Rayan Nasser
email: rayan.nasser94@hotmail.com

) in the n-dimensional case, where n ≥ 1. First, they showed that the system is strongly stable. Next, they showed that the system is not uniformly stable. Hence, they established a polynomial energy decay rate of type t -2 3 . But this result is not optimal. Consequently, our main interest in this

The set of non negative real numbers. R *

The set of non zero real numbers.

N

The set of natural numbers. N *

The set of non zero natural numbers.

Z

The set of integer numbers. Z *

The set of non zero integer numbers.

Q

The set of rational numbers. Q +

The set of non negative rational numbers. Q *

The set of non zero rational numbers.

C

The set of complex numbers. i

The imaginary unit.

ℜ

The real part.

Im

The imaginary part.

L p

The Lebesgue space.

H m

The sobolev space. C 0

The space of continuous function. C 1 the space of continuously differentiable functions.

E ′

The dual space.

(•, •) E ′ ,E
The scalar product in the duality E ′ .

|•|

The modulus.

•

The norm. max

The maximum. min

The minimum. sup

The supreme. inf

The infimum.

f x = ∂ x f
The partial derivative of f with respect of x. f xx = ∂ xx f The second partial derivative of f with respect of x. g ′ (s)

The derivative of g with respect to s.

Asymptotic Notation

Let f and g be two functions defined on an interval (x 0 , ∞) of the real line. We use the following notations:

• f (x) = O (g(x)) as x → ∞ to state that there exist a positive number M and a real number

x 1 > x 0 such that | f (x)| ≤ M |g (x)| ∀ x ≥ x 1 .
• f (x) = o (g(x)) as x → ∞ to state that lim x→∞ f (x) g(x) = 0.

• f ∼ g as x → ∞ to state that lim x→∞ f (x) g(x) = 1.

Introduction

Control theory deals with the behavior of dynamical systems with inputs, and how their behavior is modified when adding a feedback. So, the objective of control theory is to control a system. To do this, a controller; often called a damping or a feedback, is designed and activated on a given output. In fact, the field of control theory in PDEs has been extended as more practical models have been presented and investigated. It is beneficial in many areas, such as engineering, biology, computer science and biomedical research. On the other hand, numerical analysis is considered to be an essential component of computational science. It is utilized to develop and implement numerical methods for solving mathematical problems that arise in science and engineering. In this thesis, we depend on the semigroup theory combined with numerical analysis in order to study the stabilization of some wave equations.

The following thesis is divided into five chapters.

Taking into consideration that the majority of studies in this thesis are based on the semigroup theory, in Chapter 1, we will state some well known results on the semigroup; especially the results and theorems related to the strong, exponential and polynomial stability of a C 0 -semigroup. Also, we will recall some geometric conditions used in our work. Finally, we will present some theorems attached to the numerical part, and will be used to prove our main results.

In Chapter 2, we consider the stabilization of an elastic wave equation subjected to an internal viscoelastic damping of Kelvin-Voigt type in the multidimensional case; with or without geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. Let Ω ⊂ R N be a nonempty bounded open set with Lipschitz boundary Γ. We consider the wave equation with locally distributed Kelvin-Voigt type damping given in the following equation:

     ρ(x)u tt (x,t) -div(a(x)∇u + b(x)∇u t ) = 0 in Ω × R + , u(x,t) = 0 on Γ × R + ,
(u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x))

in Ω, (Wave Equation 1)

where the coefficient functions ρ, a, b ∈ L ∞ (Ω) and we assume that ρ(x) ≥ ρ 0 > 0, a(x) ≥ a 0 > 0, b(x) ≥ 0, ∀ x ∈ Ω.

shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system (see [START_REF] Zhang | On the lack of exponential stability for an elasticâ Ȃ Şviscoelastic waves interaction system[END_REF]). So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. However, to our knowledge, several important geometric situations are not covered by previous works. For example, in the case where the damped region {b > 0} satisfies or does not satisfy the GCC condition, the problem of the energy decay rate is still open. So, we are interested in solving this open problem. In this chapter, we consider the stabilization of the wave equation with viscoelastic damping of Kelvin-Voigt type localized via non smooth coefficient in a bounded domain Ω ⊂ R N . The energy of a solution u of System (Wave Equation 1) is given by

E(u,t) = 1 2 Ω ρ(x)|u t | 2 + a(x)|∇u| 2 dx.
and a direct calculation gives

d dt E(u,t) = - Ω b(x)|∇u t | 2 dx ≤ 0
as b is nonnegative, which implies that the system is dissipative in the sense that its energy is decreasing with respect to time t.

We proceed by introducing the Hilbert energy space H by

H = H 1 0 (Ω) × L 2 (Ω),
equipped with the usual inner product

U, Ũ H = Ω (a∇u • ∇ ũ + ρv ṽ)dx,
where U = (u, v) ∈ H and Ũ = ( ũ, ṽ) ∈ H . We use U H to denote the corresponding norm. We next define the linear unbounded operator A : D(A ) ⊂ H -→ H by

D(A ) = {(u, v) ∈ H | v ∈ H 1 0 (Ω), div(a∇u + b∇v) ∈ L 2 (Ω)} and A (u, v) = v, 1 ρ div(a∇u + b∇v) , ∀(u, v) ∈ D(A ).
If (u, u t ) is a regular solution of System (Wave Equation 1), then we transform this system into the following evolution equation

U t = A U, U(0) = U 0 ,
where U 0 = (u 0 , u 1 ) ∈ H . According to [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF], the operator A is m-dissipative in the energy space H . Therefore, thanks to Lumer-Philips theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that A generates a C 0 -semigroup of contractions e tA in H and therefore the Problem (Wave Equation 1) is well-posed.

In Subsection 2.2, we study the strong stability of System (Wave Equation 1) in the sense that its energy E(u,t) converges to zero when t goes to infinity for all initial data in H . For this aim, we assume that there exist a nonempty open set ω ⊂ Ω and b 0 > 0 such that b(x) ≥ b 0 ∀x ∈ ω.

(LA)

We use a general criteria of Arendt-Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF] to show that the C 0 -semigroup e tA associated to System (Wave Equation 1) is strongly stable in the absence of compactness of the resolvent of A .

In Section 2.3, we move to our main interest which falls under studying the energy decay rate of System (Wave Equation 1). Assuming that b ≥ b 0 > 0 in a nonempty open subset ω of Ω, we establish first a polynomial energy decay estimate of type t -1 for smooth initial data provided that the damping region ω satisfies the Geometric Control Condition GCC (see Definition 1.4.3) and meas(ω ∪ Γ) > 0, or ω satisfies Strictly Geometric Control Condition SGCC (see Definition 2.3.1). To prove Theorem (2.3.4) in this section, we use the result of Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] , [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF]).

However, in Section 2.4, in the case where ω does not satisfy the GCC condition, i.e. in the presence of trapped rays that do not meet the damped region ω, we focus on the 2-dimensional square and we prove that the energy of smooth initial data decays polynomially. Also, we refer to the result of Borichev and Tomilov to prove the result in this section (see Theorem 2.4.1). Eventually, we finalize our work by applying Theorem 2.4.1 on a square where the damping is localized in a vertical strip.

In fact, the frequency domain approach and new multiplier technics are employed to prove the main results in this chapter. Finally, we would like to note that the geometric situations covered by the results in this chapter are richer than that considered in all previous literature (see for instance [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF], [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF], [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF]) and include in particular an example where the damping region is faraway from the boundary and an example where the damping region does not satisfy the GCC condition.

The results of this chapter are published in "MCRF" Journal in an article titled "Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions". (see [START_REF] Wehbe | Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions[END_REF])

In Chapter 3, we study the stabilization of a 1-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin-Voigt type in a bounded domain. Unlike [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF][START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF], we consider only one damping Kelvin-Voigt mechanism acting in the internal of the body. The system is given by the following:

          
ρ U tt (x,t) -κ U x (x,t) + δ χ (α,β ) U xt (x,t) x = 0, (x,t) ∈ (0, L) × (0 + ∞), U(0,t) = U(L,t) = 0, t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), (Wave Equation 2)

where δ > 0, 0 < α < β < L, U = uχ (0,α) + vχ (α,β ) + wχ (β ,L) and χ (a,b) is the characteristic function of the interval (a, b). Here, ρ and κ are discontinuous positive functions, i.e., assume that there exist positive constant numbers ρ 1 , ρ 2 , ρ 3 , κ 1 , κ 2 and κ 3 such that

ρ =        ρ 1 , x ∈ (0, α), ρ 2 , x ∈ (α, β ), ρ 3 , x ∈ (β , L),
and κ =        κ 1 , x ∈ (0, α), κ 2 , x ∈ (α, β ), κ 3 , x ∈ (β , L).
chapter is to study the optimal energy decay rate of the transmission problem of wave equations with locally internal Kelvin-Voigt type damping acting via non smooth coefficient at the interface, i.e., when the damping is in the middle of the material without any dissipation in the other parts.

In Section 3.2, we formulate System (Wave Equation 2) as an abstract Cauchy problem. System (Wave Equation 2) is equivalent to the following system

                                                
ρ 1 u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞),

ρ 2 v tt -(κ 2 v x + δ v xt ) x = 0, (x,t) ∈ (α, β ) × (0, +∞),
ρ 3 w tt -κ 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞), u(0,t) = w(L,t) = 0, t ∈ (0, +∞), u(α,t) = v(α,t), v(β ,t) = w(β ,t), t ∈ (0, +∞), κ 2 v x (α,t) + δ v xt (α,t) = κ 1 u x (α,t), t ∈ (0, +∞), κ 2 v x (β ,t) + δ v xt (β ,t) = κ 3 w x (β ,t), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ (0, α),

(v(x, 0), v t (x, 0)) = (v 0 (x), v 1 (x)) , x ∈ (α, β ), (w(x
, 0), w t (x, 0)) = (w 0 (x), w 1 (x)) , x ∈ (β , L).

(Transmission Problem 1)

The energy of solutions of the above system is defined by:

E(t) = κ 1 2 α 0 |u x | 2 dx + ρ 1 2 α 0 |u t | 2 dx + κ 2 2 β α |v x | 2 dx + ρ 2 2 β α |v t | 2 dx + κ 3 2 L β |w x | 2 dx + ρ 3 2 L β |w t | 2 dx.
A straight forward computation gives

d dt E (t) = -δ β α |v xt | 2 dx ≤ 0.
Thus, System (Transmission Problem 1) is dissipative in the sense that its energy is non increasing with respect to the time t.

Let us define

H m = H m (0, α) × H m (α, β ) × H m (β , L), m = 1, 2, L 2 = L 2 (0, α) × L 2 (α, β ) × L 2 (β , L), H 1 L = {(u, v, w) ∈ H 1 | u(0) = w(L) = 0, u(α) = v(α), v(β ) = w(β )}.
The Hilbert energy space is given by where U = (u, v, w, y, z, φ ) ∈ H and Ũ = ( ũ, ṽ, w, ỹ, z, φ ) ∈ H . We use U H to denote the corresponding norm. We define the linear unbounded operator A : D(A ) ⊂ H -→ H by Introduction and for all U = (u, v, w, y, z, φ ) ∈ D(A),

H = H 1 L × L
A U = y, z, φ , κ 1 ρ 1 u xx , 1 ρ 2 (κ 2 v x + δ z x ) x , κ 3 ρ 3 w xx .
If U = (u, v, w, u t , v t , w t ) is a regular solution of System (Transmission Problem 1), then we transform this system into the following initial value problem

U t = A U, U(0) = U 0 ,
where U 0 = (u 0 , v 0 , w 0 , u 1 , v 1 , w 1 ) ∈ H . Similar to [START_REF] Muní | Transmission Problems in (Thermo)Viscoelasticity with Kelvin-Voigt Damping: Nonexponential, Strong, and Polynomial Stability[END_REF], we can prove that the operator A generates a C 0 -semigroup of contractions e tA in H and the energy E(t) of System (Transmission Problem 1) converges to zero as t goes to infinity for all initial data in H . Therefore, Problem (Transmission Problem 1) is well-posed and strongly stable.

In Section 3.3, we prove that System (Transmission Problem 1) is non-uniformly stable and we estimate the upper bound of the polynomial decay rate. According to Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Bátkai | Polynomial stability of operator semigroups[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF]), it is sufficient to find a sequence of (λ n ) n ⊂ R * + with λ n → +∞ and a sequence of vectors (U n ) n ⊂ D (A ) and

(F n ) n ⊂ H such that (iλ n I -A )U n = F n is bounded in H and lim n→∞ λ -1 2 +ε n U n H = ∞.
Finally, in Section 3.4, we establish an optimal polynomial energy decay rate of type t -4 (see Theorem 3.4.1).

The results of this chapter are published in "MMAS" Journal in an article titled "Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface". (see [START_REF] Nasser | Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface[END_REF])

In Chapter 4, we study the stabilization of a wave equation with non smooth localized internal Kelvin-Voigt damping and with boundary or localized internal delay feedback in the one-dimensional case, regarding that the Kelvin-Voigt and the delay damping are both localized via non smooth coefficients.

