M Afilal

A Guesmia

A Soufyane

NEW STABILITY RESULTS FOR A LINEAR THERMOELASTIC BRESSE SYSTEM WITH SECOND SOUND

Keywords: Bresse system, Heat conduction, Well-posedness, Asymptotic behavior, Semigroup theory, Energy method, Frequency domain approach. AMS Classification: 35B40, 35L45, 74H40, 93D20, 93D15. ∼

In this paper, we consider a linear one-dimensional thermoelastic Bresse system with second sound consisting of three hyperbolic equations and two parabolic equations coupled in a certain manner under mixed homogeneous Dirichlet-Neumann boundary conditions, where the heat conduction is given by Cattaneo's law. Only the longitudinal displacement is damped via the dissipation from the two parabolic equations, and the vertical displacement and shear angle displacement are free. We prove the well-posedness of the system and some exponential, non exponential and polynomial stability results depending on the coefficients of the equations and the smoothness of initial data. Our method of proof is based on the semigroup theory and a combination of the energy method and the frequency domain approach.

Introduction

In this paper, we consider the following linear Bresse system with second sound: (1.1)

                 ρ 1 ϕ tt -k (ϕ x + ψ + l w)
x -lk 0 (w x -lϕ) = 0 in (0, 1) × (0, ∞) , ρ 2 ψ tt -bψ xx + k (ϕ x + ψ + l w) = 0 in (0, 1) × (0, ∞) , ρ 1 w tt -k 0 (w x -lϕ) x + lk (ϕ x + ψ + l w) + δθ x = 0 in (0, 1) × (0, ∞) , ρ 3 θ t + q x + δw xt = 0 in (0, 1) × (0, ∞) , τ q t + βq + θ x = 0 in (0, 1) × (0, ∞)

with the initial data (1.2)

            
ϕ (x, 0) = ϕ 0 (x) , ϕ t (x, 0) = ϕ 1 (x) in (0, 1) , ψ (x, 0) = ψ 0 (x) , ψ t (x, 0) = ψ 1 (x) in (0, 1) , w (x, 0) = w 0 (x) , w t (x, 0) = w 1 (x) in (0, 1) , θ (x, 0) = θ 0 (x) , q (x, 0) = q 0 (x) in (0, 1)

and mixed homogeneous Dirichlet-Neumann boundary conditions

(1.3) ϕ (0, t) = ψ x (0, t) = w x (0, t) = θ (0, t) = 0 in (0, ∞) , ϕ x (1, t) = ψ (1, t) = w (1, t) = q (1, t) = 0 in (0, ∞) ,
where ρ 1 , ρ 2 , ρ 3 , b, k, k 0 , τ, β, δ and l are positive constants, the initial data ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 , θ 0 and q 0 belong to a suitable Hilbert space, and the unknowns 1 of (1.1) -(1.3) are the following variables:

(ϕ, ψ, w, θ, q) : (0, 1) × (0, ∞) → R 5 .

The Bresse system [START_REF] Bresse | Cours de Mécanique Appliquée[END_REF] is consisting of three coupled hyperbolic equations (1.4)

       ρ 1 ϕ tt -k (ϕ x + ψ + l w) x -lk 0 (w x -lϕ) = F 1 in (0, L) × (0, ∞) , ρ 2 ψ tt -bψ xx + k (ϕ x + ψ + l w) = F 2 in (0, L) × (0, ∞) , ρ 1 w tt -k 0 (w x -lϕ) x + lk (ϕ x + ψ + l w) = F 3 in (0, L) × (0, ∞) ,
where L > 0,

F i : (0, L) × (0, ∞) → R
are the external forces (controllers) and w, ϕ and ψ represent, respectively, the longitudinal, vertical and shear angle displacements. For more details, we refer to [START_REF] Lagnese | Modelling of dynamic networks of thin thermoelastic beams[END_REF] and [START_REF] Lagnese | Modelling Analysis and Control of Dynamic Elastic Multi-Link Structures[END_REF]. For the last few years, many researchers studied the well-posedness and the stability of Bresse systems (1.4). Under different types of controls F i , various stability results have been obtained depending on the nature and the number of controls, the regularity of initial data and the following parameters:

(1.5)

s 1 = k ρ 1 , s 2 = b ρ 2 and s 3 = k 0 ρ 1 ;
for this purpose, we refer the reader to [START_REF] Alabau-Boussouira | Stability to weak dissipative Bresse system[END_REF], [START_REF] Alves | Stability and optimality of decay rate for weakly dissipative Bresse system[END_REF], [START_REF] Charles | Decay rates for Bresse system with arbitrary nonlinear localized damping[END_REF], [START_REF] Fatori | The optimal decay rate for a weak dissipative Bresse system[END_REF], [START_REF] Noun | Weakly locally internal stabilization of elastic Bresse system[END_REF], [START_REF] Soriano | Asymptotic stability for Bresse systems[END_REF], [START_REF] Soriano | Bresse system with indefinite damping[END_REF], [START_REF] Soufyane | The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system[END_REF] and [START_REF] Wehbe | Exponential and polynomial stability of an elastic Bresse system with two locally distributed feedbacks[END_REF] in case of (local or global, linear or nonlinear) frictional damping, and [START_REF] Lima Santos | Asymptotic behavior to Bresse system with past history[END_REF], [START_REF] Guesmia | Bresse system with infinite memories[END_REF], [START_REF] Guesmia | Uniform and weak stability of Bresse system with two infinite memories[END_REF] and [START_REF] Guesmia | Asymptotic stability of Bresse system with one infinite memory in the longitudinal displacements[END_REF] in case of memories. In some papers, it was proved that, when each equation of (1.4) is directly damped; that is

F 1 F 2 F 3 = 0,
the stability of (1.4) holds regardless to s 1 , s 2 and s 3 . However, when at least one equation in (1.4) is free; that is F 1 F 2 F 3 = 0 and (F 1 , F 2 , F 3 ) = (0, 0, 0), system (1.4) is still stable depending on the relation between the coefficients s 1 , s 2 and s 3 like:

s i = s j , i, j ∈ {1, 2, 3}.
When (F 1 , F 2 , F 3 ) = (0, 0, 0), system (1.4) is conservative, which means that the energy is conserved and equal to the energy of initial data along the trajectory of solutions. When the Bresse system is indirectly damped via the coupling (in a certain manner) with other equations, we mention here the work [START_REF] Liu | Energy decay rate of the thermoelastic Bresse system[END_REF], where the authors studied the stability of a thermoelastic Bresse system consisting of the following equations:

(1.6)                  ρ 1 ϕ tt -k (ϕ x + ψ + l w)
x -lk 0 (w x -lϕ) + lδθ = 0 in (0, L) × (0, ∞) , ρ 2 ψ tt -bψ xx + k (ϕ x + ψ + l w) + δq x = 0 in (0, L) × (0, ∞) , ρ 1 w tt -k 0 (w x -lϕ) x + lk (ϕ x + ψ + l w) + δθ x = 0 in (0, L) × (0, ∞) , ρ 3 θ t -θ xx + β (w x -lϕ) t = 0 in (0, L) × (0, ∞) , ρ 3 q t -q xx + βψ xt = 0 in (0, L) × (0, ∞)

with homogeneous Dirichlet-Neumann-Neumann boundary conditions

(1.7) ϕ(x, t) = ψ x (x, t) = w x (x, t) = θ(x, t) = q(x, t) = 0, x = 0, L, t ∈ (0, ∞)

or homogeneous Dirichlet-Dirichlet-Dirichlet boundary conditions (1.8) ϕ(x, t) = ψ(x, t) = w(x, t) = θ(x, t) = q(x, t) = 0, x = 0, L, t ∈ (0, ∞) .

They proved that the norm of solutions in the energy space decays exponentially to zero at infinity if (1.9)

s 1 = s 2 = s 3 .
Otherwise, the norm of solutions decays polynomially to zero with rates depending on the regularity of the initial data. For the classical solutions, these rates were t -1 4 + in case (1.7), and t -1 8 + in case (1.8), where is an arbitrary positive constant. In [START_REF] Fatori | Rates of decay to weak thermoelastic Bresse system[END_REF], the authors considered the following coupled Bresse system with only one heat equation:

(1.10)

             ρ 1 ϕ tt -k (ϕ x + ψ + l w) x -lk 0 (w x -lϕ) = 0 in (0, L) × (0, ∞) , ρ 2 ψ tt -bψ xx + k (ϕ x + ψ + l w) + δθ x = 0 in (0, L) × (0, ∞) , ρ 1 w tt -k 0 (w x -lϕ) x + lk (ϕ x + ψ + l w) = 0 in (0, L) × (0, ∞) , ρ 3 θ t -θ xx + (βψ t ) x = 0 in (0, L) × (0, ∞)
with (1.7) or (1.8). They proved that the exponential stability of (1.10) is equivalent to (1.9). On the other hand, when (1.9) is not satisfied, the obtained decay rate in [START_REF] Fatori | Rates of decay to weak thermoelastic Bresse system[END_REF] for classical solutions is t -1 6 + in general, and t -1 3 + when s 1 = s 2 and s 1 = s 3 . The results of [START_REF] Fatori | Rates of decay to weak thermoelastic Bresse system[END_REF] were extended in [START_REF] Najdi | Weakly locally thermal stabilization of Bresse systems[END_REF] to the case where the thermal dissipation is locally distributed; that is δ and β are non negative functions on x such that theirs minimums on some open interval I ⊂ (0, L) are positive. Moreover, when (1.9) is not satisfied, the authors of [START_REF] Najdi | Weakly locally thermal stabilization of Bresse systems[END_REF] improved the polynomial stability estimates of [START_REF] Fatori | Rates of decay to weak thermoelastic Bresse system[END_REF] by getting the decay rates t -1 4 and t -1 2 instead of t -1 6 + and t -1 3 + , respectively. In [START_REF] Keddi | Exponential and polynomial decay in a thermoelastic-Bresse system with second sound[END_REF], the authors considered the following coupled system:

(1.11)                  ρ 1 ϕ tt -k (ϕ x + ψ + l w) x -lk 0 (w x -lϕ) = 0 in (0, 1) × (0, ∞) , ρ 2 ψ tt -bψ xx + k (ϕ x + ψ + l w) + δθ x = 0 in (0, 1) × (0, ∞) , ρ 1 w tt -k 0 (w x -lϕ) x + lk (ϕ x + ψ + l w) = 0 in (0, 1) × (0, ∞) , ρ 3 θ t + q x + δψ xt = 0 in (0, 1) × (0, ∞) , τ q t + βq + θ x = 0 in (0, 1) × (0, ∞) .
They proved that (1.11) is exponentially stable if

s 1 = s 3 , ρ 1 k - ρ 2 b 1 - τ kρ 3 ρ 1 = τ δ 2 b
and l small, and (1.11) is not exponentially stable if

s 1 = s 3 or ρ 1 k - ρ 2 b 1 - τ kρ 3 ρ 1 = τ δ 2 b .
Moreover, when

s 1 = s 3 , ρ 1 k - ρ 2 b 1 - τ kρ 3 ρ 1 = τ δ 2 b
and l small, the polynomial stability for (1.11) was proved in [START_REF] Keddi | Exponential and polynomial decay in a thermoelastic-Bresse system with second sound[END_REF] with the decay rate t -1 2 . In (1.6) and (1.10), the heat equations are governed by Fourier's law of heat conduction. However, the heat conduction in (1.1) and (1.11) is given by Cattaneo's law (for more details, see [START_REF] Keddi | Exponential and polynomial decay in a thermoelastic-Bresse system with second sound[END_REF]).

In [START_REF] Dell'oro | Asymptotic stability of thermoelastic systems of Bresse type[END_REF], the author considered the following coupled system:

(1.12)

       ρ 1 ϕ tt -k (ϕ x + ψ + l w) x -lk 0 (w x -lϕ) = 0, ρ 2 ψ tt -bψ xx + k (ϕ x + ψ + l w) + δθ x = 0, ρ 1 w tt -k 0 (w x -lϕ) x + lk (ϕ x + ψ + l w) = 0, ρ 3 θ t -k 1 ∞ 0 g(s)θ xx (t -s)ds + γψ xt = 0, with homogeneous Dirichlet-Neumann boundary conditions (1.13) ϕ(x, t) = ψ x (x, t) = w x (x, t) = θ(x, t) = 0, x = 0, L, t ∈ (0, ∞)
He proved that (1.12) is exponentially stable if and if

(1.14) k = k 0 , ( ρ 1 ρ 3 k - 1 g(0)k 1 )( ρ 1 k - ρ 2 b ). - 1 g(0)k 1 ρ 1 γ 2 ρ 3 kb = 0.
On the other hand if (1.14) is not satisfied no decay rates was derived in [START_REF] Dell'oro | Asymptotic stability of thermoelastic systems of Bresse type[END_REF]. We need to mention here, that the coupling (through the second equation) and the boundary conditions considered in [START_REF] Dell'oro | Asymptotic stability of thermoelastic systems of Bresse type[END_REF] are not the same as the one considered in this paper. Notice that, when the three hyperbolic equations in Bresse system are (all or some of them) directly damped; that is (F 1 , F 2 , F 3 ) = (0, 0, 0), system (1.4) is dissipative. However, systems (1.1), (1.6), (1.10) and (1.11) are consisting of coupled conservative three hyperbolic equations with one or two parabolic equations, so the stability of the overall system is preserved thanks to the dissipation generated by the parabolic equations. On the other hand, we remark that in (1.6), the second and third hyperbolic equations are indirectly damped by the coupling with the heat equations, and the first hyperbolic one is only weakly damped through the coupling with the second and the third hyperbolic equations.

