Abstract

We prove the existence of a pullback -attractor in for the nonautonomous suspension bridge equations.

1. Introduction

In this paper, we consider the following nonautonomous suspension bridge equation: where is a bounded domain of with a smooth boundary , is an unknown function, which could represent the deflection of the road bed in the vertical plane, represents the restoring force, denotes the spring constant, represents the viscous damping, and is a given positive constant.

Suspension bridge equations have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and Mckenna in 1990. There are many results for the problem (1) (cf. [1–8]), for instance, the existence, multiplicity, and properties of the travelling wave solutions, and so forth. About the long-time behavior of suspension bridge equations, for the autonomous case, in [9, 10], the authors have discussed long-time behavior of the solutions of the problem on and obtained the existence of global attractors in the space and .

Caraballo et al. advanced the concept of the pullback -attractor in [11], and the existence of the pullback attractors was proved under the assumptions of asymptotic compactness and existence of a family of absorbing sets. Recently, Park and Kang [12] studied the pullback -attractor for suspension bridge equations in the weak space . Motivated by the ideas of [11, 13], we study the existence of a strong pullback -attractor for the nonautonomous suspension bridge equations in the strong topological space .

The nonlinear functions satisfy the following assumptions: where constant .

With the usual notation, we introduce the spaces , , where . We equip these spaces with inner product and norm  , , and , respectively: Obviously, we have where is dual space of , respectively; the injections are continuous and each space is dense in the following one.

Choosing , by the Poincaré inequality, we have

We introduce the Hilbert spaces and endow this space with norm

This paper is organized as follows. At first, in Section 2, we recall some preliminaries and results concerning the pullback attractor. Then, in Section 3, we prove our main result about the existence of pullback -attractor for the nonautonomous dynamical system generated by the solution of (1).

2. Notation and Preliminaries

Let be a complete metric space, be a metric space which will be called the parameter space. We define a nonautonomous dynamical system by a cocycle mapping which is driven by an autonomous dynamical system acting on a parameter space . Specifically, is a dynamical system on ; that is, is a group of homeomorphisms under composition on with the properties that(i) for all ;(ii) for all ;(iii)the mapping is continuous.

Definition 1. A mapping is said to be a cocycle on with respect to group , if (i) for all ;(ii) for all and all .
Let denote the family of all nonempty subsets of , let be the set of all bounded subsets of , and let be the class of all families . We consider a nonempty subclass .

Definition 2 (see [11]). Let be a nonautonomous dynamical system on . is said to be pullback -asymptotically compact if, for any , any , and any sequences , , the sequence possesses a convergent subsequence.

Definition 3 (see [11]). A family is said to be pullback -absorbing if, for each and , there exists such that

Definition 4 (see [11]). A family is said to be pullback -attracting if where is the Hausdorff semidistance between and .

Definition 5 (see [11]). A family is called a global pullback -attractor if it satisfies:(i) is compact for any ;(ii) is pullback -attracting;(iii) is invariant; that is, for all .

Definition 6 (see [14]). Let be a nonautonomous dynamical system on . is said to be satisfying pullback -Condition () if, for any , , and any , there exist a and a finite dimensional subspace of such that (i) is bounded;(ii), where is a bounded projector.

Lemma 7 (see [14]). Let be a nonautonomous dynamical system on . possesses a global pullback -attractor satisfying if it (i)has a pullback -absorbing set ;(ii)satisfies pullback -Condition ().

Theorem 8 (see [12]). Suppose that and the assumption (2)–(4) hold. satisfies (17). Then there exists a unique global pullback -attractor in for the nonautonomous dynamical system defined by (15).

We need the following lemmas in order to prove the main result.

Lemma 9 (see [14]). Let be an infinite dimensional Hilbert space and let the family be an orthonormal of . Suppose and, for any , for some . Then where is the orthogonal projector.

Lemma 10 (see [10]). Suppose that and satisfying (3). Then are continuous compact.

Lemma 11 (see [10]). Let , satisfying (3), and . Then is continuous compact.

3. Pullback -Attractors for Nonautonomous Suspension Bridge Equations

First, we give the following result.

Theorem 12. Suppose that , satisfying (2)–(4), if and . Then system (1) has a unique solution: where . If, in addition, and , then Moreover, the mapping is continuous in .

We can construct the nonautonomous dynamical system generated by problem (1) in or . We consider , and define The uniqueness of solution to problem (1) implies that Also, for all , the mapping defined by (15) is continuous. Consequently, the mapping defined by (15) is a continuous cocycle on .

Now, we assume that and for any , where . Let be the set of all functions such that and denotes the class of all families such that for some , where is the closed ball in centered at 0 with radius .

3.1. Pullback -Attractors in

In this subsection, we assume that and for any , Let be the set of all functions , which satisfies (18) with , and denotes the class of all families such that for some , where is the closed ball in centered at with .

Theorem 13. Suppose that satisfy (19). Then, there exists a unique global pullback -attractor in for the nonautonomous dynamical system defined by (15).

Proof. By Lemma 7, we need to prove the existence of a pullback -absorbing set belonging to and then show that the cocycle defined by (15) satisfies pullback -Condition ().
Multiplying (1) by and integrating over , we have
Using the Hölder and Young inequalities, we obtain
We can easily see that
According to (3), Theorem 8, and the Sobolev embedding theorem, we know that are uniformly bounded in . That is, there exists a constant , such that
In view of the Hölder inequality and (23), we can know
We choose small enough, such that ; we get
On the other hand, by the Hölder and Young inequalities, (7) and (23), it follows that
Therefore, combining (26)–(29), we get
Set
Thus, denote
We have
By the Gronwall lemma, we have
Set
Then
From Theorem 8, we have where and then
Set
Let . Combining (36) and (39), we have for all , , and .
Set and consider the family of closed balls in defined by From (18) and (41), is a pullback -absorbing for the cocycle in .
Next, we show that the cocycle satisfies the pullback -Condition ().
We assume that , are eigenvalue of operator in , satisfying denotes eigenvector corresponding to eigenvalue , which forms an orthogonal basis in , and at the same time they are also a group of canonical bases in or and satisfy
Let and is an orthogonal projector. For any , we write where .
Taking the scalar product with for (1) in , we have
Similar to the estimate of (21) and (22), we have
Moreover, we obtain where .
Combining (48)–(51), we obtain from (47)
Like for (26)–(29), using the Hölder and Young inequalities, we get where , , and .
By the Gronwall lemma, we have where .
Then, given any , we have for any and .
Now we estimate , and one by one. Given any and any , first, by the definition of , it is easy to see that there exists such that, for , .
Second, it is easy to see that By Theorem 8, there exists such that , . Lemma 10, we can choose large enough such that , for , .
Third, by Lemmas 10 and 11, we know that there exist and such that , for , .
Finally, by Lemma 9, we can choose large enough so that for .
By the above analysis and (55), we know that, for any , there exist and ; then which implies the pullback -Condition ().
We complete the proof.