In the first part of this chapter, we study the stability of elastic wave equation with local Kelvin-Voigt damping, boundary feedback and time delay term at the boundary, i.e., we consider the following system

                  
U tt (x,t) -κ U x (x,t) + δ 1 χ (α,β ) U xt (x,t) x = 0, (x,t) ∈ (0, L) × (0, +∞), U(0,t) = 0, t ∈ (0, +∞), U x (L,t) = -δ 3 U t (L,t) -δ 2 U t (L,t -τ), t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L),

(see Example 3.5) proved that System (Wave with boundary delay) is unstable for an arbitrary small value of τ. Later, when δ 3 = 0 and in the absence of delay (i.e., δ 2 = 0), they proved in [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Ammari | Stability of the wave equations on a tree with local Kelvin-Voigt damping[END_REF] that the energy of System (Wave with boundary delay) is polynomially stable. Indeed, when δ 2 = 0 and δ 3 = 0, the stability of System (Wave with boundary delay) is still an open problem. In this part, we assume that

δ 3 > 1 2κ 3 and |δ 2 | < 1 κ 3 2κ 3 δ 3 -1. (H)
In Section 4.2, like in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the auxiliary unknown η(L, ρ,t) = U t (L,t -ρ τ), ρ ∈ (0, 1), t > 0.

Thus, Problem (Wave with boundary delay) is equivalent to

                        
U tt (x,t) -κ U x (x,t) + δ 1 χ (α,β ) U xt (x,t) x = 0, (x,t) ∈ (0, L) × (0, +∞), τη t (L, ρ,t) + η ρ (L, ρ,t) = 0, (ρ,t) ∈ (0, 1) × (0, +∞), U(0,t) = 0, t ∈ (0, +∞), U x (L,t) = -δ 3 U t (L,t) -δ 2 η(L, 1,t), t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), η(L, ρ, 0) = f 0 (L, -ρ τ), ρ ∈ (0, 1).

(Wave Equation 3)

Under assumption (H), let us define the energy of a solution of System (Wave Equation 3) as

E(t) = 1 2 L 0 |U t | 2 + κ|U x | 2 dx + τ 2 1 0 |η| 2 dρ. (E(t))
Multiplying the first equation of (Wave Equation 3) by U t , integrating over (0, L) with respect to x, then using integration by parts and the boundary conditions in (Wave Equation 3) at x = 0 and at x = L, we get 1 2

d dt L 0 |U t | 2 + κ|U x | 2 dx = -δ 1 β α |U xt | 2 dx -κ 3 δ 3 |U t (L,t)| 2 -κ 3 δ 2 η(L, 1,t)U t (L,t).
Multiplying the second equation of (Wave Equation 3) by η, integrating over (0, 1) with respect to ρ, then using the fact that η(L, 0,t) = U t (L,t), we get

τ 2 d dt 1 0 |η| 2 dρ = - 1 2 |η(L, 1,t)| 2 + 1 2 |U t (L,t)| 2 .
Adding the above two equations, we get

d E(t) dt = -δ 1 β α |U xt | 2 dx -κ 3 δ 3 - 1 2 |U t (L,t)| 2 -κ 3 δ 2 η(L, 1,t)U t (L,t) - 1 2 |η(L, 1,t)| 2 . (E ′ (t))
For all p > 0, we have

-κ 3 δ 2 η(L, 1,t)U t (L,t) ≤ κ 3 |δ 2 | |η(L, 1,t)| 2 2p + κ 3 |δ 2 | p |U t (L,t)| 2 2 .
Inserting the above equation in (E ′ (t)), we get

d E(t) dt ≤ -δ 1 β α |U xt | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1,t)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |U t (L,t)| 2 .
Under assumption (H), we can easily check that there exists a strictly positive number p satisfying

κ 3 |δ 2 | < p < 2 κ 3 |δ 2 | κ 3 δ 3 - 1 2 , (Frame of p) such that 1 2 - κ 3 |δ 2 | 2p > 0 and κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 > 0,
so that the energies of the strong solutions satisfy E ′ (t) ≤ 0. Hence, System (Wave Equation 3) is dissipative in the sense that its energy is non increasing with respect to the time t.

In Subsection 4.2.1, we consider the first case, when the Kelvin-Voigt damping is localized in the internal of the body, i.e., α > 0. In this case, System (Wave Equation 3) is equivalent to the following system

                                                                   u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞), v tt -(κ 2 v x + δ 1 v xt ) x = 0, (x,t) ∈ (α, β ) × (0, +∞),
w tt -κ 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞), τη t (L, ρ,t) + η ρ (L, ρ,t) = 0, (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u(α,t) = v(α,t), v(β ,t) = w(β ,t), t ∈ (0, +∞), w x (L,t) = -δ 3 w t (L,t) -δ 2 η(L, 1,t), t ∈ (0, +∞), κ 2 v x (α,t) + δ 1 v xt (α,t) = κ 1 u x (α,t), t ∈ (0, +∞), κ 2 v x (β ,t) + δ 1 v xt (β ,t) = κ 3 w x (β ,t), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, α), (v(x, 0), v t (x, 0)) = (v 0 (x), v 1 (x)), x ∈ (α, β ), (w(x, 0), w t (x, 0)) = (w 0 (x), w 1 (x)), x ∈ (β , L), η(L, ρ, 0) = f 0 (L, -ρ τ), ρ ∈ (0, 1), (Transmission Problem 2) where the initial data (u 0 , u 1 , v 0 , v 1 , w 0 , w 1 , f 0 ) belongs to a suitable Hilbert space. So, using (E(t)), the energy of System (Transmission Problem 2) is given by

E(t) = 1 2 α 0 |u t | 2 + κ 1 |u x | 2 dx + 1 2 β α |v t | 2 + κ 2 |v x | 2 dx + 1 2 L β |w t | 2 + κ 3 |w x | 2 dx + τ 2 1 0 |η| 2 dρ.
Similarly,

d E(t) dt = -δ 1 β α |v xt | 2 dx - 1 2 |η(L, 1,t)| 2 -κ 3 δ 2 η(L, 1,t)w t (L,t) -κ 3 δ 3 - 1 2 |w t (L,t)| 2 , ≤ -δ 1 β α |v xt | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1,t)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |w t (L,t)| 2 ,
where p is defined in (Frame of p). Thus, under hypothesis (H), System (Transmission Problem 2) is dissipative in the sense that its energy is non increasing with respect to the time t.

Introduction

First, we formulate System (Transmission Problem 2) as an abstract Cauchy problem. For this aim, let us define

H m = H m (0, α) × H m (α, β ) × H m (β , L), m = 1, 2, L 2 = L 2 (0, α) × L 2 (α, β ) × L 2 (β , L), H 1 L = {(u, v, w) ∈ H 1 | u(0) = 0, u(α) = v(α), v(β ) = w(β )}.
The Hilbert space L 2 is equipped with the norm:

(u, v, w) 2 L 2 = α 0 |u| 2 dx + β α |v| 2 dx + L β |w| 2 dx.
Also, it is easy to check that the space H 1 L is Hilbert space over C equipped with the norm:

(u, v, w) 2 H 1 L = κ 1 α 0 |u x | 2 dx + κ 2 β α |v x | 2 dx + κ 3 L β |w x | 2 dx.
Moreover, by Poincaré inequality we can easily verify that there exists C > 0 depending on κ 1 , κ 2 , κ 3 , α, β and L, such that

(u, v, w) L 2 ≤ C (u, v, w) H 1 L , ∀(u, v, w) ∈ H 1 L .
We now define the Hilbert energy space H by where U = (u, v, w, y, z, φ , η(L, •)) ∈ H and Ũ = ( ũ, ṽ, w, ỹ, z, φ , η(L, •)) ∈ H . We use U H to denote the corresponding norm. We define the linear unbounded operator A : D(A ) ⊂ H -→ H by: to show that the C 0 -semigroup e tA associated to System (Transmission Problem 2) is strongly stable in the absence of compactness of the resolvent of A . Finally, using frequency domain approach combined with piecewise multiplier techniques, we establish a polynomial energy decay rate of type t -4 (see Theorem 4.2.7).

H = H 1 L × L 2 × L 2 (0,
However, in Subsection 4.2.2, we consider the second case where α = 0, i.e., when the Kelvin-Voigt damping is distributed near the boundary and the time delay feedback is effective at L. In fact, we can easily adapt the proof in Subsection 4.2.1 to show the well-posedness and strong stability of System (Wave Equation 3) with α = 0. Eventually, we prove that this system is exponentially stable (see Theorem 4.2.14) using the result obtained by Huang and Prüss in [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF].

In the second part of this chapter (Section 4.3), we study the stability of the elastic wave equation with local KelvinVoigt damping and local internal time delay. This system takes the following form

            
U tt (x,t) -[κ U x (x,t) + a(x) (δ 1 U xt (x,t) + δ 2 U xt (x,t -τ))] x = 0, (x,t) ∈ (0, L) × (0, +∞), U(0,t) = U(L,t) = 0, t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), U t (x,t) = f 0 (x,t), (x,t) ∈ (0, L) × (-τ, 0), (Wave with internal delay)

where L, τ and δ 1 are strictly positive constant numbers, δ 2 is a non zero real number and the initial data (U 0 ,U 1 , f 0 ) belongs to a suitable space. Here 0 < α < L, a(x) = χ (α,L) and U = uχ (0,α) + vχ (α,L) , whereχ (a,b) is the characteristic function of the interval (a, b). We assume that there exist strictly positive constant numbers κ 1 , κ 2 , such that κ = κ 1 χ (0,α) + κ 2 χ (α,L) . In fact, here we will divide the bar into 2 pieces; the first piece is an elastic part, while in the second piece the KelvinVoigt damping and the time delay are effective. So, the KelvinVoigt damping and the time delay are effective on (α, L). Regarding System (Wave with internal delay), in the absence of delay (i.e., δ 2 = 0), they proved in [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF][START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF] that the energy of this system is polynomially stable. Indeed, when δ 2 = 0, the stability of System (Wave with internal delay) is still an open problem. In this part, we assume that

|δ 2 | < δ 1 . (H1)
Simalrly, like in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the auxiliary unknown η(x, ρ,t) = v t (x,t -ρ τ), x ∈ (α, L), ρ ∈ (0, 1), t > 0.

In this case, System (Wave with internal delay) is equivalent to the following system

                                            
u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞), v tt -(κ 2 v x + δ 1 v xt (x,t) + δ 2 η x (x, 1,t)) x = 0, (x,t) ∈ (α, L) × (0, +∞), τη t (x, ρ,t) + η ρ (x, ρ,t) = 0, (x, ρ,t) ∈ (α, L) × (0, 1) × (0, +∞), u(0,t) = v(L,t) = η(L, ρ,t) = 0, t ∈ (0, +∞), ρ ∈ (0, 1), u(α,t) = v(α,t), t ∈ (0, +∞),

Under assumption (H1), let us define the energy of a solution of System (Transmission Problem 3) as

E(t) = 1 2 α 0 |u t (x,t)| 2 + κ 1 |u x (x,t)| 2 dx + 1 2 L α |v t (x,t)| 2 + κ 2 |v x (x,t)| 2 dx + τ|δ 2 | 2 L α 1 0 |η x (x, ρ,t)| 2 dρ dx.
Multiplying the first and second equations of System (Transmission Problem 3) and the derivative with respect to x of the third equation of (Transmission Problem 3) by u t , y t and |δ 2 |η x , integrating over (0, α), (α, L) and (α, L) × (0, 1) respectively, taking the sum, then using integration by parts and the boundary conditions in (Transmission Problem 3), we get

E ′ (t) = -δ 1 + |δ 2 | 2 L α |v xt (x,t)| 2 dx - |δ 2 | 2 L α |η x (x, 1,t)| 2 dx -δ 2 L α v xt (x,t) η x (x, 1,t)dx.
Using Young's inequality for the third term in the right, we get

E ′ (t) ≤ (-δ 1 + |δ 2 |) L α |v xt (x,t)| 2 dx.
Under assumption (H1), the energies of the strong solutions satisfy E ′ (t) ≤ 0. Hence, System (Transmission Problem 3) is dissipative in the sense that its energy is non increasing with respect to the time t.

