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Djamel Ouchenane, Zineb Khalili, Fares Yazid, Mohamed Abdalla, Bahri Belkacem Cherif, Ibrahim Mekawy, "A New Result of Stability for Thermoelastic-Bresse System of Second Sound Related with Forcing, Delay, and Past History Terms", Journal of Function Spaces, vol. 2021, Article ID 9962569, 15 pages, 2021. https://doi.org/10.1155/2021/9962569
A New Result of Stability for Thermoelastic-Bresse System of Second Sound Related with Forcing, Delay, and Past History Terms
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.
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
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 , 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., ).
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  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,  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:
Then, we have
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
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
From (39), we define
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 ), we have the following well-posedness result such that (25) is satisfied.
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
The proof of the stability for our system is based on the following lemmas: