New general decay results in a finite-memory bresse system

This paper is concerned with the following memory-type Bresse system \begin{document}$ \begin{array}{ll} \rho_1\varphi_{tt}-k_1(\varphi_x+\psi+lw)_x-lk_3(w_x-l\varphi) = 0,\\ \rho_2\psi_{tt}-k_2\psi_{xx}+k_1(\varphi_x+\psi+lw)+ \int_0^tg(t-s)\psi_{xx}(\cdot,s)ds = 0,\\ \rho_1w_{tt}-k_3(w_x-l\varphi)_x+lk_1(\varphi_x+\psi+lw) = 0, \end{array} $\end{document} with homogeneous Dirichlet-Neumann-Neumann boundary conditions, where \begin{document}$ (x,t) \in (0,L) \times (0, \infty) $\end{document} , \begin{document}$ g $\end{document} is a positive strictly increasing function satisfying, for some nonnegative functions \begin{document}$ \xi $\end{document} and \begin{document}$ H $\end{document} , \begin{document}$ g'(t)\leq-\xi(t)H(g(t)),\qquad\forall t\geq0. $\end{document} Under appropriate conditions on \begin{document}$ \xi $\end{document} and \begin{document}$ H $\end{document} , we prove, in cases of equal and non-equal speeds of wave propagation, some new decay results that generalize and improve the recent results in the literature.

where t represents time and x is a space variable. The unknowns ϕ = ϕ(x, t), ψ = ψ(x, t), w = w(x, t) denote the vertical angle, the shear angle and the longitudinal displacements, respectively. Here, the constant parameters are given by ρ 1 = ρA, ρ 2 = ρI, l = R −1 , where ρ is the material density, A is its cross-sectional area, I is the second moment of the area of the cross-section and R is the radius of curvature. The axial force, the bending moment and the shear force are respectively denoted by N, S and M . In this work, we consider the viscoelastic-type Bresse system whose constitutive laws are where k 1 = κGA, k 2 = EI, k 3 = EA, E is the modulus of elasticity, G is the shear modulus and κ is the shear factor. Many researchers have established several results dealing with the well-posedness and asymptotic stability of the Bresse system using different types of dissipation mechanisms acting on each (or some) of the equation(s) in the system. It is a well known fact that if the dissipation mechanisms are acting on only one or two of the equations, then the asymptotic behavior of the system depends completely on the speeds of wave propagation given by The reader is referred to [1,2,4,6,5,8,10,11,12] and the references therein for results related to stabilization of Bresse systems using different types of damping terms.
To the best of our knowledge, there is no result in the literature that dealt with the stability of Bresse system with finite memory. In this work, we study the following finite-memory Bresse system: ϕ(x, 0) = ϕ 0 (x), ϕ t (x, 0) = ϕ 1 (x), ψ(x, 0) = ψ 0 (x), ψ t (x, 0) = ψ 1 (x), w(x, 0) = w 0 (x), w t (x, 0) = w 1 (x), (P ) where (x, t) ∈ (0, L) × (0, ∞), l, k 1 , k 2 , k 3 , ρ 1 , ρ 2 are positive constants, ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 are given data and g is a relaxation function satisfying some conditions to be specified in the next section. Under a smallness condition on l, we prove some general decay results for the energy associated to this system in the case of equal and non-equal speeds of wave propagation. Our results will give an optimal decay rate, in the sense that the decay rate of the energy is the same as that of the relaxation function, in the case of equal speeds of wave propagation. This paper is organized as follows: in Section 2, we state some preliminary results. In Section 3, we state and prove some technical lemmas. The statement and proof of our main results are given in Sections 4 and 5. Through out this work we use c to represent a generic positive constant, independent of t but may depend on the initial data.

2.
Preliminaries. In this section, we introduce our assumptions, present some useful lemmas and state the existence theorem. Assumptions: We assume that the relaxation function g satisfies the following hypotheses: Remark 1.
(1) Assumptions (A.1) and (A.2) entail that, there exists t 0 > 0 such that The non-increasing property of g gives A combination of this with the continuity of H yields for two constants a, b > 0. Consequently, for any t ∈ [0, t 0 ], we have and, hence, (2) If H is a strictly increasing and strictly convex C 2 −function on (0, r], with H(0) = H (0) = 0, then there is a strictly convex and strictly increasing C 2 −functionH : [0, +∞) −→ [0, +∞) which is an extension of H. For instance, we can defineH, for any t > r, bȳ Now, integrating both sides of the second and third equations in (P ) over (0, L) and using the boundary conditions, we get Solving these ODEs simultaneously yields L 0 ψ(x, t)dx = a 1 cos(a 0 t) + a 2 sin(a 0 t) + a 3 t + a 4 Therefore, we perform the following change of variables Furthermore, (ϕ, ψ, w) satisfies the equations and the boundary conditions in (P ) with the initial data From now on, we work with ψ, w and, respectively, write ψ, w for convenience. We also introduce the following spaces, For completeness, we state, without proof the global existence and regularity result which can be easily established by a standard Galerkin argument.

Lemma 2.3 ([9]
). Assume that conditions (A.1) holds. Then for any v ∈ L 2 loc [0, +∞); L 2 (0, L) , we have 3. Technical lemmas. In this section, we state and prove some lemmas needed to establish our main results. All the computations are done for regular solutions but they still hold for weak and strong solutions by a density argument. satisfies, along the solution of (P ), the estimates Proof. Differentiating I 1 , using equations in (P ) and integrating by parts, we get Next, we estimate the terms on the right-hand side of the above equation. Using Young's inequality, Lemma 2.3 and Poicaré's inequality, we obtain, for any 0 < δ < 1, A combination of these estimates gives the desired result.
Lemma 3.2. Assume that the hypotheses (A.1) and (A.2) hold. Then, for any ε 0 , δ 1 > 0, the functional I 2 defined by satisfies, along the solution of (P ), the estimate Proof. Differentiation of I 2 , using equations in (P ) and integration by parts yield Using Young's inequality, we get, for any ε 0 , δ 1 > 0, satisfies, along the solution of (P ) and for any ε 0 > 0, the estimate

SALIM A. MESSAOUDI AND JAMILU H. HASSAN
Proof. Differentiating I 3 , using equations in (P ) and integrating by parts, we have Use of Young's inequality for the first term in the right-hand side gives (9).
satisfies, along the solution of (P ), the estimate Proof. Differentiation of I 4 , using equations of (P ) gives Repeating the above computations yields the desired result.
Lemma 3.5. Assume that conditions (A.1) and (A.2) hold. Then for any 0 < δ < 1 and δ 2 > 0, the functional I 5 defined by satisfies, along the solution of (P ), the estimate Proof. Using equations of (P ) and repeating similar computations as above, we arrive at Poincaré's inequality for the third term yields (11).
Lemma 3.6. Assume that the hypotheses (A.1) and (A.2) hold. Then, for any ε 0 , ε 1 , ε 2 > 0 and 0 < δ < 1, the functional I 6 defined by satisfies, along the solution of (P ), the estimate Proof. Use of equations of (P ) and integration by parts lead to Now, we estimate the terms in the right-hand side of the above equation. Exploiting Young's inequality, we get
As in [9], let us define the functional where f (t) := ∞ t g(s)ds and state a lemma whose proof is similar to that in [9] taking into consideration the nature of our problem.
where g 0 = ∞ 0 g(s)ds. Lemma 3.8. The functional L defined by satisfies, for a suitable choice of N, N j ≥ 0 for j = 1, 2, · · · , 6 with N 3 = N 6 = 1, and, for l small enough, and the estimate where m is a fixed positive constant and t 0 has been introduced in Remark 1.

NEW GENERAL DECAY RESULTS IN A FINITE-MEMORY BRESSE SYSTEM
1649 Finally, we choose N so large that L ∼ E and N 2 > c, therefore we have, ∀ t ≥ t 0 , As αg 2 αg−g < g, it follows from (A1) and the Lebesgue Dominated Convergence Theorem that Consequently, there exists α 0 ∈ (0, 1) such that Then, choose N even larger (if needed) so that This yields Hence, we arrive at the required estimate.
4. General decay rates for equal speeds of wave propagation. In this section, we state and prove a new general decay result in the case of equal speeds of wave propagation.
. Assume that (A.1) and (A.2) hold and that Then for l small enough, there exist two positive constants C and λ (independent of t but may depend on the initial data) such that the solution of (P ) satisfies where H 1 (t) = r t 1 sH (s) ds and t 0 = g −1 (r).
Proof. We start by using the non-increasing property of ξ and estimates (3) and (5) to deduce, for any t ≥ t 0 , Exploiting this last estimate and (16), the estimate (15) becomes, for any t ≥ t 0 , By setting F := L + cE ∼ E, we obtain Case I H is linear: Multiplying (18) by ξ(t), then exploiting (A.2) and (5), we get Using the non-increasing property of ξ, we have ξF + cE ∼ E and A simple integration over (t 0 , t) yields, for two positive constants C and λ, Continuity of E gives Case II H is nonlinear: First, we use Lemmas 3.7 and 3.8 to conclude that, the functional L defined by is nonnegative and satisfies, for some β > 0 and for any t ≥ t 0 , From this estimate, we deduce that Now, we define a functional η by where (19) allows us to choose 0 < γ < 1 so that We further assume that η(t) > 0, for any t > t 0 , otherwise we get an exponential decay from (18). Also, we define another functional θ by and observe that In addition, it follows from the strict convexity of H and the fact that H(0) = 0 that H(sτ ) ≤ sH(τ ), for 0 ≤ s ≤ 1 and τ ∈ (0, r].

This fact, hypothesis (A.2), (20) and Jensen's inequality lead to
whereH is a C 2 extension of H that is strictly increasing and strictly convex on (0, ∞). This implies that and (18) becomes For 0 < ε 1 < r, we define the functional F 1 by Then, using estimate (22) and the fact that E ≤ 0,H > 0 andH > 0, we deduce that F 1 ∼ E and also, for any t > t 0 , we have LetH * be the convex conjugate ofH in the sense of Young (see [3, pp. 61-64]), which is given byH and satisfies the following generalized Young inequality By taking A =H ε 1 and combining (5), (23) and (24), we arrive at Multiplying this estimate by ξ(t) and using ε 1 and inequality (21), we get Take ε 1 smaller, if needed, to get, for some positive constant k, Consequently, by setting F 2 = ξF 1 + cE, we obtain, for two constants α 1 , α 2 > 0, and where H 2 (τ ) = τ H (ε 1 τ ). Since it follows from the strict convexity of H on (0, r] that H 2 , H 2 > 0 on (0, 1]. Then, set R(τ ) = α 1 F 2 (τ ) E(0) and exploit (25) and (26) to get R ∼ E and, for some λ > 0, An integration over (t 0 , t) gives where H 1 (t) := r t 1 sH (s) ds. A combination of this estimate with the fact that R ∼ E gives This completes the proof of Theorem 4.1.
Remark 2. Routine calculations show that the decay rate deduced from estimate (17) is optimal in the sense that it agrees with the decay rate of g deduced from (2). For the details, see [9,Remark 2.3].

Estimate (27) entails that
(3) Consider the following relaxation function, for ν > 1, and a is chosen so that hypothesis (A.1) remains valid. Then with where b is a fixed constant, p = 1+ν ν and it satisfies 1 < p < 2. Then, we deduce from (27) that For more examples, see [9].

5.
General decay rate for different speeds of wave propagation. In this section, we state and prove a generalized decay result in the case of non-equal speeds of wave propagation. We start by differentiating both sides of the differential equations in (P ) with respect to t and use the fact that to obtain the following system (P * ) The energy functional associated to (P * ) is given by, for any t ≥ 0, Using similar arguments as in [7,Lemma 3.11] we have the following result.
Lemma 5.1. Let (ϕ, ψ, w) be the strong solution of (P ). Then, the energy of (P * ) satisfies, for all t ≥ 0, and From estimates (29) and (30), we deduce that, for any t ≥ 0, where c 1 is some fixed positive constant.
Now, we state and prove a general decay result in the case of nonequal speeds of wave propagation.
. Assume that conditions (A.1), (A.2) hold and that Then for l small enough, there exist some positive constants C and λ that depend on the initial data but independent of t and t 1 ≥ t 0 = g −1 (r) such that the energy functional associated to problem (P ) satisfies the estimate where H 2 is given by H 2 (τ ) = τ H (τ ).
Proof. Using Lemma 5.2 in estimate (15), we have, for some m > 0, After fixing ε small enough, we arrive at where m 1 is a fixed positive constant. By setting F := L + cE ∼ E, we obatain, for any t ≥ t 0 , Case I H is linear: Multiplying both sides of estimate (34) by ξ(t), then using hypothesis (A.2) and Lemma 5.3 we get, for any t ≥ t 0 , .
From the non-increasing property of ξ, we have, for some fixed positive constant c 2 , An integration over (t 0 , t), exploitation of the non-increasing property of E and estimate (30) yield, for any t > t 0 , Thus, we have, for some fixed positive constant C, Case II H is nonlinear: First, we use estimates (3), (5) and (32) to get, for any where c 2 is a fixed positive constant. Inserting this estimate in (34), we obtain, for a fixed constant c 3 > 0 and any t ≥ t 0 , Next, we introduce a functional η defined by η(t) := γ t − t 0 t t0 ψ x (t) − ψ x (t − s) 2 2 + ψ xt (t) − ψ xt (t − s) 2 2 ds, ∀ t > t 0 .