On the other hand, in (1.10) and (1.11), only the second hyperbolic equation is effectively damped by the dissipation coming from the parabolic equations.

In our case (1.1), only the third hyperbolic equation is indirectly damped through the coupling with the heat equations. Our objective, first is to consider (1.1)-(1.3), we prove the well-posedness and we establish some decay rates for the solutions (like: exponential stability, non exponential stability and polynomial stability) depending on the relationship between the coefficients of (1.1) and the smoothness of the initial data.

Without loss of generality, we consider the domain (0, 1) instead of (0, L). The proof of the well-posedness is based on the semigroup theory. However, the stability results are proved using the energy method combining with the frequency domain approach.

The paper is organized as follows. In section 2, we prove the well-posedness of (1.1) -(1.3). In sections 3 and 4, we show, respectively, our non exponential and exponential stability results for (1.1) -(1.3). The proof of our polynomial decay for (1.1) -(1.3) is proved in section 5.

2. Well-posedness of (1.1) - (1.3) In this section, we prove the existence, uniqueness and smoothness of solutions for (1.1) -(1.3) using the semigroup theory. In order to transform (1.1) -(1.3) into a first order evolution system on a suitable Hilbert space, we introduce the vector functions Φ = ϕ, φ, ψ, ψ, w, w, θ, q T and Φ 0 = (ϕ 0 , ϕ 1 , ψ 0 , ψ 1 , w 0 , w 1 , θ 0 , q 0 ) T , where φ = ϕ t , ψ = ψ t and w = w t . System (1.1) with initial data (1.2) can be written as

(2.1) Φ t = AΦ in (0, ∞) , Φ (0) = Φ 0 ,
where A is a linear operator defined by

(2.2) AΦ =                            φ k ρ 1 (ϕ x + ψ + l w) x + lk 0 ρ 1 (w x -lϕ) ψ b ρ 2 ψ xx - k ρ 2 (ϕ x + ψ + l w) w k 0 ρ 1 (w x -lϕ) x - lk ρ 1 (ϕ x + ψ + l w) - δ ρ 1 θ x - 1 ρ 3 q x - δ ρ 3 wx     
. Now, we introduce the following spaces:

               H 1 * (0, 1) = f ∈ H 1 (0, 1) : f (0) = 0 ,
and the energy space is given by L 2 (0,1) = 0, then ψ = 0, ϕ = -c sin (lx) and w = c cos (lx), where c is a constant such that c = 0 or l = π 2 + mπ, for some m ∈ N. Furthermore, we get ϕ = ψ = w = 0 if

(2.3) l = π 2 + mπ, ∀m ∈ N.
Here and after we assume that (2.3) is satisfied. Thus, H is a Hilbert space and D (A) is dense in H. If the domain (0, 1) is replaced by (0, L), then (2.3) becomes

lL = π 2 + mπ, ∀m ∈ N.
Now, we prove that the operator A generates a C 0 semigroup of contractions on H. For this purpose, it is sufficient to prove that A is maximal monotone. A direct calculation gives

(2.4) AΦ, Φ H = -β q 2 L 2 (0,1) ≤ 0.
Hence, A is dissipative in H. On the other hand, it is easy to show that 0 ∈ ρ (A); that is, for any

F = (f 1 , • • • , f 8 ) T ∈ H, there exists Z = (z 1 , • • • , z 8 ) T ∈ D (A) satisfying (2.5) AZ = F.
Indeed, from the first, third and fifth equations in (2.5), we get (2.6)

z 2 = f 1 , z 4 =
       k (z 1x + z 3 + l z 5 ) x + lk 0 (z 5x -lz 1 ) = ρ 1 f 2 , bz 3xx -k (z 1x + z 3 + l z 5 ) = ρ 2 f 4 , k 0 (z 5x -lz 1 ) x -lk (z 1x + z 3 + l z 5 ) = δz 7x + ρ 1 f 6 .
To prove that (2.10) admits a solution satisfying (2.11)

z 1 ∈ H 2 * (0, 1) , z 3 , z 5 ∈ ∼ H 2 * (0, 1
) and z 1x (1) = z 3x (0) = z 5x (0) = 0, we define the following bilinear form:

G 1 ((v 1 , v 2 , v 3 ) , (w 1 , w 2 , w 3 )) = k v 1x + v 2 + lv 3 , w 1x + w 2 + lw 3 L 2 (0,1) +b v 2x , w 2x L 2 (0,1) +k 0 v 3x -lv 1 , w 3x -lw 1 L 2 (0,1) , ∀ (v 1 , v 2 , v 3 ) T , (w 1 , w 2 , w 3 ) T ∈ H 0 × H 0 ,
and the following linear form:

G 2 (v 1 , v 2 , v 3 ) = v 1 , ρ 1 f 2 L 2 (0,1) + v 2 , ρ 2 f 4 L 2 (0,1) + v 3 , δz 7x + ρ 1 f 6 L 2 (0,1) , ∀ (v 1 , v 2 , v 3 ) T ∈ H 0 ,
where

H 0 = H 1 * (0, 1) × ∼ H 1 * (0, 1) × ∼ H 1
* (0, 1) Thus, the variational formulation of (2.10) is given by

(2.12) G 1 ((z 1 , z 3 , z 5 ) , (w 1 , w 2 , w 3 )) = G 2 (w 1 , w 2 , w 3 ) , ∀ (w 1 , w 2 , w 3 ) T ∈ H 0 .
From Lax-Milgram theorem, it follows that (2.12) has a unique solution

(z 1 , z 3 , z 5 ) ∈ H 0 .
Therefore, using classical elliptic regularity arguments, we conclude that (z 1 , z 3 , z 5 ) solves (2.10) and satisfies the regularity and boundary conditions (2.11). This proves that (2.5) has a unique solution Z ∈ D (A). By the resolvent identity, we have λI -A is surjective, for any λ > 0 (see [START_REF] Liu | Semigroups associated with dissipative systems[END_REF]), where I denotes the identity operator. Consequently, the Lumer-Phillips theorem implies that A is the infinitesimal generator of a linear C 0 semigroup of contractions on H. Thus, the well-posedness result for (2.1) is stated in the following (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]):

Theorem 2.1. Assume that (2.3) holds. For any p ∈ N and Φ 0 ∈ D(A p ), system (2.1) admits a unique solution

(2.13) Φ ∈ ∩ p j=0 C p-j R + ; D A j , where D A j is endowed by the graph norm • D(A j ) = j r=0 A r • H .
In the next three sections, we will show some exponentiel, non exponential and polynomial stability results for (2.1). The proof of these results is based on the following frequency domain theorems: Theorem 2.2. ( [START_REF] Huang | Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and [START_REF] Pruss | On the spectrum of C 0 semigroups[END_REF]) A C 0 semigroup of contractions on a Hilbert space H generated by an operator A is exponentially stable if and only if

(2.14) iR ⊂ ρ (A) and sup λ∈R (iλI -A) -1 L(H) < ∞.
Theorem 2.3. ( [START_REF] Liu | Characterization of polymomial decay rate for the solution of linear evolution equation[END_REF]) If a bounded C 0 semigroup e tA on a Hilbert space H generated by an operator A satisfies, for some j ∈ N * ,

(2.15) iR ⊂ ρ (A) and sup |λ|≥1 1 λ j (iλI -A) -1 L(H) < ∞.
Then, for any p ∈ N * , there exists a positive constant c p such that

(2.16) e tA z 0 H ≤ c p z 0 D(A p ) ln t t p j ln t, ∀z 0 ∈ D (A p ) , ∀t > 0.
3. Lack of exponential stability of (1.1)

-(1.3)
Our objective here is to show that the semigroup associated with our Bresse system with second sound (2.1) is not exponentially stable depending on the following relations:

(3.1) (k -k 0 ) ρ 3 - ρ 1 τ k -δ 2 = bρ 1 -kρ 2 = 0 and (3.2) l 2 = ρ 2 k 0 + ρ 1 b ρ 2 k 0 π 2 + mπ 2 + ρ 1 k ρ 2 (k + k 0 )
, ∀m ∈ Z.

Theorem 3.1. We assume that (2.3) holds, and (3.1) or (3.2) does not hold. Then the semigroup associated with (2.1) is not exponentially stable.

Proof. We use Theorem 2.2 by proving that the first or second condition in (2.14) is not satisfied. First, we prove that the first condition in (2.14) is equivalent to (3.2). Note that, according to the fact that 0 ∈ ρ (A) (see section 2), A -1 is bounded and it is a bijection between H and D(A). Since D(A) has a compact embedding into H, so it follows that A -1 is a compact operator, which implies that the spectrum of A is discrete. Let λ ∈ R * . We will prove that the unique

Φ = ϕ, φ, ψ, ψ, w, w, θ, q T ∈ D(A) satisfying (3.3) A Φ = i λ Φ is Φ = 0 if and only if (3.2) holds; that is the fact that iλ is not an eigenvalue of A is equivalent to (3.2). But equation (3.3) is equivalent to (3.4)                                  φ = iλϕ, ψ = iλψ, w = iλw, k ρ 1 (ϕ x + ψ + l w) x + lk 0 ρ 1 (w x -lϕ) = iλ φ, b ρ 2 ψ xx - k ρ 2 (ϕ x + ψ + l w) = iλ ψ, k 0 ρ 1 (w x -lϕ) x - lk ρ 1 (ϕ x + ψ + l w) - δ ρ 1 θ x = iλ w, - 1 ρ 3 q x - δ ρ 3 wx = iλθ, - β τ q - 1 τ θ x = iλq.
Using (2.4), we find

-β q L 2 (0,1) = Re AΦ, Φ H = Re iλ Φ, Φ H = Re iλ Φ 2 H = 0. Then (3.5) q = 0.
Taking into account that θ ∈ H 1 * (0, 1), using (3.5) and the eight equation in (3.4), we deduce that

             φ = iλϕ, ψ = iλψ, k (ϕ x + ψ) x -l 2 k 0 ϕ = -ρ 1 λ 2 ϕ, bψ xx -k (ϕ x + ψ) = -ρ 2 λ 2 ψ, -k 0 ϕ x -k (ϕ x + ψ) = 0, which is equivalent to φ = iλϕ, ψ = iλψ and (3.11)          l 2 k 0 -ρ 1 λ 2 ϕ -k (ϕ x + ψ) x = 0, -ρ 2 λ 2 ψ -bψ xx + k (ϕ x + ψ) = 0, ϕ x + ψ = - k 0 k ϕ x .
By deriving (3.11) 3 and combining with (3.11) 1 , we see that ϕ satisfy the following equation:

(3.12)

ϕ xx + αϕ = 0, where α = l 2 k0-ρ1λ 2 k0
. At this stage, we distinguish three cases.

Case 1: Consequently, we get

λ 2 = l 2 k0 ρ1 . Then ϕ(x) = c 1 x + c 2 , for c 1 , c 2 ∈ C.
(3.17) Φ = 0. Case 2: λ 2 > l 2 k0 ρ1 . Then ϕ(x) = c 1 e √ -αx + c 2 e - √ -αx .
Using again the boundary conditions (3.13), we find (3.14), and similarly as before, we arrive at (3.17).

Case 3:

λ 2 < l 2 k0 ρ1 . Then ϕ(x) = c 1 cos √ αx + c 2 sin √ αx .
Using the boundary conditions (3.13), we deduce that c 1 = 0, and

(3.18) c 2 = 0 or ∃ m ∈ Z : α = π 2 + mπ 2 .
If c 2 = 0, then (3.14) holds, and as before, we find (3.17).

