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

Djamel Ouchenane, Zineb Khalili, Fares Yazid, Mohamed Abdalla , Bahri Belkacem Cherif , and Ibrahim Mekawy Laboratory of Pure and Applied Mathematics, University of Laghouat, Algeria Mathematics Department, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt Preparatory Institute for Engineering Studies in Sfax, Tunisia Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Saudi Arabia


Introduction
In this work, we considered with the following problem: ðx, tÞ ∈ ð0, 1Þ × ð0,∞Þ, with initial-boundary conditions with τ > 0 is a time delay and μ 1 and μ 2 are positive real numbers. The function θ is the temperature difference, q is the heat flux, and ρ 1 , ρ 2 , ρ 3 , k, l, k 0 , b, γ, κ, α, β are positive constants. We use the energy method and assume that the relaxation function g satisfies the following hypotheses: (G1) g : ℝ + ⟶ ℝ + is a C 1 function such that (G2) Let ζ be a positive constant with and we suppose that the forcing term f ðψðx, tÞÞ satisfies some hypotheses.
(A1) f : ℝ ⟶ ℝ such that for all ψ 1 , ψ 2 ∈ ℝ, where k 0 > 0,θ > 0: withf Depending on some of the following parameters, we considerη It is well known that, in the single wave equation, if μ 2 = 0 , that is, in the absence of a delay, the energy of system exponentially decays (see, e.g., ). On the contrary, if μ 1 = 0, that is, there exists only the delay part in the interior, the system becomes unstable.
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][43][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.

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

Journal of Function Spaces
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 ξ be positive constants such that where τ is a real number with 0 < τ and μ 1 , μ 2 are a positive constants, and the initial data are ðφ 0 , φ 1 , ψ 0 , ψ 1 , w 0 , w 1 , f , θ 0 , q 0 , η 0 Þ. If we set then Therefore, problems (19)- (21) can be written as where the operator A is defined by We consider the following spaces: 3 Journal of Function Spaces where L 2 g ðℝ + , H 1 0 ð0, 1ÞÞ denotes the Hilbert space of H 1 0 − valued functions on ℝ + , endowed with the inner product We will show under the assumption (22) that A generates a C 0 semigroup on H . Now, we consider the vectors and we define the inner product where the domain of A 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 A is dissipative and satisfies, for any
Proof. For any U ∈ DðAÞ, using the inner product, Then, By the fact that −β and using Young's inequality, we find Keeping in mind condition (22), the desired result yields.
Proof. We need to show that for all that is, From (39), we define so θð0, tÞ = 0: where 5 Journal of Function Spaces Furthermore, by (39), we can find as zðx, 0Þ = uðxÞ for x ∈ ð0, 1Þ: Following the same last approach, we obtain by using equation for z in (39) From (39), we obtain Then, such that We note that the last equation in (41) with ϕðx, 0Þ = 0 has a unique solution In order to solve (42), we consider where a : is the bilinear form given by is the linear form defined by Lφ,ψ,w,q ð Þ= It is easy to verify that a is continuous and coercive, and L is continuous. So applying the Lax-Milgram theorem, we deduce that for all ðφ,ψ,w,qÞ ∈ H 1 * ð0, 1Þ ×H ϖ, θ, q, ϕÞ T , then we have By using (6), Holder's and Poincaré's inequalities, we can obtain which gives us Then, the operator F is locally Lipschitz in H . The proof is hence complete.

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 6 Journal of Function Spaces The proof of the stability for our system is based on the following lemmas: Lemma 5. Let ðφ, ψ, w, θ, q, z, η t Þ be the solution of (19)- (21). Then, the energy functional, defined by (55), satisfies such that C > 0.

Journal of Function Spaces
Proof. A simple differentiation of F 6 , using the first and third equations in (1), leads to and using Young's and Cauchy-Schwarz inequalities, with the fact that k = k 0 , gives (76).
Proof. Taking the derivative of (90) with respect to t, we have Then, by using the first equation in (1), we find Consequently, we arrive at Applying Young's inequality and Poincaré's inequality, we find (90).
where c is a positive constant.

Journal of Function Spaces
Proof. Taking the deviate of (95) with respect to t and using the equation (16), we get d dt Making use of the estimate above, implies that there exists a positive constant c 1 such that (96) holds.

Theorem 15.
Assume that η = 0 and k = k 0 : Then, ðφ, ψ, w, θ, q, z, η t Þ the solution of (19)- (21) satisfies where the positive constant c 0 is directly depending on initial data and the uniform constant c 1 is depending only on the coefficients of the system. For N, Then, from (56) At this point, we have to choose our constants very carefully. First, choosing ε i i = 1, :::10 small enough such that Moreover, we pick N 9 large enough so that and we take ε 11 small enough such that 11 Journal of Function Spaces Next, choosing N 5 large enough such that After that, we can choose N large enough such that Thus, the relation (100) becomes which leads by (55) that there exists also η 2 , such that Lemma 16. For N large enough, there exist two positive constants β 1 and β 2 depending on N i , i = 1, ⋯, 9 and ε i , i = 1, ⋯ , 11 such that Proof. We consider the functional and show that By using, the trivial relation Young's and Poincaré's inequalities, we get where α 1 , ⋯, α 6 are the positive constants as follows: forĈ = max α 1 , α 2 , α 3 , α 4 , α 5 , α 6 f g min ρ 1 , ρ 2 , ρ 3 , k, b, κ, γ, δ, τ 0 f g : Therefore, we get Then, we can choose N large enough so that β 1 = N −Ĉ > 0. Then, (108) holds true for β 2 = N +Ĉ > 0, and this concludes the proof of the Lemma.

Conclusion and Perspective
In this current study, a one-dimensional linear thermoelastic Bresse system with delay term, forcing, and infinity history acting on the shear angle displacement is considered. According to an appropriate assumption between the weight of the delay and the weight of the damping, the wellposedness of the problem using the semigroup method is proved, where an asymptotic stability result of global solution is obtained. In next article, we will generalize this result to convex bounded domain with a holomorphic map, and let x and y be two distinct fixed points for our problem. We will suppose there is at least one complex geodesics passing through two distinct variables. We will see that this method of proof cannot be generalized to the case of a bounded domain of a complex Banach space. Also, in the last part of the next article, we will study the fixed points of the analytical automorphisms of the open unit-ball B of a complex Banach space. More precisely, we will assume that B is homogeneous and we will show that, if the right hand side is an analytical automorphism of B, there exists a complex geodesic which we will specify formed of fixed points of the right hand. We will see that the set of fixed points of the right hand can be much larger by using the studied algorithm in ( [46][47][48][49][50][51]).

Data Availability
No data were used to support the study.

Conflicts of Interest
The authors declare that they have no competing interests.