Uniform Attractors for Nonclassical Diffusion Equations with Memory

Xie, Yongqin; Li, Yanan; Zeng, Ye

doi:https://doi.org/10.1155/2016/5340489

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Volume 2016 | Article ID 5340489 | https://doi.org/10.1155/2016/5340489

Uniform Attractors for Nonclassical Diffusion Equations with Memory

Yongqin Xie,¹Yanan Li,¹and Ye Zeng¹

Academic Editor: Simone Secchi

Received26 Mar 2016

Accepted28 Apr 2016

Published30 May 2016

Abstract

We introduce a new method (or technique), asymptotic contractive method, to verify uniform asymptotic compactness of a family of processes. After that, the existence and the structure of a compact uniform attractor for the nonautonomous nonclassical diffusion equation with fading memory are proved under the following conditions: the nonlinearity satisfies the polynomial growth of arbitrary order and the time-dependent forcing term is only translation-bounded in .

1. Introduction

In this paper, we consider dynamical behavior of solutions for the following nonclassical diffusion equation with a fading memory term: The problem is supplemented with the boundary condition,and initial condition,where is a bounded smooth domain in and is a given external time-dependent forcing.

Concerning the memory kernel , as in [1–6], we assume the following hypotheses:and there is a constant , such thatFrom (4) and (5), we getWe also denote

As Wang and Zhong [5], we introduce the past history of , that is,as a new variable of the system, which will be ruled by a supplementary equation. Denote and (1) transforms into the following system:with initial-boundary conditionsThe past history of the variable satisfies the condition as follows: there exist positive constants and (from (5)), such that

The nonlinearity fulfills , along with arbitrary order polynomial growth restrictionand the dissipation conditionwhere and are positive constants.

This equation appears as an extension of the usual nonclassical diffusion equation in fluid mechanics, solid mechanics, and heat conduction theory (see [7–10]). Equation (1) with a one-time derivative appearing in the highest order term is called pseudoparabolic or Sobolev-Galpern equations [11–13]. Aifantis [7] proposed a general frame for establishing the equations. For certain classes of materials such as polymer and high-viscosity liquids, the diffusive process is nontrivially influenced by the past history of , which is represented in (1) by the convolution term against a suitable memory kernel characterizing the diffusive species [14].

Proving the existence of uniform attractors for (1) may be a hard task, mainly due to the fact that the nonlinearity satisfies arbitrary polynomial growth condition instead of critical, so Sobolev compact embedding theorems are no longer useful. The asymptotic compactness of solutions cannot be obtained by the usual method (used, e.g., in [2, 5, 6, 15, 16]).

When is a Dirac measure at some fixed time instant or when it vanishes, (1) reduces to the usual nonclassical diffusion equation, which has been investigated extensively by many authors, especially about the asymptotic behavior of solutions; see, for example, [16–22] and the references therein. In [21], the author has proved the existence of global attractor in , when under the assumptions that satisfies critical exponent growth condition corresponding to and some additional condition for nonlinearity, which essentially requires that the nonlinearity is subcritical. Recently, the authors in [23] obtained the existence of a global attractor for only under critical nonlinearity, and when , they proved the asymptotic regularity and the existence of (nonautonomous) exponential attractor. The asymptotic behavior of solutions of this equation has received considerably less attention in the literature under the assumption that the nonlinearity satisfies arbitrary polynomial growth condition. In the ordinary case for some recent results on this equations the reader can refer to Sun et al. [24] and Anh and Toan [25, 26]. Hereafter in [23] the authors mention that these are some mistakes in the coauthor’s earlier paper [24]. In [25], they proved that the nonautonomous dynamical system generated by this class of solutions has a pullback attractor. In [27], they proved the existence of global attractor in with the initial data and .

For convenience, hereafter let be the modular (or absolute value) of and let be the norm of . Let be the norm of . Denote by the dual space of and let be the norm of .

denotes any positive constant which may be different from line to line even in the same line.

