# Uniform attractors for non-autonomous suspension bridge-type equations

Xuan Wang12*, Lu Yang2 and Qiaozhen Ma1

Author Affiliations

1 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

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Boundary Value Problems 2014, 2014:75  doi:10.1186/1687-2770-2014-75

 Received: 5 July 2013 Accepted: 18 March 2014 Published: 28 March 2014

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

We discuss the long-time dynamical behavior of the non-autonomous suspension bridge-type equation, where the nonlinearity is translation compact and the time-dependent external forces only satisfy Condition () instead of being translation compact. By applying some new results and the energy estimate technique, the existence of uniform attractors is obtained. The result improves and extends some known results.

MSC: 34Q35, 35B40, 35B41.

##### Keywords:
non-autonomous suspension bridge equation; uniform Condition (C); uniform attractor

### 1 Introduction

Consider the following equations:

(1.1)

Suspension bridge equations (1.1) have been posed as a new problem in the field of nonlinear analysis [1] by Lazer and McKenna in 1990. This model has been derived as follows. In the suspension bridge system, the suspension bridge can be considered as an elastic and unloaded beam with hinged ends. denotes the deflection in the downward direction; represents the viscous damping. The restoring force can be modeled owing to the cable with one-sided Hooke’s law so that it strongly resists expansion but does not resist compression. The simplest function to model the restoring force of the stays in the suspension bridge can be denoted by a constant k times u, the expansion, if u is positive, but zero if u is negative, corresponding to compression; that is, , where

Besides, the right-hand side of (1.1) also contains two terms: the large positive term l corresponding to gravity, and a small oscillatory forcing term , possibly aerodynamic in origin, where ϵ is small.

There are many results for the problem (1.1) (cf.[1-7]), for instance, the existence, multiplicity and properties of the traveling wave solutions, etc.

In the study of equations of mathematical physics, the attractor is a proper mathematical concept as regards the depiction of the behavior of the solutions of these equations when time is large or tends to infinity, which describes all the possible limits of solutions. In the past two decades, many authors have proved the existence of an attractor and discussed its properties for various mathematical physics models (e.g., see [8-10] and the references therein). For the long-time behavior of suspension bridge-type equations, for the autonomous case, in [11,12] 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 .

It is well known that, for a model to describe the real world which is affected by many kinds of factors, the corresponding non-autonomous model is more natural and precise than the autonomous one, moreover, it always presents a nonlinear equation but not just a linear one. Therefore, in this paper, we will discuss the following non-autonomous suspension bridge-type equation: Let Ω be an open bounded subset of with smooth boundary, , and we add the nonlinear forcing term (which is dependent on the deflection u and time t) to (1.1) and neglect gravity, then we can obtain the following initial-boundary value problem:

(1.2)

where is an unknown function, which could represent the deflection of the road bed in the vertical plane; and are time-dependent external forces; represents the restoring force, k denotes the spring constant; represents the viscous damping, α is a given positive constant.

To our knowledge, this is the first time for one to consider the non-autonomous dynamics of equation (1.2). At the same time, in mathematics, we only assume that the force term satisfies the so-called Condition () (introduced in [13]), which is weaker than the assumption of being translation compact (see [8] or Section 2 below).

This paper is organized as follows. At first, in Section 2, we give (recall) some preliminaries, including the notation we will use, the assumption on nonlinearity and some general abstract results for a non-autonomous dynamical system. In Section 3 we prove our main result about the existence of a uniform attractor for the non-autonomous dynamical system generated by the solution of (1.2).

### 2 Notation and preliminaries

With the usual notation, we introduce the spaces , , , where . We equip these spaces with an inner product and a norm , , , and , , respectively,

Obviously, we have

where , is the dual space of H, V, respectively, the injections are continuous and each space is dense in the following one.

In the following, the assumption on the nonlinearity g is given. Let g be a function from to ℝ and satisfy

(2.1)

where , and there exists , such that

(2.2)

Suppose that γ is an arbitrary positive constant, and

(2.3)

(2.4)

where δ is a sufficiently small constant.

As a consequence of (2.1)-(2.2), if we denote , then there exist two positive constants , such that

(2.5)

(2.6)

where , and we can take m sufficiently small.

By virtue of (2.3), we can get

(2.7)

When , the problem (1.2) is equivalent to the following equations in H:

(2.8)

From the Poincaré inequality, there exists a proper constant , such that

(2.9)

We introduce the Hilbert spaces

and endow this space with the norm

To prove the existence of uniform attractors corresponding to (2.8), we also need the following abstract results (e.g., see [8]).

