₁ )–(H ₃ ) hold. For a fixed external term , take the symbol space , the set contains all translations of in . Take the the closure of in . Denote by the translation semigroup, for all or , , . It is evident that is continuous on in the topology of and on in the topology of , respectively,

(214)

In Section 3, we will show that for every and , , problem (2.8)–(2.10) has a unique global solution Thus, there exists a family of processes from to . In order to obtain the uniform attractor of the family of processes, we suppose the external term is locally asymptotic smallness (see Definition 4.5). Let be a Banach space which the processes acting in, for a given symbol space , the uniform (w.r.t. ) -limit set of is defined by

(215)

The first result of this paper is stated in the following, which will be proved in Section 4.

Theorem 2 A.

Assume that be locally asymptotic smallness and hold. Then the process corresponding to problems (2.8)–(2.10) with external term possesses compact uniform attractor in which coincides with uniform (w.r.t. attractor for the family of processes , , that is,

(216)

where is the uniform absorbing set in , and is kernel of the process . The uniform attractor uniformly attracts the bounded set in .

We also consider finite-dimensional approximation to the infinite-dimensional systems (1.2)-(1.3) on finite lattices. For every positive integer , let , consider the following ordinary equations with initial data in :

(217)

In Section 5, we will show that the finite-dimensional approximation systems possess a uniform attractor in , and these uniform attractors are upper semicontinuous when . More precisely, we have the following theorem.

Theorem 2 B.

Assume that and hold. Then for every positive integer , systems (2.17) possess compact uniform attractor . Further, is upper semicontinuous to as , that is,

(218)

where

(219)

3. Processes and Uniform Absorbing Set

In this section, we show that the process can be defined and there exists a bounded uniform absorbing set for the family of processes.

Lemma 3.1.

Assume that and hold. Let , and . Then the solution of (2.8)–(2.10) satisfies

(31)

where , .

Proof.

Taking the inner product of (2.8) with in , by we get

(32)

Similarly, taking the inner product of (2.9) with in , we get

(33)

Note that

(34)

Summing up (3.2) and (3.3), from (3.4), we get

(35)

Thus, by ,

(36)

Since , from [12, Proposition V.4.2.], we have

(37)

From (3.6)-(3.7), applying Gronwall's inequality of generalization (see [12, Lemma II.1.3]), we get (3.1). The proof is completed.

It follows from Lemma 3.1 that the solution of problem (2.8)–(2.10) is defined for all . Therefore, there exists a family processes acting in the space , , , , where is the solution of (2.8)–(2.10), and the time symbol belongs to and , respectively. The family of processes satisfies multiplicative properties:

(38)

Furthermore, the following translation identity holds:

(39)

The kernel of the processes consists of all bounded complete trajectories of the process , that is,

(310)

denotes the kernel section at a times moment :

(311)

Lemma 3.1 also shows that the family of processes possesses a uniform absorbing set in .

Lemma 3.2.

Assume that and hold. Let . Then, there exists a bounded uniform absorbing set in for the family of processes , that is, for any bounded set , there exists ,

(312)

Proof.

Let , from (3.1) we have

(313)

where

(314)

Let . The proof is completed.

4. Uniform Attractor

In this section, we establish the existence of uniform attractor for the non-autonomous lattice systems (2.8)–(2.10). Let be a Banach space, and let be a subset of some Banach space.

Definition 4.1.

is said to be   weakly continuous, if for any , the mapping   is weakly continuous from to .

A family of processes , is said to be uniformly-limit compact if for any and bounded set , the set is bounded for every and is precompact set as . We need the following result in [14].

Theorem 4.2.

Let be the weak closure of . Assume that , is weakly continuous, and

• (i) has a bounded uniformly (w.r.t. ) absorbing set ,

• (ii) is uniformly (w.r.t. ) -limit compact.

Then the families of processes , , possess, respectively, compact uniform (w.r.t. , , resp.) attractors and satisfying

(41)

Furthermore, is nonempty for all .

Let be a Banach space and , denote the space of functions , with values in that are locally -power integral in the Bochner sense, it is equipped with the local -power mean convergence topology. Recall the Propositions in [12].

Proposition 4.3.

A set is precompact in if and only if the set is precompact in for every segment . Here, denotes the restriction of the set to the segment .

Proposition 4.4.

A function is tr.c. in if and only if

• (i) for any the set is precompact in ;

• (ii) there exists a function , such that

  

  (42)

Now, one introduces a class of function.

Definition 4.5.

A function is said to be locally asymptotic smallness if for any , there exists positive integer such that

(43)

Denote by the set of all locally asymptotic smallness functions in . It is easy to see that . The next examples show that there exist functions in but not in , and a function belongs to is not necessary a tr.c. function in .

Example 4.6.

Let ,

(44)

For every , ,

(45)

Thus,

(46)

and . However, for every positive integer , and for any positive ,

(47)

Therefore, .

Example 4.7.

,

(48)

for ,

(49)

Here, denote the positive integer set.

For every positive integer , , and for ,

(410)

which implies that

(411)

Therefore, . Note that for any ,

(412)

From Proposition 4.4, is not translation compact in .

Remark 4.8.