Let be the Hilbert spaces of functions , endowed with the inner product and norm, respectively. Consider

We also define the product space : endowed with the norm

For convenience, we first show the preliminary result as follows.

Lemma 1 (see [5]). Let , and let the memory kernel satisfy (4) and (5). Then for any the following estimate holds, where is from (5).

For the time-dependent forcing , we assume the following hypotheses: Here is the space under the local weak convergence topology. Recall that a sequence converges to as in if and only if for all and every holds.

Let (translation-bounded in ), and introduce a set of functions obtained by time-translation in : We define the hull of , denoted as , as the closure of with respect to the local weak convergence topology of . If , then ; that is, where means the norm of .

Let , , be a family of processes acting in a Banach space with symbol space ; then for any The operators are the translation semigroup on ; a family of processes , , is called to satisfy the translation identity if

We recall (e.g., see [28]) also that the kernel of the process consists of all bounded complete trajectories of the process; that is,and is denoted by the kernel section at a time moment .

It is well-known that the key point is to obtain certain asymptotic compactness for the solution operator in the study of the long time behavior, especially for attractors. The nonlinearity having an arbitrary polynomial growth brings a difficulty here even for the autonomous and without memory case; see, for example, [22, 24, 27]. The main contribution of this paper is to extend the method in [15, 22] to overcome the difficulty caused by a lack of Sobolev compact embedding theorems. The conception, asymptotic contractive function, and new a priori estimates for verifying uniform asymptotic compactness of the family of processes are devised. We also prove some weak continuity for the family of processes and then obtain the structure of the compact uniform attractors.

The main results of this paper are given expression to in the following two theorems, which will be proved in Sections 2 and 3, respectively.

Theorem 2. Let be a Banach space and let , , be a family of processes on . Assume further that and is the hull of in . Then , , has a uniform attractor in provided that the following conditions hold true: (i), has a bounded absorbing set in .(ii)For any , there exist and an asymptotic contractive function on such that where depend on .

Theorem 3 (uniform attractor). Let be a bounded domain in with smooth boundary, and and satisfy (4)–(6) and (13)-(14), respectively. Assume further that and is the hull of in . Then the family of processes , , corresponding to (10)-(11), has a compact uniform (with respect to ) attractor in . Moreover, this attractor can be decomposed as follows:where is the kernel of the process and is the kernel section at time .

This theorem gives the existence of the uniform attractor and its structure as the union of the kernel sections (see [29]) of the nonautonomous process.

It is worth noting that Theorem 3 is also interesting in the autonomous case (see, e.g., [5]) or in the unbounded domain case (see, e.g., [22, 23]). This is the basis for further considering the asymptotic behavior, such as, to construct the nonautonomous (pullback) exponential attractor [2, 25]. On the other hand, from Theorem 16, the existence of the uniform attractor is obtained directly. However, since the external forcing term is only assumed to be translation-bounded, consequently the symbol space is only weak compact with respect to the local weak convergence topology. So, in order to obtain the structure equality (29), we need to verify some weak continuity for the family of processes, which is different from the usual strong continuity. This may be the reason why some authors (e.g., see [1]) have to assume further that the external forcing term is translation-compact.

2. Preliminaries

Definition 4 (see [15]). Let be a Banach space, be a bounded subset of and be a symbol (or parameter) space. We call a function , defined on , to be a contractive function on , if, for any sequence , and any , there is a subsequence and respectively, such that

Definition 5. Let be a Banach space and let be a bounded subset of and let be a symbol (or parameter) space. We call a function , defined on , to be an asymptotic contractive function on , if for any and , , there is a contractive function on such that

We denote the set of all asymptotic contractive functions on by .

In the following theorem, we present a new method (or technical) to verify the asymptotic compactness for the family of processes generated by evolutionary equations, which will be used in our later discussion.

Theorem 6. Let be a Banach space; , , is a family of processes on . Assume further that the following conditions hold true: (i), , has a bounded absorbing set in .(ii)For any and , , there exist and such that Then , , is uniform (with respect to ) asymptotically compact in , where depend on .

