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Existence and general stabilization of the Timoshenko system with a thermo-viscoelastic damping and a delay term in the internal feedback

In this paper, we consider a Timoshenko system with a thermo-viscoelastic damping and a delay term in the internal feedback together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and {\mu}1, {\mu}2 are positive constants. Under an hypothesis between the weight of the delay term in the feedback and the the weight of the friction damping term, using the Faedo-Galerkin approximations together with some energy estimates, we prove the global existence of the solutions. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the weights of the two feedbacks and the coefficients, we establish the general energy decay result from which the exponential and polynomial types of decay are only special cases.

System (1.1) arises in the theory of the transverse vibration of a beam which was first introduced by Timoshenko. In 1921, Timoshenko [25] considered the following system of coupled hyperbolic equations: where ϕ is the transverse displacement of the beam and ψ is the rotation angle of the filament of the beam. The coefficients ρ, I ρ , E, I and K are the density, the polar moment of inertia of a cross section, Young's modulus of elasticity, the moment of inertia of a cross section, and the shear modulus, respectively.
Many mathematicians have studied system (1.3) and some results concerning the existence and asymptotic behavior of solutions have been established, see for instance [2,10,22,23] and the references therein. Kim and Renardy [5] considered ( and used the multiplier techniques to establish an exponential decay result for the energy of (1.3). They also provided numerical estimates to the eigenvalues of the operator associated with the system (1.3). Soufyane and Wehbe [24] considered They proved that the uniform stability of (1.4) holds if and only if the wave speeds are equal K ρ = EI Iρ ; Otherwise only the asymptotic stability has been proved. For more results related to system (1.3), we refer the readers to [4,16,21,18] and references therein.
Kirane and Said-Houari [6] examined the system of viscoelastic wave equations with a linear damping and a delay term where Ω is a regular and bounded domain of R N (N ≥ 1), µ 1 , µ 2 are positive constants, τ > 0 represents the time delay and u 0 , u 1 , f 0 are given functions belonging to suitable spaces.
They proved that the energy of problem (1.7) decreases exponentially as t tends to infinity provided that 0 < µ 2 ≤ µ 1 and g decays exponentially. Liu [9] considered the viscoelastic wave equation with a linear damping and a time-varying delay term in the feedback. Under suitable assumptions, he established a general decay result.
Motivated by above research, we consider the well-posedness and the general energy decay for problem (1.1). First, using the Faedo-Galerkin approximations together with some energy estimates, and under some restriction on the parameters µ 1 and µ 2 , the system is showed to be well-posed. Then, under the hypothesis µ 2 ≤ µ 1 , we prove a general decay of the total energy of our problem by using energy method. Our method of proof uses some ideas developed in [6] for the wave equation with a viscoelastic damping and a delay term, enabling us to obtain suitable Lyapunov functionals, from which we derive the desired results. We recall that for µ 1 = µ 2 , Nicaise and Pignotti showed in [14] that some instabilities may occur. Here, due to the presence of the viscoelastic damping, we prove that our energy still decays generally even The remaining part of this paper is organized as follows. In Section 2, we present some materials and recall some useful lemmas needed for our work and state our main results. In Section 3, we will prove the well-posedness of the solution. We will prove several technical lemmas and the general decay result under the two cases: µ 2 < µ 1 and µ 2 = µ 1 in Section 4.

Preliminaries and main results
In this section, we present some assumptions and state the main results. We use the standard Lebesgue space L 2 (0, 1) and the Sobolev space H 1 0 (0, 1) with their usual scalar produces and norms and define the following space X as First, in order to exhibit the dissipative nature of system (1.1), as in [13], we introduce the new variables Φ = ϕ t and Ψ = ψ t . Then, as in [15], we introduce the function Then, we have Therefore, problem (1.1) is equivalent to with the following initial datum and boundary conditions: Next, we denote by * the usual convolution term and the binary operators ⋄ and •, respectively, by The following lemma was introduced in [6]. It will be used in Section 4 to prove the general energy decay result for problem (1.1)-(1.2).
Lemma 2.1 For any function g ∈ C 1 (R) and any h ∈ H 1 (0, 1), we have 3) The proof of this lemma follows by differentiating the term g ⋄ h.
Lemma 2.2 ([1]) For any function g ∈ C([0, ∞), R + ) and any h ∈ L 2 (0, 1), we have Now, we assume that the kernel g satisfies the following assumptions: (H1) g : R + → R + is a differential function such that (H2) There exists a non-increasing differential function ζ : To state our decay result, we introduce the energy functional associated to problem (2.1) where ξ is a positive constant such that Our main results read as follows.