Let E be a Banach space, and let a two-parameter family of mappings on E:

Definition 2.1 ([8])

Let Σ be a parameter set. , is said to be a family of processes in Banach space E, if for each , is a process; that is, the two-parameter family of mappings from E to E satisfy

(2.10)

(2.11)

where Σ is called the symbol space and is the symbol.

Note that the following translation identity is valid for a general family of processes , , if a problem has unique solvability and for the translation semigroup satisfying :

A set is said to be a uniformly (w.r.t. ) absorbing set for the family of processes , if for any and , there exists such that for all . A set is said to be uniformly (w.r.t. ) attracting for the family of processes , , if for any fixed and every ,

(2.12)

Definition 2.2 ([8])

A closed set is said to be the uniform (w.r.t. ) attractor of the family of processes , if it is uniformly (w.r.t. ) attracting (attracting property) and contained in any closed uniformly (w.r.t. ) attracting set of the family of processes , : (minimality property).

Now we recall the results in [14].

Definition 2.3 ([14])

A family of processes , , is said to be satisfying the uniform (w.r.t. ) Condition (C) if for any fixed , and , there exist a and a finite dimensional subspace of E such that

(i) is bounded; and

(ii) , ,

where and is abounded projector.

Theorem 2.4 ([14])

Let Σ be a complete metric space, and letbe a continuous invariantsemigroup on Σ satisfying the translation identity. A family of processes, , possess a compact uniform (w.r.t. ) attractorinEsatisfying

(2.13)

if it

(i) has a bounded uniformly (w.r.t. ) absorbing set; and

(ii) satisfies uniform (w.r.t. ) Condition (C),

where. Moreover, ifEis a uniformly convex Banach space, then the converse is true.

Let X be a Banach space. Consider the space of functions , with values in X that are 2-power integrable in the Bochner sense. is a set of all translation compact functions in , is the set of all translation bound functions in .

In [13], the authors have introduced a new class of functions which are translation bounded but not translation compact. In Section 3, let the forcing term satisfy Condition (); we can prove the existence of compact uniform (w.r.t. , ) attractor for a non-autonomous suspension bridge equation in .

Definition 2.5 ([13])

Let X be a Banach space. A function is said to satisfy Condition () if, for any , there exists a finite dimensional subspace of X such that

where is the canonical projector.

Denote by the set of all functions satisfying Condition (). From [13], we can see that .

Remark 2.6 In fact, the function satisfying Condition () implies the dissipative property in some sense, and Condition () is very natural in view of the compact condition, and the uniform Condition (C).

Lemma 2.7 ([13])

If, then for anyandwe have

whereis the canonical projector andδis a positive constant.

In order to define the family of processes of the equations (2.8), we also need the following results:

Proposition 2.8 ([8])

IfXis reflexive separable, then

(i) for all, ;

(ii) the translation groupis weakly continuous on;

(iii) for all.

Proposition 2.9 ([8])

Let, then

(i) for all, , and the setis bound in;

(ii) the translation groupis continuous onwith the topology of;

(iii) for all.

### 3 Uniform attractors in

To describe the asymptotic behavior of the solutions of our system, we set and , where denotes the closure of a set in topological space . If , then ; this is to be

where denotes the norm in H.

#### 3.1 Existence and uniqueness of solutions

At first, we give the concept of solutions for the initial-boundary value problem (2.8).

Definition 3.1 Set , for . We suppose that , , satisfying (2.1)-(2.4) and . The function is said to be a weak solution to the problem (2.8) in the time interval I, with initial data , provided

(3.1)

for all and a.e. .

Then, by using of the methods in [15] (Galerkin approximation method), we get the following result as regards the existence and uniqueness of solutions:

Theorem 3.2 (Existence and uniqueness of solutions)

Define, . Let, , satisfying (2.1)-(2.4). Then for any given, there is a unique solutionfor the problem (2.8) in. Furthermore, for, let (and) be two initial conditions, and denote bythe corresponding solutions to the problem (2.8). Then the estimates hold as follows: for all,

(3.2)

Thus, (2.8) will be written as an evolutionary system, introduced as and for brevity, as , the system (2.8) can be written in the operator form

(3.3)

where is the symbol of (3.3). If , then the problem (3.3) has a unique solution . This implies that the process given by the formula is defined in .

