# Exponential Stability and Global Attractors for a Thermoelastic Bresse System

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## 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

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.

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].

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.

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 Open image in new window , 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 Open image in new window , 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 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).

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.

for every Open image in new window .

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 .

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

Clearly, Open image in new window .

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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