If c 2 = 0, then, by (3.18),

(3.19) ∃ m ∈ Z : l 2 k 0 -ρ 1 λ 2 k 0 = π 2 + mπ 2 .
Therefore, (3.11) 3 is equivalent to

(3.20) ψ(x) = -c 2 1 + k 0 k √ α cos √ αx ,
and then the first two equations in (3.11) are reduced to

(3.21) λ 2 = k 0 kk 0 + bl 2 (k + k 0 ) (k + k 0 ) (k 0 ρ 2 + bρ 1 )
.

We see that (3.19) and (3.21) lead to

∃ m ∈ Z : l 2 = ρ 2 k 0 + ρ 1 b ρ 2 k 0 π 2 + mπ 2 + ρ 1 k ρ 2 (k + k 0 ) ; that is (3.2) does not hold. So, if (3.
2) holds, we get a contradiction, and hence, c 2 = 0 and, as before, we find (3.17). If (3.2) does not hold, then, for λ ∈ R satisfying (3.21), the function

Φ(x) = c 2 sin √ αx , iλ sin √ αx , -1 + k 0 k √ α cos √ αx , -iλ 1 + k 0 k √ α cos √ αx , 0, 0, 0, 0 T is a solution of (3.
3), for any c 2 ∈ C, and then iλ / ∈ ρ (A). Thus, we proved that i R ⊂ ρ (A) is equivalent to (3.2). Now, we show that the second condition in (2.14) does not hold if (3.1) is not satisfied, i.e. we assume that (3.1) is not satisfied and we will prove that there exists a sequence (λ n ) n ⊂ R such that

(λ n I -A) -1 L(H) -→ ∞,
which is equivalent to prove that there exists (F n ) n ⊂ H with F n H ≤ 1, for which we have

(3.22) ||(λ n I -A) -1 F n Φn || H -→ ∞,
therefore, we have

(3.23) λ n Φ n -AΦ n = F n .
Our objective is to show that the solution Φ n is not bounded when F n is bounded in H. The equation (3.23) implies that

(3.24)                            iλ n ϕ n - ∼ ϕ n = f 1n , iλ n ρ 1 ∼ ϕ n -k (ϕ nx + ψ n + l w n ) x -lk 0 (w nx -lϕ n ) + δθ nx = ρ 1 f 2n , iλ n ψ n - ∼ ψ n = f 3n , iλ n ρ 2 ∼ ψ n -bψ nxx + k (ϕ nx + ψ n + l w n ) = ρ 2 f 4n , iλ n w n - ∼ w n = f 5n , iλ n ρ 1 ∼ w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + l w n ) = ρ 1 f 6n , iλ n τ q n + βq n + θ nx = τ f 7n , iλ n ρ 3 θ n + q nx + δ ∼ w nx = ρ 3 f 8n .
We will show that, for all n ∈ N, given c 4 ∈ C * and

F n (x) = (0, 0, 0, c 4 cos (N x), 0, 0, 0, 0) T , where N = (2n+1)π 2 , there exists λ n ∈ R and Φ n = (iλ n -A) -1 F n ∈ D(A) such that lim λn→∞ Φ n H = ∞.
The system (3.24) will be written as

(3.25)                  iλ n ϕ n - ∼ ϕ n = 0, iλ n ψ n - ∼ ψ n = 0, iλ n w n - ∼ w n = 0, -λ 2 n ρ 1 ϕ n -k (ϕ nx + ψ n + l w n ) x -lk 0 (w nx -lϕ n ) = 0, -λ 2 n ρ 2 ψ n -bψ nxx + k (ϕ nx + ψ n + l w n ) = ρ 2 c 4 cos (N x), -λ 2 n ρ 1 w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + l w n ) + δθ nx = 0, iλ n ρ 3 θ n + q nx + δ ∼ w nx = 0, iλ n τ q n + βq n + θ nx = 0.
Because of the boundary conditions, one can take the following solution:

(3.26) ϕ n (x) = α 1 sin (N x), ψ n (x) = α 2 cos (N x), w n (x) = α 3 cos (N x), θ n (x) = α 4 sin (N x), q n (x) = α 5 cos (N x),
where the constants α 1 , α 2 , α 3 , α 4 and α 5 are the solution of the following system:

(3.27)            (-λ 2 n ρ 1 + N 2 k + l 2 k 0 )α 1 + kN α 2 + (k + k 0 )l N α 3 = 0, kN α 1 + (-λ 2 n ρ 2 + bN 2 + k)α 2 + kl α 3 = ρ 2 c 4 , (k 0 + k)lN α 1 + lkα 2 + (-λ 2 n ρ 1 + k 0 N 2 + l 2 k + δ(iλnτ +β)δλnN 2 (iλ 2 n ρ3τ +λnρ3β-iN 2 ) )α 3 = 0, (iλ 2 n ρ 3 τ + λ n ρ 3 β -iN 2 )α 5 + δλ n α 3 N 2 = 0, (iλ n τ + β)α 5 = -α 4 N.
We distinguish two cases.

Case 1: b ρ 2 = k 0 ρ 1 and [k -k 0 ] ρ 3 - ρ 1 τ k -δ 2 = 0. Let λ 2 n = k ρ 1 N 2 + A, where
A is a constant to be chosen later. Then form (3.27) we have

(3.28) (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 α 1 = -ρ 2 k N c 4 - l (k + k 0 ) (k 0 -k) ρ 2 ρ 1 N 3 + + kk 0 -Aρ 2 (k + k 0 ) -k 2 N l α 3 , (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 α 2 = ρ 2 l 2 k 0 -Aρ 1 c 4 + l (k + k 0 ) k N 2 -kl l 2 k 0 -Aρ 1 α 3 and α 3 must satisfy (3.29)          l 2 k 0 -Aρ 1 (k 0 -k) ρ 2 ρ 1 -k 2 - l 2 (k + k 0 ) 2 ρ 2 ρ 1 (k 0 -k) N 4 +   l 2 k -Aρ 1 l 2 k 0 -Aρ 1 (k 0 -k) ρ 2 ρ 1 -l 2 k -Aρ 1 k 2 + (k -Aρ 2 ) l 2 k 0 -Aρ 1 (k 0 -k) -(k -Aρ 2 ) l 2 (k + k 0 ) 2 + l 2 k 2 (k + k 0 )   N 2 +lk 2 N + l 2 k 0 -Aρ 1 l 2 k -Aρ 1 (k -Aρ 2 ) -l 2 k 2 l 2 k 0 -Aρ 1          α 3 (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 ) [l 2 k 0 -Aρ 1 ] -k 2 N 2 + δ 2 N 2 - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 1 - τ ρ 3 k ρ 1 N 2 -Aτ ρ 3 + iρ 3 βN k ρ 1 + A N 2 α 3 = - l 2 k 0 -Aρ 1 -(k + k 0 ) N 2 ρ 2 kl (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 ) [l 2 k 0 -Aρ 1 ] -k 2 N 2 c 4 .
Now, we distinguish four subcases.

k 0 -k = 0 and 1 -τ ρ 3 k ρ 1 = 0, then, from (3.28) and (3.29), we have

(3.30) (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 α 1 = -ρ 2 kN c 4 -kk 0 -k 2 -Aρ 2 (k + k 0 ) lN α 3 , (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 α 2 = ρ 2 l 2 k 0 -Aρ 1 c 4 + l (k + k 0 ) kN 2 -kl l 2 k 0 -Aρ 1 α 3
and α 3 satisfies (3.31)

l 2 k 2 (k + k 0 ) -l 2 k -Aρ 1 k 2 -(k -Aρ 2 ) l 2 (k + k 0 ) 2 N 2 +lk 2 N + l 2 k 0 -Aρ 1 l 2 k -Aρ 1 (k -Aρ 2 ) -l 2 k 2 l 2 k 0 -Aρ 1 α 3 ((k -Aρ 2 ) [l 2 k 0 -Aρ 1 ] -k 2 N 2 ) + δ 2 N 2 - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 1 - τ ρ 3 k ρ 1 N 2 -Aτ ρ 3 + iρ 3 βN k ρ 1 + A N 2 α 3 = - l 2 k 0 -Aρ 1 -(k + k 0 ) N 2 ([k -Aρ 2 ] [l 2 k 0 -Aρ 1 ] -k 2 N 2 ) ρ 2 klc 4 .
We choose A so that

(3.32) A = kρ 1 + ρ 2 l 2 k 0 + N 4ρ 2 ρ 1 k 2 + kρ 1 + ρ 2 l 2 k 0 2 N 2 2ρ 2 ρ 1 N k √ ρ 2 ρ 1 ,
then with (3.32), we have

(3.33) (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 = l 2 kk 0 ,
since, our concern is the asymptotic behavior of the constants, so, for N large enough, we obtain

        Aρ 1 k 2 + Aρ 2 l 2 (k + k 0 ) 2 - A 3 N 2 ρ 2 1 ρ 2 + l 2 k 2 (k + k 0 ) ((k -Aρ 2 ) [l 2 k 0 -Aρ 1 ] -k 2 N 2 ) - τ kδ 2 1 - τ ρ 3 k ρ 1 ρ 1         N 2 α 3 (3.34) - l 2 k 0 -Aρ 1 -(k + k 0 ) N 2 ([k -Aρ 2 ] [l 2 k 0 -Aρ 1 ] -k 2 N 2 ) ρ 2 klc 4 .
By using (3.32), we have

α 3 √ ρ 2 ρ 1 l (k + k 0 ) N c 4 and α 2 ρ 1 k l 2 k 0 (k + k 0 ) c 4 ,
so, we deduce with expression of α 2 that

Φ n H -→ ∞.
k 0 -k = 0 and 1 -τ ρ 3 k ρ 1 = 0, then we have from (3.28), (3.29) and (3.33)

(3.35)            (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 α 1 = -ρ 2 kN c 4 -l kk 0 -Aρ 2 (k + k 0 ) -k 2 N α 3 , (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 α 2 = ρ 2 l 2 k 0 -Aρ 1 c 4 + l (k + k 0 ) kN 2 -kl l 2 k 0 -Aρ 1 α 3 and (3.36) -l 2 k -Aρ 1 k 2 -(k -Aρ 2 ) l 2 (k + k 0 ) 2 + l 2 k 2 (k + k 0 ) N 2 +lk 2 N + l 2 k 0 -Aρ 1 l 2 k -Aρ 1 (k -Aρ 2 ) -l 2 k 2 l 2 k 0 -Aρ 1 α 3 l 2 kk 0 + δ 2 N 2 - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 -Aτ ρ 3 + iρ 3 βN k ρ 1 + A N 2 α 3 = - l 2 k 0 -Aρ 1 -(k + k 0 ) N 2 l 2 kk 0 ρ 2 klc 4 .
Using 1 -τ ρ 3 k ρ 1 = 0, (3.32), (3.35) and (3.36) when N large enough, we deduce that

                         α 3 2 - √ ρ 2 ρ 1 + iρ 2 ρ 3 β k ρ 1 l -4k + δ 2 ρ 3 + 4i β √ ρ 2 k τ N c 4 , α 2 δ 2 ρ 3 √ ρ 2 ρ 1 l 2 k -4k + δ 2 ρ 3 + 4i β √ ρ 2 k τ N c 4 ,
so, we obtain

Φ n H -→ ∞.
k 0 -k = 0 and 1 -τ ρ 3 k ρ 1 = 0, then we have from (3.28) and (3.29)

    (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 )   l 2 k 0 -Aρ 1 -k 2 N 2   α 1 = -ρ 2 kN c 4 -   l (k + k 0 ) (k 0 -k) ρ 2 ρ 1 N 3 + kk 0 -Aρ 2 (k + k 0 ) -k 2 N l   α 3 and     (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 )   l 2 k 0 -Aρ 1 -k 2 N 2   α 2 = ρ 2 l 2 k 0 -Aρ 1 c 4 + l (k + k 0 ) kN 2 -kl l 2 k 0 -Aρ 1 α 3 .
Also, we have (3.37)

         l 2 k 0 -Aρ 1 (k 0 -k) ρ 2 ρ 1 -k 2 - l 2 (k + k 0 ) 2 ρ 2 ρ 1 (k 0 -k) N 4 +   l 2 k -Aρ 1 l 2 k 0 -Aρ 1 (k 0 -k) ρ 2 ρ 1 -l 2 k -Aρ 1 k 2 + (k -Aρ 2 ) l 2 k 0 -Aρ 1 (k 0 -k) -(k -Aρ 2 ) l 2 (k + k 0 ) 2 + l 2 k 2 (k + k 0 )   N 2 +lk 2 N + l 2 k 0 -Aρ 1 l 2 k -Aρ 1 (k -Aρ 2 ) -l 2 k 2 l 2 k 0 -Aρ 1          α 3 (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 ) [l 2 k 0 -Aρ 1 ] -k 2 N 2 + δ 2 N 2 - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 -Aτ ρ 3 + iρ 3 βN k ρ 1 + A N 2 α 3 = - l 2 k 0 -Aρ 1 -(k + k 0 ) N 2 (k 0 -k) ρ 2 ρ 1 N 2 + (k -Aρ 2 ) [l 2 k 0 -Aρ 1 ] -k 2 N 2 ρ 2 klc 4 ,
Here we choose A as follow:

A =    (k 0 -k) ρ 2 N 2 -kρ 1 -ρ 2 l 2 k 0 + [(k 0 -k) ρ 2 N 2 -(kρ 1 + ρ 2 l 2 k 0 )] 2 -4ρ 2 ρ 1 l 2 k 0 (k 0 -k) ρ 2 ρ 1 -k 2 N 2    2ρ 2 ρ 1 (3.38) (k 0 -k) N 2 ρ 1 ,
then we have

(3.39) (k 0 -k) ρ 2 ρ 1 N 2 + k -Aρ 2 l 2 k 0 -Aρ 1 -k 2 N 2 = l 2 kk 0 ,
therefore, for N large enough and using (3.37), (3.38) and (3.39), we have

(3.40)        α 3 - 2lk 0 ρ 2 (k 0 -k) 2 N 2 c 4 , α 2 - ρ 2 (k 0 -k) l 2 kk 0 N 2 c 4 ,        so, we deduce that Φ n H -→ ∞. k 0 -k = 0 and 1 - τ ρ 3 k ρ 1 = 0,
       α 3 - 2lk 0 ρ 2 (k 0 -k) 2 N 2 c 4 , α 2 - ρ 2 (k 0 -k) l 2 kk 0 N 2 c 4 , so, we get Φ n H -→ ∞. Case 2: b ρ 2 = k 0 ρ 1 . Let λ 2 n = k ρ 1 N 2 + A, then from (3.25) we have (3.42)                                    -Aρ 1 + l 2 k 0 α 1 + kN α 2 + l (k + k 0 ) N α 3 = 0, kN α 1 + b - ρ 2 k ρ 1 N 2 -Aρ 2 + k α 2 + kl α 3 = ρ 2 c 4 , (k 0 -k) N 2 -Aρ 1 + l 2 k α 3 + l (k + k 0 ) N α 1 + lkα 2 + δN α 4 = 0, α 4 = δN - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 1 - τ ρ 3 k ρ 1 N 2 -τ ρ 3 A + iρ 3 βN k ρ 1 + A N 2 α 3 , α 5 = - iδλ n N 2 [N 2 -τ ρ 3 λ 2 n + iρ 3 βλ n ] α 3 ,                                    then we obtain (3.43)       b - ρ 2 k ρ 1 N 2 + (k -Aρ 2 )   l 2 k 0 -Aρ 1 -k 2 N 2     α 2 = ρ 2 l 2 k 0 -Aρ 1 c 4 + (k + k 0 ) N 2 -l 2 k 0 -Aρ 1 klα 3 , (3.44)     b - ρ 2 k ρ 1 N 2 + (k -Aρ 2 )   l 2 k 0 -Aρ 1 -k 2 N 2   α 1 = -ρ 2 kN c 4 -   (k + k 0 ) b - ρ 2 k ρ 1 N 2 + (k -Aρ 2 ) -k 2   lN α 3 and         (k 0 -k) N 2 -Aρ 1 + l 2 k +   -(k + k 0 ) b - ρ 2 k ρ 1 N 2 + k -Aρ 2 -k 2 l 2 (k + k 0 ) N 2 + (k + k 0 ) N 2 -l 2 k 0 -Aρ 1 k 2 l 2   b - ρ 2 k ρ 1 N 2 + k -Aρ 2 (l 2 k 0 -Aρ 1 ) -k 2 N 2         α 3 + δ 2 N 2 - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 1 - τ ρ 3 k ρ 1 N 2 -τ ρ 3 A + iρ 3 βN k ρ 1 + A N 2 α 3 (3.45) = - lkρ 2 l 2 k 0 -Aρ 1 c 4 -ρ 2 kN l (k + k 0 ) N c 4 b - ρ 2 k ρ 1 N 2 + k -Aρ 2 (l 2 k 0 -Aρ 1 ) -k 2 N 2
.

Now, we choose A such that

A =      ρ 1 b - ρ 2 k ρ 1 N 2 + ρ 2 l 2 k 0 + kρ 1 + ρ 1 b - ρ 2 k ρ 1 N 2 + ρ 2 l 2 k 0 + kρ 1 2 -4ρ 1 ρ 2 b - ρ 2 k ρ 1 l 2 k 0 -k 2 N 2 -B      2ρ 1 ρ 2 (3.46) b - ρ 2 k ρ 1 N 2 ρ 2 ,
where B is another constant to be chosen later. So, by using (3.46), we have

(3.47) b - ρ 2 k ρ 1 N 2 + (k -Aρ 2 ) l 2 k 0 -Aρ 1 -k 2 N 2 = l 2 kk 0 + B.
From (3.45) and by using (3.47), we have

      (k 0 -k) N 2 -Aρ 1 + l 2 k +   -(k + k 0 ) b - ρ 2 k ρ 1 N 2 + (k -Aρ 2 ) -k 2 l 2 (k + k 0 ) N 2 + (k + k 0 ) N 2 -l 2 k 0 -Aρ 1 k 2 l 2   (l 2 kk 0 + B)       α 3 + δ 2 N 2 - τ k ρ 1 N 2 -Aτ + iβN k ρ 1 + A N 2 1 - τ ρ 3 k ρ 1 N 2 -τ ρ 3 A + iρ 3 βN k ρ 1 + A N 2 α 3 (3.48) = - lkρ 2 l 2 k 0 -Aρ 1 c 4 -ρ 2 kN l (k + k 0 ) N c 4 l 2 kk 0 + B .
From (3.48) and by using (3.46), we deduce, for N large enough, the following:

    k 0 - b ρ 2 ρ 1 N 2 + k 0 + bρ 1 ρ 2 (l 2 kk 0 + B) k 2 l 2 N 2     α 3 - bτ δ 2 ρ 2 1 - τ ρ 3 b ρ 2 N 2 + iρ 3 βN b ρ 2 N 4 α 3 (3.49) = - -Aρ 1 -(k + k 0 ) N 2 (l 2 kk 0 + B) lkρ 2 c 4 .
Here, we distinguish two subcases.

1 -τ ρ 3 b ρ 2 = 0, then we have (3.50)

α 3 = -i     lkρ 2 ρ 3 β b ρ 2 ρ 1 + k 0 ρ 2 b τ δ 2 (l 2 kk 0 + B) N     c 4 and α 2 = - ρ 1 b -kρ 2 l 2 kk 0 + B N 2 c 4 .
By choosing B = 0, we deduce that

Φ n H -→ ∞. 1 - τ ρ 3 b ρ 2 = 0, then, from (3.48), we have (3.51)     k 0 - b ρ 2 ρ 1 + k 0 + bρ 1 ρ 2 l 2 kk 0 + B k 2 l 2 - bτ δ 2 ρ 2 1 - τ ρ 3 b ρ 2     N 2 α 3 = k 0 + ρ 1 b ρ 2 l 2 kk 0 + B lkρ 2 N 2 c 4 ,
here, we choose B such that

(3.52) B = ρ 2 bτ δ 2 k 0 + bρ 1 ρ 2 1 - τ ρ 3 b ρ 2 k 2 l 2 -l 2 kk 0 ,
so, by (3.52), we obtain

k 0 + bρ 1 ρ 2 l 2 kk 0 + B k 2 l 2 = bτ δ 2 ρ 2 1 - τ ρ 3 b ρ 2
, then we deduce from (3.48) and (3.52) that

α 3 = bτ δ 2 ρ 2 ρ 2 kl 1 - τ ρ 3 b ρ 2 k 0 - b ρ 2 ρ 1 c 4
and

α 1 = - bτ δ 2 ρ 2 k 0 + bρ 1 ρ 2 1 - τ ρ 3 b ρ 2 k 2 l 2     ρ 2 k + k 0 bτ δ 2 1 - τ ρ 3 b ρ 2 k 0 - b ρ 2 ρ 1     N c 4 ,
thus we have

w nx (x) -lϕ n (x) = k 0 τ 2 b 2 δ 4 ρ 2 k 2 l 1 - τ ρ 3 b ρ 2 2 k 0 - b ρ 2 ρ 1 k 0 + bρ 1 ρ 2 N c 4 sin (N x), hence Φ n H -→ ∞.
The proof of our theorem is then completed.

Exponential stability of (1.1) -(1.3)

In this section, we use again Theorem 2.2 to prove that the semigroup associated with (2.1) is exponentially stable provided that (2.3), (3.1) and (3.2) hold. Theorem 4.1. We assume that (2.3), (3.1) and (3.2) hold. Then the semigroup associated with (2.1) is exponentially stable.

Proof. In section 3, we have proved that the first condition in (2.14) is equivalent to (3.2). Now, by contradiction, we will prove the second condition in (2.14). So, we assume that the second condition in (2.14) is false, then there exist sequences

(Φ n ) n ⊂ D (A) and (λ n ) n ⊂ R satisfying (4.1) Φ n H = 1, ∀ n ≥ 0, (4.2) lim n→∞ |λ n | = ∞ and (4.3) lim n→∞ (i λ n I -A) Φ n H = 0, which implies that (4.4)                                            iλ n ϕ n - ∼ ϕ n -→ 0 in H 1 * (0, 1) , iλ n ρ 1 ∼ ϕ n -k (ϕ nx + ψ n + lw n ) x -lk 0 (w nx -lϕ n ) -→ 0 in L 2 (0, 1) , iλ n ψ n - ∼ ψ n -→ 0 in ∼ H 1 * (0, 1) , iλ n ρ 2 ∼ ψ n -bψ nxx + k (ϕ nx + ψ n + lw n ) -→ 0 in L 2 (0, 1) , iλ n w n - ∼ w n -→ 0 in ∼ H 1 * (0, 1) , iλ n ρ 1 ∼ w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + lw n ) + δθ nx -→ 0 in L 2 (0, 1) , iλ n ρ 3 θ n + q nx + δ ∼ w nx -→ 0 in L 2 (0, 1) , iλ n τ q n + βq n + θ nx -→ 0 in L 2 (0, 1) ,                                           
where the notation -→ means the limit when n goes to infinity. In the following, we will check the second condition in (2.14) by finding a contradiction with (4.1). Our proof is divided into several steps.

Step 1. Taking the inner product of (i λ n I -A) Φ n with Φ n in H and using (2.4), we get (4.5)

Re (i λ n I -A) Φ n , Φ n H = β q n 2 L 2 (0,1)
. Using (4.1) and ( 4.3), we deduce that (4.6) q n -→ 0 in L 2 (0, 1) .

Step 2. Applying triangle inequality, we have

θ nx λ n L 2 (0,1) ≤ 1 |λ n | iλ n τ q n + βq n + θ nx L 2 (0,1) + i τ q n + β λ n q n L 2 (0,1)
.

By
       ϕ n -→ 0 in L 2 (0, 1) , ψ n -→ 0 in L 2 (0, 1) , w n -→ 0 in L 2 (0, 1) .        .
Step 3. Taking the inner product of (4.4) 7 with iθ n λ n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

ρ 2 θ n 2 L 2 (0,1) -q n , iθ nx λ n L 2 (0,1) -δ ∼ w n , iθ nx λ n L 2 (0,1) -→ 0,
then, from (4.1) and (4.7), we get (4.9) θ n -→ 0 in L 2 (0, 1) .

Applying triangle inequality, we have

w nxx λ n L 2 (0,1) ≤ 1 k 0 |λ n | iλ n ρ 1 ∼ w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + lw n ) + δθ nx L 2 (0,1) + 1 k 0 iρ 1 ∼ w n + lk 0 λ n ϕ nx + lk λ n (ϕ nx + ψ n + lw n ) + δ θ nx λ n L 2 (0,1) .
Then, by (4.1), (4.2), (4.4) 6 and (4.7), we deduce that (4.10) 1 λ n w nxx n is uniformly bounded in L 2 (0, 1) .

Step 4. Taking the inner product of (4.4) 7 with iw nx λ n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

ρ 3 θ n , w nx L 2 (0,1) -q n , iw nxx λ n L 2 (0,1) -δ iλ n w nx - ∼ w nx , iw nx λ n L 2 (0,1) +δ w nx 2 L 2 (0,1) -→ 0.
Using (4.1), (4.2), (4.4) 5 , (4.6), (4.9) and (4.10), we deduce that (4.11)

w nx -→ 0 in L 2 (0, 1) ,
and from (4.4) 5 , we have (4.12)

∼ w nx λ n -→ 0 in L 2 (0, 1) . As ∼ w n in ∼ H 1 * (0, 1)
and by using (4.12), we obtain (4.13)

∼

w n λ n -→ 0 in L 2 (0, 1) .