In Subsection 4.3.1, we rewrite System (Transmission Problem 3) as an abstract Cauchy problem. For this aim, let us define

H 1 R (α, L) = v ∈ H 1 (α, L) | v(L) = 0 , L 2 = L 2 (0, α) × L 2 (α, L), H 2 = H 2 (0, α) × H 2 (α, L), H 1 = {(u, v) ∈ H 1 (0, α) × H 1 (α, L) | u(0) = 0, u(α) = v(α), v(L) = 0}.
The spaces L 2 , H 1 and H 1 R (α, L) are obviously Hilbert spaces over C equipped respectively with the norms

(u, v) 2 L 2 = α 0 |u| 2 dx + L α |v| 2 dx, (u, v) 2 H 1 = κ 1 α 0 |u x | 2 dx + κ 2 L α |v x | 2 dx and v 2 H 1 R (α,L) = L α |v x | 2 dx.
In addition, by Poincaré inequality, we can easily verify that there exist C 1 > 0 and C 2 > 0 depending on κ 1 , κ 2 , α and L, such that

(u, v) L 2 ≤ C 1 (u, v) H 1 and L α |w| 2 dx ≤ C 2 w 2 H 1 R (α,L) , ∀(u, v) ∈ H 1 , w ∈ H 1 R (α, L).
Let us define the energy Hilbert space H 2 by

H 2 = H 1 × L 2 × L 2 (0, 1), H 1 R (α, L)
equipped with the following inner product

U, Ũ H 2 = κ 1 α 0 u x ũx dx + κ 2 L α v x ṽx dx + α 0 y ỹdx + L α zzdx + τ|δ 2 | L where U = (u, v, y, z, η(•, •)) ∈ H 2 and Ũ = ( ũ, ṽ, ỹ, z, η(•, •)) ∈ H 2 .
We use U H 2 to denote the corresponding norm. We define the linear unbounded operator A 2 : D(A 2 ) ⊂ H 2 -→ H 2 by:

D(A 2 ) = (u, v, y, z, η(•, •)) ∈ H 2 | (y, z) H 1 , (u, κ 2 v + δ 1 z + δ 2 η(•, 1)) ∈ H 2 , κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1) = κ 1 u x (α), η, η ρ ∈ L 2 (0, 1), H 1 R (α, L) , η(•, 0) = z(•)
and for all

U = (u, v, y, z, η(•, •)) ∈ D(A 2 ), A 2 U = y, z, κ 1 u xx , (κ 2 v x + δ 1 z x + δ 2 η x (•, 1)) x , -τ -1 η ρ (•, •) . If U = (u, v, u t , v t , η(•, •)
) is a regular solution of System (Transmission Problem 3), then we transform this system into the following initial value problem

U t = A 2 U, U(0) = U 0 ,
where

U 0 = (u 0 , v 0 , u 1 , v 1 , f 0 (•, -• τ)) ∈ H 2 .
Similarly, we prove that the unbounded linear operator A 2 is m-dissipative in the energy space H 2 under hypothesis (H1) (see Proposition 4.3.1). Thus, thanks to Lumer-Phillips theorem in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], we deduce that A 2 generates a C 0 -semigroup of contractions e tA Finally, in Subsection 4.3.3, when a(x) = 1 ∀x ∈ (0, L) and |δ 2 | ≥ δ 1 , we show that System (Wave with internal delay) is unstable (see Theorem 4.3.8).

The results of this chapter are published in "Asymptotic Analysis" Journal in an article titled "Stability results for an elastic-viscoelastic wave equation with localized Kelvin-Voigt damping and with an internal or boundary time delay". (see [START_REF] Ghader | Stability results for an elasticâ Ȃ Şviscoelastic wave equation with localized Kelvinâ Ȃ ŞVoigt damping and with an internal or boundary time delay[END_REF])

In Chapter 5, we study the numerical solution of the transmission problem of a wave equation with localized Kelvin-Voigt damping acting faraway from the boundary via non smooth coefficient. So, we consider System (Transmission Problem 1) and we construct a numerical scheme based on Finite Volume Method (FVM) in space. In fact, Riečanová et al. [START_REF] Rieä | ą. Study of the numerical solution to the wave equation[END_REF] studied the numerical solution of a wave equation with Dirichlet boundary condition on a rectangular domain, and they obtained the stability estimates and convergence of the numerical scheme. Their method was based on FVM in space together with the average of n + 1 and n -1 time step diffusion. However, in the literature, we realize that there are no results on the convergence and stability estimates, concerning the transmission problem of elastic-viscoelastic systems where there is a discontinuity at the interface, based especially on FVM. The main objective of this chapter is to fill this gap. So, we rewrite

Introduction System (Transmission Problem 1) such that c 2 1 = κ 1 ρ 1 , c 2 2 = κ 2 ρ 2 , c 2 3 = κ 3 ρ 3 , ξ = δ ρ 2 :                                                  u tt -c 2 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞), v tt -c 2 2 v xx -ξ v xxt = 0, (x,t) ∈ (α, β ) × (0, +∞), w tt -c 2 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞), u(0,t) = w(L,t) = 0, t ∈ (0, +∞), u(α,t) = v(α,t), v(β ,t) = w(β ,t), t ∈ (0, +∞), c 2 2 v x (α,t) + ξ v xt (α,t) = c 2 1 u x (α,t), t ∈ (0, +∞), c 2 2 v x (β ,t) + ξ v xt (β ,t) = c 2 3 w x (β ,t), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (ϕ(x), ψ(x)) , x ∈ (0, α), (v(x, 0), v t (x, 0)) = (η(x), ζ (x)) , x ∈ (α, β ), (w(x, 0), w t (x, 0)) = (γ(x), θ (x)) , x ∈ (β , L). (Continuous Problem)
First of all, we introduce the same spaces defined in Chapter 3:

H 1 = H 1 (0, α) × H 1 (α, β ) × H 1 (β , L), L 2 = L 2 (0, α) × L 2 (α, β ) × L 2 (β , L), H 1 L = {(u, v, w) ∈ H 1 | u(0) = w(L) = 0, u(α) = v(α), v(β ) = w(β )}.
We say that (u, v, w) is a weak solution of (Continuous Problem) if for all T > 0, the following conditions hold:

1. (u, v, w) ∈ L 2 ([0, T ]; H 1 L ) and (u t , v t , w t ) ∈ L 2 ([0, T ]; L 2 ), 2. (u(., 0), v(., 0), w(., 0)) = (ϕ(.), η(.), γ(.)), 3.

-T 0 α 0 u t p t dxdt + c 2 1 T 0 α 0 u x p x dxdt -α 0 ψ(x)p(x, 0)dx -T 0 β α v t q t dxdt + c 2 2 T 0 β α v x q x dxdt + ξ T 0 β α v tx q x dxdt -β α ζ (x)q(x, 0)dx -T 0 L β w t z t dxdt + c 2 3 T 0 L β w x z x dxdt -L β θ (x)z(x, 0)dx = 0 for all (p, q, z) ∈ L 2 ([0, T ]; H 1 L ) and (p t , q t , z t ) ∈ L 2 ([0, T ]; L 2
) such that p(T, x) = 0, q(T, x) = 0 and z(T, x) = 0. We set Φ = (φ , η, γ) and Ψ = (ψ, ζ , θ ) and we study our problem under the following hypothesis:

Φ ∈ H 1 L , Ψ ∈ H 1 . (H ′ )
Our system is discretized using an admissible mesh T of the interval (0, L) defined by a family

{K i : i ∈ {1, • • • , N max }; N max ∈ N * } of control volumes such that K i = (x i-1 2 , x i+ 1 2
) and a family

(x i ) i=0,••• ,N max +1 being the center of (K i ) i=1,••• ,N max such that 0 = x 0 = x 1 2 < x 1 < x 3 2 < x 2 < • • • < x i-1 2 < x i < x i+ 1 2 < • • • < x N max < x N max + 1 2 = x N max +1 = L.
We discretize the intervals [0, α], [α, β ] and [β , L] into N α , N and N β points respectively such that N α , N, N β ∈ N * and N max = N α + N + N β . To be clear, let h α = α/N α , h = (β -α)/N and h β = (L -β )/N β and discretize as the following:

We discretize [0, α] such that:

• For i = 0, . . . , N α , x i+ 1 2 = ih α , which yields, x 1 2 = 0 and x N α + 1 2 = α. • For i = 1, . . . , N α , x i = i - 1 2 h α , which yields, x 1 = h α 2 and x N α = α - h α 2 .
We discretize [α, β ] such that:

• For i = N α + 1, . . . , N α + N, x i+ 1 2 = α + (i -N α )h, which yields, x N α + 3 2 = α + h and x N α +N+ 1 2 = β . • For i = N α + 1, . . . , N α + N, x i = α + i -N α - 1 2 h, which yields, x N α +1 = α + h 2 and x N α +N = β - h 2 .
We discretize [β , L] such that:

• For i = N α + N + 1, . . . , N max , x i+ 1 2 = β + (i -N α -N)h β , which yields, x N α +N+ 3 2 = β + h β and x N max + 1 2 = L. • For i = N α + N + 1, . . . , N max , x i = β + i -N α -N - 1 2 h β , which yields, x N α +N+1 = β + h β 2 and x N max = L - h β 2 . Now, we set • h i = x i+ 1 2 -x i-1 2 ; i = 1, . . . , N max , • h i+ 1 2 = x i+1 -x i ; i = 0, . . . , N max , • h i-1 2 = x i -x i-1 ; i = 1, . . . , N max + 1, • h + i = x i+ 1 2 -x i ; i = 1, . . . , N max , • h - i = x i -x i-1 2 ; i = 1, . . . , N max , • size(T ) = max{h i , i = 1, . . . , N max }.
For the discretization in time, we use constant discrete time step ∆t = T N with t n+1t n = ∆t for all n = 0, 1, • • • , N . For the time step ∆t, we set the following condition:

There exists τ 0 ∈ [0, T ] such that ∆t ≤ τ 0 .

(TA)

We designate the discrete unknowns by {u n

i ; i = 1, • • • , N α , n ∈ N}, {v n i ; i = N α + 1, • • • , N α + N, n ∈ N}, {w n i ; i = N α + N + 1, • • • , N max , n ∈ N}
where they stand for the approximation of the mean values of u, v, w over the control volumes K i respectively. However, this data is used to construct the approximations of u, v, w within the cells. In fact, we have:

u i (t) = 1 h i x i + 1 2 x i -1 2 u(x,t)dx, i = 1, . . . , N α , v i (t) = 1 h i x i + 1 2 x i -1 2 v(x,t)dx, i = N α + 1, . . . , N α + N, w i (t) = 1 h i x i + 1 2 x i -1 2 w(x,t)dx, i = N α + N + 1, . . . , N max .
We start the solution with Φ i = (ϕ i , η i , γ i ) regarding that

u 0 i = ϕ i where ϕ i = 1 h i K i ϕ(x)dx, for i = 1, • • • , N α , v 0 i = η i where η i = 1 h i K i η(x)dx, for i = N α + 1, • • • , N α + N, w 0 i = γ i where ϕ i = 1 h i K i γ(x)dx, for i = N α + N + 1, • • • , N max .
For the solution of the first time step; i.e. for n = 0, we use the imaginary time level -1 and we symbolize the values of the approximation functions at this step by:

u -1 i for i = 1, • • • , N α , v -1 i for i = N α + 1, • • • , N α + N, w -1 i for i = N α + N + 1 • • • , N max .
Also, we set

Ψ i = (ψ i , ζ i , θ i ) such that ψ i = 1 h i K i ψ(x)dx, for i = 1, • • • , N α , ζ i = 1 h i K i ζ (x)dx, for i = N α + 1, • • • , N α + N, θ i = 1 h i K i θ (x)dx, for i = N α + N + 1, • • • , N max .
Therefore, using the above notations, we can approximate the initial conditions of u t , v t and w t using centered time approximation to get

u -1 i = u 1 i -2∆tψ i , for i = 1, • • • , N α , v -1 i = v 1 i -2∆tζ i , for i = N α + 1, • • • , N α + N, w -1 i = w 1 i -2∆tθ i , for i = N α + N + 1, • • • , N max . Consequently, U -1 i = U 1 i -2∆tΨ i , for i = 1, • • • , N max .
Finally, we set u 0 = 0 and w N max +1 = 0. Now, denote the discrete norm in L 2 by

||U T || = N max ∑ i=0 (U i ) 2 h i 1/2
. Also, we denote the discrete norm in H 1 L by

||U T || 1,T = ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i ) 2 1/2 , with U 0 = U N max +1 = 0.
In Subsection 5.2.1, we develop a conditionally stable explicit numerical scheme using FVM in space. However, in Subsection 5.2.3, we set a semi-implicit numerical scheme with an average of n + 1 and n -1 time step diffusion using FVM in space.

In Subsections 5.2.2 and 5.2.4, we plan to design a discrete energy that dissipates when the control is acting and is conserved during its absence. So, the discrete energy for the explicit scheme is defined by:

• the discrete kinetic energy for U as:

E k (U n ) = 1 2 N max ∑ i=1 h i U n+1 i -U n i ∆t 2
• the discrete potential energy for U as:

E p (U n ) = 1 2 N max ∑ i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(U n i+1 -U n i )
The total discrete energy is then defined as

E n = E k (U n ) + E p (U n ).
(Energy/Explicit)

• the discrete kinetic energy for U as:

E k (U n ) = N max ∑ i=1 h i U n+1 i -U n i ∆t 2
• the discrete potential energy for u as:

E p (U n ) = 1 2 N max ∑ i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) 2 + 1 2 N max ∑ i=0 ℓ i+ 1 2 (U n i+1 -U n i ) 2
The total discrete energy is then defined as

E n = E k (U n ) + E p (U n ).
(Energy/Implicit)

In Section 5.3, we derive the stability estimates of the numerical solution under Hypothesis (H ′ ) and Condition (TA). In fact, the numerical solution is bounded independently of the space and time discretization parameters.

In Section 5.4, we prove the convergence of the numerical solution to the weak solution under Hypothesis (H ′ ) and Condition (TA).

Finally, in Section 5.5, we give some numerical examples which demonstrate the theoretical results obtained in [START_REF] Nasser | Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface[END_REF].

