Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 769514, 8 pages

http://dx.doi.org/10.1155/2013/769514

Research Article

Strong Attractor of Beam Equation with Structural Damping and Nonlinear Damping

Danxia Wang,¹ Jianwen Zhang,² and Yinzhu Wang³

¹School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

²Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

³Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China

Received 4 December 2012; Revised 18 February 2013; Accepted 27 February 2013

Academic Editor: Shueei M. Lin

Copyright © 2013 Danxia Wang 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.

Abstract.

This paper is mainly concerned with the existence of a global strong attractor for the nonlinear extensible beam equation with structural damping and nonlinear external damping. This kind of problem arises from the model of an extensible vibration beam. By the asymptotic compactness of the related continuous semigroup, we prove the existence of a strong global attractor which is connected with phase space .

1. Introduction

Global attractor is a basic concept in the study of long-time behavior of nonlinear dissipative evolution equations with various dissipation. There have been many methods to prove the existence of the global attractor. It can be proved by the theory of -contractions of the solution semigroup , such as [1–3] and the reference therein. It can also be proved by the decomposition of the solution semigroup (see Hale [4], Temam [5], etc.).

In this paper, we use the method of the asymptotically compact property of the solution semigroup which is different from the method of [1–5] to prove the existence of a strong global attractor for the Kirchhoff type equations with structural damping and nonlinear external damping which arises from the model of the nonlinear vibration beam

where , and are all positive constants, is a bounded domain of with smooth boundary , , , , and are nonlinear functions specified later, and is an external force term. represents the vertical deflection of the beam, and is a real-valued function on .

In this context of problem (1), based on the vibrating beams equation

which is proposed by Woinowsky-krieger [6]; Ma and Narciso [7] considered problem (1) without structural damping and posed a weak global attractor in weak phase space . Eden and Milani [8] considered the existence of exponential attractor for problem (1) with and a linear weak damping , being a nonlinear function and without structural damping. Ball [9] presented the existence and uniqueness of global solutions for problem (1) with are all linear functions.

On the other hand, the existence of the attractor for a related problem, with the boundary conditions of (2) replaced with , was considered by Ma and Narciso [7], Eden and Milani [8] with a linear damping or nonlinear damping without structural damping, respectively. Chueshov and Lasiecka [10] considered a kind of boundary condition which is but without structural damping.

Generally speaking, there have been many works on the long-time behavior for nonlinear beam equations [6–10]. But for the beam equation (1) with structural damping, in strong phase space , the global solutions and the strong global attractor have not still been proved until now.

The outline of this paper is arranged as follows: in Section 2 we give the existence and uniqueness of global solutions in space , in Section 3 we give the boundedness of solutions in phase space , and finally in Section 4, we give the proof of the existence of a strong global attractor in phase space .

2. Some Assumptions and Existence of Global Solution

In (1), we assume that damping term and the source term are in the form of

with

We assume that the nonlinear functions are all class , and satisfying and

The functions are also class , with , , and for all , where , and are all constants. There also exists constants such that

In addition, nonlinear function also satisfies

where , , and are all constants, .

Our analysis is based on the following Sobolev spaces: , , with the usual inner products and norms as follows, respectively:

Consider with the inner products and the norms .

Take and with the inner products and norms as follows, respectively:

Note that assumption (6) implies that , with or .

Finally, we assume that are the first eigenvalue of and , respectively; then we have

In the following, we state the result of the existence and uniqueness of the solutions for systems (1)–(3).

Theorem 1. Assume that , , and the assumptions of these functions , and hold; then problems (1)–(3) have unique solutions depending continuously on initial data in .

By virtue of Galerkin method, we may prove Theorem 1 combined with the priori estimates of Section 3.

According by Theorem 1, for any , we may introduce the mapping

It maps into itself, and it enjoys the usual semigroup properties as follows:

And it is obvious that the map , for all , is continuous in space . In the following, we will introduce the existence of bounded absorbing set and global attractor in space for map .

3. The Existence of Bounded Absorbing Set in Space

In this section, we will show boundedness of the solutions for systems (1)–(3).

Theorem 2. Assume that these assumptions of Theorem 1 hold then for the dynamic system determined by problems (1)–(3), there exists the boundary absorbing set in space .

Proof. Taking the inner products of with both sides of (1) and then making summation, we have

where , and is fixed at arbitrary time, and here the energy function is defined on by

Considering the assumption , we have

With , we have

With the assumptions , and and by using Mean Value Theorem and Mean Value inequality, we have

where among and . Set

Consider

So (15) is transformed into

Considering the assumptions , , , , and , and letting , we have

Substituting (23) into (22), we have

On the one hand, applying the Gronwall inequality to (24), we get

Note that and are bounded; then there exists a positive constant such that is bounded; so

On the other hand, considering that , fixing , and assuming that , then as , we have

that is,

Take the inner products by in both sides of (1); then make summation to get

Considering the continuity of the functions and , we have

where are all positive constants. Also

In addition, with , there exists a constant such that ; so

Also by using Schwarz and Mean Value inequalities and Mean Value Theorem, we have

where among and . Set

and write ; then (29) is transformed into

Here the function

is obtained by the energy function being changed slightly.

Let , we have , and so

Then an application of the Gronwall inequality leads to

If , there exists a positive constant such that .

Putting satisfing , then as , we get

So

The global estimate (40) shows the existence of an absorbing set of .

4. The Existence of Global Attractor in Space

The general theory [11] indicates that the continuous semigroup defined on a Banach space has a global attractor which is connected when the following conditions are satisfied.

(i) There exists a bounded absorbing set such that for any bounded set ,

(ii) is asymptotically compact; that is, for any bounded sequence in and tending to , there exists a subsequence such that is convergent as .

Theorem 3. Under the assumptions of Theorem 1, the continuous semigroup has a global attractor which is connected to .

Proof. Let be two solutions of Problems (1)–(3) in space as shown above corresponding to the initial data and with , respectively. Then satisfies

Taking the inner products in both sides of (42) by , , and , respectively, we have

Equation (46)(45)(44) yields

Consider that

where is among 0 and is among and , and

where is among 0 and , is among and .

Also considering for all , , for all , and , , by Hölder inequality we have

Setting

then substituting (48)–(56) into (47), by Schwarz inequality and Young inequality, and taking and , we have

Again setting , and considering that , we have . On the one hand, from (58) we have

Applying the Gronwall inequality to (59), we get

On the other hand, with and setting , we get

Hence

Now, let be a bound sequence in , and the corresponding solutions of problems (1)–(3) in . We assume . Let and . Then, applying estimate (62) to , we have

By taking in the above, we have

By Sobolev embedding Theorem, for any , we can extract a subsequence which is convergent in for any . For any , we first fix such that

And, next, taking large , we have

Then by (62) we have that

We conclude that is asymptotically compact on . The theorem is now proved.

Acknowledgments

This work is partially supported by the Natural Science Foundation of China (11172194) and Shanxi Province (2011021002-2 and 2010011008). The authors also wish to give their thanks to the referees for their comments to improve the presentation of this paper.

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