Proof. For any , we assume that is any bounded subset of , and , and satisfy as . It is enough to show that, from the assumptions, there is such that and for each large enough and and furthermore that where , and . So we only need to consider the case as .
In the following, the existence of a Cauchy subsequence of is proved by the diagonal method.
Take with as .
At first, for , by the assumptions, there exist and such thatSince , for some fixed we assume that is so large that for each and .
Let ; then from (23)–(26) we havewhere .
Due to the definition of and , we know that, for from (35), there is a contractive function on and has a subsequence such that one gets the following estimation:And similar to [15, 22, 30], we havewhich, combined with (35) and (37), implies thatTherefore, there is such thatBy induction, we obtain that, for each , there is a subsequence of and certain such thatholds for all . Now, we consider the diagonal subsequence . Since, for each , is a subsequence of , thenwhich, combined with as , implies that is a Cauchy sequence in . This shows that is precompact in . Then the proof is complete.

Proof of Theorem 2. From Theorem 6, the family of processes , , on satisfy the following conditions:(i), , has a bounded absorbing set in ;(ii), , is asymp(iii)totically compact in .Then , , has a uniform attractor in .

Lemma 7 (see [31]). Let be Banach spaces, with reflexive. Suppose that is a sequence that is uniformly bounded in and is uniformly bounded in , for some . Then there is a subsequence of that converges strongly in .

3. Uniform Attractors in

3.1. A Priori Estimates

We start with the following general existence and uniqueness of solutions for the nonclassical diffusion equations with fading memory which can be obtained by the approximation methods; here we formulate only the results.

Lemma 8. Assume that and and satisfies (13)-(14). Then, for any initial data and any , there exists a unique solution for problem (10)-(11). Moreover, we have the following Lipschitz continuity: for any , denote by the corresponding solutions of (10); then for all where is a monotonically increasing function.

By Lemma 8, we can define a process in as the following: and , , is a family of processes on . See [5] for more details.

Lemma 9. Let (13)-(14) hold, and . Then there exists a positive , which depends only on , such that, for any bounded subset , there is such that

Proof. Multiplying (10) by and then integrating over , it follows thatObserve thatUsing Lemma 1, we haveUsing the Hölder inequality, combining with (47) and (48), then (46) can be reformulated as follows:According to Poincaré inequality, we getLet ; again, we haveUsing the Gronwall Lemma, it follows from (51) thatWe know that is bounded, and by (12) and (6) So is bounded.
For any , we infer from (52) thatFor any , we getwhere . Then we get The proof is complete.

Combining with (43), we know that, for any , maps the bounded set of into a bounded set for all that is as follows.

Corollary 10. Let (13)-(14) hold, and . Then, for any bounded (in ) subset , there is such that

Lemma 11. Let (13)-(14) hold, and . Then, for any bounded (in ) subset , there exists a positive constant , such that hold for any (from Lemma 9).

Proof. Combining with (49), let ; it follows thatFor any (from Lemma 9) and integrating the inequality above from to , we can getBy Lemma 9, we haveSetting , then Let ; then we have that hold for any .

Let ; from assumption (3), there are positive constants , , , , such that

Lemma 12. There is a positive constant , for any (from Lemma 9); the following estimate holds for any .

Proof. Multiplying (10) by and then integrating in , we getBy the Hölder inequality and the Young inequality, we havehere is from (7). AndLetThenSo we get the following inequality:Integrating the inequality above to from to , we haveand integrating (72) to from to ,Besides, by the Hölder inequality, Young inequality, and (13), we can infer from (69) thatOn the contrary,Combining (73) and (74), we haveBy Lemma 11, it follows from (76) thatand by (75)–(77), we haveLet Then the proof is complete.

Lemma 13 (bounded uniformly absorbing set). Let (13)-(14) hold, and . Then there exist positive , , which depend only on , such that, for any bounded (in ) subset , there is such thathold for all and all .

For brevity, in the sequel, let be the bounded absorbing set obtained in Lemmas 9 and 12; that is,

Lemma 14. There is a positive constant , for any (from (81)); the following estimate holds for any .

