Uniform and weak stability of Bresse system with one infinite memory in the shear angle displacements

In this paper, we consider a one-dimensional linear Bresse system in a bounded open interval with one infinite memory acting only on the shear angle equation. First, we establish the well posedness using the semigroup theory. Then, we prove two general (uniform and weak) decay estimates depending on the speeds of wave propagations and the arbitrary growth at infinity of the relaxation function.


Introduction
In this paper, we consider a Bresse system in one-dimensional open bounded interval subjected to homogeneous Dirichlet-Neumann-Neumann boundary conditions and with the presence of one infinite memory acting on the shear angle equation. Precisely, we are concerned with the following problem: where (x, t) ∈]0, L[×R + , g : R + → R + is a given function, and L , l, ρ i , i = 1, 2, and k j , j = 1, 2, 3, are positive constants. The integral term in system (1.1) represents the infinite memory, and the state (unknown) is Our objective is to establish the well posedness and the asymptotic stability of this problem in terms of the growth of g at infinity and the speeds of wave propagations given by (1. 2) The Bresse system is known as the circular arch problem and is given by the following equations: where ρ 1 , ρ 2 , l, k, k 0 and b are positive physical constants, N , Q and M denote, respectively, the axial force, the shear force and the bending moment, and w, ϕ and ψ represent, respectively, the longitudinal, vertical and shear angle displacements. Here, such that ρ, E, G, k , A, I and R are positive constants and denote, respectively, the density, the modulus of elasticity, the shear modulus, the shear factor, the cross-sectional area, the second moment of area of the crosssection and the radius of curvature. Finally, F 1 , F 2 and F 3 are the external forces defined in ]0, L[×]0, +∞[. For more reading about this matter, we refer to Lagnese et al. [18,19]. It is worth noting that the system considered by Bresse [3] is obtained by taking with γ > 0.
To stabilize the Bresse system, various dampings have been employed and several decay results have been established. Alabau-Boussouira et al. [1] considered the case (1.3) and proved that the exponential stability is equivalent to (1.4) When (1.4) is not satisfied, they showed that the norm of solutions decays polynomially to zero with rates depending on the regularity of the initial data. These latter results were extended and improved in [22] by considering a locally distributed dissipation (that is, γ in (1.3) is replaced by a non-negative function a : ]0, L[→ R + which is positive only on a part of ]0, L[). In their work, the authors of [22] obtained a better decay rate when (1.4) does not hold. The exponential stability result of [1] was also established by Soriano et al. [29] for the case of indefinite damping. That is, when γ = a( is small enough. In such a situation, a may change sign in ]0, L[. Also, some optimal polynomial decay rates for Bresse systems for the case (1.3) were proved in [7] when (1.4) does not hold. Wehbe and Youcef [31] treated the case where a i :]0, L[→ R + are non-negative functions which can vanish on some part of ]0, L[, and proved that the exponential stability holds if and only if s 1 = s 2 . When s 1 = s 2 , a polynomial decay rate depending on the regularity of the initial data was obtained. This rate, in the case of classical solutions, is t − 1 2 + . When only the first and second equations are controlled by means of linear frictional dampings; that is, with γ i > 0, the equivalence between the exponential stability and the equality s 1 = s 3 was established in [2]. In addition, a polynomial stability was also shown when s 1 = s 3 , where the decay rate depends on the regularity of the initial data. In the particular case of classical solutions, the polynomial decay of [2] is of the rate t − 1 2 and it is optimal. Soufyane and Said-Houari [30] looked into the case of three frictional dampings in the whole space R (instead of ]0, L[) and established some polynomial stability estimates. For stabilization via nonlinear frictional dampings, we refer the readers to [4,28].
Concerning the stabilization via heat effect, one of the earliest results concerning the asymptotic behavior of the Bresse system is due to Liu and Rao [20], where a Bresse system of the form in a bounded interval, together with initial and boundary conditions, has been considered. In that work, Liu and Rao [20] proved that the norm of solutions decays exponentially if and only if (1.4) holds. Otherwise, the solutions decay polynomially with rates depending on the regularity of the initial data. For the classical solutions, with boundary conditions of Dirichlet-Neumann-Neumann or Dirichlet-Dirichlet-Dirichlet type, these rates are of the form t − 1 4 + or t − 1 8 + , respectively, where > 0 is an arbitrary "small" constant. Other results similar to those of [20] were obtained in [8] in a bounded interval, where ϕ , ψ and w are, respectively, the vertical, shear angle and longitudinal displacements, θ and q denote the temperature difference and the heat flux, and ρ 1 , ρ 2 , ρ 3 , k, k 0 , b, β, γ and τ are positive constants. Under suitable relations between the constants, the authors of [16] showed exponential and optimal polynomial decay rates. The same system was treated by Said-Houari and Hamadouche [25] in the whole space R, where they showed that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and they proved that the decay rate of the solution in the L 2 -norm is of the rate t −1/12 . For more problems of thermoelastic Bresse systems, we refer the reader to [24], where a global existence was proved using two heat equations, and to [26,27], where Cauchy thermoelastic Bresse problems were treated. Concerning the stability of Bresse systems via memories, there are only very few results. For instance, Guesmia and Kafini [10] discussed, without restrictions on the speeds, the stability issue for the case when the three equations are controlled via infinite memories of the form where g i : R + → R + are differentiable, non-increasing and integrable functions on R + . Their decay estimate depends only on the growth of the relaxation functions g i at infinity, which are allowed to have a decay rate at infinity arbitrary close to 1 s . The same stability estimate of [10] was later established in [11] when only two infinite memories are considered, that is When (1.9) does not hold, a weak stability estimate was given in [11], where the decay rate depends also on the smoothness of the initial data. Similar results were obtained in [15] when the memory term acts on the longitudinal displacements. However, when the memory term acts on the vertical displacements, it was proved in [14] that the system can not be exponentially stable even if the speeds of wave propagations are equal, but it is still polynomially stable.
To the best of our knowledge, the only known stability results for Bresse systems with only one infinite memory acting on the shear angle displacements are the ones obtained in [6] in case where g : R + → R + is differentiable, non-increasing and integrable function on R + . In [6], it was assumed that g satisfies, for α 1 , α 2 > 0, 11) and was shown that the exponential stability holds if and only if (1.4) is satisfied. Otherwise, only the polynomial stability with a decay rate of type t − 1 2 and its optimality were obtained. Notice that the condition (1.11) implies that g converges exponentially to zero at infinity and satisfies g(0)e −α 2 s ≤ g(s) ≤ g(0)e −α 1 s , ∀s ∈ R + . (1.12) Our goal in this work is to study the well posedness and asymptotic stability of system (1.1) in terms of the arbitrary growth at infinity of the kernel g and the speeds of wave propagations (1.2). We prove that the systems is well posed and its energy converges to zero when time goes to infinity and provide two general decay estimates: a uniform stability estimate under (1.4), and another weak stability result in general. Our results generalize those of [6] and allow a wider class of relaxation functions. See Remark 3.3 below.
The proof of the well posedness is based on the semigroup theory. For the stability estimates, we use the energy method and an approach introduced by the present authors in [12,13].
The paper is organized as follows. In Sect. 2, we present our assumptions on the relaxation function g and state and prove the well posedness of (1.1). In Sect. 3, we present our stability results. The proof of our uniform and weak decay estimates are given, respectively, in Sects. 4 and 5.

