Advances in Difference Equations

, 2010:748789 | Cite as

Exponential Stability and Global Attractors for a Thermoelastic Bresse System

Open Access
Research Article

Abstract

We consider the stability properties for thermoelastic Bresse system which describes the motion of a linear planar shearable thermoelastic beam. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. We use multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of the longitudinal or of the shear angle displacement. Moreover, the existence of the global attractor is firstly achieved.

Keywords

Wave Speed Exponential Stability Global Attractor Longitudinal Displacement Global Weak Solution 

1. Introduction

In this paper, we will consider the following system:
together with initial conditions
and boundary conditions

where Open image in new window , Open image in new window , and Open image in new window are the longitudinal, vertical, and shear angle displacement, Open image in new window and Open image in new window are the temperature deviations from the Open image in new window along the longitudinal and vertical directions, Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window are positive constants for the elastic and thermal material properties.

From this seemingly complicated system, very interesting special cases can be obtained. In particular, the isothermal system is exactly the system obtained by Bresse [1] in 1856. The Bresse system, (1.1)–(1.3) with Open image in new window , Open image in new window removed, is more general than the well-known Timoshenko system where the longitudinal displacement Open image in new window is not considered. If both Open image in new window and Open image in new window are neglected, the Bresse thermoelastic system simplifies to the following Timoshenko thermoelastic system:
which was studied by Messaoudi and Said-Houari [2]. For the boundary conditions

they obtained exponential stability for the thermoelastic Timoshenko system (1.8) when Open image in new window ; later, they proved energy decay for a Timoshenko-type system with history in thermoelasticity of type III [3], and this paper is similar to [2] with an extra damping that comes from the presence of a history term; it improves the result of [2] in the sense that the case of nonequal wave speed has been considered and the relaxation function g is allowed to decay exponentially or polynomially. We refer the reader to [4, 5, 6, 7, 8, 9, 10] for the Timoshenko system with other kinds of damping mechanisms such as viscous damping, viscoelastic damping of Boltzmann type acting on the motion equation of Open image in new window or Open image in new window . In all these cases, the rotational displacement Open image in new window of the Timoshenko system is effectively damped due to the thermal energy dissipation. In fact, the energy associated with this component of motion decays exponentially. The transverse displacement Open image in new window is only indirectly damped through the coupling, which can be observed from (1.2). The effectiveness of this damping depends on the type of coupling and the wave speeds. When the wave speeds are the same ( Open image in new window ), the indirect damping is actually strong enough to induce exponential stability for the Timoshenko system, but when the wave speeds are different, the Timoshenko system loses the exponential stability. This phenomenon has been observed for partially damped second-order evolution equations. We would like to mention other works in [11, 12, 13, 14, 15] for other related models.

Recently, Liu and Rao [16] considered a similar system; they used semigroup method and showed that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial-type decay rate can be obtained; their main tools are the frequency-domain characterization of exponential decay obtained by Prüss [17] and Huang [18] and of polynomial decay obtained recently by Muñoz Rivera and Fernández Sare [5]. For the attractors, we refer to [19, 20, 21, 22, 23, 24].

In this paper, we consider system (1.1)–(1.5); that is, we use multiplier techniques to prove the exponential stability result only for Open image in new window . However, from the theory of elasticity, Open image in new window and Open image in new window denote Young's modulus and the shear modulus, respectively. These two elastic moduli are not equal since

where Open image in new window is the Poisson's ratio. Thus, the exponential stability for the case of Open image in new window is only mathematically sound. However, it does provide useful insight into the study of similar models arising from other applications.

2. Equal Wave Speeds Case: Open image in new window 

Here we state and prove a decay result in the case of equal wave speeds propagation.

Define the state spaces
The associated energy term is given by
By a straightforward calculation, we have

From semigroup theory [25, 26], we have the following existence and regularity result; for the explicit proofs, we refer the reader to [16].

Lemma 2.1.

Let Open image in new window be given. Then problem (1.1)–(1.5) has a unique global weak solution Open image in new window verifying

We are now ready to state our main stability result.

Theorem 2.2.

Suppose that Open image in new window and Open image in new window . Then the energy Open image in new window decays exponentially as time tends to infinity; that is, there exist two positive constants Open image in new window and μ independent of the initial data and Open image in new window , such that

The proof of our result will be established through several lemmas.

where Open image in new window is the solution of

Lemma 2.3.

Proof.

By using the inequalities

and Young's inequality, the assertion of the lemma follows.

Lemma 2.4.

Proof.

Using (1.4) and (1.1), we get

The assertion of the lemma then follows, using Young's and Poincaré's inequalities.

Lemma 2.5.

Proof.

Using (1.3) and (1.5), we have

Then, using Young's and Poincaré's inequalities, we can obtain the assertion.