Preliminaries

In this thesis, we focus our study mainly on semigroup theory and partially on numerical analysis. For ease of reading, we collect, in this chapter, some basic concepts and results on semigroup and numerical analysis which will be used throughout the following chapters. For the semigroup theory, we present some well known theorems related especially to the results of strong, exponential and polynomial stability of a C 0 -semigroup. Moreover, we review some geometric conditions employed in our work. Finally, we show some theorems related to the numerical part that will be used to demonstrate our main results. For the details and proofs we refer to [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF][START_REF] Kato | Perturbation Theory for Linear Operators[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Liu | Semigroups associated with dissipative systems[END_REF][START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Liu | Semigroups associated with dissipative systems[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF][START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF][START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF][START_REF] Rauch | Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains[END_REF][START_REF] Bardos | Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[END_REF][START_REF] Lebeau | Équation des ondes amorties[END_REF][START_REF] Robertsson | Numerical Methods, Finite Difference[END_REF][START_REF] Eymard | Study of a finite volume scheme for the regularized mean curvature flow level set equation[END_REF][START_REF] Eymard | Handbook of numerical analysis[END_REF].

Bounded and Unbounded linear operators

In this section, we list some important results about bounded and unbounded operators (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF][START_REF] Kato | Perturbation Theory for Linear Operators[END_REF]).

Let (E, • E ), (F, • F ) be two Banach spaces over C and H be a Hilbert space endowed with the scalar product (•, •) H and the induced norm • H .

A linear operator T : E -→ F is a transformation which maps linearly E in F, that is

T (αu + β v) = αT (u) + β T (v), ∀ u, v ∈ E and α, β ∈ C. Definition 1.1.1. A linear operator T : E → F is said to be bounded if there exists C ≥ 0 such that Tu F ≤ C u E , ∀ u ∈ E.
The set of all bounded linear operators from E into F is denoted by L (E, F). Moreover, the set of all bounded linear operators from E into E is denoted by L (E).

Definition 1.1.2. A bounded operator T ∈ L (E, F) is said to be compact if for each sequence (x n ) n∈N ∈ E such that x n E = 1 for each n ∈ N, the sequence (T x n ) n∈N has a convergent subsequence in F.
The set of all compact operators from E into F is denoted by K (E, F). For simplicity, one writes • We define the graph of T by • The resolvent set of T is defined by

K (E) = K (E, E).
G (T ) = {(u, Tu) : u ∈ D (T )} ⊂ E × F. Definition 1.1.7. A map T is said to be closed if G (T ) is closed in E × F.
ρ (T ) = {λ ∈ C : λ I -T is bijective from D (T ) onto F} .
• The resolvent of T is defined by

R (λ , T ) = (λ I -T ) -1 : λ ∈ ρ (T ) .
• The spectrum set of T is the complement of the resolvent set in C, denoted by

σ (T ) = C \ ρ (T ) .
Definition 1.1.9. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator. We can split the spectrum σ (T ) of T into three disjoint sets: the ponctuel spectrum denoted by σ p (T ), the continuous spectrum denoted by σ c (T ) and the residual spectrum denoted by σ r (T ), where these sets are defined as follows:

• σ p (T ) = {λ ∈ C : ker(λ I -T ) = {0}}; in this case λ is called an eigenvalue of T . • σ c (T ) = λ ∈ C : ker(λ I -T ) = 0, R(λ I -T ) = F and (λ I -T ) -1 is not bounded . • σ r (T ) = {λ ∈ C : ker(λ I -T ) = 0 and R(λ I -T ) is not dense in F} .
Definition 1.1.10. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator and let λ be an eigenvalue of T . A non-zero element e ∈ E is called a generalized eigenvector of T associated with the eigenvalue λ , if there exists n ∈ N * such that (λ I -T ) n e = 0 and (λ I -T ) n-1 e = 0.

Moreover, if n = 1, then e is called an eigenvector.

Definition 1.1.11. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator. We say that T has compact resolvent, if there exists λ 0 ∈ ρ (T ) such that (λ 0 I -T ) -1 is compact.

Theorem 1.1.12. [42, Theorem 6.7] Let (T, D (T )) be a closed unbounded linear operator on H, then ρ (T ) is an open set of C.

Definition 1.1.13. An unbounded operator (T,

D (T )) on H is • Positive if (Tu, u) H ≥ 0, ∀ u ∈ D (T ) . • Coercive if ∃ C > 0, such that (Tu, u) H ≥ C u 2 H , ∀ u ∈ D (T ) .

Semigroups, Existence and uniqueness of solution

In this section, we begin by presenting few fundamental concepts concerning the semigroups. The vast majority of evolution equations can be reduced to the form

U t (x,t) = AU(x,t), t > 0, U(x, 0) = U 0 (x), (1.2.1)
where A is the infinitesimal generator of a C 0 -semigroup S (t) over a Hilbert space H. Let's start by basic definitions and theorems (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF][START_REF] Liu | Semigroups associated with dissipative systems[END_REF]).

Let (X, • X ) be a Banach space and H be a Hilbert space equipped with the inner product

Au := lim t→0 + S (t) u -u t , u ∈ D (A)
exists. Then, A is called the infinitesimal generator of the semigroup (S (t)) t≥0 .

Proposition 1.2.3 (Theorem 2.2 [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Let (S (t)) t≥0 be a C 0 -semigroup in X. Then there exist a constant M ≥ 1 and ω ≥ 0 such that S (t) L (X) ≤ Me ωt , ∀t ≥ 0.

If ω = 0, then the corresponding semigroup is uniformly bounded. Moreover, if M = 1, then (S (t)) t≥0 is said to be a C 0 -semigroup of contractions.

Definition 1.2.4. An unbounded linear operator (A, D (A)) on X is said to be dissipative if

(λ I -A) x X ≥ λ x X , ∀ x ∈ D (A) and ∀ λ > 0.
Proposition 1.2.5. Let (A, D (A)) be an unbounded linear operator on H, then

A is dissipative if and only if ℜ (Ax, x) H ≤ 0, ∀ x ∈ D (A) . Definition 1.2.6. An unbounded linear operator (A, D (A)) on X is said to be maximal dissipative (m-dissipative) if • A is a dissipative operator. • ∃ λ 0 > 0 such that R (λ 0 I -A) = X.
Theorem 1.2.7. [67, Theorem 4.5] Let A be an m-dissipative operator, then

• R (λ I -A) = X, ∀ λ > 0. • ]0, ∞[ ⊆ ρ (A) .
Theorem 1.2.8 (Hille-Yosida Theorem 3.1 [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). An unbounded linear operator (A, D (A)) on X is the infinitesimal generator of a C 0 -semigroup of contractions (S (t)) t≥0 if and only if

• A is closed and D (A) = X.
• The resolvent set ρ (A ) of A contains R + and for all λ > 0,

(λ I -A) -1 L (X) ≤ λ -1 .
Theorem 1.2.9 (Lummer-Phillips Theorem 4.3 [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Let (A, D (A)) be an unbounded linear operator on X with dense domain D (A) in X. A is the infinitesimal generator of a C 0 -semigroup of contractions if and only if it is an m-dissipative operator.

Theorem 1.2.10 (Theorem 4.6 [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Let (A, D (A)) be an unbounded linear operator on X.

If A is dissipative with R (I -A) = X and X is reflexive, then D (A) = X.
Corollary 1.2.11. Let (A, D (A)) be an unbounded linear operator on H. A is the infinitesimal generator of a C 0 -semigroup of contractions if and only if it is m-dissipative operator.

Theorem 1.2.12 (Theorem 1.2.4 [START_REF] Liu | Semigroups associated with dissipative systems[END_REF]). Let A be a linear operator with dense domain D (A) in a Hilbert space H. If A is dissipative and 0 ∈ ρ (A), then A is the infinitesimal generator of a C 0 -semigroup of contractions on H.

Consequently, A is maximal dissipative operator on a Hilbert space H if and only if it generates a C 0 -semigroup of contractions (S(t)) t≥0 on H. Then the existence of solution is justified by the following corollary which follows from Hille-Yosida Theorem.

Corollary 1.2.13 (Hille-Yosida Theorem 7.4 [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]). Let (A, D (A)) be an unbounded linear operator on H. Assume that A is the infinitesimal generator of a C 0 -semigroup of contractions (S (t)) t≥0 . 1. For U 0 ∈ D (A), problem (1.2.1) admits a unique strong solution

U (t) = S(t)U 0 ∈ C 0 (R + ; D (A)) ∩C 1 (R + ; H) .
2. For U 0 ∈ H, problem (1.2.1) admits a unique weak solution

U (t) ∈ C 0 (R + ; H) .

Stability of a semigroup

In this section, we list some definitions and theorems about strong, exponential and polynomial stability of a C 0 -semigroup. For more details see [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF][START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Liu | Semigroups associated with dissipative systems[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF][START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF].

Let (X, • X ) be a Banach space and H be a Hilbert space equipped with the inner product (•, •) H and the induced norm • H .

Definition 1.3.1. Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on X. We say that the C 0 -semigroup (S (t)) t≥0 is:

• Strongly stable if lim t→+∞ S (t) u X = 0, ∀ u ∈ X.
• Uniformly stable if lim t→+∞ S (t) L (X) = 0.

• Exponentially stable if there exist two positive constants M and ε such that

S (t) u X ≤ Me -εt u X , ∀ t > 0, ∀ u ∈ X.
• Polynomially stable if there exist two positive constants C and α such that

S (t) u X ≤ Ct -α u D(A) , ∀ t > 0, ∀ u ∈ D(A).
Proposition 1.3.2. Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on X. The following statements are equivalent • (S (t)) t≥0 is uniformly stable.

• (S (t)) t≥0 is exponentially stable. Now, we look for the necessary conditions of strong stability of a C 0 -semigroup. To this end, we will recall two methods. The first result obtained by Arendt and Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF].

Theorem 1.3.3 (Arendt and Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF]). Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on a reflexive Banach space X. If • A has no pure imaginary eigenvalues • σ (A) ∩ iR is countable then, S (t) is strongly stable. Remark 1.3.4. If the resolvent (I -A) -1 of A is compact, then σ (A) = σ p (A); i.e., the spectrum of A becomes exactly the set of eigenvalues of A. Thus, the state of Theorem 1.3.3 reduces to σ p (A ) ∩ iR = / 0.

The second one is a classical method based on Arendt and Batty theorem and the contradiction argument ([57, page 25]).

Lemma 1.3.5 (Liu and Zheng page 25 [START_REF] Liu | Semigroups associated with dissipative systems[END_REF]). Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on a Hilbert space H. Furthermore, assume that 0 ∈ ρ(A).

If iR ⊆ ρ (A), then there exists {(λ n ,U n )} n≥1 ⊂ R × D (A) , with λ n → ω as n → ∞, |λ n | < |ω| and U n H = 1, such that (iλ n I -A)U n → 0 in H, as n → ∞.
We will check condition iR ⊂ ρ(A) by finding a contradiction with U n H = 1 such as U n H = o(1).

Also, we will state the following unique continuation theorem (see [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]), which can be a helpful tool in proving strong stability.

Theorem 1.3.6 (Calderón theorem in [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]). Let g be such that g(y

) ≤ C|y|. Let Ω be a connected open set in R N and let ω ⊂ Ω with ω = / 0. If u ∈ H 2 (Ω) satisfies Pu = g(u)
in Ω and u(x) = 0 in ω, then u vanishes in Ω.

Next, after showing that the C 0 -semigroup is strongly stable, we aim to look for the necessary and sufficient conditions of exponential stability of a C 0 -semigroup. In fact, the results of exponential stability are obtained by using several methods like: multiplier method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them. In this thesis, we will recall only the frequency domain approach method which was obtained by Huang-Prüss in [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF].

Theorem 1.3.7 (Huang-Prüss in [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF][START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF]). Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on H. S (t) is uniformly stable if and only if

• iR ⊂ ρ (A) . • sup β ∈R (iβ I -A) -1 L (H) < +∞.
In the case that the C 0 -semigroup is not exponentially stable, we look for polynomial stability. In general, polynomial stability results are also obtained using different methods like: multiplier method, frequency domain approach, Riesz basis approach, Fourier analysis or a combination of them. In this thesis, we will only mention the frequency domain approach method that was obtained by Batty in [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF], A.Borichev and Y.Tomilov in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], Z. Liu and B. Rao in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF].

Theorem 1.3.8 (Batty in [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF], A.Borichev & Y.Tomilov in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], Z. Liu & B. Rao in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF]). Assume that A is the generator of a strongly continuous semigroup of contractions (S (t)) t≥0 on H. If iR ⊂ ρ (A), then for a fixed ℓ > 0, the following conditions are equivalent 1. sup

λ ∈R (iλ I -A) -1 L (H) = O |λ | ℓ . 2. S (t)U 0 H ≤ C t 1 ℓ U 0 D(A) ∀ t > 0, U 0 ∈ D (A), for some C > 0.

Geometric conditions

This section is devoted to recall the geometric conditions used in our thesis (see [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF][START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF][START_REF] Rauch | Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains[END_REF][START_REF] Bardos | Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[END_REF][START_REF] Lebeau | Équation des ondes amorties[END_REF]).

We start by introducing the Piecewise Multiplier Geometric condition (PMGC in short) introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF].