Well posedness
In this section, we discuss the well posedness of (1.1) using the semigroup approach. Following the method of [5], we consider the functional and Then, the system (1.1) takes the following abstract form: where A is the linear operator defined by and and that is, according to the definition of H and A, More generally, for n ∈ N,

Remark 2.1
As in [11], by integrating on ]0, L[ the second and third equations in (1.1), and using the boundary conditions, we get Substituting (2.14) into (2.13), we get Using the initial data of ψ and w in (1.1), we see that Then, from (2.16) and (2.17), one can check that 20) and, hence, Therefore, Poincaré's inequality In addition, (ϕ,ψ,w) satisfies the boundary conditions and the first three equations in (1.1) with initial data instead of ψ 0 , ψ 1 , w 0 and w 1 , respectively. In the sequel, we work withψ,w andη instead of ψ, w and η, but, for simplicity of notation, we use ψ, w and η instead ofψ,w andη, respectively. Now, to prove the well posedness of (2.6), we make the following hypothesis: (H1) The function g : R + → R + is differentiable, non-increasing and integrable on R + such that there exists a positive constant k 0 such that, for any On the other hand, thanks to (2.22) applied for ψ and w, and Poincaré's inequality applied for ϕ, there exists a positive constantk 0 such that, for any Thus, from (2.25) and (2.27), we deduce that the right hand side of the inequality (2.25) defines a norm on 2. As in [11], we conclude from (2.23) that Indeed, for the choice ϕ = w = 0, (2.23) gives This inequality implies, for By letting λ go to +∞, we deduce (2.28).
According to Remark 2.2, we notice that, under the hypothesis (H1), the sets L 2 and H are Hilbert spaces equipped, respectively, with the inner products that generate the norms, and

Now, a simple computation implies that, for any
Since g is non-increasing, we deduce from (2.31) that This implies that A is dissipative. Notice that, according to (2.24) and the fact that g is non-increasing, we see that, for v ∈ L 2 , so the integral in the right hand side of (2.31) is well defined.
Next, we follow the proof given in [11] to prove that I d − A is surjective, where I d is the identity operator. (2.33) The first three equations in (2.33) take the form (2.34) Using (2.34), the last equation in (2.33) is equivalent to By integrating (2.35) and using the fact that v 7 (0) = 0 (from (2.11)), we get On the other hand, using Fubini theorem, Hölder's inequality and noting that f 7 ∈ L 2 , we get and therefore, (2.36) implies that v 7 ∈ L 2 . Moreover, ∂ s v 7 ∈ L 2 by (2.35). Therefore, to prove that (2.33) admits a solution V ∈ D(A), it is enough to show that and (v 1 , v 2 , v 3 ) exists and satisfies the required regularity and boundary conditions in D(A), that is and Let us assume that (2.37)-(2.40) hold. Multiplying the fourth, fifth and sixth equations in (2.33) by ρ 1ṽ1 , ρ 2ṽ2 and ρ 1ṽ3 , respectively, integrating their sum over ]0, L[, using the boundary conditions (2.37) and (2.40), and inserting (2.34) and (2.36), we get that (v 1 , v 2 , v 3 ) solves the variational problem (2.43) We note that, as before, using again Fubini theorem, Hölder's inequality and the fact that On the other hand,g 0 ≤ g 0 < k 2 (by (2.28)). Then, by virtue of (2.23) and (2.27), we have a 1 is a bilinear, continuous and coercive form on andã 1 is a linear and continuous form on Consequently, using the Lax-Milgram theorem, we deduce that (2.41) has a unique solution Therefore, using classical elliptic regularity arguments, we conclude that the forth, fifth and sixth equations in ( [23]), we deduce from (2.32) and (2.33) that A generates a C 0 -semigroup of contractions in H. This gives the following well-posedness results of (2.6) (see [17,23]).

