Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

Ma, Honglv; Wang, Jing; Xie, Jun

doi:https://doi.org/10.1155/2021/7575078

Advances in Mathematical Physics

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Abstract Introduction Preliminaries Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 7575078 | https://doi.org/10.1155/2021/7575078

Pullback Attractors for Nonautonomous Degenerate Kirchhoff Equations with Strong Damping

Honglv Ma,¹Jing Wang,¹and Jun Xie²

Academic Editor: Sergey Shmarev

Received31 Jul 2021

Accepted15 Nov 2021

Published20 Dec 2021

Abstract

In this paper, we obtain the existence of pullback attractors for nonautonomous Kirchhoff equations with strong damping, which covers the case of possible generation of the stiffness coefficient. For this purpose, a necessary method via “the measure of noncompactness” is established.

1. Introduction

Let be a bounded domain with smooth boundary . We consider the following Kirchhoff wave model with strong damping:

where is a time-dependent external force term, and are initial data, and and are nonlinear functions specified later.

To describe small vibrations of an elastic stretched string, Kirchhoff [1] introduced the equation

where is the lateral deflection, the space coordinate, the time, the Young’s modulus, the mass density, the cross-section area, the length, the initial axial tension, and the external force. It has been called the Kirchhoff equation since then. In general, we call the Kirchhoff equation nondegenerate if the stiffness satisfies the strict hyperbolicity condition and degenerate if on . Obviously, the degenerate stiffness coefficient in (1) corresponds to the case that the initial axial tension equals zero.

From the mathematical point of view, global existence of the model like (2) has been proven in a multitude of special situations in . We refer to [2–5] for the analytic data, [6–9] for the dispersive equations and small data, and [10–15] for the weak damped equations.

Introducing the strong damping term provides an additional a priori estimate. Certainly, from the physical point of view, the dissipative plays an important spreading role for the energy gathered arising from the nonlinearity in a real process. Concerning Kirchhoff equations with strong dissipation, the first result on the well-posedness we are aware of was obtained by Nishihara [16]. He proved the global existence of the solution for the model . In recent years, many mathematicians and physicists paid their attentions to this type of problem and obtained the well-posedness under different types of hypotheses, such as the absent source term [17] and the subcritical source term [18–23]. In general, the exponent is called to be critical when someone studies the problem in . Assuming the stiffness factor is nondegenerate (), References [18–24] also proved the existence of the attractor. In the case of possible degeneration of the stiffness coefficient and the case of supercritical source term (), the first result on the well-posedness we are aware of is given by Chueshov [25]. However, when he proved the existence of a global attractor for problem (1) in the natural energy space endowed with a partially strong topology (in the sense, if with a partially strong topology, then strongly in and weakly in ), he assumed that

Under this condition, one can conclude that if is bounded for . Recently, Ma et al. [26] proved the existence of the global attractor in the case of degeneration for the autonomous Kirchhoff system.

The pullback attractor is a basic concept to study the longtime dynamics of nonautonomous evolution equations (see [27–32] and references therein). It is worth mentioning that there are only a few recent results devoted to the pullback attractor for nonautonomous systems like (1). In 2013, Wang and Zhong [33] investigated the upper semicontinuity of pullback attractors for problem (1) with () and . Recently, Li and Yang [34] studied the robustness of pullback attractors with . We notice that all these publications assume that the stiffness factor is nondegenerate, or more precisely, and is nondecreasing.

In this paper, we consider the problem (1) under the degenerate hyperbolicity condition . We do not assume that is monotone and allow , such as (degenerate and monotone) or (degenerate and nonmonotone) with . Based on the result in [25, 26], we prove the existence of pullback attractors in if is really degenerate. To overcome the difficulties caused by the degeneration, we first established a method (condition (-PC)) via “the measure of noncompactness” (some ideas come from [35, 36]) to prove that the process is pullback -asymptotically compact.

The paper is organized as follows. In Section 2, we introduce some preliminaries and establish a necessary abstract result (see Theorem 5). In Section 3, we discuss the existence of pullback attractors for the equation (1) (see Theorem 12).

2. Preliminaries

In this section, we will give some notations and results. As usual, we denote by and the norm and the inner product in , respectively. Let . We define the norms in by

Let be a Banach space and be a process acting on . In the following, we recall some definitions and results related to the pullback attractors; more details can be found in [27, 29, 33].