Proof. By Lemma 12, for any , we integrate (70) to from to ; then we haveAccording to Lemma 13, Corollary 10, and Lemma 12, we know that is bounded for any , so there is a positive constant , and the conclusion is true.

3.2. Uniform Attractors

In the following, we will prove the existence of uniform attractors for system (10) with initial-boundary conditions (11) in by using the method of asymptotic contractive function.

Lemma 15. Let satisfy (13)-(14); is the sequence of solutions of (10) with initial data and ; then there is a subsequence of that converges strongly in and

Proof. By Lemmas 13 and 14, the sequence is uniformly bounded in and is uniformly bounded in . Since is reflexive, is . Let , , and ; then there is a subsequence of that converges strongly in by Lemma 7. One can write these asThen by using Hölder inequality, we have

Theorem 16. Let be a bounded domain in with smooth boundary, and let and satisfy (4)-(5) and (13)-(14), respectively. Assume further that and is the hull of in . , , is the family of processes generated by the solutions of (10) with initial data . Then , , is uniformly (with respect to ) asymptotic compact in .

Proof. For any and , let be the corresponding solution with respect to initial data and ; that is, is the solution of (10) with initial-boundary conditions (11).
For convenience, we denote and ; then satisfies the following equation:with initial-boundary conditionsMultiplying (87) by and integrating in , we obtainBy Lemma 1 and assumption (14), we have Using the Poincaré inequality, it follows thatLet ; then we get By the Gronwall Lemma, we haveFor any , there exists , such thatholds for any ; here Hereafter, we verify that there is such that is an asymptotic contractive function on . In fact where is from Corollary 10. Letwhere and . For any , let be fixed; then Combining Lemma 15 and Definition 5, we know that is the contractive function on and is the asymptotic contractive function on .
Then , , is uniform (with respect to ) asymptotic compact in by Theorem 6.

We also recall the following useful result, whose proof is simple and we omit it.

Lemma 17 (see [15]). Let be a reflexive Banach space and in . Then, for each compact (in ) subset , the uniform convergence holds: for any , there is , depending only on , such that

Proof of Theorem 3. Theorem 2 implies that the family of processes , , corresponding to (1), has a compact (in ) uniform (with respect to ) attracting set which is bounded in ; consequently, as a direct application of the abstract theorem [29, Chapter IV, Theorem 3.1], we obtain the existence of a compact uniform (with respect to ) attractor , and .
We remark that the above existence does not require any continuity of the family of processes (e.g., see [29]). However, in order to obtain the explicit form of , that is, equality (29), we need some continuity. Moreover, since the symbol space now has only weak compactness, we need to verify the corresponding weak continuity.
Hence, in order to obtain (29), we need to verify the following: for any fixed and , if in and with respect to the local weak convergence topology of , then In the following, we will verify that the continuity above holds on , which obviously is a compact uniformly attracting set for the corresponding skew-product flow semigroup (generated by the family processes , , and the shift semigroup ). We will complete the proof of the week continuity.
Let , let be the solutions of (10) with initial data , and let the time-dependent forcing , be the solutions of (10) with initial data and the time-dependent forcing ; we assume the following hypotheses:Since (10) holds as an equality in ( is conjugate to from (13)), this means that for any fixedwhere , and is the projector from to . Since is bounded in , following Corollary 10, we know that there is such thatHence, Then by the compactness lemma (Lemma 7) we know thatIt is Denote , , , and , , . The proof of the following inequality is similar to (93): Due to (107) and (102), we only need to show that if in , thenuniformly on -compact set. But this is a direct application of Lemma 17. Therefore we complete the proof of the continuity on .
Based on the continuity claim above, and by constructing a skew-product flow on and applying [29, Chapter IV, Theorem 5.1], we obtain the structure equality (29). This completes the proof of Theorem 3.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by National Science and Technology Major Projects of China (no. 2012ZX10001001-006) and National Natural Science Foundation of China (nos. 11101053 and 71471020).

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Copyright © 2016 Yongqin Xie 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.

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