Step 5. Taking the inner product of (4.4) 6 with i ∼ w n λ n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

ρ 1 ∼ w n 2 L 2 (0,1) + k 0 (w nx -lϕ n ) , i ∼ w nx λ n L 2 (0,1) +lk (ϕ nx + ψ n + lw n ) , i ∼ w n λ n L 2 (0,1) + δ θ nx λ n , i ∼ w n L 2 (0,1)
-→ 0.

Using (4.1), (4.7), (4.12) and (4.13), we obtain

(4.14) ∼ w n -→ 0 in L 2 (0, 1) ,
and with (4.4) 5 , we find (4.15) λ n w n -→ 0 in L 2 (0, 1) .

Step 6. Taking the inner product of k (ϕ nx + ψ n + lw n ) with θ nx in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

k (ϕ nx + ψ n + lw n ) , θ nx = -k (ϕ nx + ψ n + lw n ) x , θ n L 2 (0,1) = iλ n ρ 1 ∼ ϕ n -k (ϕ nx + ψ n + lw n ) x -lk 0 (w nx -lϕ n ) , θ n L 2 (0,1) -λ n ρ 1 i ∼ ϕ n , θ n L 2 (0,1)
+ lk 0 (w nx -lϕ n ) , θ n L 2 (0,1) , then, by using (4.1), (4.4) 2 and (4.9), (4.16) k

(ϕ nx + ψ n + lw n ) , θ nx L 2 (0,1) + λ n ρ 1 i ∼ ϕ n , θ n L 2 (0,1) -→ 0.
Taking the inner product of (ϕ nx + ψ n + lw n ) with iλ n ∼ w n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

(ϕ nx + ψ n + lw n ) , iλ n ∼ w n L 2 (0,1) = -iλ n ϕ nx , ∼ w n L 2 (0,1) -iλ n ψ n , ∼ w n L 2 (0,1) -l iλ n w n , ∼ w n L 2 (0,1) - ∼ ψ n , ∼ w n L 2 (0,1) -l iλ n w n - ∼ w n , ∼ w n L 2 (0,1) -l ∼ w n 2 L 2 (0,1) = -iλ n ϕ nx - ∼ ϕ nx , ∼ w n L 2 (0,1) + ∼ ϕ n , ∼ w nx L 2 (0,1) -iλ n ψ n - ∼ ψ n , ∼ w n L 2 (0,1) - ∼ ψ n , ∼ w n L 2 (0,1) -l iλ n w n - ∼ w n , ∼ w n L 2 (0,1) -l ∼ w n 2 L 2 (0,1)
. Then, by using (4.1), (4.4) 1 , (4.4) 3 , (4.4) 5 and (4.14), we deduce that (4.17)

(ϕ nx + ψ n + lw n ) , iλ n ∼ w n L 2 (0,1) - ∼ ϕ n , ∼ w nx L 2 (0,1) -→ 0.
Taking the inner product of

∼ ϕ n with ∼ w nx in L 2 (0, 1), we get ∼ ϕ n , ∼ w nx L 2 (0,1) = ∼ ϕ n , ∼ w nx - ∼ ϕ n L 2 (0,1) + ∼ ϕ n 2 L 2 (0,1) = - ∼ ϕ n , iλ n w nx - ∼ w nx L 2 (0,1) + ∼ ϕ n , iλ n ϕ n - ∼ ϕ n L 2 (0,1) + ∼ ϕ n , iλ n (w nx -ϕ n ) L 2 (0,1) + ∼ ϕ n 2 L 2 (0,1)
, then, by (4.1), (4.4) 1 and (4.4) 5 , we have

(4.18) λ n ∼ ϕ n , i (w nx -ϕ n ) L 2 (0,1) - ∼ ϕ n , ∼ w nx L 2 (0,1) + ∼ ϕ n 2 L 2 (0,1) -→ 0.
Taking the inner product of (4.4) 2 with (w nx -lϕ n ) in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

iλ n ρ 1 ∼ ϕ n , (w nx -lϕ n ) L 2 (0,1) + k (ϕ nx + ψ n + lw n ) , (w nx -lϕ n ) x L 2 (0,1) -lk 0 (w nx -lϕ n ) 2 L 2 (0,1) -→ 0, which implies that λ n ρ 1 i ∼ ϕ n , (w nx -lϕ n ) L 2 (0,1) - k k 0 (ϕ nx + ψ n + lw n ) , iλ n ρ 1 ∼ w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + lw n ) + δθ nx L 2 (0,1) + kρ 1 k 0 (ϕ nx + ψ n + lw n ) , iλ n ∼ w n L 2 (0,1) + lk 2 k 0 (ϕ nx + ψ n + lw n ) 2 L 2 (0,1) + δk k 0 (ϕ nx + ψ n + lw n ) , θ nx L 2 (0,1) -lk 0 (w nx -lϕ n ) 2
L 2 (0,1) -→ 0. Using (4.1), (4.4) 6 , (4.8) and (4.11), we get

(4.19) -λ n ρ 1 ∼ ϕ n , i (w nx -lϕ n ) L 2 (0,1) + kρ 1 k 0 (ϕ nx + ψ n + lw n ) , iλ n ∼ w n L 2 (0,1) + lk 2 k 0 (ϕ nx + ψ n + lw n ) 2 L 2 (0,1) + δk k 0 (ϕ nx + ψ n + lw n ) , θ nx L 2 (0,1) -→ 0,
then, by (4.16), (4.17), (4.18) and (4.19), we obtain

(4.20) k k 0 -1 ρ 1 ∼ ϕ n , ∼ w nx L 2 (0,1) - δ k 0 λ n ρ 1 i ∼ ϕ n , θ n L 2 (0,1) + lk 2 k 0 (ϕ nx + ψ n + lw n ) 2 L 2 (0,1) + ρ 1 ∼ ϕ n 2 L 2 (0,1)
-→ 0.

Step 7. Taking the inner product of (4.4) 8 with (ϕ nx + ψ n + lw n ) in L 2 (0, 1), we get iλ n τ q n , ϕ nx L 2 (0,1) -τ q n , iλ n ψ n L 2 (0,1) -lτ q n , iλ n w n L 2 (0,1)

+ βq n , (ϕ nx + ψ n + lw n ) L 2 (0,1) + θ nx , (ϕ n,x + ψ n + lw n ) L 2 (0,1) -→ 0, then iλ n τ q n , ϕ nx L 2 (0,1) -τ q n , iλ n ψ n - ∼ ψ n L 2 (0,1) -τ q n , ∼ ψ n L 2 (0,1) -lτ q n , iλ n w n - ∼ w n L 2 (0,1) -lτ q n , ∼ w n L 2 (0,1) + βq n , (ϕ nx + ψ n + lw n ) L 2 (0,1) + θ nx , (ϕ nx + ψ n + lw n ) L 2 (0,1) -→ 0.
By using (4.1), (4.4) 3 , (4.4) 5 , (4.6) and (4.16), we have

iλ n τ q n , ϕ nx L 2 (0,1) - λ n ρ 1 k θ n , i ∼ ϕ n L 2 (0,1)
-→ 0, integrating by parts and using the boundary conditions, we obtain

-λ n τ iq nx , ϕ n L 2 (0,1) - λ n ρ 1 k θ n , i ∼ ϕ n L 2 (0,1) -→ 0, therefore -λ n τ i iλ n ρ 3 θ n + q nx + δ ∼ w nx , ϕ n L 2 (0,1) -λ n τ λ n ρ 3 θ n , ϕ n L 2 (0,1) +λ n τ δ i ∼ w nx , ϕ n L 2 (0,1) - λ n ρ 1 k θ n , i ∼ ϕ n L 2 (0,1) -→ 0, hence τ iλ n ρ 3 θ n + q nx + δ ∼ w nx , iλ n ϕ n - ∼ ϕ n L 2 (0,1) +τ iλ n ρ 3 θ n + q nx + δ ∼ w nx , ∼ ϕ n L 2 (0,1) -λ n τ iρ 3 θ n , iλ n ϕ n - ∼ ϕ n L 2 (0,1) -λ n τ iρ 3 θ n , ∼ ϕ n L 2 (0,1) +τ δ ∼ w n , iλ n ϕ n - ∼ ϕ n x L 2 (0,1) -τ δ ∼ w nx , ∼ ϕ n L 2 (0,1) - λ n ρ 1 k θ n , i ∼ ϕ n L 2 (0,1)
-→ 0, so, using (4.1), (4.4) 1 , (4.4) 7 , we get

(4.21) τ ρ 3 - ρ 1 k λ n θ n , i ∼ ϕ n L 2 (0,1) -τ δ ∼ w nx , ∼ ϕ n L 2 (0,1) -λ n τ iρ 3 θ n , iλ n ϕ n - ∼ ϕ n L 2 (0,1)
-→ 0.

On the other hand, integrating by parts and using the boundary conditions, we find that

λ n iρ 3 θ n , iλ n ϕ n - ∼ ϕ n L 2 (0,1) = iλ n ρ 3 θ n + q nx + δ ∼ w nx , iλ n ϕ n - ∼ ϕ n L 2 (0,1) -q nx , iλ n ϕ n - ∼ ϕ n L 2 (0,1) -δ ∼ w nx , iλ n ϕ n - ∼ ϕ n L 2 (0,1) = iλ n ρ 3 θ n + q nx + δ ∼ w nx , iλ n ϕ n - ∼ ϕ n L 2 (0,1) + q n , iλ n ϕ nx - ∼ ϕ nx L 2 (0,1) + δ ∼ w n , iλ n ϕ nx - ∼ ϕ nx L 2 (0,
τ ρ 3 - ρ 1 k λ n i ∼ ϕ n , θ n L 2 (0,1) -τ δ ∼ ϕ n , ∼ w nx L 2 (0,1)
-→ 0, and then, multiplying (4.23) by ρ1 τ δ k k0 -1 and adding (4.20), we obtain

ρ 1 λ n k 0 δ (k -k 0 ) ρ 3 - ρ 1 τ k -δ 2 i ∼ ϕ n , θ n L 2 (0,1) + lk 2 k 0 (ϕ nx + ψ n + lw n ) 2 L 2 (0,1) +ρ 1 ∼ ϕ n 2 L 2 (0,1)
-→ 0.

Here we use the fact that (k

-k 0 ) ρ 3 - ρ 1 τ k -δ 2 = 0 (condition (3.1)), we deduce that lk 2 k 0 (ϕ nx + ψ n + lw n ) 2 L 2 (0,1) + ρ 1 ∼ ϕ n 2 L 2 (0,1)
-→ 0, then, from (4.8), we have Step 8. Taking the inner product of (4.4) 4 with (ϕ nx + ψ n + lw n ) in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

iλ n ρ 2 ∼ ψ n , ϕ nx L 2 (0,1) + iλ n ρ 2 ∼ ψ n , ψ n L 2 (0,1) + l iλ n ρ 2 ∼ ψ n , w n L 2 (0,1) +b ψ nx , (ϕ nx + ψ n + lw n ) x L 2 (0,1) + k (ϕ nx + ψ n + lw n ) 2 L 2 (0,1) -→ 0, then -λ n ρ 2 ∼ ψ n , iϕ nx L 2 (0,1) -ρ 2 ∼ ψ n , iλ n ψ n - ∼ ψ n L 2 (0,1) -ρ 2 ∼ ψ n 2 L 2 (0,1) -lρ 2 ∼ ψ n , iλ n w n - ∼ w n L 2 (0,1) -lρ 2 ∼ ψ n , ∼ w n L 2 (0,1) - b k ψ nx , iλ n ρ 1 ∼ ϕ n -k (ϕ nx + ψ n + lw n ) x -lk 0 (w nx -lϕ n ) L 2 (0,1) + b k ψ nx , iλ n ρ 1 ∼ ϕ n L 2 (0,1) - lk 0 b k ψ nx , (w nx -lϕ n ) L 2 (0,1) +k ϕ nx + ψ n + lw n 2 L 2 (0,
-ρ 2 ∼ ψ n 2 L 2 (0,1) + bρ 1 k λ n ψ nx , i ∼ ϕ n L 2 (0,1)
-→ 0. Now, we use that

λ n ψ nx , i ∼ ϕ n L 2 (0,1) = - iλ n ψ nx - ∼ ψ nx , ∼ ϕ n L 2 (0,1) - ∼ ψ nx , ∼ ϕ n L 2 (0,1)
, and by integrating by parts and using the boundary conditions, we have

λ n ψ nx , i ∼ ϕ n L 2 (0,1) = -iλ n ψ nx - ∼ ψ nx , ∼ ϕ n L 2 (0,1) + ∼ ψ n , ∼ ϕ nx L 2 (0,1) = - iλ n ψ nx - ∼ ψ nx , ∼ ϕ n L 2 (0,1) - ∼ ψ n , iλ n ϕ nx - ∼ ϕ nx L 2 (0,1) + ∼ ψ n , iλ n ϕ nx L 2 (0,1)
, therefore, from (4.1), (4.4) 1 and (4.4) 3 , we see that (4.29)

λ n ψ nx , i ∼ ϕ n L 2 (0,1) -λ n ∼ ψ n , iϕ nx L 2 (0,1)
-→ 0, so, inserting (4.29) into (4.28), we obtain (4.30)

λ n k (bρ 1 -kρ 2 ) ψ nx , i ∼ ϕ n L 2 (0,1) -ρ 2 ∼ ψ n 2 L 2 (0,1) -→ 0.
At this stage, we use the fact that bρ 1 -kρ 2 = 0 (condition (3.1)), then we have from (4.30)

(4.31) ∼ ψ n -→ 0 in L 2 (0, 1) ,
and by (4.4) 3 , we deduce that (4.32) λ n ψ n -→ 0 in L 2 (0, 1) .

