General decay of solutions of a Bresse system with viscoelastic boundary conditions

In this paper we are concerned with a multi-dimensional Bresse system, in a bounded domain, where the memory-type damping is acting on a portion of the boundary. We establish a general decay results, from which the usual exponential and polynomial decay rates are only special cases.


1.
Introduction. In their study on networks of flexible beams, Lagnese, Leugering and Schmidt [15] derived a general model for 3-d nonlinear beams. A special case of this model is a linear planar, shearable beam whose motion is governed by the following system of partial differential equations: where (x, t) ∈ (0, L) × (0, ∞) and u, v and w represent the vertical deplacement, rotation angle, and longitudinal displacement,respectively, of the point x of the beam at the instant t. The coefficients ρ 1 , ρ 2 , E and I denote respectively the mass per unit length, the mass moment of inertia of a cross-section of the beam, Young's modulus and the moment of inertia of a cross-section of the beam. The coefficient κ 0 , κ and l are equals to EA, κ GA and R −1 respectively and where G is the modulus of elasticity in shear, A is the cross sectional area, κ is the shear factor and R denotes the radius of the curvature. We recall that systems (1) are also known as a circular arch problem, consisting of three coupled wave type equations, originally derived by Bresse [4]. A more recent discussion about mathematical modeling of the Bresse system can be found in Lagnese et al. [15], see also Alabau et al. [1].
Let us mention some known results on the decay rates for the Bresse model. In [16], Liu and Rao studied the stabilization for the Bresse system with two different temperature mechanisms affecting the longitudinal displacement and shear angle displacement. Under the equal wave speeds, they established an exponential energy decay rate. Otherwise, they showed that the smooth solution decays polynomially to zero with rates t −1/2 or t −1/4 provided the boundary conditions are Dirichlet-Neumann-Neumann or Dirichlet-Dirichlet-Dirichlet type, respectively.
An important problem in the Bresse system is to find a minimum dissipation by which their solutions decay uniformly to zero in time. In this direction we have the paper of Fatori and Rivera [8], which improved the paper by Liu and Rao [16]. They showed that, in general, the Bresse system is not exponentially stable but that there exists polynomial stability with rates that depend on the wave speeds and the regularity of the initial data. Moreover, they introduced a necessary condition to dissipative semigroup decay polynomially. This result allowed them to show some optimality to the polynomial rate of decay.
In [1], the Bresse system with frictional damping was considered by Alabau et al.. The authors showed that the Bresse system is exponentially stable if and only if the velocities of waves propagations are equal. Also, they showed that when the wave speeds are not the same, the system is not exponentially stable, and they proved that the solution in this case goes to zero polynomially, with rates that can be improved by taking more regular initial data. The rate of polynomial decay was improved by Fatori and Monteiro [7]. The indefinite damping acting on the shear angle displacement was considered by Palomino et al. in [9].
In [20], Noun and Wehbe extended the results from Alabau-Boussouira et al. [1], by taking into consideration the important case when the dissipation law is locally distributed and improved the polynomial energy decay rate. The authors studied the energy decay rate of the Bresse system with one locally internal distributed dissipation law acting on the equation about the shear angle displacement. Under the equal speed wave propagation condition, they showed that the system is exponentially stable. On the contrary, they established a new polynomial energy decay rate.
Related to this subject, we can mention the work of Khemmoudj and Hamadouche [14]. In that paper, the authors obtained an asymptotic stability of a class of Bressetype system with three boundary dissipations and with a rigid body attached to its free end. They showed that exponential stabilization can be achieved by applying force and moment feedback boundary controls on the shear, longitudinal and transverse displacement velocities at the point of contact between the mass and the beam.
Concerning the asymptotic behavior of the Bresse system with past memory acting on the three equations we cite the work of Guesmia et al [12]. In that paper the authors showed, under suitable conditions on the initial data and the memories, that the Bresse system converges to zero when time goes to infinity, and they provided a connection between the decay rate of energy and the growth of memories at infinity.
In [2], Santos et al. considered the Bresse system with past history acting only on the shear angle displacement. They show the exponential decay of the solution if and only if the wave speeds are the same. If not, they show that the Bresse system is polynomial stable with optimal decay rate.
(ii) Assumptions on the nonlinear functions. For the coupling terms f i , i = 1, 2, 3, we suppose that Additionally, we assume that there exists a nonnegative function F (u, v, w) ∈ C 2 (R 3 ) such that 3. Further, we assume that F is homogeneous of order p + 1 : Since F is homogeneous, the Euler Homogeneous Function theorem yields the following useful identity:

