On a Thermoelastic Laminated Timoshenko Beam: Well Posedness and Stability

Abstract

In this paper, we are concerned with a linear thermoelastic laminated Timoshenko beam, where the heat conduction is given by Cattaneo’s law. We firstly prove the global well posedness of the system. For stability results, we establish exponential and polynomial stabilities by introducing a stability number .

1. Introduction

In this paper, we address the following thermoelastic laminated Timoshenko beam in :which subject to the following boundary conditions:and initial conditionswhere , and are positive constants. represents the difference temperature and is the heat flux.

Laminated beam, which is a relevant research subject due to the high applicability of such materials in the industry, was firstly introduced by Hansen and Spies, see, for instance [1, 2]. They introduced a mathematical model for two-layered beams with structural damping due to the interfacial slip which is given bywhere the coefficients are positive constants and represent density, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness, and adhesive damping parameter, respectively. The function denotes the transversal displacement, represents the rotational displacement, and is proportional to the amount of slip along the interface at time t and longitudinal spatial variable x. The third equation describes the dynamics of the slip.

Up till now, there are some results concerning laminated beam equations, which are mainly concerned with global existence and stability of the related system. By adding suitable damping effects, such as internal damping, (boundary) frictional damping, and viscoelastic damping, it was shown that if the linear damping terms are added in two of the three equations, system (4) is exponentially stable under the “equal wave speeds” assumption . But if the damping terms are added in the three equations, then the system decays exponentially without the equal wave speeds assumption, see, for example, [3–17]. For thermoelastic laminated Timoshenko beam, there are few published works, we can mention the results due to Liu and Zhao [18] and Apalara [19]. In [18], the authors considered the following laminated beams with past historytogether with the following boundary conditions:

They firstly proved the global well posedness of solutions to the system. The main results are the stability of the system. If , they proved the exponential and polynomial stabilities depending on the behavior of the kernel function only. If , they established exponential stability in case of equal wave speeds assumption and lack of exponential stability in case of nonequal wave speeds assumption. Apalara [19] considered a laminated beam with second sound of the formtogether with the following boundary conditions:and proved the global well posedness and established exponential and polynomial stabilities depending on the parameter

One can also refer to two recent results of laminated beams with thermal damping in [20, 21], and a result of a coupled hyperbolic equations with a heat equation of second sound in [22].

When , system (4) reduces to the well-known Timoshenko system, which have been widely studied. There are so many papers on the Timoshenko system in the literature, most of those results recover the global well posedness, stability, and long-time dynamics by adding some kinds of damping. Here, we recall some works on the thermoelastic Timoshenko system. Muñoz Rivera and Racke [23] considered a Timoshenko system with thermoelastic dissipation and established exponential stability in case of equal wave speed assumption and polynomial stability if wave speeds are nonequal. Almeida Júnior et al. [24] studied a thermoelastic Timoshenko beam acting on shear force. They obtained the same stability results as in [23]. In addition, they proved that the polynomial decay is optimal. Fernández Sare and Racke [25] considered a Timoshenko system with second sound. They proved that the system is not exponentially stable even if the propagation speeds are equal. The results were generalized by Guesmia et al. [26]. Recently, Santos et al. [27] introduced a stability number for the system in [25] and established the exponential decay result for and polynomial decay for by using the semigroup method. One can also find a stability result for the Timoshenko system with second sound in Apalara et al. [28]. Feng [29] considered a Timoshenko-Coleman-Gurtin system and studied the long-time dynamics of the system. We at last mention the contribution of Hamadouche and Messaoudi [30] and Aouadi and Boulehmi [31], where the authors considered two classes of nonuniform thermoelastic Timoshenko systems and proved global well posedness and established some stability results.

Our goals in the present work are to study the global well posedness and stability of systems (1)–(3). The main points are summarized as follows:(i)We prove the global well posedness of systems (1)–(3) by using Lumer–Philips theorem. The main result is presented in Theorem 1.(ii)We introduce a new stability number denoted by and we show that the system is exponential stable when and polynomial stable when . The main results are presented in Theorems 1 and 2.(iii)The proof of stability results is based on the multiplier method. Since the boundary conditions here we considered are different from those in Apalara [19], so the multipliers we will define are greatly different from the multipliers in Apalara [19].

It follows, from (1), that

If we denotewe easily verify that satisfies (1) and in addition,

Hence, Poincaré’s inequality holds for . In the following, we work with and but write ω and q for convenience.

The remaining paper is planned as follows. In Section 2, we study the well posedness of the system. In Section 3, we establish the stability results. Throughout this paper, is a generic constant that changes from one inequality to another.

2. Well Posedness

We start by denoting the vector-valued function by U:

Then, systems (1)–(3) can be written aswhere the operator is defined by

We consider the following spaces:

Letbe the Hilbert space equipped with the inner product

The domain of is given by

The well posedness result can be stated in the following theorem.

Theorem 1. Let , then problems (1)–(3) admit a unique weak solution . In addition, if , then .