Definition 1.4.1. A subset ω of Ω satisfies the Piecewise Multiplier Geometric Condition if there exist Ω j ⊂ Ω having Lipschitz boundary Γ j = ∂ Ω j and x j ∈ R N , j = 1, ..., J such that Ω j ∩ Ω i = / 0 for j = i and ω contains a neighborhood in Ω of the set ∪ J j=1 γ j (x j ) ∪ Ω \ ∪ J j=1 Ω j where γ j (x j ) = {x ∈ Γ j : (x -x j ) • ν j (x) > 0} and ν j is the outward unit normal vector to Γ j . Remark 1.4.2. The PMGC is a generalization of the Γ-condition introduced by J-L. Lions in [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF], saying that ω satisfies the Γ-condition if it contains a neighborhood in Ω of the set {x ∈ Γ : (xx 0 ) • ν(x) > 0}, for some x 0 ∈ R N , where ν is the outward unit normal vector to Γ = ∂ Ω.

Next, we recall the Geometric Control Condition (GCC in short) introduced by J. Rauch and M. Taylor in [START_REF] Rauch | Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains[END_REF] for manifolds without boundaries and by C. Bardos, G. Lebeau and J. Rauch in [START_REF] Bardos | Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[END_REF] (see also [START_REF] Lebeau | Équation des ondes amorties[END_REF]) for domains with boundaries. Definition 1.4.3. For a subset ω of Ω and T > 0, we shall say that (ω, T ) satisfies the Geometric Control Condition if every geodesic traveling at speed one in Ω meets ω in time t < T. Remark 1.4.4. The PMGC is much more restrictive than the GCC. For example, in Figure 1., we consider the case where Ω is a disk and we draw three different subsets in Ω. The Γ-condition is only satisfied by ω 0 . The PMGC is satisfied by ω 0 and ω 1 . However, ω 2 doesn't satisfy neither the PMGC nor the Γ-condition. Finally, the GCC is satisfied by the three different subsets of Ω.

Numerical analysis results

Many numerical methods can be used to solve the wave equations; for instance, the Finite Difference Method (FDM) [START_REF] Minkoff | Spatial Parallelism Of A 3d Finite Difference, Velocity-Stress Elastic Wave Propagation Code[END_REF], the Finite Element Method (FEM) [START_REF] Zhang | STABILITY FOR IMPOSING ABSORB-ING BOUNDARY CONDITIONS IN THE FINITE ELEMENT SIMULATION OF ACOUSTIC WAVE PROPAGATION[END_REF], the Finite Volume Method (FVM) [START_REF] Dumbser | Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems[END_REF][START_REF] Eymard | Handbook of numerical analysis[END_REF], the Pseudospectral Method (PSM) [START_REF] Fornberg | A Practical Guide to Pseudospectral Methods[END_REF], the Spectral Element Method (SEM) [START_REF] Komatitsch | Introduction to the spectral element method for three-dimensional seismic wave propagation[END_REF] and the Discontinuous Galerkin Method (DGM) [START_REF] Chung | Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions[END_REF]. Each numerical method has its own merits and limitations. However, in this thesis, taking into consideration that our system is a second order differential equation, the obtained results depend in particular on the FVM and partly on the FDM. In this section, we will gather and collect the basic definitions and theorems used in our thesis, concerning the numerical part.

The FVM is a very popular method used for representing and evaluating partial differential equations in the form of algebraic equations. FVM is a discretization method that is well suited for the numerical simulation of various types, such as, elliptic, parabolic or hyperbolic. It has been widely used in several engineering fields, such as fluid mechanics, heat and mass transfer or petroleum engineering. This method evaluates exact expressions for the average value of the solution over some volume and uses this data to construct approximations of the solution within cells. FVM has important common features with FEM, especially that it is capable to be applied on arbitrary geometries, using structured or unstructured meshes. An additional advantage is the local conservativity of the numerical fluxes, that is the numerical flux is conserved from one discretization cell to its neighbor. For more details see [START_REF] Eymard | Handbook of numerical analysis[END_REF].

Furthermore, in numerical analysis, FDM is a classical method for solving differential or partial differential equations by approximating derivatives with finite differences. Both spatial domain and time interval are discretized into a finite number of steps, and the value of the solution at these discrete points is approximated by solving algebraic equations containing finite differences and values from nearby points. For more details see [START_REF] Robertsson | Numerical Methods, Finite Difference[END_REF]. Now, given a partial differential equation Pu = f and a finite difference scheme P ∆t,∆x v = f with ∆t being the time step parameter and ∆x is the space discretization parameter. Definition 1.5.1. A finite difference scheme approximating a partial differential equation is said to be convergent if for any solution to the partial differential equation u(t, x), and solutions to the finite difference scheme v n i , such that v 0 i converges to u 0 (x) as i∆x converges to x, v n i converges to u(t, x) as (n∆t, i∆x) converges to (t, x) as ∆t, ∆x converge to zero. Definition 1.5.2. The L 2 norm of a grid function u, denoted by ||u|| ∆x , is defined by

||u|| ∆x = ∆x +∞ ∑ m=-∞ (u m ) 2 1/2
. Definition 1.5.3. A finite difference scheme P ∆t,∆x v n i = 0 for a first order equation is stable in a stability region ℧, if there exists an integer J, such that for any positive time T , there exists a constant C T such that

||v n || ∆x ≤ C T J ∑ j=0 ||v n j || ∆x ,
for 0 ≤ n∆t ≤ T and (∆t, ∆x) ∈ ℧.

Also, we consider the following elliptic problem.

   -u xx = f (x, u(x)), x ∈ (0, 1), u(0) = u(1) = 0, (Elliptic Problem)
where f : (0, 1) × R → R is a function satisfying the following conditions 1. f (x, s) is measurable with respect to x ∈ (0, 1), for all s ∈ R.

2. f (x, s) is continuous with respect to s ∈ R, for a.e. x ∈ (0, 1).

3. f ∈ L ∞ ((0, 1) × R). It is possible to prove that there exists at least one weak solution to Problem (Elliptic Problem), u ∈ H 1 0 (0, 1), satisfying

1 0 u x v x dx = 1 0 f v dx, for all v ∈ H 1 0 (0, 1). (Weak Variational Problem) Definition 1.5.4. A mesh is said to be admissible if • For all k = k ′ , T k ∩ T k ′ is
either empty or consists of exactly one node or of one entire edge.

• No T k is of zero measure.

The second condition means that no triangle or rectangle is degenerated, that is its vertices are not aligned. However, to understand the first condition, we will demonstrate some figures in which this condition doesn't apply.

Figure 2. Forbidden meshes according to first condition

Let T be an admissible mesh of [0, 1] consisting of N cells (or control volumes), denoted by K i ; i = 1, . . . , N, and N points of discretization, denoted by x i ; i = 1, . . . , N. We consider the following finite volume scheme of Problem (Elliptic Problem)

-u i+1 -u i x i+1 -x i -u i -u i-1 x i -x i-1 = m(K i ) f i (u i ), i = 1, . . . , N, u 0 = u N+1 = 0, (Numerical Scheme) with f i (u i ) = 1 m(K i ) K i f (x, u i ) dx, i = 1, . . . , N and m(K i ) = meas(K i ).
Theorem 1.5.5 (Existence and stability results in [START_REF] Eymard | Handbook of numerical analysis[END_REF]). Let f : (0, 1) × R → R satisfy assumptions (1), (2), (3) and T be an admissible mesh of (0, 1). Then, there exists (u 1 , . . . , u N ) ⊤ ∈ R N , a solution of (Numerical Scheme) satisfying

N ∑ i=0 (u i+1 -u i ) 2 x i+1 -x i ≤ C, ( Stability Estimate)
for some C ≥ 0 depending only on f . Theorem 1.5.6 (Compactness in [START_REF] Eymard | Handbook of numerical analysis[END_REF]). For an admissible mesh T of (0, 1), let (u 1 , . . . , u N ) ⊤ ∈ R N satisfy (Stability Estimate) for some C ∈ R independent of T , and let u T : (0, 1) → R be defined by u T (x) = u i if x ∈ K i ; i = 1, . . . , N. Then, the set {u T , T admissible mesh of (0, 1)} is relatively compact in L 2 (0, 1). Further more, if u T n → u in L 2 (0, 1) and size(T n ) → 0 as n → ∞, then u ∈ H 1 0 (0, 1).

Remark that size(T ) = max i=1,...,N m(K i ).

Theorem 1.5.7 (Convergence in [START_REF] Eymard | Handbook of numerical analysis[END_REF]). Let f : (0, 1) × R → R satisfy assumptions (1), ( 2), (3) and T be an admissible mesh of (0, 1). Let (u 1 , . . . , u N ) ⊤ ∈ R N be a solution to (Numerical Scheme) and let u T : (0, 1) → R be defined by u T (x) = u i if x ∈ K i ; i = 1, . . . , N. Then, for any sequence (T n ) n∈N of the admissible mesh such that size(T n ) → 0 as n → ∞, there exists a subsequence, still denoted by (T n ) n∈N , such that u T n → u in L 2 (0, 1), where u ∈ H 1 0 (0, 1) is a weak solution to (Elliptic Problem), i.e., a solution to (Weak Variational Problem).

We also mention the following convergence theorem from [START_REF] Eymard | Study of a finite volume scheme for the regularized mean curvature flow level set equation[END_REF].

Theorem 1.5.8 (A variant of Ascoli's theorem in [START_REF] Eymard | Study of a finite volume scheme for the regularized mean curvature flow level set equation[END_REF]). Let Ω be a polyhedral open bounded connected subset of R N , with N ∈ N * and T > 0. M is a finite family of nonempty connected open disjoint subsets of Ω. Let u 0 ∈ H 1 0 (Ω) be given. Let (u m , D m , τ m ) m∈N be a sequence, such that for all m ∈ N, (D m , τ m ) is a space-time discretization of Ω × (0, T ) in the sense of Definition 2.3 in [START_REF] Eymard | Study of a finite volume scheme for the regularized mean curvature flow level set equation[END_REF], u m ∈ H D m ,τ m (see definition of space in [START_REF] Eymard | Study of a finite volume scheme for the regularized mean curvature flow level set equation[END_REF]) and the space and time discretization parameters; h D m and τ m , tend to zero as m → ∞. For all m ∈ N, setting D = D m and τ = τ m , we define the functions u D,τ (x,t), for all t ∈ [0, T ] and a.e. x ∈ Ω by We assume that there exists C > 0 (independent of m), such that ||u n+1 m || 1,D m ≤ C for all n = 0, . . . , N T m and ||z D m ,τ m || 2 L 2 (Ω×(0,T )) ≤ C. Then, there exists u ∈ C 0 (0, T ; L 2 (Ω)) with u(., 0) = u 0 and a subsequence of (u m , D m , τ m ) m∈N , again denoted by (u m , D m , τ m ) m∈N , such that lim

u D,τ (x, 0) = u 0 p = 1 |p| p u 0 (x)dx,
m→∞ sup t∈[0,T ] ||u D m ,τ m (t) -u(t)|| L 2 (Ω) = 0.

Stability of N-d transmission problem

In this chapter, we investigate a multidimensional transmission problem between viscoelastic system with localized Kelvin-Voigt damping and purely elastic system under different types of geometric conditions. The Kelvin-Voigt damping is localized via non smooth coefficient in a suitable subdomain. It was shown that the discontinuity of the material coefficient along the interface elastic/viscoelastic can't assure an exponential stability of the total system. So, it is natural to hope for a polynomial stability result under certain geometric conditions on the damping region. For this aim, using frequency domain approach combined with a new multiplier technic, we will establish a polynomial energy decay estimate of type t -1 for smooth initial data. This result is obtained if either one of the geometric assumptions (A1) or (A2) holds (see below). Also, we establish a general polynomial energy decay estimate on a bounded domain where the geometric conditions on the localized viscoelastic damping are violated and we apply it on a square domain where the damping is localized in a vertical strip. However, the energy of our system decays polynomially of type t -2/5 if the strip is localized near the boundary. Else, it's of type t -1/3 . The main novelty in this work is that the geometric situations covered here are richer and less restrictive than those considered in [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF], [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF], [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF] and include in particular an example where the damping region is localized faraway from the boundary.