Definition 1. A family of compact sets is said to be a pullback attractor for process if
(i) is invariant, that is, , for all
(ii) is pullback attracting, i.e., for all bounded subset of , where is the Hausdorff semidistance

Definition 2. A family of sets is said to be a pullback absorbing family for process , if for all and all bounded , there exists , such that , for all . In addition, the family is said to be pullback -absorbing, if for any , there exists such that for .

Definition 3. A process is said to be pullback -asymptotically compact in , if for any , any sequences and ; the sequence is relatively compact in .

Lemma 4 (see [29]). Let the family be pullback absorbing and be continuous and pullback -asymptotically compact in . Then, the family defined by is a pullback attractor for .

To verify the pullback -asymptotically compact property in , it suffices to check the following condition.

2.1. -Pullback Condition (-PC)

For any and , there exist and a finite dimensional space of such that

where is a bounded projector.

Theorem 5. Let the family be a pullback -absorbing family of the process . If the -pullback condition (-PC) holds, then is pullback -asymptotically compact in .

Proof. By Definition 3, the result will be proven if we can show that for any , any sequences and , is relatively compact in .
For every , condition (-PC) implies that there exist and the finite dimensional space , such that (5) holds. Then, we have where is the measure of noncompactness defined as On the other hand, the properties of give that there exists , such that for , and Then, we can find , such that , which means that has a finite -net for any . The proof is complete.

3. Existence of Pullback Attractors

In this section, we will prove the existence of the pullback attractor when is really degenerate and is subcritical. We assume that , , and satisfy the following conditions.

Assumption 6. The function , for , and some constants , . Moreover, there exists such that where .

Assumption 7. is a function, , , and , and the following properties hold:
(i) if , then is arbitrary
(ii) if , then (iii) if , then where and are positive constants and is the first eigenvalue of .

Assumption 8. and

Remark 9. (1) or satisfies Assumption 6. It indicates that we include into the consideration the case of possibly degenerate since . Moreover, because in this case, becomes . If , then is a constant, and equation (1) is the nonlinear wave equation with strong damping.
(2) Assumptions 6 and 7 imply that there exist constants , with , such that where .
The well-posedness of the problem has been established by Chueshov [25] in the autonomous case. Noticing that the conditions of are more general than the above Assumptions 6–8, we can obtain the following Proposition 10 by a similar argument as in [25], except for the treatment of . The reader is referred to the Appendix for a detailed proof of these facts.

Proposition 10. Let Assumptions 6–8 be in force. Then, for and , problem (1) has a unique weak solution with and
(1) for every , there exists such that where ,
(2) for every , there exists such that (3) the Lipschitz stability holds for , where are two weak solutions of problem (1) with initial data ,

We define the solution operator associated to problem (1) as

where is the weak solution of problem (1) corresponding to initial data . Then, we know from Proposition 10 that is a continuous evolution process. For convenience, we denote by for any function . As , we also denote by .

Lemma 11. Let Assumptions 6–8 be valid. Then, the process defined in (21) has a pullback -absorbing family . Moreover, is bounded in for every .

Proof. As usual, the argument below can be justified by considering Galerkin approximations. Using the multiplier in Equation (1), we have that where for which is small enough, is a positive constant, and are independent of .
Since Assumption 6 implies that there exists such that combining with (15), we have that Then, we can find small enough such that By (22) and (28), we get that According to the Gronwall inequality, we have Then, (23), (26), and yield that where is a monotone positive function on . Let Obviously, is a pullback absorbing family of the process in . Moreover, for every , there exists a such that Let . By a standard procedure (see, e.g., Theorem 3.1 of [34]), we know that is a pullback absorbing family. Moreover, is bounded in for every , and there exists a such that for .

For simplicity, we assume that and in the following.

Theorem 12. Let Assumptions 6–8 be in force. Then, the process possesses a pullback attractor as shown in (4). Moreover, is bounded in for every .

Proof. According to Lemma 4, Theorem 5, Lemma 11, and the continuity of , it suffices to show that satisfies the condition (-PC). Let be an orthonormal basis and be the corresponding eigenvalues of which consists of eigenvectors of , i.e., . Let in and be an orthogonal projector. Denote , , and with , .
Let and be given. Without loss of generality, we assume .
For every and every , let Denote . It is easy to see that Since , we find Thus, where is independent of . Then, there exists such that On the other hand, for every , using (16) and (18), we get that where . Using (), one can find (without loss of generality, we assume ), such that for every , By the Sobolev embedding theorem, we know that the embedding is compact. Then, the boundedness of in implies that is compact in . Therefore, for , there exists , such that for every , where , , and .