Example 4.7 shows that a locally asymptotic function is not necessary translation compact in .

In the following, we give some properties of locally asymptotic smallness function.

Lemma 4.9.

is a closed subspace of .

Proof.

Let such that

(413)

Then, for any , there exists positive integer such that for every ,

(414)

Since , there exist such that for all ,

(415)

Let , we get that

(416)

Therefore, . This completes the proof.

Lemma 4.10.

Every translation compact function in is locally asymptotic smallness.

Proof.

Since is tr.c. in , we get that is precompact in . By Proposition 4.3, we get that is precompact in . Thus, for any , there exists finite number such that for every , there exist some , , such that

(417)

For the given above, implies that there exists positive integer such that

(418)

Therefore,

(419)

which implies is locally asymptotic smallness. This completes the proof.

We now establish the uniform estimates on the tails of solutions of (2.8)–(2.10) as .

Lemma 4.11.

Assume that hold and is locally asymptotic smallness. Then for any , there exist positive integer and such that if , , satisfies

(420)

Proof.

Choose a smooth function such that for , and

(421)

and there exists a constant such that for . Let be a suitable large positive integer, . Taking the inner product of (2.8) with and (2.9) with in , we have

(422)

From , we have

(423)

where is same as in Lemma 3.1

(424)

as in (3.10)

(425)

Summing up (4.22), from (4.23)–(4.25) we get

(426)

Thus,

(427)

We now estimate the integral term on the right-hand side of (4.27).

(428)

Similarly,

(429)

Since is locally asymptotic smallness, from (4.27)–(4.29) we get that for any , if , there exist and sufficient large positive integer such that

(430)

The proof is completed.

Lemma 4.12.

Assume that hold, let , . If in and weakly in , then for any ,

(431)

Proof.

Let , . Since is bounded in , by Lemma 3.2, we get that

(432)

Therefore, for all , ,

(433)

Note that is the solution of (2.8) and (2.9) with time symbol , it follow from (4.32) that

(434)

In the following, we show that . By the fact that is the solution of (2.8) and (2.9), for any , we get that

(435)

Note that weakly in . Let in (4.35), by (4.34) we get that is the solution of (2.8) and (2.9) with the initial data . By the unique solvability of problem (2.8)–(2.10), we get that . This completes the proof.

Proof of Theorem A.

From Lemmas 3.2, 4.11 and 4.12, and Theorem 4.2, we get the results.

5. Upper Semicontinuity of Attractors

In this section, we present the approximation to the uniform attractor obtained in Theory A by the uniform attractor of following finite-dimensional lattice systems in :

(51)

with the initial data

(52)

and the periodic boundary conditions

(53)

Similar to systems (2.8)–(2.10), under the assumption , the approximation systems (5.1)–(5.2) with possess a unique solution , which continuously depends on initial data. Therefore, we can associate a family of processes which satisfy similar properties (3.8)–(3.9). Similar to Lemma 3.2, we have the following result.

Lemma 5.1.

Assume that , and hold. Let . Then, there exists a bounded uniform absorbing set for the family of processes , that is, for any bounded set , there exists , for ,

(54)

In particular, is independent of and .

Since (5.1) is finite-dimensional systems, it is easy to know that under the assumption of Lemma 5.1, the family of processes , is uniformly (w.r.t. ) -limit compact. Similar to Lemma 4.12, if in , weakly in , then for any , ,

(55)

Lemma 5.2.

Assume that and hold. Then the process corresponding to problems (5.1)-(5.2) with external term possesses compact uniform (w.r.t. ) attractor in which coincides with uniform (w.r.t. attractor for the family of processes , , that is,

(56)

where is the uniform absorbing set in , and is kernel of the process . The uniform attractor uniformly attracts the bounded set in .

Proof of Theorem B.

If , it follows from Lemma 5.2 that there exist and a bounded complete solution such that

(57)

Since , there exist and a subsequence of , which is still denote by , such that

(58)

From Lemma 5.1, we get that

(59)

which imply that

(510)

Thus,

(511)

Let be a sequence of compact intervals of such that and . From (5.9) and (5.11), using Ascoli's theorem, we get that for each , there exists a subsequence of (still denoted by ) and such that

(512)

Proceeding as in the proof of Lemma 4.11, we get that the weak convergence is actually strong convergence, and therefore is precompact in for each . Then we infer that there exists a subsequence of and such that converges to . Using Ascoli's theorem again, we get, by induction, that there is a subsequence of such that converges to in , where is an extension of to . Finally, taking a diagonal subsequence in the usual way, we find that there exist a subsequence of and such that for any compact interval

(513)

From (5.9) we get that

(514)

Next, we show that is the solution of (2.8)–(2.10). It follows from (5.10) that

(515)

For fixed , let . Since is the solution of (5.1)–(5.2) with , we have

(516)

Thus, for each , we have

(517)

Letting , by (5.8), (5.13), (5.15) and (5.17) we find that satisfies

(518)

Since is arbitrary, we note that (5.18) are valid for all . From (5.14) we find that is a bounded complete solution of (2.8)–(2.10). Therefore, . By (5.13) we get that

(519)

The proof is complete.

Remark 5.3.

All the result of this paper is valid for the systems in [20, 21].