Now we define the symbol space. A fixed symbol can be given, where is in , the function satisfying (2.1)-(2.4), and ℳ is a Banach space,

endowed with the following norm:

Obviously, the function is in . We define , where denotes the closure of a set in topological space (or ). So, if , then and all satisfy Condition ().

Applying Proposition 2.8, Proposition 2.9, and Theorem 3.2, we can easily know that the family of processes , , , are defined. Furthermore, the translation semigroup satisfies , , and the following translation identity:

holds.

Then for any , the problem (3.3) with σ instead of possesses a corresponding process acting on .

Consequently, for each , (here , satisfying (2.1)-(2.4)), we can define a process

and , , is a family of processes on .

#### 3.2 Bounded uniformly absorbing set

Before we show the existence of bounded uniformly absorbing set, we firstly make a prior estimate of solutions for equations (2.8) in .

Lemma 3.3Assume thatis a solution of (2.8) with initial data. If the nonlinearitysatisfies (2.1)-(2.4), , , , then there is a positive constantsuch that for any bounded (in) subsetB, there existssuch that

(3.4)

Proof Now we will prove to be bounded in .

We assume that ϱ is positive and satisfies

(3.5)

Multiplying (2.8) by and integrating over Ω, we have

(3.6)

We easily see that

(3.7)

(3.8)

Then, substituting (3.7)-(3.8) into (3.6), we obtain

(3.9)

In view of (2.4) and (2.6), we know

(3.10)

and

(3.11)

Consequently,

(3.12)

We introduce the functional as follows:

(3.13)

Setting , we choose proper positive constants m and δ, such that

(3.14)

hold, then .

We define , then

(3.15)

where , .

Analogous to the proof of Lemma 2.1.3 in [8], we can estimate the integral and obtain

(3.16)

where .

By virtue of (2.3), we get

Choosing , we obtain from (3.13)

(3.17)

In consideration of (2.7) and , we see

(3.18)

and

(3.19)

Combining (3.16), (3.17), and (3.19), we deduce that

Assuming that , as , we have

(3.20)

We thus complete the proof. □

And then, combining Theorem 3.2 with Lemma 3.3, we get the result as follows.

Theorem 3.4 (Bounded uniformly absorbing set)

Presume thatand. Letsatisfy (2.1)-(2.4), , and, be the family of processes corresponding to (2.8) in, thenhas a uniformly (w.r.t. ) absorbing setin. That is, for any bounded subset, there existssuch that

#### 3.3 The existence of uniform attractor

We will show the existence of uniform attractor to the problem (2.8) in .

Theorem 3.5 (Uniform attractor)

Letbe the family of processes corresponding to the problem (2.8). Ifsatisfying (2.1), (2.2), (2.5), and (2.6), , and, thenpossesses a compact uniform (w.r.t. ) attractorin, which attracts any bounded set inwith, satisfying

(3.21)

whereis the uniformly (w.r.t. ) absorbing set in.

Proof From Theorem 2.4 and Theorem 3.4, we merely need to prove that the family of processes , satisfy the uniform (w.r.t. ) Condition (C) in . We assume that , are eigenvalue of operator A in , satisfying

denotes eigenvector corresponding to eigenvalue ,  , which forms an orthogonal basis in , at the same time they are also a group of canonical basis in V or H, and they satisfy

Let , is an orthogonal projector. For any , we write

where .

Choosing and . Taking the scalar product with for (2.8) in H, we have

(3.22)

where

(3.23)

(3.24)

Clearly, we get

(3.25)

(3.26)

Combining (3.23)-(3.26), we obtain from (3.22)

(3.27)

We define the functional

and we set , then

(3.28)

By Gronwall’s lemma, we obtain

(3.29)

Obviously, there exists a constant , such that

so

(3.30)

Since , , from Lemma 2.7, we can know for any , there exists a constant m large enough such that

(3.31)

(3.32)

where .

Let , then

So for every , we get

(3.33)

where .

Therefore, the family of processes , satisfy uniformly (w.r.t. ) Condition (C) in . Applying Theorem 2.4, we can obtain the existence of a uniform (w.r.t. ) attractor of the family of processes , in , which satisfies (3.21).

We thus complete the proof. □

So we can draw the conclusion: when the nonlinearity is translation compact and the time-dependent external forces only satisfies Condition () instead of translation compact, the uniform attractors in exist.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

The authors read and approved the final manuscript.

### Acknowledgements

This work is partly supported by NSFC (11361053, 11201204, 11101134, 11261053, 11101404) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-11-5).

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