Introduction

Local viscoelastic damping is a natural phenomena of bodies arising from a solid that have one part made of viscoelastic material, and the other made of elastic material. Let Ω ⊂ R N be a nonempty bounded open set with Lipschitz boundary Γ. We consider the wave equation with locally distributed Kelvin-Voigt type damping given in the following equation:

     ρ(x)u tt (x,t) -div(a(x)∇u + b(x)∇u t ) = 0 in Ω × R + , u(x,t) = 0 on Γ × R + , (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) in Ω, (2.1.1)
where the coefficient functions ρ, a, b ∈ L ∞ (Ω) and we assume that

ρ(x) ≥ ρ 0 > 0, a(x) ≥ a 0 > 0, b(x) ≥ 0, ∀ x ∈ Ω.
In 1988, F. Huang proved that when the Kelvin-Voigt damping div(b(x)∇u t ) is globally distributed, i.e. b(x) ≥ b 0 > 0 for almost every x in Ω, the corresponding semigroup of System (2.1.1) is not only exponentially stable, but also is analytic (see [START_REF] Huang | On the Mathematical Model for Linear Elastic Systems with Analytic Damping[END_REF]). Thus, Kelvin-Voigt damping is stronger than the viscous damping b(x)u t in this case. Indeed, in [START_REF] Lebeau | Équation des ondes amorties[END_REF], it was proved that the semigroup corresponding to the system of wave equations with global viscous damping is exponentially stable but not analytic. However, the exponential stability is still true even if the viscous damping is localized; via a smooth or a non smooth damping coefficient, in a suitable subdomain satisfying the Geometric Control Condition (GCC in short) introduced by C. Bardos, G. Lebeau and J. Rauch in [START_REF] Bardos | Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[END_REF] (see also [START_REF] Lebeau | Équation des ondes amorties[END_REF] and Definition 1.4.3 below). Nevertheless, when viscoelastic damping is distributed locally, the situation is more delicate and such comparison between viscous and viscoelastic damping is not valid anymore. In fact, in 1998, K. Liu and Z. Liu considered a one-dimensional wave equation with Kelvin-Voigt damping distributed locally on any subinterval of the region occupied by the beam, where the damping coefficient is the characteristic function of the subinterval. They proved that the semigroup associated with the equation for the transversal motion of the beam is exponentially stable, although the semigroup associated with the equation for the longitudinal motion of the beam is not (see [START_REF] Liu | Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping[END_REF]). This shows that Kelvin-Voigt damping does not obey the GCC. This surprising result, due to the discontinuity of the materials and the unboundedness of viscoelastic damping, motivated the study of elastic system with local Kelvin-Voigt damping.

Later, in the one-dimensional case, it was found that the smoothness of the damping coefficient at the interface is a critical factor for the stability and the regularity of the solutions (see [START_REF] Liu | Spectrum and Stability for Elastic Systems with Global or Local Kelvin-Voigt Damping[END_REF][START_REF] Liu | Exponential decay of energy of vibrating strings with local viscoelasticity[END_REF][START_REF] Liu | Eventual differentiability of a string with local Kelvin-Voigt damping[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Liu | Stability of a String with Local Kelvin-Voigt Damping and Nonsmooth Coefficient at Interface[END_REF][START_REF] Zhang | Exponential stability of an elastic string with local Kelvin-Voigt damping[END_REF]). However, there are only a small number of publications on the corresponding N-dimensional case. In 2006, K. Liu and B.

Rao considered this problem in the N-dimensional space where the damping region is a neighborhood (in Ω) of the entire boundary Γ (see [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF]). They proved that the energy of the system goes exponentially to zero as t goes to infinity for all usual initial data by assuming that the damping coefficient b satisfies

b ∈ C 1,1 (Ω), ∆b ∈ L ∞ (Ω) and |∇b(x)| 2 ≤ M 0 b(x)
for almost every x in Ω, where M 0 is a positive constant. In 2012, L. Tebou studied the stabilization of the wave equation with Kelvin-Voigt damping (see [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF]). He established polynomial energy decay of type t -1 provided that the damping region is localized and verifies the Piecewise Multiplier Geometric Condition (PMGC in short) introduced by K. Liu in [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF] [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF]). Later on, in 2017, M. Cavalcanti, V. Cavalcanti and L. Tebou showed the exponential decay of the energy of a wave equation with two types of locally distributed mechanisms; a frictional damping and a Kelvin-Voigt type damping where the location of each damping is such that none of them alone is able to exponentially stabilize the system (see [START_REF] Cavalcanti | Stabilization of the wave equation with localized compensating frictional and kelvin-voigt dissipating mechanisms[END_REF]). Under the condition that the damping region satisfyies the PMGC geometric conditions, they proved that the energy of the system decays polynomially as type t -1 in the absence of regularity of the Kelvin-Voigt damping coefficient b. However, they established exponential stability when this coefficient is smooth. Recently, in 2018, K. Ammari, F. Hassine and L. Robbianio considered a wave equation with Kelvin-Voigt damping localized in a subdomain ω faraway from the boundary without geometric conditions (see [START_REF] Ammari | Stabilization for the wave equation with singular kelvinâ Ȃ Şvoigt damping[END_REF]). They established a logarithmic energy decay rate for smooth initial data. Finally, in 2018, Q. Zhang considered the wave equation with Kelvin-Voigt damping in a nonempty bounded convex domain Ω with partition Ω = Ω 1 ∪ Ω 2 where the viscoelastic damping is localized in Ω 1 (see [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF]). Under the condition that the damping coefficient b is non smooth, she established a polynomial energy decay rate of type t -1 for smooth initial data in the following two cases:

Case 1. The damping region Ω 1 is a neighborhood of the entire boundary Γ of Ω. Case 2. The domain Ω ⊂ R N (N = 2 or 3), ∂ Ω 1 and ∂ Ω 2 are either convex curvilinear polygons or curved plane polyhedron, the damping region Ω 1 is a neighborhood of a part Γ 1 = / 0 of the boundary Γ and m(x) • ν 2 ≤ 0 where m(x) = xx 0 for x 0 fixed in R N (N = 2, 3) for all x ∈ Γ 2 = Γ \ Γ 1 .

In conclusion, several important geometric situations are not covered by all previous cited papers. For example, in the case where the damped region {b > 0} satisfies or does not satisfy the GCC condition (see for instance Fig. 1-c, Fig. 2, Fig. 3 and Fig. 4), the problem of the energy decay rate is still open. So, our aim is to answer this open problem.

In this chapter, we consider the stabilization of the wave equation with Kelvin-Voigt damping in a bounded domain Ω ⊂ R N . The damping is localized in Ω via non smooth coefficient. The energy of a solution u of System (2.1.1) is given by

E(u,t) = 1 2 Ω ρ(x)|u t | 2 + a(x)|∇u| 2 dx.
Then, a straight forward computation gives [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF], [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF], [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF]) and include in particular an example where the damping region is faraway from the boundary and an example where the damping region does not satisfy the GCC condition.

d dt E(u,t) = - Ω b(x)|∇u t | 2 dx.
This chapter is organized as follows: In Section 2.2, we study the Well-Posedness and the strong stability of System (2.1.1). Section 2.3 is devoted to study the energy decay rate of System (2.1.1) under different types of geometric conditions, where we prove that the energy of our system has a polynomial decay rate of type t -1 (see Theorem 2.3.4). Finally, in section 2.4, we study the stabilization of a wave equation with localized Kelvin-Voigt damping in the absence of GCC, and we obtain a general polynomial decay rate of the energy. In particular, we give applications on a bounded square with internal Kelvin-Voigt damping localized in a vertical strip which does not verify any geometric condition. We establish a polynomial decay rate of the energy of type t -1/3 or t -2/5 under conditions (LC1) or (LC2) respectively.

Well-Posedness and strong stability

This section is devoted to the study of existence, uniqueness and asymptotic behavior of the solution of System (2.1.1).

Well-Posedness of the problem

In this part, by using semigroup theory, we give the well-posedness results for Problem (2.1.1). For this aim, we introduce the Hilbert energy space H by

H = H 1 0 (Ω) × L 2 (Ω),
which is endowed with the usual inner product

U, Ũ H = Ω (a∇u • ∇ ũ + ρv ṽ)dx,
where U = (u, v) ∈ H and Ũ = ( ũ, ṽ) ∈ H . We use U H to denote the corresponding norm. We next define the linear unbounded operator A :

D(A ) ⊂ H -→ H by D(A ) = {(u, v) ∈ H | v ∈ H 1 0 (Ω), div(a∇u + b∇v) ∈ L 2 (Ω)} and A (u, v) = v, 1 ρ div(a∇u + b∇v) , ∀(u, v) ∈ D(A ).
If (u, u t ) is a regular solution of System (2.1.1), then we transform this system into the following evolution equation

U t = A U, U(0) = U 0 , (2.2.1)
where U 0 = (u 0 , u 1 ) ∈ H . According to [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF], we have the following proposition:

Proposition 2.2.1. The unbounded linear operator A is m-dissipative in the energy space H .

Thanks to Lumer-Philips theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that A generates a C 0 -semigroup of contractions e tA in H and therefore Problem (2.1.1) is well-posed. Then we have the following result:

Theorem 2.2.2. For any U 0 ∈ H , Problem (2.2.1) admits a unique weak solution

U ∈ C 0 (R + , H ). Moreover, if U 0 ∈ D(A ), then U ∈ C 1 (R + , H ) ∩C 0 (R + , D(A )).

Strong stability

In this subsection we study the strong stability of System (2.1.1) in the sense that its energy E(u,t) converges to zero when t goes to infinity for all initial data in H . For this aim, we assume that there exist a nonempty open set ω ⊂ Ω and b 0 > 0 such that

b(x) ≥ b 0 ∀x ∈ ω. (LA)
Since the resolvent of A is not compact, we have to discuss the full spectrum on the imaginary axis. We use a general criteria of Arendt-Batty in [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF] which states that in a reflexive Banach space a C 0 -semigroup e tA is strongly stable if e tA is bounded, A has no eigenvalues on the imaginary axis and σ (A ) ∩ iR is countable. So, we will prove the following result:

I -A ) = {0}, ∀β ∈ R.
Proof. From Proposition 2.2.1, 0 ∈ ρ(A ). We still need to show the result for β ∈ R * . Suppose that there exist a real number β = 0 and U = (u, v) ∈ D(A ) such that

A U = iβU. (2.2.
3)

It follows that 0 = ℜ(iβ ||U|| 2 H ) = ℜ(A U,U) H = - Ω b(x)|∇v| 2 dx. ( 2
ρβ 2 u + div(a∇u) = 0 in Ω. (2.2.9) Now, let x 0 ∈ ω, so there exists R > 0 such that B(x 0 , R) ⊂ ω. Consider a cut-off function η ∈ C ∞ c (Ω) such that 0 ≤ η ≤ 1 and η(x) = 1 if x ∈ B (x 0 , R/2), 0 if x ∈ Ω \ B(x 0 , R).
Multiply Equation (2.2.9) by ηu and use integration by parts over Ω to obtain

β 2 Ω ρη|u| 2 dx - Ω au∇η • ∇udx - Ω aη|∇u| 2 dx = 0.
Using Equation (2.2.8) and the fact that B(x 0 , R) ⊂ ω, β = 0 and ρ ≥ ρ 0 > 0 in the above equation, we obtain

u = 0 in B (x 0 , R/2) . ( 2
     ∆u = -ρ a β 2 u -1 a ∇a • ∇u in Ω, u = 0 in B (x 0 , R/2) , u = 0 on Γ.
Thus, we define the following elliptic operator P by

P : H 2 (V ) → L 2 (V ) u → ∆u
and the function g by

g : L 2 (V ) → L 2 (V ) u → -ρ a β 2 u -1 a ∇a • ∇u.
Using Calderón theorem (see [START_REF] Rousseau | On Carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations[END_REF]) and taking ω = B (x 0 , R/2), we get finally that u = 0 in Ω and thus U = 0 in Ω. Hence, ker(iβ I -A ) = 0 and the proof is thus complete. Lemma 2.2.5. Under the same assumptions of Theorem 2.2.3, we have

R(iβ I -A ) = H , ∀β ∈ R.
Proof. Given F = ( f , g) ∈ H , we solve equation

(iβ I -A )U = F. (2.2.11)
Equivalently, we consider the following system

v = iβ u -f , (2.2
.12)

β 2 u + 1 ρ div(a∇u + iβ b∇u) = -(g + iβ f ) + 1 ρ div(b∇ f ). (2.2.13)
Now, define the linear operator L :

H 1 0 (Ω) -→ H -1 (Ω) by L u = - 1 ρ div(a∇u + iβ b∇u). ( 2

.2.14)

Using Lax-Milgram's theorem (see [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]), it is easy to show that L is an isomorphism from H 1 0 (Ω) onto H -1 (Ω). Then we transform (2.2.13) into the following form

(β 2 I -L )u = -(g + iβ f ) + 1 ρ div(b∇ f ). (2.2.15)
Since the operator L is an isomorphism from H 1 0 (Ω) onto H -1 (Ω) and I is a compact operator from H 1 0 (Ω) onto H -1 (Ω), then using Fredholm's Alternative theorem, problem (2.2.15) admits a unique solution in H 1 0 (Ω) if and only if the operator β 2 I -L is injective. For that purpose, let u ∈ ker(β 2 I -L ). Then, if we set v = iβ u, we deduce that U = (u, v) ∈ D(A ) is a solution of

(iβ -A )U = 0.
Using Lemma 2.2.4, we deduce that u = v = 0. This implies that equation (2.2.15) admits a unique solution u ∈ H 1 0 (Ω) and div(a∇u

+ iβ b∇u -b∇ f ) = -(g + iβ f ) ∈ L 2 (Ω)
. By setting v = iλ uf , we deduce that U = (u, v) ∈ D(A ) is the unique solution of equation (2.2.11) and the proof is thus complete.

Proof of Theorem 2.2.3. Using Lemma 2.2.4, the operator A has no pure imaginary eigenvalues and by Lemma 2.2.5, we have R(iβ I -A ) = H , for all β in R. Therefore, the closed graph theorem implies that σ (A ) ∩ iR = / 0. Following Arendt-Batty (see [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF]), the C 0 -semi group of contractions (e tA ) t≥0 is strongly stable and the proof is complete.