Step 9. Taking the inner product of (4.4) 4 with ψ n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

-ρ 2 ∼ ψ n , iλ n ψ n L 2 (0,1) + b ψ nx 2 L 2 (0,1) + k (ϕ nx + ψ n + lw n ) , ψ n L 2 (0,
1 λ 8 (iλI -A) -1 H < ∞.
We establish (5.2) by contradiction. So, if (5.2) is false, then there exist sequences

(Φ n ) n ⊂ D (A) and (λ n ) n ⊂ R satisfying (5.3) Φ n H = 1, ∀ n ∈ N, (5.4
                                                       λ 8 n iλ n ϕ n - ∼ ϕ n → 0 in H 1 * (0, 1) , λ 8 n iλ n ρ 1 ∼ ϕ n -k (ϕ nx + ψ n + lw n ) x -lk 0 (w nx -lϕ n ) → 0 in L 2 (0, 1) , λ 8 n iλ n ψ n - ∼ ψ n → 0 in ∼ H 1 * (0, 1) , λ 8 n iλ n ρ 2 ∼ ψ n -bψ nxx + k (ϕ nx + ψ n + lw n ) → 0 in L 2 (0, 1) , λ 8 n iλ n w n - ∼ w n → 0 in ∼ H 1 * (0, 1) , λ 8 n iλ n ρ 1 ∼ w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + lw n ) + δθ nx → 0 in L 2 (0, 1) , λ 8 n iλ n ρ 3 θ n + q nx + δ ∼ w nx → 0 in L 2 (0, 1) , λ 8 n (iλ n τ q n + βq n + θ nx ) → 0 in L 2 (0, 1) .                                                        .
Our goal is to derive Φ n H → 0 as a contradiction with (5.3). This will be established through several steps.

Step 1. Taking the inner product of λ 8 n (i λ n I -A) Φ n with Φ n in H, we get (as for (4.5))

Re λ 8 n (i λ n I -A) Φ n , Φ n L 2 (0,1) = β λ 4 n q n 2 L 2 (0,1
) , so we have (5.7) λ 4 n q n -→ 0 in L 2 (0, 1) .

Step 2. Applying triangle inequality, we obtain

λ 3 n θ nx L 2 (0,1) ≤ λ 3 n (iλ n τ q n + βq n + θ nx ) L 2 (0,1) + iλ 4 n τ q n + βλ 3 n q n L 2 (0,1)
, then, using (5.4), (5.6) 8 and (5.7), we have (5.8) λ 3 n θ nx -→ 0 in L 2 (0, 1) . Knowing that θ n in H 1 * (0, 1), then we have (5.9) λ 3 n θ n -→ 0 in L 2 (0, 1) .

Step 3. Using (5.3), (5.4), (5.6) 1 , (5.6) 3 and (5.6) 5 , we obtain (5.10)

ϕ n , ψ n w n -→ 0 in L 2 (0, 1) , (λ n ϕ n ) n , (λ n ψ n ) n , (λ n w n ) n are uniformly bounded in L 2 (0, 1) . .
Step 4. By triangle inequality, we have

w nxx λ n ≤ 1 k 0 λ n iλ n ρ 1 ∼ w n -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + lw n ) + δθ nx L 2 (0,1) + 1 k 0 iρ 1 ∼ w n + lk 0 ϕ nx λ n + lk λ n (ϕ nx + ψ n + lw n ) + δ λ n θ nx L 2 (0,1)
, then we deduce from (5.3), (5.4), (5.6) 6 and (5.8) that (5.11) w nxx λ n n is uniformly bounded in L 2 (0, 1) .

integrating by parts and using the boundary conditions, we have

λ 2 n w nx 2 L 2 (0,1) = λ 4 n w nx , w nx L 2 (0,1) = λ 3 n iw nx , iλ n w nx - ∼ w nx L 2 (0,1) + λ 3 n iw nx , ∼ w nx L 2 (0,1) = iw nx , λ 3 n iλ n w nx - ∼ w nx L 2 (0,1) + 1 δ iw nx , λ 3 n iλ n ρ 3 θ n + q nx + δ ∼ w nx L 2 (0,1) + ρ 3 δ λ n w n , λ 3 n θ nx L 2 (0,1) + 1 δ i w nxx λ n , λ 4 n q n L 2 (0,1)
, then, by using (5.3), (5.4), (5.6) 5 , (5.6) 7 , (5.7), (5.8), (5.10) and (5.11), we find (5.12) λ 2 n w nx → 0 in L 2 (0, 1) .

As w n in ∼ H 1 * (0, 1), we deduce from (5.12) that (5.13) λ2 n w n → 0 in L 2 (0, 1) , and using (5.4) and (5.6) 5 , we see that Also, dividing (5.6) 6 by λ 8 n and using (5.3), (5.4), (5.8) and (5.15), we deduce that (5.16) (w nxx ) n is uniformly bounded in L 2 (0, 1) .

Step 5. Taking the inner product of (5.6) 7 with iw nx λ 4 n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

-ρ 3 iλ 3 n θ nx , λ 2 n w n L 2 (0,1) -δ λ 4 n iλ n w nx - ∼ w nx , iw nx L 2 (0,1) -λ 4 n q n , iw nxx L 2 (0,1) + δλ 5 n w nx 2
L 2 (0,1) → 0. Using (5.3), (5.4), (5.6) 5 , (5.7), (5.8), (5.13) and (5.16), we obtain (5.17

∼

w nx → 0 in L 2 (0, 1) .

Step 6. Applying again triangle inequality, we have

ϕ nxx λ n L 2 (0,1) ≤ 1 k 1 λ n iλ n ρ 1 ∼ ϕ n -k (ϕ nx + ψ n + lw n ) x -lk 0 (w nx -lϕ n ) L 2 (0,1) + 1 k iρ 1 ∼ ϕ n - k λ n (ψ nx + lw nx ) - lk 0 λ n (w nx -lϕ n ) L 2 (0,1)
, and using (5.3), (5.4) and (5.6) 2 , we deduce that (5.19) ϕ nxx λ n n is uniformly bounded in L 2 (0, 1) .

Taking the inner product of (5.6) 6 with ϕ nx λ 8 n in L 2 (0, 1), integrating by parts and using the boundary conditions, we obtain

ρ 1 iλ n ∼ w n , ϕ nx L 2 (0,1) + k 0 λ n w nx , ϕ nxx λ n L 2 (0,1) + l (k + k 0 ) ϕ nx 2 L 2 (0,1) +lk (ψ n + lw n ) , ϕ nx L 2 (0,1) + δ θ nx , ϕ nx L 2 (0,1) -→ 0,
then, from (5.3), (5.4), (5.8), (5.10), (5.12), (5.15) and (5.19), we have (5.20) ϕ nx -→ 0 in L 2 (0, 1) .

Step 7. Taking the inner product of (5.6) 6 with ϕ nx λ 7 n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

-ρ 1 ∼ w n , λ n iλ n ϕ nx - ∼ ϕ nx L 2 (0,1) + ρ 1 λ n ∼ w nx , ∼ ϕ n L 2 (0,1) +k 0 λ 2 n w nx , ϕ nxx λ n L 2 (0,1) + l (k + k 0 ) λ n ϕ nx +lk λ n (ψ n + lw n ) , ϕ nx L 2 (0,1) + δ λ n θ nx , ϕ nx L 2 (0,1) -→ 0,
hence, using (5.3), (5.4), (5.6) 1 , (5.8), (5.10), (5.12), (5.19) and (5.20), we obtain (5.21) λ n ϕ nx 2 L 2 (0,1) -→ 0.

Taking the inner product of (5.6) 2 with ϕ n λ 7 n in L 2 (0, 1), integrating by parts and using the boundary conditions, we get

-ρ 1 λ n ∼ ϕ n , iλ n ϕ n - ∼ ϕ n L 2 (0,1) -ρ 1 λ n ∼ ϕ n 2 L 2 (0,1)
+kλ n (ϕ nx + ψ n + lw n ) , ϕ nx L 2 (0,1) -lk 0 λ n (w nx -lϕ n ) , ϕ n L 2 (0,1) -→ 0, which implies

-ρ 1 ∼ ϕ n , λ n iλ n ϕ n - ∼ ϕ n L 2 (0,1) -ρ 1 λ n ∼ ϕ n 2 L 2 (0,1)
+kλ n ϕ nx 2 L 2 (0,1) + k (λ n ψ n + lλ n w n ) , ϕ nx L 2 (0,1) -lk 0 (λ n w nx -lλ n ϕ n ) , ϕ n L 2 (0,1) -→ 0, so, using (5.3), (5.4), (5.6) 1 , (5.10), (5.12) and (5.21), we deduce that -→ 0 in L 2 (0, 1) .

On the other hand, by integrating by parts and using the boundary conditions, we see that λ n w nxx , iλ n ϕ nx L 2 (0,1) = λ 2 n iw nx , ϕ nxx L 2 (0,1)

= λ n iλ n w nx -∼ w nx , ϕ nxx L 2 (0,1)

+ λ n ∼ w nx , ϕ nxx L 2 (0,1) = λ 2 n iλ n w nx - ∼ w nx , ϕ nxx λ n L 2 (0,1) + λ n |λ n | 1 2 ∼ w nx , ϕ nxx |λ n | 1 2
L 2 (0,1) , then, using (5.4), (5.6) 5 , (5.18) and (5.24), we obtain (5.25) λ n w nxx , iλ n ϕ nx L 2 (0,1) -→ 0.

Furthermore, integrating by parts and using the boundary conditions, -→ 0, then, using (5.26), we obtain

λ n (ϕ nx + ψ n + lw n ) x , ∼ ϕ n L 2 (
ρ 1 i λ n ∼ ϕ n 2 L 2 (0,1)
+ ik 0 λ n ϕ nx By triangle inequality, we deduce from (5.3) and (5.4) that (5.29) ψ nxx λ n n is uniformly bounded in L 2 (0, 1) .

Taking the inner product of (5.6) 2 with ψ nx λ 8 n in L 2 (0, 1), we get

ρ 1 iλ n ∼ ϕ n , ψ nx L 2 (0,1)
-k ϕ nxx , ψ nx L 2 (0,1) -k ψ nx 2 L 2 (0,1)

-l(k + k 0 ) w nx , ψ nx L 2 (0,1) + l 2 k 0 ϕ n , ψ nx L 2 (0,1) → 0, then, integrating by parts and using the boundary conditions, we obtain

ρ 1 iλ n ∼ ϕ n , ψ nx L 2 (0,1) + k λ n ϕ nx , ψ nxx λ n L 2 (0,1) -k ψ nx 2 L 2 (0,1)
-l(k + k 0 ) w nx , ψ nx L 2 (0,1) + l 2 k 0 ϕ n , ψ nx L 2 (0,1) → 0, so, using (5.3), (5.4), (5.10), (5.12), (5.27), (5.28) and (5.29), we deduce that (5.30) ψ nx -→ 0 in L 2 (0, 1) .

Taking the inner product of (5. + k (ϕ nx + ψ n + lw n ) , ψ n L 2 (0,1) -→ 0, hence, using (5.3), (5.4), (5.6) 3 , (5.10) and (5.30), we get (5.31) ∼ ψ n -→ 0 in L 2 (0, 1) .

