# Exponential Stability and Global Attractors for a Thermoelastic Bresse System

- Zhiyong Ma
^{1}Email author

**2010**:748789

https://doi.org/10.1155/2010/748789

© Zhiyong Ma. 2010

**Received: **13 September 2010

**Accepted: **29 October 2010

**Published: **21 November 2010

## 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.

## 1. Introduction

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

they obtained exponential stability for the thermoelastic Timoshenko system (1.8) when ; 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 [4–10] 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 or . In all these cases, the rotational displacement 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 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 ( ), 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 [11–15] 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 [19–24].

where is the Poisson's ratio. Thus, the exponential stability for the case of 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:

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

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.

We are now ready to state our main stability result.

Theorem 2.2.

*μ*independent of the initial data and , such that

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

Lemma 2.3.

Proof.

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

Lemma 2.4.

Proof.

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

Lemma 2.5.

Proof.

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

Lemma 2.6.

Proof.

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

Lemma 2.7.

Proof.

Then, noticing , again, from the above two equalities and Young's inequality, we can obtain the assertion.

Lemma 2.8.

Proof.

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

Lemma 2.9.

Proof.

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

We are now ready to prove Theorem 2.2.

Proof of Theorem 2.2.

Integrating (2.39) over 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).

Recall that the global attractor of acting on is a compact set enjoying the following properties:

(1) is fully invariant for , that is, for every ;

(2) is an attracting set, namely, for any bounded set ,

where denotes the Hausdorff semidistance on .

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

Remark 3.1.

it is clear that is still a bounded absorbing set which is also invariant for , that is, for every .

where stands for -inner product on .

is a connected and compact global attractor of . Therefore, we have proved the following result.

Theorem 3.2.

Under the assumption of , problem (3.1) possesses a unique global attractor .

## Authors’ Affiliations

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