Stability
In this section, we study the stability of (2.6), where the obtained two (uniform and weak) decay rates of solution depend on the speeds of wave propagations (1.2) and the growth of g at infinity characterized by the following additional hypothesis: (H2) Assume that g(0) > 0 and there exists a non-increasing differentiable function ξ : R + → R * + such that g (s) ≤ −ξ(s)g(s), ∀s ∈ R + .
2. When ξ ≡ constant, condition (3.1) implies that g converges exponentially to zero at infinity. However, when ξ = constant, condition (3.1) allows s → g(s) to have a decay rate arbitrarily close to 1 s at infinity, which represents the critical limit, since g is integrable on R + . To illustrate our general stability estimates, we give here some particular examples of g satisfying (3.1), and show the specific corresponding decay rates given by (3.4) and (3.7). (i) Let g(t) = de −(1+t) q with 0 < q < 1 and d > 0 (g converges to zero at infinity faster than any polynomial). Then, (3.1) holds with ξ(t) = q(1 + t) q−1 , and consequently, (3.4) and (3.7) give, respectively, for two positive constants c 1 and c 2 , (ii) Let g(t) = d(1+t) −q with q > 1 and d > 0 (g has at most a polynomial decay at infinity). Assumption (3.1) holds with ξ(t) = q(1+t) −1 , and consequently, (3.4) and (3.7) give, respectively, for two positive constants c 1 and c 2 , To prove (3.4) and (3.7), we will consider suitable multipliers and construct appropriate Lyapunov functionals satisfying some differential inequalities, for any U 0 ∈ D(A) and t ∈ R + ; so all the calculations are justified. By integrating these differential inequalities, we get (3.4

) and (3.7), for any U 0 ∈ D(A). By simple density arguments (D(A) is dense in H), (3.4) remains valid, for any U 0 ∈ H.
We will use c, throughout the rest of this paper, to denote a generic positive constant which depends continuously on the initial data U 0 and the fixed parameters in (1.1), (2.22) and (2.26), and can be different from step to step. When c depends on some new constants y 1 , y 2 , . . ., introduced in the proof, the constant c is noted c y 1 , c y 1 ,y 2 , . . ..
Let us consider the energy functional E associated to (2.6) defined by From (2.6) and (2.31), we see that Recalling that g is non-increasing, (3.11) implies that E is non-increasing, and consequently, (2.6) is dissipative.

Proof of uniform decay (3.4)
First, we consider the following functional: Proof First, we note that Second, using Young's and Hölder's inequalities, we get the following inequality: for all λ > 0, there exists

Lemma 4.2 Let
Then, for any δ 0 , 0 , 1 , 2 > 0, there exist c δ 0 , c 0 > 0 such that (4.7) Proof First, notice that Now, by exploiting the first two equations in (1.1), integrating by parts, using (4.8) and the boundary conditions, we get By applying (4.4), (4.5) and Young's inequality for the last four terms of the above equality, we deduce (4.7).

Lemma 4.3 Let
Then, for any 0 > 0, there exists c 0 > 0 such that (4.10) Proof Using the first and third equations in (1.1), integrating by parts, recalling (4.8) and using the boundary conditions, we find By applying Young's inequality for the last four term of the above equality, we obtain (4.10).

Lemma 4.5 Let
Then, for any δ 0 > 0, there exists c δ 0 > 0 such that (4.17) Proof By exploiting the first three equations in (1.1), integrating by parts and using the boundary conditions, we find By applying (4.4) for the last term in this equality, we arrive at (4.17).
(5.1) System (5.1) is well posed for initial data U 0 ∈ D(A) thanks to Theorem 2.3, where U t ∈ C(R + ; H). Let U 0 ∈ D(A) andẼ be the energy of (5.1) defined bỹ Similarly to (3.11), we haveẼ soẼ is non-increasing. We use an idea introduced in [9] to get the following lemma. Comments.
1. This work generalizes the results of [6] and allows a wider class of relaxation functions. 2. Note that when w = l = 0, we obtain the Timoshenko system and our results reduce to those of [12]. 3. It would be very interesting to obtain these general decay results without conditions (3.2) and (3.3).