A combination of (5.4) and all the above convergence leads to

Φ n H -→ 0,
which is a contradiction with (5.3). Consequently, the proof of our Theorem 5.1 is completed.

1 *

 1 (0, 1), we have (3.9) w = w = 0.Using (3.5), (3.6) and (3.9), then the system (3.4) is reduced into:(3.10)

  which implies that, using the first two equations in (3.10) and the last one in (3.11),

  then, using (3.28), (3.29) and (3.39), we obtain the same result as before(3.41)

(4. 24 )

 24 ϕ nx -→ 0 in L 2 (0, 1) and (4.25) ∼ ϕ n -→ 0 in L 2 (0, 1) , and using (4.2), (4.4) 1 and (4.25), we have (4.26) λ n ϕ n -→ 0 in L 2 (0

8 n

 8 (iλ n I -A) Φ n H = 0, which implies that(5.6)

  → 0 in L 2 (0, 1) .

) |λ n | 5 2

 5 w nx → 0 in L 2 (0, 1) , and from (5.6) 5 , we get(5.18) |λ n | 3 2

2 L 2

 22 (0,1) -lk 0 (λ n w nx -lλ n ϕ n ) , ∼ ϕ n L 2 (0,1)-→ 0, and from (5.3), (5.4), (5.6) 1 , (5.12), (5.22) and (5.23), we deduce that (5.27)λ n ∼ ϕ n -→ 0 in L 2 (0, 1) and(5.28) λ n ϕ nx -→ 0 in L 2 (0, 1) . + ψ n + lw n ) → 0 in L 2 (0, 1) .

6 ) 4 with ψ n λ 8 n 2 ∼

 682 in L 2 (0, 1), integrating by parts and using the boundary conditions, we get -ρ ψ n , iλ n ψ n -

  f 3 and z 6 = f 5 , -βz 8 -τ f 8 and z 8x = -δf 5x -ρ 3 f 7 .

	and then
				∼
	(2.7)	z 2 ∈ H 1 * (0, 1) and z 4 , z 6 ∈	H 1 * (0, 1) .
	Substituting z 2 into the seventh equation in (2.5), we conclude from the last two
	equations in (2.5) that
	(2.8) z 7x = By a direct integration, we see that (2.8) has a unique solution satisfying
				∼
	(2.9)	z 7 ∈ H 1 * (0, 1) and z 8 ∈	H 1 * (0, 1) .
	Finally, the second, fourth and sixth equations in (2.5) become
	(2.10)

  Theorem 5.1. We assume that (2.3) and (3.2) hold. Then, for each p ∈ N * , there exists a constant c p > 0 such that(5.1) ∀Φ 0 ∈ D (A p ) , ∀t > 0, e tA Φ 0 H ≤ c p Φ 0 D(A p )

	A combination of (4.6), (4.8), (4.9), (4.11), (4.14), (4.24), (4.25), (4.31) and (4.33)
	leads to
		Φ n H -→ 0,
	which is a contradiction with (4.1). Hence, the proof of Theorem 4.1 is completed.
		5. Polynomial stability of (1.1) -(1.3)
	In this section, we prove the polynomial decay of the solutions of (2.1) using
	Theorem 2.3. Our main result is stated as follow:
			p
		ln t t	8 ln t.
	Proof. In section 3, we have proved that the first condition in (2.15) is satisfied if
	(3.2) holds. Now, we need to show that
	(5.2)	sup
		|λ| ≥ 1
			1) -→ 0,
	and by using (4.8), (4.24), (4.31) and (4.32), we obtain
	(4.33)	ψ nx -→ 0 in L 2 (0, 1) .

  0,1) = -λ n (ϕ nx + ψ n + lw n ) , -k 0 (w nx -lϕ n ) x + lk (ϕ nx + ψ n + lw n ) + δθ nx , + δθ nx , λ n iλ n ϕ nxnx -lϕ n ) x , λ n iλ n ϕ nxnx , iϕ nx L 2 (0,1) -k 0 λ n lk w nxx , iλ n ϕ nx L 2 (0,1)k 0 λ 2 , then, using (5.6) 1 , (5.6) 6 , (5.8), (5.15), (5.16), (5.18), (5.23) and (5.25), we find (5.26)λ n (ϕ nx + ψ n + lw n ) x , -kλ n (ϕ nx + ψ n + lw n ) x , ∼ ϕ n L 2 (0,1)-lk 0 (λ n w nx -lλ n ϕ n ) ,

	= -	1 lk	λ 2 n iλ n ρ 1		∼ ϕ nx λ n L 2 (0,1)
						-	1 lk	iλ n ρ 1	∼ w n ∼ ϕ nx L 2 (0,1)
	+	k 0 lk	(w ∼ ϕ nx L 2 (0,1)	-	λ 3 n lk	iρ 1	∼ w nx , iϕ n	L 2 (0,1)
	+	δ lk	λ 2 n θ n k L 2 (0,1) ∼ 2 i ϕ nx ϕ n L 2 (0,1) + k 0 k i λ n ϕ nx 2 L 2 (0,1) -→ 0.
	Taking the inner product of (5.6) 2 with	∼ ϕ n n λ 7	in L 2 (0, 1), we get
	ρ 1 i λ n	∼ ϕ n	2 L 2 (0,1)
									∼ ϕ nx L 2 (0,1)

∼ w n ∼ ϕ n L 2 (0,1)

L 2 (0,1)


Abderrahmane Youkana
email: abder.youkana@yahoo.fr

Salim A Messaoudi
email: messaoud@kfupm.edu.sa

Aissa Guesmia
email: aissa.guesmia@univ-lorraine.fr

A GENERAL DECAY AND OPTIMAL DECAY RESULT IN A HEAT SYSTEM WITH A VISCOELASTIC TERM *

Keywords:

We consider a quasilinear heat system in the presence of an integral term and establish a general and optimal decay result from which improves and generalizes several stability results in the literature.

Introduction

In this work, we consider the following problem:

           A(t)|u t | m-2 u t -∆u + t 0 g(t -s)∆u(x, s)ds = 0 Ω × (0, +∞), u(x, t) = 0 ∂Ω × R + , u(x, 0) = u 0 (x) Ω, (1.1)
where m ≥ 2, Ω is a bounded domain of R n , n ∈ N * := {1, 2, • • • }, with a smooth boundary ∂Ω, g : R + → R + is a positive nonincreasing function, and

A : R + → M n (R)
is a bounded square matrix satisfying A ∈ C(R + ) and, for some positive constant c 0 ,

(A(t)v, v) ≥ c 0 |v| 2 , ∀t ∈ R + , ∀v ∈ R n , (1.2)
where (•, •) and | • | are the inner product and the norm, respectively, in R n . The equation in consideration arises from various mathematical models in engineering and physics. For instance, in the study of heat conduction in materials with memory, the classical Fourier law is replaced by the following form (cf. [START_REF] Nohel | Nonlinear Volterra equations for the heat flow in materials with memory//Integral and Functional Differential Equations[END_REF]):

q = -d∇u - t -∞ ∇ (k(x, t)u(x, τ )) dτ,
where u is the temperature, d the diffusion coefficient and the integral term represents the memory effect in the material. This type of problems has considered by a number of researchers; see [START_REF] Da Prato | Existence and regularity for a class of integro-differential equations of parabolic type[END_REF][START_REF] Nohel | Nonlinear Volterra equations for the heat flow in materials with memory//Integral and Functional Differential Equations[END_REF][START_REF] Yin | On parabolic Volterra equations in several space dimensions[END_REF] and the references therein. From a mathematical point of view, we expect that the integral term would be dominated by the leading term in the equation, so that the theory of parabolic equation can be applied. In fact, this has been confirmed by the work of Yin [START_REF] Yin | On parabolic Volterra equations in several space dimensions[END_REF] , in which he considered a general equation of the form

u t = divA(x, t, u, u x ) + a(x, t, u, u x ) + t 0 divB(x, t, τ, u, u x )dτ
and proved the existence of a unique weak solution under suitable conditions on A, B and a.

See more results concerning global existence and asymptotic behavior in Nakao and Ohara [START_REF] Nakao | Gradient estimates for a quasilinear parabolic equation of the mean curvature type[END_REF], Nakao and Chen [START_REF] Nakao | Global existence and gradient estimates for the quasilinear parabolic equations of m-Laplacian type with a nonlinear convection term[END_REF], and Engler et al. [START_REF] Englern | Gradient estimates for solutions of parabolic equations and systems[END_REF]. Pucci and Serrin [START_REF] Pucci | Asymptotic stability for nonlinear parabolic systems[END_REF] discussed the following system:

A(t)|u t | m-2 u t = ∆u -f (x, u),
for m > 1 and f satisfying (f (x, u), u) ≥ 0 and showed that strong solutions tend to the rest state as t → +∞, however, no rate of decay has been given. Berrimi and Messaoudi [START_REF] Berrimi | A decay result for a quasilinear parabolic system[END_REF] showed that, if A satisfies (1.2), then solutions with small initial energy decay exponentially for m = 2 and polynomially if m > 2. Messaoudi and Tellab [START_REF] Messaoudi | A general decay result in a quasilinear parabolic system with viscoelastic term[END_REF] considered (1.1), under condition (1.2) and for relaxation function g satisfying a general decay condition of the form

g ′ (t) ≤ -ξ(t)g(t), ∀t ∈ R + ,
for some nonincreasing differentiable function ξ : R + → R + , and established a general decay result, from which the exponential and polynomial decay rates of [START_REF] Berrimi | A decay result for a quasilinear parabolic system[END_REF] are only special cases. Recently, Liu and Chen [4] investigated (1.1), with a nonlinear source term, and established a general decay result under suitable conditions on g and the nonlinear source term. They also proved a blow-up result for the solution with both positive and negative initial energy.

In this work, we discuss (1.1) when g is of a more general decay, and establish a general and optimal decay result, which improves those of Berrimi and Messaoudi [START_REF] Berrimi | A decay result for a quasilinear parabolic system[END_REF], Liu and Chen [START_REF] Liu | Global and blow-up of solutions for a quasilinear parabolic system with viscoelastic and source terms[END_REF], and Messaoudi and Tellab [START_REF] Messaoudi | A general decay result in a quasilinear parabolic system with viscoelastic term[END_REF].

Preliminaries

In this section, we present some material needed in the proof of our result. For the relaxation function g we assume that (G1) The function g : R + → R + is a differentiable function satisfying g(0) > 0 and 1 -+∞ 0 g(s)ds = l > 0.

(G2) There exist a constant p ∈ [1, 3/2) and a nonincreasing differentiable function ξ :

R + → R + satisfying g ′ (t) ≤ -ξ(t)g p (t), ∀t ∈ R + .
(G3) We also assume that

2 ≤ m ≤ 2n n -2 if n ≥ 3, m ≥ 2 if n = 1, 2.
Remark 2.1 There are many functions satisfying (G1) and (G2). Examples of such functions are, for b > 0, α > 0, ν > 1, and a > 0 small enough,

g 1 (t) = ae -b(t+1) α and g 2 (t) = a (1 + t) ν .
We will also be using the embedding H 1 0 (Ω) ֒→ L q (Ω), L r (Ω) ֒→ L q (Ω), for 2 ≤ q ≤ r < +∞, and Poincaré's inequality. The same embedding constant C * will be used, and C denotes a generic positive constant.

We introduce the following:

E(t) = 1 2 (g • ∇u)(t) + 1 2 1 - t 0 g(s)ds ||∇u(t)|| 2 2 , ∀t ∈ R + , (2.1)
where || • || q = || • || (L q (Ω)) n , for 1 ≤ q < +∞, and

(g • ∇u)(t) = t 0 g(t -τ )||∇u(•, t) -∇u(•, τ )|| 2 2 dτ, ∀t ∈ R + . (2.2)
Similarly to [START_REF] Pucci | Asymptotic stability for nonlinear parabolic systems[END_REF], we give the definition of a strong solution of (1.1).

Definition 2.2 A strong solution of (1.1) on [0, T ] is a function u ∈ C [0, T ); (H 1 0 (Ω)) n ∩ C 1 ((0, T ); (L m (Ω)) n ) which satisfies t 0 Ω ∇u(x, s) - s 0 g(s -τ )∇u(x, τ )dτ • ∇φ(x, s)dxds + t 0 A(s)|u t | m-2 u t (x, s) • φ(x, s)dxds = 0, for all t in [0, T ) and all φ in C [0, T ); (H 1 0 (Ω)) n . Remark 2.3
Similarly to [START_REF] Pucci | Asymptotic stability for nonlinear parabolic systems[END_REF], we assume the existence of a solution. For the linear case (m = 2), one can easily establish the existence of a weak solution by the Galerkin method. In the one-dimensional case (n = 1), the existence is established in a more general setting by Yin [START_REF] Yin | On parabolic Volterra equations in several space dimensions[END_REF] .