Next, we set

Lemma 2.6.

Proof.

Letting Open image in new window , Open image in new window , then using (1.2), (1.3), we have

Then, using Young's inequality, we can obtain the assertion.

Lemma 2.7.

Proof.

Let Open image in new window , Open image in new window , then using (1.1), (1.2), we have

Then, noticing Open image in new window , again, from the above two equalities and Young's inequality, we can obtain the assertion.

Next, we set

Lemma 2.8.

Proof.

Using (1.1), (1.2), we have
Noticing (2.3) and (2.4), we have that Open image in new window satisfy the following:
Similarly,

Then, insert (2.28) and (2.29) into (2.27), and the assertion of the lemma follows.

Now, we set

Lemma 2.9.

Proof.

Using (1.5), we have

Then, using Young's and Poincaré's inequalities, we can obtain the assertion.

Now, letting Open image in new window , we define the Lyapunov functional Open image in new window as follows:
By using (2.4), (2.9), (2.13), (2.16), (2.19), (2.23), (2.26), and (2.31), we have
We can choose Open image in new window big enough, Open image in new window small enough, and
Then Open image in new window are all negative constants; at this point, there exists a constant Open image in new window , and (2.34) takes the form

We are now ready to prove Theorem 2.2.

Proof of Theorem 2.2.

Firstly, from the definition of Open image in new window , we have
which, from (2.37) and (2.38), leads to

Integrating (2.39) over Open image in new window and using (2.38) lead to (2.6). This completes the proof of Theorem 2.2.

3. Global Attractors

In this section, we establish the existence of the global attractor for system (1.1)–(1.5).

Setting Open image in new window , Open image in new window , Open image in new window , Open image in new window , then, (1.1)–(1.5) can be transformed into the system
We consider the problem in the following Hilbert space:

Recall that the global attractor of Open image in new window acting on Open image in new window is a compact set Open image in new window enjoying the following properties:

(1) Open image in new window is fully invariant for Open image in new window , that is, Open image in new window for every Open image in new window ;

(2) Open image in new window is an attracting set, namely, for any bounded set Open image in new window ,

where Open image in new window denotes the Hausdorff semidistance on Open image in new window .

More details on the subject can be found in the books [23, 26, 27].

Remark 3.1.

The uniform energy estimate (2.6) implies the existence of a bounded absorbing set Open image in new window for the Open image in new window semigroup Open image in new window . Indeed, if Open image in new window is any ball of Open image in new window , then for any bounded set Open image in new window , it is immediate to see that there exists Open image in new window such that

for every Open image in new window .

Moreover, if we define

it is clear that Open image in new window is still a bounded absorbing set which is also invariant for Open image in new window , that is, Open image in new window for every Open image in new window .

In the sequel, we define the operator Open image in new window as Open image in new window with Dirichlet boundary conditions. It is well known that Open image in new window is a positive operator on Open image in new window with domain Open image in new window . Moreover, we can define the powers Open image in new window of Open image in new window for Open image in new window . The space Open image in new window turns out to be a Hilbert space with the inner product

where Open image in new window stands for Open image in new window -inner product on Open image in new window .

In particular, Open image in new window , Open image in new window , and Open image in new window . The injection Open image in new window is compact whenever Open image in new window . For further convenience, for Open image in new window , introduce the Hilbert space

Clearly, Open image in new window .

Now, let Open image in new window , where Open image in new window is the invariant, bounded absorbing set of Open image in new window given by Remark 3.1, and take the inner product in Open image in new window of (3.1) and Open image in new window to get
Here, the boundary term of integration by parts is neglected since we are working with more regular functions. We denote
Then, introduce the functional
By repeating similar argument as in the proofs of Lemmas 2.3–2.9 and (3.8), choosing our constants very carefully and properly, we get
On the other hand,
which gives
Let Open image in new window be the ball of Open image in new window ; from the compact embedding Open image in new window , Open image in new window is compact in Open image in new window . Then, due to the compactness of Open image in new window , for every fixed Open image in new window and every Open image in new window , there exist finitely many balls of Open image in new window of radius Open image in new window such that Open image in new window belongs to the union of such balls, for every Open image in new window . This implies that
where Open image in new window is the Kuratowski measure of noncompactness, defined by
Since the invariant, connected, bounded absorbing set Open image in new window fulfills (3.15), exploiting a classical result of the theory of attractors of semigroups (see, e.g., [28]), we conclude that the Open image in new window -limit set of Open image in new window , that is,

is a connected and compact global attractor of Open image in new window . Therefore, we have proved the following result.

Theorem 3.2.

Under the assumption of Open image in new window , problem (3.1) possesses a unique global attractor Open image in new window .

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Copyright information

© Zhiyong Ma. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.College of ScienceShanghai Second Polytechnic UniversityShanghaiChina

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