Now, we will consider two situations. Without loss of generality, we assume .

Case 1. For every , the inequality holds for any , where .

Multiplying (1) by , we have that

Let . Since in this case, the above inequality implies that

By Gronwall’s inequality, we obtain that

Since

(37) yields that

Combining (45), we have

If , by the Hölder inequality, we have that

On the other hand, if , we get that

The above inequalities guarantee that And because we get that

i.e., .

Case 2. There exist and such that In this case, we claim that the following inequality is true, i.e., for every , In fact, if this claim is not true, the continuity of gives that is not an empty set. Let . It is easy to prove that . Moreover, by the definition of , we have that

According to the intermediate value theorem, we know that the set is not empty. Denoting , we can conclude from the definition of supremum that

Thus,

Notice that and for ; we have that . Then, integrating (43) on , we have that

It implies that

Combing (40), (41), and (59),we get

Thus, , which is in contradiction with (59), and condition (-PC) holds. This completes the proof.

Appendix

A. Proof of Proposition 10

We prove the well-posedness of Problem (1) using the same method as in [25].

Step 1. We start with the case when and assume that with some . We seek for the approximate solutions of the form satisfying the finite-dimensional projections of (1). Moreover, we have that

We omit the superscript in the sequel. Now, we use the multiplier and get that

Similarly, multiplying (1) by , we have that

Let

From (A3) and (A4),

Using (14), (24), and , we find that

where is independent of . Obviously,

By (13) and (14), there exists , for any ,

Combing (A8) and the above inequalities, we have that

Therefore, using Gronwall’s inequality, we obtain which means that

Now, multiplying (1) by , we have

Since when and for any when , when , we easily obtain that

It follows that

for every . Let

We can choose , such that

Thus, combing (A3), (A12), and (A15), we have that

This implies that

The above a priori estimates show that is bounded in for every . Moreover, using the equation for , we can show for some . Thus, there exists a subsequence, stilled denoted , and , such that

as . Moreover, by the Lions lemma (see Lemma 1.3 in [37]) we have that as . Then, making a limit transition in the nonlinear term, we prove the existence of a weak solution under the additional condition . One can see that this solution satisfies (14) and (15).

Step 2. Now, let and be weak solutions to (1) with different initial data such that for some . Notice that we do not assume here. Since , we conclude from (60) that

We can see that solves the equation

where with . By the definition of a weak solution, we can multiply (A25) by in and reduce that

Using for every when and when , we have that

Therefore, combining with we can conclude that

Now consider the multiplier . Since , , and , we can multiply (A26) by and obtain

Similar to (A29), we can get

Similar to (A27), we have

Therefore, we can conclude from Young’s inequality that

Let

for small enough. Then, there exists a positive constants such that

From (A29) and (A33), we have the estimation

Using Gronwall’s inequality, we get that

for all , which implies the desired conclusion in (20). By this inequality, we can prove the existence of weak solutions for initial data . Indeed, we can choose a sequence such that in . Owing to (20), the corresponding solutions converge to functions in . From the boundedness for in we also have weak convergence of to in the space . This implies that is a weak solution of Problem (1). By (19), this solution is unique.

Step 3. For the proof of smoothness properties stated in (18), we use the same method as [18, 38]. As usual, the argument below can be justified by considering Galerkin approximations. Set and differentiate (1) with respect to time. This yields

Multiplying the above equation by , we obtain that

This implies that

Multiplying the above equation by with and using Young’s inequality, we obtain that

Denote then we have that

for some positive constants depending on . Due to (A40) and (A41), it is apparent that

Multiplying (A44) by , we get that

It is easy to know

Since one can see that . Using and Young’s inequality, we get

By Gronwall’s inequality and (16), one can find

This implies (18). The proof is completed.

Data Availability

All data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was partly supported by the NSFC Grant 11801071 and the Scientific Research Project of Nanjing Xiaozhuang University (2017NXY53).

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Copyright © 2021 Honglv Ma 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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