Finally, we state an important lemma [START_REF] Messaoudi | General and optimal decay for a quasilinear viscoelastic equation[END_REF].

Lemma 2.4 Assume that g satisfies (G1) and (G2) and u is the solution of (1.1), then there exists a positive constant k 0 such that

ξ(t)(g • ∇u)(t) ≤ k 0 (-E ′ (t)) 1 2p-1 , ∀t ∈ R + , (2.3)
where E ′ is given in (3.1) below.

We also recall the following particular case of the well-known Jensen inequality which will be of essential use in obtaining our result: let f : Ω → R + and h : Ω → R + be integrable functions on Ω such that

Ω h(x)dx = k > 0.
Then, for any p > 1, we have

1 k Ω (f (x)) 1 p h(x)dx ≤ 1 k Ω f (x)h(x)dx 1 p
.

(2.4)

Decay Result

In this section, we state and prove our main result. We start with a lemma.

Lemma 3.1 Let u be the solution of (1.1). Then the energy satisfies

E ′ (t) = - Ω A(t)|u t | m dx - 1 2 g(t)||∇u(t)|| 2 2 + 1 2 (g ′ • ∇u)(t) ≤ 0, ∀t ∈ R + . (3.1)
Proof By multiplying the first equation in (1.1) by u t , integrating over Ω we get (3.1), after routine manipulations. Lemma 3.2 Let u be a solution of problem (1.1). Then, for any δ > 0, we have

∇u(t) 2 2 ≤ c 4 δE(t) - C δ c 0 E ′ (t) + c 5 (g • ∇u)(t), ∀t ∈ R + , (3.2)
where c 0 is introduced in (1.2), c 4 and c 5 are two positive constants, and C δ is a positive constant depending on δ.

Proof Multiplying the first equation in (1.1) by u and integrating over Ω, we get

∇u(t) 2 2 = - Ω A(t)|u t | m-2 u t u(x, t)dx + Ω t 0 g(t -s)∇u(x, s) • ∇u(x, t)dsdx. (3.3)
Now, we estimate the right-hand side of (3.3). By using Young's and Poincaré's inequalities, the boundedness of A, conditions (G1) and (G3), and the fact that

E(t) ≤ E(0),
we find, for any δ > 0,

- Ω A(t)|u t | m-2 u t udx ≤ δ u(•, t) m m + C δ u t (•, t) m m ≤ δC m * ∇u(•, t) m 2 + C δ u t (•, t) m m ≤ δC m * 2E(0) l m-2 2 2 l E(t) + C δ u t (•, t) m m .
On the other hand, using condition (1.2) and identity (3.1), we have

u t (•, t) m m = Ω |u t | m 2 2 dx ≤ 1 c 0 Ω A(t)|u t | m dx ≤ -1 c 0 E ′ (t).
By combining the above two inequalities, we conclude that

- Ω A(t)|u t | m-2 u t udx ≤ c 1 δE(t) - C δ c 0 E ′ (t). ( 3
≤ (1 + η) Ω t 0 g(t -s)|∇u(t)|ds 2 dx + 1 + 1 η Ω t 0 g(t -s)|∇u(s) -∇u(t)|ds 2 dx ≤ (1 + η)(1 -l) 2 ∇u(•, t) 2 2 + 1 + 1 η (1 -l)(g • ∇u)(t). (3.6)
Substuting (3.6) in (3.5), we get

Ω ∇u(x, t) t 0 g(t -s)∇u(x, s)dsdx ≤ 1 2 1 + (1 + η)(1 -l) 2 ∇u(•, t) 2 2 + 1 2 1 + 1 η (1 -l)(g • ∇u)(t). (3.7)
Combining (3.3), (3.4) and (3.7), we find

∇u(•, t) 2 2 ≤ c 1 δE(t) - C δ c 0 E ′ (t) + 1 2 1 + (1 + η)(1 -l) 2 ∇u(•, t) 2 2 + 1 2 1 + 1 η (1 -l)(g • ∇u)(t). (3.8)
We then choose 0

< η < l(2 -l)/(1 -l) 2 , which makes c 2 = 1 2 1 + (1 + η)(1 -l) 2 < 1,
and, therefore, (3.8) takes the form

∇u(•, t) 2 2 ≤ c 1 δE(t) - C δ c 0 E ′ (t) + c 2 ∇u(•, t) 2 2 + c 3 (g • ∇u)(t),
where

c 3 = 1 2 (1 + 1 η )(1 -l).
This yields (3.2) with c 4 = c1 1-c2 and c 5 = c3 1-c2 . Theorem 3.3 Let u be the solution of (1.1). Then, there exist strictly two positive constants λ 0 and λ 1 such that the energy satisfies, for all t ∈ R + ,

E(t) ≤ λ 0 e -λ1 t 0 ξ(s)ds if p = 1,
(3.9)

E(t) ≤ λ 0 1 + t 0 ξ 2p-1 (s)ds -1 2p-2 if p > 1. (3.10) Moreover, if ξ and p in (G 2 ) satisfy +∞ 0 1 + t 0 ξ 2p-1 (s)ds -1 2p-2
dt < +∞, (3.11) then, for all t ∈ R + , Proof From (3.1) and for any κ > 0, we have

E(t) ≤ λ 0 1 + t 0 ξ p (s)ds -1 p-1 if p > 1. ( 3
E ′ (t) ≤ 0 = -κE(t) + κE(t) ≤ -κE(t) + κ 1 2 (g • ∇u)(t) + 1 2 1 - t 0 g(s)ds ∇u(•, t) 2 2 ≤ -κE(t) + κ 2 (g • ∇u)(t) + κ 2 ∇u(•, t) 2 2 . Recalling Lemma 3.2, we get E ′ (t) ≤ -κE(t) + κ 2 (g • ∇u)(t) + κ 2 c 4 δE(t) - C δ c 0 E ′ (t) + c 5 (g • ∇u)(t) ≤ -κ 1 - c 4 2 δ E(t) - κC δ 2c 0 E ′ (t) + κ(1 + c 5 ) 2 (g • ∇u)(t).
Then we have

1 + κC δ 2c 0 E ′ (t) ≤ -κ 1 - c 4 2 δ E(t) + κ(1 + c 5 ) 2 (g • ∇u)(t).
By choosing δ small enough, we obtain, for two positive constants λ and γ,

E ′ (t) ≤ -λE(t) + γ(g • ∇u)(t). (3.14)
Case of p = 1 Multiplying (3.14) by ξ(t) and exploiting (G 2 ), we get

ξ(t)E ′ (t) ≤ -λξ(t)E(t) + γ(ξg • ∇u)(t) ≤ -λξ(t)E(t) -γ(g ′ • ∇u)(t) ≤ -λξ(t)E(t) -γE ′ (t). (3.15)
We then set L = (ξ + γ)E ∼ E to obtain, from (3.15),

L ′ (t) ≤ -λξ(t)E(t) ≤ -λ 1 ξ(t)L(t). (3.16)
A simple integration of (3.16) leads to

L(t) ≤ Ce -λ1 t 0 ξ(s)ds .
This gives (3.9), by virtue of L ∼ E.

Case of p > 1 To establish (3.10), we again consider (3.14) and use Lemma 2.1 to get

ξ(t)E ′ (t) ≤ -λξ(t)E(t) + C (-E ′ (t)) 1 2p-1 .
Multiplication of the last inequality by ξ α E α (t), where α = 2p -2 > 0, gives

1 α + 1 ξ α+1 d dt E α+1 (t) ≤ -λξ α+1 (t)E α+1 (t) + c (ξE) α (t) (-E ′ (t)) 1 α+1 .
Use of Young's inequality, with q = α + 1 and q * = α+1 α , yields, for any ε > 0,

1 α + 1 ξ α+1 d dt E α+1 (t) ≤ -λξ α+1 (t)E α+1 (t) + C εξ α+1 (t)E α+1 (t) -C ε E ′ (t) = -(λ -εC)ξ α+1 (t)E α+1 (t) -C ε E ′ (t).
We then choose 0 < ε < λ C and recall that ξ ′ ≤ 0, to obtain, for c 6 > 0,

ξ α+1 E α+1 (t) ′ (t) ≤ ξ α+1 d dt E α+1 (t) ≤ -c 6 ξ α+1 (t)E α+1 (t) -CE ′ (t); which implies ξ α+1 E α+1 + CE ′ (t) ≤ -c 6 ξ α+1 (t)E α+1 (t). Let W = ξ α+1 E α+1 + CE ∼ E. Then W ′ (t) ≤ -Cξ α+1 (t)W α+1 (t) = -Cξ 2p-1 (t)W 2p-1 (t).
Integrating over (0, t) and using the fact that W ∼ E, we obtain, for some λ 0 > 0,

E(t) ≤ λ 0 t 0 ξ 2p-1 (s)ds + 1 -1 2p-2
; so (3.10) holds To establish (3.12), we put

η(t) = t 0 ∇u(t) -∇u(t -s)

.

This completes the proof of our main result.

The following examples illustrate our result and show the optimal decay rate in the polynomial case: 2p-2 dt < +∞, and hence, by (3.12), we get

E(t) ≤ C(1 + t) -1 p-1 = C(1 + t) -ν ,
which is the optimal decay.

Example 3.6 Let g(t) = ae -(1+t) ν , where 0 < ν ≤ 1, and a > 0 is chosen so that (3.20) holds. Then g ′ (t) = -aν(1 + t) ν-1 e -(1+t) ν .

Therefore (G2) holds with p = 1 and ξ(t) = ν(1 + t) ν-1 . Consequentely, we can use (3.9) to deduce E(t) ≤ Ce -λ(1+t) ν .

Example 3 . 5

 35 Let g(t) = a(1 + t) -ν , where ν > 2, and a > 0 so that +∞ 0 g(t)dt < 1.(3.20)We haveg ′ (t) = -aν(1 + t) -ν-1 = -b a(1 + t) -ν ν+1 ν, where b = νa -1 ν . Then (G2) holds with ξ(t) = b and p = ν+1 ν ∈

  (3.19) by ξ α (t)E α (t), for α = p -1, and repeating the same computations as in above, we arrive at, for some λ 0 > 0,

	Therefore, using (3.17) we obtain
	p (t) ξ p-1 (0) ξ(t)E ′ (t) ≤ -λξ(t)E(t) + γη 1-1	0	t	ξ(s)g p (s) ∇u(t) -∇u(t -s) 2 2 ds	1 p
	≤ -λξ(t)E(t) + C(-g ′ • ∇u)	1 p (t),
	and then
		ξ(t)E ′ (t) ≤ -λξ(t)E(t) + C(-E ′ (t))	1 p .	(3.19)
	If η(t) = 0, then s → ∇u(s) is a constant function on [0, t]. Therefore
							(g • ∇u)(t) = 0,
	and hence we have, from (3.14),
							E ′ (t) ≤ -λE(t),
	which implies (3.19).
	Now, multiplying E(t) ≤ λ 0			t	ξ p (s)ds + 1	-1 p-1
									0
										2 2 ds.
	Using Remark 3.1, we have
			t
	η(t) ≤ 2	0	∇u(t) 2 2 + ∇u(t -s) 2 2 ds
	≤	4 1 -l	0	t	(E(t) + E(t -s)) ds
	=	8 1 -l	0	t	E(s)ds <	8 1 -l	0	+∞	E(s)ds < +∞.
	This implies that
						sup	η 1-1 p (t) < +∞.	(3.17)
					t∈R +
	Assume that η(t) > 0. Then, from (3.14), we find
	ξ(t)E ′ (t) ≤ -λξ(t)E(t) + γξ(t)(g • ∇u)(t)
	= -λξ(t)E(t) + γ	η(t) η(t)	0	t	(ξ p (s)g p (s))	1 p	∇u(t) -∇u(t -s) 2 2 ds.	(3.18)
	Applying Jensen's inequality (2.4) for the second term of the right-hand side of (3.18), with
	Ω = [0, t], f (s) = ξ p (s)g p (s) and h(s) = ∇u(t) -∇u(t -s) 2 2 ,
	to get
	ξ(t)E ′ (t) ≤ -λξ(t)E(t) + γη(t)	1 η(t)	0	t	ξ p (s)g p (s) ∇u(t) -∇u(t -s) 2 2 ds	1 p	.

Acknowledgements The authors thank KFUPM and Lorraine-Metz uinversity for their continuous support. This work has been finalized during the visit of the third author to KFUPM in December 2016 and during the scholarship of the first author in Lorraine-Metz in 2016-2017.

This work has been partially funded by KFUPM under Project # IN161006.