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 [25] for the analytic data, [69] for the dispersive equations and small data, and [1015] 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 [1823]. In general, the exponent is called to be critical when someone studies the problem in . Assuming the stiffness factor is nondegenerate (), References [1824] 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 [2732] 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 68, 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 68 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 68 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 68 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


(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


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