Polynomial stability

In this section, we study the energy decay rate of System (2.1.1). Indeed, if the material parameter b is smooth enough at the interface, then the energy of System (2.1.1) decays exponentially to zero as t goes to infinity under appropriate geometric conditions imposed on the damped region (see [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF], [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF], [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF], [START_REF] Cavalcanti | Stabilization of the wave equation with localized compensating frictional and kelvin-voigt dissipating mechanisms[END_REF]). However, Q. Zhang proved in [START_REF] Zhang | On the lack of exponential stability for an elasticâ Ȃ Şviscoelastic waves interaction system[END_REF] that the exponential decay fails in any geometry if the damping coefficient b is discontinuous along the interface. Note that, in this case, the lack of exponential stability still holds even in the 1d case (see [START_REF] Liu | Spectrum and Stability for Elastic Systems with Global or Local Kelvin-Voigt Damping[END_REF]). So, it is natural to hope for a polynomial stability result under some geometric considerations not covered previously which represents the main goal of this work. To this end, in addition to the well-known geometric conditions; for instance, Γ-condition ( [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF]), PMGC ( [START_REF] Liu | Locally distributed control and damping for the conservative systems[END_REF]) and GCC ( [START_REF] Rauch | Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains[END_REF][START_REF] Bardos | Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary[END_REF][START_REF] Lebeau | Équation des ondes amorties[END_REF]), we introduce a new geometric condition. Definition 2.3.1. For a subset ω of Ω, we shall say that ω satisfies Strictly the Geometric Control Condition (SGCC in short), if there exists an open subset ω included strictly in ω (i.e. ω ⊂ ω) and satisfying the GCC.

Remark 2.3.2. It is easy to see that if ω verifies the SGCC, then it verifies the GCC. The converse of this implication is false (see Fig. 1-c).

Remark 2.3.3. It is easy to see that, Figure 4 does not satisfy any geometry.

For the study of the energy decay rate we need the following geometric assumptions:

The open subset ω verifies the GCC and meas(ω ∩ Γ) > 0.

(

A1)

The open subset ω verifies the SGCC. (A2)

There are several geometries that verify the previous assumptions. For example: We are now in position to state our main result of this part. Theorem 2.3.4. Let a, ρ ∈ C 2 (Ω) and assume that the boundary Γ is of class C 3 . Assume that condition (LA) holds. Assume also assumption (A1) or assumption (A2) holds. Then there exists a constant C > 0 such that for all initial data U 0 ∈ D(A ), the energy of System (2.1.1) satisfies the following estimation

ω 0 (a)
E(U,t) ≤ C t ||U 0 || 2 D(A ) , ∀t > 0. (2.3.1)
Proof. Following Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF] , [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF]), a C 0 -semigroup of contractions (e tA ) t≥0 on a Hilbert space H verifies (2.3.1) if

iR ⊂ ρ(A ) (S1)
and lim sup

|λ |→∞ 1 λ 2 (iλ -A ) -1 L (H ) < ∞ (S2).
Since the resolvent of the operator A is not compact in the energy space H (see [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF]) and 0 ∈ ρ(A ), then to prove iR ⊂ ρ(A ) is equivalent to prove that iβ I -A is bijective in the energy space H for all β ∈ R ⋆ . This last is proven in Subsection 2.2.2 according to a unique continuation theorem and Fredholm's alternative. Then, we still need to prove condition (S2). This is checked by using a contradiction argument.

Indeed, suppose there exists

{(λ n ,U n := (u n , v n ))} n≥1 ⊂ R * + × D (A ), such that λ n → +∞, ||U n || H = 1, (2.3.2) λ 2 n (iλ n I -A )U n = ( f n , g n ) → 0 in H . (2.3.3)
Our aim is to show that ||(u n , v n )|| H → 0. This condition permits to conclude a contradiction with (2.3.2). By detailing equation (2.3.3), we get the following system

iλ n u n -v n = λ -2 n f n in H 1 0 (Ω), (2.3.4
)

iρλ n v n -div(a∇u n + b∇v n ) = λ -2 n ρg n in L 2 (Ω). (2.3.5)
The proof of our theorem is divided into several Lemmas.

Lemma 2.3.5. Assume that assumption (LA) holds. Then the solution

(u n , v n ) ∈ D(A ) of Equations (2.3.4)- (2.3.5
) satisfies the following asymptotic behavior estimations

||u n || L 2 (Ω) = O λ -1 n , (2.3.6
)

||∇v n || L 2 (ω) = o λ -1 n . ( 2
= (u n , v n ) in H , then using the fact that U n is uniformly bounded in H , we get Ω b(x)|∇v n | 2 dx = ℜ (iλ n I -A )U n ,U n H = o λ -2 n .
It follows from the localization condition (LA) that

ω |∇v n | 2 = o λ -2 n .
Lemma 2.3.6. Assume that assumptions (LA) and (A1) hold. Then the solution (u n , v n ) ∈ D(A ) of Equations (2.3.4)-(2.3.5) satisfies the following asymptotic behavior estimation

ω |u n | 2 = o λ -2 n . (2.3.8)
Proof. Since assumption (A1) holds, then using Poincaré's inequality and Equation (2.3.7), we obtain

λ 2 n ω |v n | 2 dx = o(1). (2.3.9)
Multiplying Equation (2.3.4) by -iλ n u n and integrating over ω, we get

λ 2 n ω |u n | 2 dx + iλ n ω v n u n dx = -iλ -1 n ω u n f n dx.
Using estimations (2.3.6), (2.3.9) and the fact that f n converges to zero in L 2 (Ω), we deduce from the above equation that

λ 2 n ω |u n | 2 dx = o(1).
The desired estimation (2.3.8) is established. (2.3.10)

Proof. Since assumption (A2) holds, then there exists a nonempty open subset ω of ω such that ω ⊂ ω.

Hence, ω ∩ (Ω \ ω) = / 0, and thus we define the function η

∈ C ∞ c (R N ) by 0 ≤ η(x) ≤ 1 and η(x) = 0, if x ∈ Ω \ ω, 1, if x ∈ ω.
Multiplying Equation (2.3.5) by ηv n and integrating over Ω, we obtain

iλ n Ω ρη|v n | 2 dx + Ω (a∇u n + b∇v n ) • (η∇v n + v n ∇η) dx = λ -2 n Ω ρ η g n v n dx.
Using estimations (2.3.6), (2.3.7), then taking into consideration that Suppη ⊂ ω and the fact that v n , ∇u n are uniformly bounded in L 2 (Ω) and g n converges to zero in L 2 (Ω), we deduce that

• Ω aη∇u n • ∇v n dx ≤ a ∞ Ω η|∇u n | 2 dx 1/2 Ω η|∇v n | 2 dx 1/2 = o λ -1 n , • Ω bη|∇v n | 2 dx =≤ η ∞ Ω b|∇v n | 2 dx = o λ -2 n , • Ω bv n ∇v n • ∇ηdx ≤ ∇η ∞ b ∞ Ω |v n | 2 dx 1/2 Ω b|∇v n | 2 dx 1/2 = o λ -1 n , • Ω ρ η g n v n dx ≤ ρ ∞ η ∞ Ω |g n | 2 dx 1/2 Ω |v n | 2 dx 1/2 = o(1).
It follows that

iλ n Ω ρη|v n | 2 dx + Ω av n ∇u n • ∇η dx = o λ -1 n . (2.3.11)
Now, taking the imaginary part and applying Young's inequality on the second term of (2.3.11), we deduce that

λ n Ω ρη|v n | 2 dx ≤ ||a∇η|| ∞ 2 Ω |∇u n | 2 dx + ||a∇η|| ∞ 2 Ω |v n | 2 dx + o λ -1 n .
Using the fact that (u n , v n ) is uniformly bounded in H and ρ(x) ≥ ρ 0 > 0, we get

Ω η|v n | 2 dx = o(1).
Using the definition of η, we deduce that

ω |v n | 2 dx = o(1).
Finally, multiply Equation (2.3.4) by iηλ n u n to get the desired estimation (2.3.10).

Lemma 2.3.8. Let U be a nonempty open subset of Ω satisfying the GCC such that U ⊆ ω. Then, for any

λ n ∈ R * + , the solution ϕ n ∈ H 2 (Ω) ∩ H 1 0 (Ω) of system ρλ 2 n ϕ n + div(a∇ϕ n ) -iλ n (1 U b)(x)ϕ n = u n in Ω, ϕ n = 0 on Γ (2.3.12)
satisfies the following estimation

||λ n ϕ n || L 2 (Ω) + ||∇ϕ n || L 2 (Ω) ≤ M||u n || L 2 (Ω) (2.3.13)
where M is positive constant independent of n.

Proof. We consider the following auxiliary problem, namely the wave equation with local viscous damping:

ρ(x)ϕ tt (x,t) -div(a(x)∇ϕ) + (1 U b)(x)ϕ t = 0 in Ω × R + , ϕ = 0 on Γ, (2.3.14)
where a, ρ ∈ C 2 (Ω). Since U satisfies the GCC, then the wave equation with local viscous damping (2.3.14) is exponentially stable (see [START_REF] Burq | Contrôlabilité exacte des ondes dans des ouverts peu réguliers[END_REF]) in the associated energy space H 1,a = H 1 0 (Ω) × L 2 (Ω). So, following Huang [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and Pruss [START_REF] Prüss | On the spectrum of C 0 -semigroups[END_REF], the resolvent set of its associated operator A a defined by

D(A a ) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω), A a (ϕ, ψ) = ψ, 1 ρ (div(a∇ϕ) -(1 U b)(x)ψ)
contains iR and the resolvent (iλ n I -A a ) -1 of A a is uniformly bounded on the imaginary axis. Consequently, there exists a positive constant M > 0 independent of n such that sup

λ n ∈R ||(iλ n I -A a ) -1 || L (H 1,a ) ≤ M. (2.3.15) Now, since u n ∈ L 2 (Ω), then there exists a unique (ϕ n , ψ n ) ∈ D(A a ) solution of (iλ n I -A a )(ϕ n , ψ n ) = 0, - 1 ρ u n , equivalently iλ n ϕ n -ψ n = 0, iλ n ρψ n -div(a∇ϕ n ) + (1 U b)(x)ψ n = -u n .
Finally, from (2.3.15), we deduce that

||(ϕ n , ψ n )|| H 1,a ≤ M||u n || L 2 (Ω)
which gives the desired estimation.

Lemma 2.3.9. Assume that assumption (LA) and assumptions (A1) or (A2) hold. Then the solution

(u n , v n ) ∈ D(A ) of (2.3.4)-(2.3.5) satisfies the following asymptotic behavior estimation Ω |λ n u n | 2 dx = o(1).
(2.3.16)

Proof. Case 1. Under assumptions (LA) and (A1), ω satisfies the GCC. So, we consider System (2.3.12) where U ≡ ω and ϕ n is its solution. Multiplying Equation (2.3.4) by iλ 3 n ρϕ n and Equation (2.3.5) by λ 2 n ϕ n and using Green's formula to obtain

-λ 4 n Ω ρu n ϕ n dx -iλ 3 n Ω ρv n ϕ n dx = iλ n Ω ρϕ n f n dx (2.3.17)
and

iλ 3 n Ω ρv n ϕ n dx + λ 2 n Ω (a∇u n + b∇v n ) • ∇ϕ n dx = Ω ρϕ n g n dx.
(2.3.18)

By using (2.3.6), (2.3.13) and the fact that

f n → 0, g n → 0 in L 2 (Ω), we get iλ n Ω ρϕ n f n dx = o(1)
and

Ω ρϕ n g n dx = o(1). (2.3.19)
Now, taking the second term of the left hand side of (2.3.18), we obtain

λ 2 n Ω (a∇u n + b∇v n ) • ∇ϕ n dx = λ 2 n Ω a∇u n • ∇ϕ n dx + λ 2 n Ω b∇v n • ∇ϕ n dx. (2.3.20)
We have

λ 2 n Ω b∇v n • ∇ϕ n dx ≤ λ n Ω b|∇v n | 2 dx 1/2 λ n b ∞ Ω |∇ϕ n | 2 dx 1/2
.

We continue by multiplying Estimation (2.3.13) by λ n and taking into consideration Estimation (2.3.6), we obtain

ϕ n L 2 (Ω) = O(λ -2 n ) and ∇ϕ n L 2 (Ω) = O(λ -1 n ). ( 2
λ 2 n Ω (a∇u n + b∇v n ) • ∇ϕ n dx = λ 2 n Ω a∇u n • ∇ϕ n dx + o(1) = -λ 2 n Ω div(a∇ϕ n )u n dx + o(1
λ 4 n Ω ρu n ϕ n dx + λ 2 n Ω div(a∇ϕ n )u n dx = o(1). (2.3.24)
By inserting the first equation of (2.3.12) in (2.3.24), we get

λ 2 n Ω |u n | 2 dx -iλ 3 n Ω (1 ω b)(x)ϕ n u n dx = o(1). (2.3.25)
We have

λ 3 n Ω (1 ω b)(x)ϕ n u n dx ≤ λ 2 n b ∞ Ω |ϕ n | 2 dx 1/2 λ n Ω (1 ω b)(x)|u n | 2 dx 1/2 .
So, from (2.3.8) (or (2.3.10) for assumption (A2)) and (2.3.21), we deduce that

iλ 3 n Ω (1 ω b)(x)ϕ n u n dx = o(1)
which together with (2.3.25) give the desired Estimation (2.3.16). Case 2. Under assumptions (LA) and (A2), there exists ω contained strictly in ω and satisfying the GCC. Similarly, we consider System (2.3.12) where U ≡ ω and ϕ n is its solution. Following the same technique as in Case 1, we get

iλ 3 n Ω (1 ω b)(x)ϕ n u n dx = o(1)
to continue in the same way and get Estimation (2.3.16).