Proof of Theorem 2.3
In this section, we will use the Faedo-Galerkin approximations together with some energy estimates, to prove the existence of the unique solution of problem (2.1)-(2.2) as stated in Proof. We divide the proof of Theorem 2.3 in two steps: the construction of approximations and then thanks to certain energy estimates, we pass to the limit.
We define now the approximations: and where (Φ n (t), Ψ n (t), θ n (t), z n (t)) satisfies the following problem: According to the standard theory of ordinary differential equations, the finite . Then a priori estimates that follow imply that in fact t n = T.
The proof now can be completed arguing as in [8].

Proof of Theorem 2.4
In this section, under the hypothesis µ 2 ≤ µ 1 , we show that the energy of the solution of problem (2.1)-(2.2) decreases generally as t tends to infinity by using the energy method and suitable Lyapunov functionals. We will separately discuss the two cases which are the case µ 2 < µ 1 and the case µ 2 = µ 1 since the proofs are slightly different.

4.1
The case µ 2 < µ 1 Our goal now is to prove the above energy E(t) is a non-increasing functional along the trajectories. More precisely, we have the following result: Then we have Proof. Multiplying Eq. (2.1) 1 by γΦ t , (2.1) 2 by γΨ t and (2.1) 3 by βθ t , integrating over (0, 1) and performing an integration by parts, we find Now, multiplying the Eq. (2.1) 4 by ξ τ z, integrating the result over (0, 1) × (0, 1) with respect to ρ and x, respectively, we obtain and define the functional The derivative of this functional will provide us with the term Lemma 4.2 Let (Φ, Ψ, θ, z) be a solution of (2.1)-(2.2). For any ε 1 > 0, we have where C p is the Poincaré constant.
Proof. By differentiating I 1 with respect to t and using Eqs. (2.1) and (2.2) we conclude that By exploiting the inequalities and Young's inequality, we find that Thus, which is exactly (4.7).
In order to obtain the negative term of 1 0 θ 2 x dx, we define the functional Proof. A simple differentiation leads to By using Eq. (2.1) 3 and (2.2), we arrive at the last three terms can be estimated, using Young's inequality and Lemma 2.2, as follows, for We deduce that The choice of α = λ 4 gives the result. In order to get the negative terms of 1 0 Φ 2 t dx and 1 0 Ψ 2 t dx, we define the functional Lemma 4.4 Let (Φ, Ψ, θ, z) be a solution of (2.1)-(2.2). Then we have (4.10)
In order to absorb the boundary terms, appearing in (4.12), we exploit as in [17], the following function: We will also introduce the functionals J 1 and J 2 defined by and The Young's inequality applied to the last term gives the result.
Differentiating J 2 (t) along solutions of (2.1), we find By integration by part, we obtain and similarly Moreover, and This completes the proof of the lemma.
Now, let us define the following functional Lemma 4.7 Let (Φ, Ψ, θ, z) be a solution of (2.1)-(2.2). For any ε 2 > 0 and η 1 > 0, we have Proof. Differentiating the functional I 5 and using Eqs. (2.1), we obtain (4.28) Using Young's and Poincaré's inequalities, we find (4.29) We can estimate terms in the right side as follows and The proof is completed.
Finally, as in [6], we introduce the functional Then the following result holds: Lemma 4.8 Let (Φ, Ψ, θ, z) be a solution of (2.1)-(2.2). Then, we have where c is a positive constant.
Proof. Differentiating the functional I 6 , we have The above equality implies that there exists a positive constant c such that (4.31) holds. Now, we define the Lyapunov functional L as follows we arrive at . At this point, we need to choose our constants very carefully. First, let us pick η 1 = N 5 ε 4 υ and choose Second, we select N 1 sufficiently large such that then we choose ε 1 so small that Next, we choose N 5 sufficiently large so that and also select N 2 sufficiently large so that Furthermore, we select ε 2 satisfies Finally, we choose N large enough so that (4.34) remains valid and (4.36) takes the form where C 1 , C 2 , C 3 , and C are positive constants.

Proof. A simple differentiation leads to
Terms in the right side of (4.42) are estimated as follows. Using Young's inequalities and Lemma 2.2, we obtain, for all η 2 > 0, Finally, Therefore, the assertion of the lemma follows by combining all the above estimates. Now, we define the following Lyapunov functional L as: L(t) = N E(t) + N 1 I 1 (t) + N 2 I 2 (t) + υ 4 I 3 (t) + υI 4 (t) + N 5 I 5 (t) + N 6 I 6 (t) + N 7 I 7 (t) (4.45) where N, N 1 , N 2 , N 5 , N 6 , N 7 are positive real numbers which will be chosen later.
Since g is continuous and g(0) > 0, then for any t ≥ t 0 > 0, we have and algebraic inequality (4.35), we get Now, our goal is to choose our constants in (4.46) in order to get the negative coefficients on the right-hand side of (4.46). To this end, let us pick η 1 = N 5 ε 4 υ , η 2 = 1 4N 7 and we pick ε 4 , N 1 , ε 1 , N 5 , N 2 , ε 2 in the same order with the same values as the case µ 2 < µ 1 , respectively.