AMMAR KHEMMOUDJ AND TAKLIT HAMADOUCHE
The homogeneity of F implies that there exists a constant M > 0 such that Remark 1. There is a large class of functions satisfying the assumptions (5)- (7). For instance functions of the form where a , b , c are positive constants, satisfy assumptions (5)-(7) with p ≥ 3. Indeed, a quick calculation shows that there exists c 0 > 0 such that Moreover, it is easy to compute and find that In what follows, we are going to assume that there exists x 0 ∈ R n such that Also we assume that ρ i ∈ C 1 (Ω), i = 1, 2, 3 , are positive functions satisfying the following hypothesis We consider the functions b j (x) ∈ W 1,∞ (Γ 1 ), j = 1, 2, given by The boundary condition of memory type for Timoshenko system, has been studied by Santos [24]. By considering k i to be the resolvent kernels of (−h i /h i (0)) for i = 1, 2, he showed that the energy of the solution decays exponentially (polynomially) when k i and −k i , i = 1, 2, decay exponentially (polynomially). The same result has been established by Messaoudi and Soufyane [17] without assuming the exponential (polynomial) decay of k 1 and k 2 but only that their norms are small enough. In [19] the general decay for the same system has been proved.
Models with boundary conditions including a memory term which produces damping were proposed in [5], [6], [18] and [23] for the study of wave propagation, in [22] and [25] for the von Karman plate system and in [13], [11] and [26] in the context of Kirchhoff equations.
The main goal of this paper is to investigate the asymptotic behavior of the system (2-4) for resolvent kernels of general type decay and obtain a more general and explicit energy decay formula, from which the usual exponential and polynomial decay rates are only special cases of our result. The proof is mainly based on the use of a multiplier method and the introduction of a suitable Lyapounov functional.
Our paper is organized as follows. In section 2 we establish the existence and uniqueness for regular and weak solutions of system (2)(3)(4). In section 3 we state and prove the general decay of the solutions of our studied system.

Existence and regularity.
In this section, we study the existence and regularity of solutions for the Bresse system (2)-(4). The scalar product and norm of the real Hilbert space L 2 (Ω) are denoted by (u, v) and | u |, respectively. By V the Hilbert space is represented as First, we will use the second equation, the third equation and the fourth equation in (3) to estimate the terms ∂u ∂ν , ∂v ∂ν and ∂w ∂ν . Defining the convolution product operator by and differentiating the second, the third and the fourth equations in (3), we arrive to the following Volterra equations: ∂v ∂ν Applying the Volterra's inverse operator, we get where the resolvent kernels satisfy Denoting by τ 1 = 1 h1(0) , τ 2 = 1 h2(0) and τ 3 = 1 h3(0) , the normal derivatives of u, v and w can be written as Reciprocally, taking initial data such that u 0 = v 0 = w 0 = 0 on Γ 1 , the identities (17), (18) and (19) imply the second, the third and the fourth equations in (3) respectively. Since we are interested in relaxation functions of more general decay and the boundary conditions (17), (18) and (19) involving the resolvent kernels k i , i = 1, 2, 3, we want to know if k i has the same decay properties. The following lemma answers this question. Let h be a relaxation function and k its resolvent kernel, that is, where γ : [0, +∞) → R + ,is a nonincreasing function satisfying, for some positive constant ε < 1, Then k satisfies for γ > c 0 . Then, there exists a positive constant ε < 1 such that where β > 0 is a constant.
for c 0 < p − 1. Then, there exists a positive constant ε < 1 such that where β > 0 is a constant.
Based on Lemma 2.1, we will use the boundary relations (17), (18) and (19) instead of the second, the third and the fourth equations in (3). Let us define and By using Hölder's inequality, we have The next lemma gives an identity for the convolution product.
The well-posedness of system (2) -(4) is given by the following theorem.
Then there exists only one strong solution (u, v, w) of the Bresse system (2)-(4) satisfying Proof. The theorem can be proved, by making use of standard semi group arguments (see for instance, P. Pei et al. [21]).
By multiplying the first equation in (2) by u t , the second equation by v t and the third equation by w t , integrating over Ω using integration by parts, the boundary conditions, and (17)- (19), one can easily find that the first order energy of system (2) is given by

Lemma 3.2.
Under the assumption of Theorem 3.1, the energy of the solution of (2)-(4), satisfies Proof. Multiplying the first equation in (2) by u t and integrating by parts over Ω we obtain 1 2 Using Gauss's Theorem, we get and Plugging the estimates (38) and (39) into (37), we find that The second equation in (2) multiplied by v t in L 2 (Ω), and integration by parts give Finally, the third equation in (2) multiplied by w t in L 2 (Ω), and integrating by parts over yield
Let ε 0 > 0 be a small constant and define the following functional: The following lemma plays an important role for the construction of the Lyapunov functional. Lemma 3.3. Under the assumption of Theorem 3.1, the solution of (2)-(4), satisfies Proof. We multiply the first equation in (2) by 2m.∇u + (n − ε 0 )u to obtain d dt Integrating by parts and using the relation divm = n, we get d dt Similarly, multiplying the second equation in (2) by (2m.∇v + (n − ε 0 )v) and integrating over Ω, using integration by parts, we arrive at d dt
Next, we use the fact that there exists a positive constant c such that to obtain d dt Using Poincaré inequality and taking ε 0 small enough, we get d dt The proof of Lemma 3.3 is completed.