A New Result of Stability for Thermoelastic-Bresse System of Second Sound Related with Forcing, Delay, and Past History Terms

Abstract

We consider a one-dimensional linear thermoelastic Bresse system with delay term, forcing, and infinity history acting on the shear angle displacement. Under an appropriate assumption between the weight of the delay and the weight of the damping, we prove the well-posedness of the problem using the semigroup method, where an asymptotic stability result of global solution is obtained.

1. Introduction

In this work, we considered with the following problem:

, with initial-boundary conditions with is a time delay and and are positive real numbers. The function is the temperature difference, is the heat flux, and are positive constants. We use the energy method and assume that the relaxation function satisfies the following hypotheses:

(G1) is a function such that

(G2) Let be a positive constant with and we suppose that the forcing term satisfies some hypotheses.

(A1) such that for all

where

(A2) with

Depending on some of the following parameters, we consider

It is well known that, in the single wave equation, if , that is, in the absence of a delay, the energy of system exponentially decays (see, e.g., [1–22]). On the contrary, if , that is, there exists only the delay part in the interior, the system becomes unstable.

Bresse system is a mathematical model that describes the vibration of a planar, linear shearable curved beam. The model was first derived by Bresse [23], and it consists of three coupled wave equations given by where We use , and to denote the axial force, the shear force, and the bending moment. By , and , we are denoting the longitudinal, vertical, and shear angle displacements. Here, , , , , and (see, e.g., [23]).

The Bresse system (10) is more general than the well-known Timoshenko system where the longitudinal displacement is not considered . The reader may refer to, for example, [24–34].

System (10) is an undamped system, and its associated energy remains constant when the time evolves. To stabilize system (10), many damping terms have been considered by several authors (see, e.g., [1, 35–40]).

In the succeeding text, we will present some works, which studied the stability of the dissipatif Bresse system. The paper [41] was concerned with asymptotic stability of a Bresse system with two frictional dissipations.

Under the condition of equal speeds of wave propagation, the authors proved that the system is exponentially stable. Otherwise, they show that Bresse system is not exponentially stable. Then, they proved that the solution decays polynomially to zero with optimal decay rate, depending on the regularity of initial data.

There are several works dedicated to the mathematical analysis of the Bresse system. They are mainly concerned with decay rates of solutions of the linear system. This is done by adding suitable damping effects that can be of thermal, viscous, or viscoelastic nature (see for instance [42–44]), among others.

Concerning thermoelastic Bresse system, [37] considered together with initial and specific boundary conditions and proved an exponential and only polynomial-type decay stabilities results.

2. Preliminaries and Well-Posedness

Firstly, we assume the following hypothesis:

Using semigroup theory, we will prove that systems (1)–(3) are well posed by introducing the following new variable [17].

Then, we have

Further, let

For this reason, we observe that

Therefore, problem (1) takes the form

The following are with the boundary conditions: The initial conditions are as follows:

Let be positive constants such that where is a real number with and are a positive constants, and the initial data are .

If we set then

Therefore, problems (19)–(21) can be written as where the operator is defined by

We consider the following spaces: where denotes the Hilbert space of valued functions on endowed with the inner product

We will show under the assumption (22) that generates a semigroup on .

Now, we consider the vectors and we define the inner product where the domain of is defined by

Important properties of the above metrics are stated in the following lemmas. Although most of these results are followed straightforwardly from the known results, they are crucial for what follows. So for the convenience of the reader, we give their proofs here.

Lemma 1. The operator is dissipative and satisfies, for any

Proof. For any , using the inner product, Then, By the fact that and using Young’s inequality, we find Keeping in mind condition (22), the desired result yields.

Lemma 2. The operator is surjective.

Proof. We need to show that for all , there exists such that that is, From (39), we define so
Inserting and (39) into (40), we get where Furthermore, by (39), we can find as for Following the same last approach, we obtain by using equation for in (39) From (39), we obtain Then, such that We note that the last equation in (41) with has a unique solution In order to solve (42), we consider where is the bilinear form given by is the linear form defined by It is easy to verify that is continuous and coercive, and is continuous. So applying the Lax-Milgram theorem, we deduce that for all , problem (48) admits a unique solution Since is dense in consequently, using Lemmas 1 and 2, we conclude that is a maximal monotone operator. Hence, by Hille-Yosida theorem (see [45]), we have the following well-posedness result such that (25) is satisfied.

Theorem 3. Let , then there exists a unique weak solution of problems (1)–(3). Moreover, if then

Lemma 4. The operator defined in (26) is locally Lipschitz in

Proof. Let then we have By using (6), Holder’s and Poincaré’s inequalities, we can obtain which gives us Then, the operator is locally Lipschitz in . The proof is hence complete.

3. Exponential Stability

Here, we present our stability result for the energy of the solution of systems (1)–(3), by using the multiplier technique. So we define the energy of our system by

The proof of the stability for our system is based on the following lemmas:

Lemma 5. Let be the solution of (19)-(21). Then, the energy functional, defined by (55), satisfies such that .

Proof. Multiplying , , , , and by and , respectively, and after simplification, we have (56).
With the fact it gives us (56).

Lemma 6. Let be the solution of (19)–(21). We have satisfies, for any the estimate

Proof. Taking the derivative of , using the fourth and fifth equations in (1) and performing integration by parts, we get According to Cauchy–Schwarz and Young’s inequalities with , we get (59).

Lemma 7. Let be the solution of (19)–(21). We have satisfies, for any , the estimate

Proof. For differentiation of , using equations in (1) and integration by parts, we obtain Estimate (62) follows by using Cauchy–Schwarz, Young’s, and Poincaré’s inequalities that

Lemma 8. Let be the solution of (19)–(21). Then, the energy functional satisfies the estimate

Proof. Using (1)–(3) gives Using Young’s and Poincaré’s inequalities, estimate (66) is established.

Lemma 9. Let be the solution of (19)–(21). Then, the energy functional satisfies for any the estimate

Proof. Taking the derivative of and using the second equation in (1), it follows that Young’s and Poincaré’s inequalities for (70) yield (69).

Lemma 10. Let be the solution of (19)–(21). Then, the energy functional satisfies the estimate

Proof. For differentiation of , using and , we arrive at Young’s inequality for (74) yields (73).

Lemma 11. Let be the solution of (19)-(21) and let . Then, the functional satisfies the estimate