Proof. It is easy to obtain that, for any ,which implies the operator is a dissipative operator.
In what follows, we shall show the operator is surjective. In other words, given , we will seek a solution ofWe rewrite (21) aswhich implies thatWe infer from (27) thatReplacing (24)–(26) and (28) in (23), we see thatWe multiply (29) by , , , and , respectively, and integrate their sum over to get the following variational formulation:where the bilinear form is given byand the linear form is defined byWe denote the Hilbert space byequipped with the normIt is easy to get that and are bounded. Moreover there exists a positive constant m such thatThus, is coercive on . Consequently, using Lax–Milgram theorem, we conclude that (30) has a unique solution:Substituting , and into (24)–(26) and (28), respectively, we haveLet and denotewhich gives us . Now we replace by in (30) to obtaini.e.,which yieldsThus,Moreover, (39) also holds for any . Then, by using integration by parts, we obtainThen, we get for any ,From (28), we obtainSince ϕ is arbitrary, we get that Hence, . Using similar arguments as above, we can obtainThus, and is maximal. By using Lumer–Philips theorem, see, for example, Liu and Zheng [32] and Pazy [33], we end the proof of the theorem.

3. Stability

In this section, we study the stability of systems (1)–(3). More precisely, we establish exponential and polynomial decay results depending on χ defined by

The energy functional of systems (1)–(3) is defined by

Now we give our stability results.

Theorem 2 (exponential decay). Suppose that . For any initial data , there exist two positive constants μ and η such that the energy functional (48) satisfies

Theorem 3 (polynomial decay). Suppose that . For any initial data , there exists positive constant such that the energy functional (48) satisfiesTo prove Theorems 1 and 2, we need the following technical lemmas.

3.1. Technical Lemmas

Lemma 1. It holds that the energy functional is nonincreasing and satisfies

Proof. Multiplying (1) by , , , θ, and q, respectively, integrating the results by parts and using boundary condition (1), we easily get (51).

Lemma 2. Define the functional by

Then, we have for any ,where is the Poincaré constant.

Proof. It follows from (1) thatUsing integration by parts and boundary condition (1), we arrive atThen, by using Hölder’s, Young’s, and Poincaré’s inequalities, we can get (53) from (57).

Lemma 3. The functional defined bysatisfies for any ,

Proof. Differentiating with respect to t and using (1), we see thatUsing integration by parts, we obtainThen, by using Young’s inequality and Hölder’s inequality, we can get (57).

Lemma 4. Define the functional by

Then, we can get for any ,

Proof. Differentiating with respect to t and using (1), we obtainIntegration by parts gives usBy using Young’s inequality and Hölder’s inequality, we can get (61).

Lemma 5. The functional defined bysatisfies for any ,

Proof. We take the derivative of and use (1) and integrate by parts to obtainThen, using Young’s inequality, we can get (65).

Lemma 6. Define the functional by

Then, we have for any ,where are positive constants.

Proof. By differentiating with respect to t, we haveUsing equation (1) and integrating by parts, we see thatInserting (70)–(75) into (69), we can obtainRecalling and using Young’s inequality, we conclude thatand for any ,which, together with (76)–(78), gives us (68).

Lemma 7. The functional defined bysatisfieswhere is a positive constant.

Proof. follows from (1) thatYoung’s inequality gives us (82).

3.2. Exponential Stability: Proof of Theorem 1

Proof. We define the functional bywhere N and are positive constants that will be chosen later.
Note thatReplacing (84) in (57) and then combining (51)–(53), (57)–(68), and (82), we obtainTakingwe obtainAt this point, we first choose large enough such thatFor fixed , we take so large thatThen, we pick large so thatAnd then we choose so large thatAt last, we take large enough so that the functional is equivalent to the energy functional , i.e., there exist two positive constants:and further so thatRecalling (48), we infer that there exists a positive constant such that, for any ,which, along with (92), impliesIntegrating (95) over , we have, for any which, using (95) again, gives us (49). The proof of Theorem 1 is done.

3.3. Polynomial Stability: Proof of Theorem 2

In this section, we consider the case to prove Theorem 2.

Differentiating system (1) with respect to time, we obtain the following system:which subject to the following boundary conditions:

For any initial data , system (97) is well posed. Next, we introduce second-order energy functional by

By using the same arguments as in Lemma 3, we can get the second-order energy defined by (99) is nonincreasing and satisfies

In Lemma 6, we have proved that, for any ,

Thanks to (1) and Young’s inequality, we derive that

Then, for any ,

Therefore, the derivative of satisfies

Proof. We define the functional byIt follows from (51)–(53), (57)–(65), and (100)–(104) thatWith the same choice of constants as in Section 3.2, we further take so large thatNoting that (48), we know that there exists a positive constant such that, for any ,Since the energy functional is positive and nonincreasing, we infer (108) that, for any ,which gives usHere, . The proof is complete.

Remark 1. We point out that the functional is inequivalent to the energy functional . That is to say, (92) does not hold true.

Data Availability

No data were used during this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 11701465).