Lemma 2.3.10. Assume that assumption (LA) and assumptions (A1) or (A2) hold. Then the solution (u n , v n ) ∈ D(A ) of (2.3.4)-(2.3.5) satisfies the following asymptotic behavior estimation

Ω |∇u n | 2 dx = o(1). (2.3.26)
Proof. Multiplying Equation (2.3.5) by u n and applying Green's formula, we obtain

iλ n Ω ρv n u n dx + Ω (a∇u n + b∇v n ) • ∇u n dx = λ -2 n Ω ρu n g n dx.
Taking into consideration (2.3.2), (2.3.7), (2.3.16) and the fact that g n converges to zero in L 2 (Ω), we deduce that

Ω a|∇u n | 2 dx = o(1)
which together with the fact that a(x) ≥ a 0 > 0 for almost every x ∈ Ω give the desired estimation (2. In fact, the result of Theorem 2.3.4 generalizes that of [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF], [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF] and [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF]. Indeed, the geometric situations covered by this theorem are richer than that considered in the previous references. For instance, in these literature mentioned, the damping is a neighborhood of the entire boundary, localized via smooth damping coefficient(see [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF]) or a part of the boundary satsifying the PMGC and localized via smooth or nonsmooth coefficient (see [START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF]). Also, in [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF], the author considered Ω to be a bounded convex domain, such that the damping region is either a neighborhood of the whole boundary or near a part of the boundary. However, in the second case, the partitions of Ω; Ω 1 and Ω 2 , are either convex curvilinear polygons or curved plane polyhedron. Also, the damping region Ω 1 is a neighborhood of a part Γ 1 = / 0 of the boundary Γ and satisfies the Γcondition. The difference is that we obtain our result when the damping region is localized internally faraway from the boundary (see assumption (A2)) or internally having an intercept with the boundary (see assumption (A1) and it obeys the GCC which is less restrictive than the PMGC or the Γcondition. Also, in [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF], it was necessary to have more viscoelastic wave than the elastic wave in order for the condition

(m • ν 2 )| Γ 2 ≤ 0; where m(x) = x -x 0 such that x 0 is a fixed point in R n , (n = 2, 3
), to apply (see Fig. 5). In our case, we do not care about the previous condition and we get the result despite the quantity of the viscoelastic wave, whether it's more or less than the elastic part (see Fig. 6). In addition, unlike the result of Theorem 4.1 in [START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF], our result holds for all n ≥ 2 and for non-convex domains. (ii) There is a relation between the geometric condition affecting the damping region and the smoothness of the boundary and regularity of the density and metric coefficients (ρ, a). Namely, if the Γcondition applies, then it is enough to have Lipschitz boundary conditions and coefficients of class C 1 . Although, knowing that the GCC is an optimal condition, so it costs more and thus needs a minimal regularity of C 2 coefficients and C 3 boundary (see [START_REF] Burq | Contrôlabilité exacte des ondes dans des ouverts peu réguliers[END_REF]). Remark that Burq-Dehman-Le Rousseau have dropped the previous condition to a boundary smoothness of class C 2 and coefficients of class C 1 (Burq-Dehman-Le Rousseau; Control for wave equation with rough coefficients, oral communication).

iii) It is unknown whether the polynomial decay rate obtained in (2.3.1) is optimal in the sense that, for any ε > 0, we can not expect the decay rate of type t -1-ε for all initial data U 0 ∈ D(A ). From our point of view, the energy decay rate (2.3.1) is not optimal, and we conjecture an optimal decay of type t -2 .

γ wave Γ 2 Γ 1 Fig.5 : A model satisfying (m • ν 2 )| Γ 2 ≤ 0 wave γ viscoelastic wave Γ 1 Γ 2 Fig. 6 : A model not satisfying (m • ν 2 )| Γ 2 ≤ 0 2.

Polynomial stability in the absence of GCC

As we have seen in Section 2.3, the GCC condition (also the PMGC and the SGCC) is sufficient to obtain a polynomial energy decay rate. Thus, in this section, we are interested in studying the nature of the energy decay rate whenever the geometric conditions are violated. We consider the following auxiliary problem, namely the wave equation on a nonempty open bounded domain Ω of Lipschitz boundary Γ with local viscous damping:

ρ(x)ϕ tt (x,t) -div(a∇ϕ) + (1 ω b)(x)ϕ t = 0 in Ω × R + , ϕ = 0 on Γ, (2.4.1)
where ω is an open subset of Ω not satisfying any geometry. Assume that the energy of the above wave equation with local viscous damping (2.4.1) decays polynomially as t -2/β ; β > 0, for smooth initial data, i.e. there exists C > 0 such that

E(t) ≤ C t 2/β ||U 0 || 2 D(A a ) , ∀t > 0. (LE)
Now we are in position to state our main result Theorem 2.4.1. Assume that assumptions (LA) and (LE) hold. Then there exists a constant C > 0 such that for all initial data U 0 ∈ D(A ), the energy of System (2.1.1) satisfies the following estimation

E(t) ≤ C t 1/β +1 ||U 0 || 2 D(A ) , β > 0, ∀t > 0. (2.4.2)
As mentioned in the proof of Theorem 2.3.4, it is enough for the proof of Theorem 2.4.1, according to Borichev and Tomilov, to show that sup

λ ∈R (iλ I -A ) -1 L (H ) = O |λ | ℓ , where ℓ = 2β + 2. (2.4.3)
This is checked by using a contradiction argument. Indeed, suppose there exists

{(λ n ,U n := (u n , v n ))} n≥1 ⊂ R * + × D (A ), such that λ n → +∞, ||U n || H = 1, (2.4.4) λ ℓ n (iλ n I -A )U n = ( f n , g n ) → 0 in H . (2.4.5)
Our aim is to show that ||(u n , v n )|| H → 0. This condition permits to conclude a contradiction with (2.4.4). By detailing equation (2.4.5), we get the following system

iλ n u n -v n = λ -ℓ n f n in H 1 0 (Ω), (2.4.6
)

iρλ n v n -div(a∇u n + b∇v n ) = λ -ℓ n ρg n in L 2 (Ω). (2.4.7)
The proof of Theorem (2.4.1) is divided into several Lemmas.

||u n || L 2 (Ω) = O λ -1 n , (2.4.8
)

||∇v n || L 2 (ω) = o λ -ℓ/2 n .
(2.4.9)

Proof. Using Equations (2.4.4) and (2.4.6), we deduce directly the first estimation (2.4.8). Now, taking the inner product of (2.4.5) with U n = (u n , v n ) in H , then using the fact that U n is uniformly bounded in H , we get

Ω b(x)|∇v n | 2 dx = ℜ (iλ n I -A )U n ,U n H = o λ -ℓ n .
It follows from the localization condition (LA) that

ω |∇v n | 2 = o λ -ℓ n . Lemma 2.4.3. Assume that conditions (LA) and (LE) hold. Then the solution (u n , v n ) ∈ D(A ) of Equations (2.4.6)-(2.4.7
) satisfies the following asymptotic behavior estimation

ω |u n | 2 = o λ -ℓ-2 n .
(2.4.10)

Proof. Using Poincaré's inequality and Equation (2.4.9), we obtain

λ ℓ n ω |v n | 2 dx = o(1). (2.4.11)
Multiplying Equation (2.4.6) by -ib(x)u n , and using estimation (2.4.11) and the fact that f n converges to zero in L 2 (Ω), we deduce from the above equation that

λ ℓ+2 n ω |u n | 2 dx = o(1).
The desired estimation (2.4.10) is established.

Lemma 2.4.4. Assume that conditions (LA) and (LE) hold. Then, for any

λ n ∈ R * + , the solution ϕ n ∈ H 2 (Ω) ∩ H 1 0 (Ω) of system ρλ 2 n ϕ n + div(a∇ϕ n ) -iλ n (1 ω b)(x)ϕ n = u n in Ω, ϕ n = 0 on Γ (2.4.12)
satisfies the following estimation

|λ n |||ϕ n || L 2 (Ω) + ||∇ϕ n || L 2 (Ω) ≤ M|λ n | β ||u n || L 2 (Ω) (2.4.13)
where M is positive constant independent of n.

Proof. As stated in the beginning of Section 2.4, the energy of System (2.4.1) decays polynomially as t -2/β with β > 0 (see condition (LE)). The associated energy space is given by H a = H 1 0 (Ω) × L 2 (Ω). So, following Borichev and Tomilov in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], the resolvent set of its associated operator A a defined by

D(A a ) = (H 2 (Ω) ∩ H 1 0 (Ω)) × H 1 0 (Ω), A (ϕ, ψ) = ψ, 1 ρ (div(a∇ϕ) -(1 ω b)(x)ψ) contains iR and |λ n | -β (iλ n I -A a ) -1
is uniformly bounded on the imaginary axis. Consequently, there exists a positive constant M > 0 independent of n such that sup

λ n ∈R |λ n | -β ||(iλ n I -A a ) -1 || L (H a ) ≤ M. (2.4.14)
Now, since u n ∈ L 2 (Ω), then there exists a unique (ϕ n , ψ n ) ∈ D(A a ) solution of

(iλ n I -A a )(ϕ n , ψ n ) = 0, - 1 ρ u n .
Equivalently, we have

iλ n ϕ n -ψ n = 0, iλ n ρψ n -div(a∇ϕ n ) + (1 ω b)(x)ψ n = -u n .
Finally, from (2.4.14), we deduce that

|λ n | -β ||(ϕ n , ψ n )|| H a ≤ M||u n || L 2 (Ω)
which gives the desired estimation. (2.4.15)

Proof.

Let ϕ n be the solution of System (2.4.1). Multiplying Equation (2.4.6) by iλ 3 n ρϕ n and Equation (2.4.7) by λ 2 n ϕ n and using Green's formula, we obtain

-λ 4 n Ω ρu n ϕ n dx -iλ 3 n Ω ρv n ϕ n dx = iλ 3-ℓ n Ω ρϕ n f n dx (2.4.16)
and

iλ 3 n Ω ρv n ϕ n dx + λ 2 n Ω (a∇u n + b∇v n ) • ∇ϕ n dx = λ 2-ℓ n Ω
ρϕ n g n dx.

(2.4.17)

Multiply Estimation (2.4.13) by λ n to get after using (2.4.8)

ϕ n L 2 (Ω) = O(λ β -2 n ) and ∇ϕ n L 2 (Ω) = O(λ β -1 n ). (2.4.18)
Now, taking into consideration Estimation (2.4.18), and the fact that

f n → 0 in H 1 0 (Ω), g n → 0 in L 2 (Ω), we get i Ω ρϕ n f n dx = o(λ -ℓ+β +1 n ) and i Ω ρϕ n g n dx = o(λ -ℓ+β n ).
(

λ 2 n Ω (a∇u n + b∇v n ) • ∇ϕ n dx = λ 2 n Ω a∇u n • ∇ϕ n dx + o(λ -ℓ/2+β +1 n ).
Applying Greens formula on the above equation gives

λ 2 n Ω (a∇u n + b∇v n ) • ∇ϕ n dx = -λ 2 n Ω div(a∇ϕ n )u n dx + o(λ -ℓ/2+β +1 n ). ( 2
λ 4 n Ω ρu n ϕ n dx + λ 2 n Ω div(a∇ϕ n )u n dx = o(λ -ℓ/2+β +1 n ). (2.4.21)
Inserting the first equation of (2.4.1) in (2.4.21), we get

λ 2 n Ω |u n | 2 dx -iλ 3 n Ω (1 ω b)(x)ϕ n u n dx = o(λ -ℓ/2+β +1 n ). (2.4.22)
We have

λ 3 n Ω (1 ω b)(x)ϕ n u n dx ≤ λ 3 n b ∞ Ω |ϕ n | 2 dx 1/2 Ω (1 ω b)(x)|u n | 2 dx 1/2 .
Using Estimations (2.4.10) and (2.4.18), we deduce from the above inequality that

iλ 3 n Ω (1 ω b)ϕ n u n dx = o(λ -ℓ/2+β n ) which together with (2.4.22) give λ 2 n Ω |u n | 2 dx = o(λ -ℓ/2+β +1 n ).
Thus, taking ℓ ≥ 2β + 2 gives the desired estimation (2.4.15). The proof of Lemma 2.4.3 is thus complete.

Lemma 2.4.6. Assume that conditions (LA) and (LE) hold. Then the solution

(u n , v n ) ∈ D(A ) of (2.4.6)- (2.4.7
) satisfies the following asymptotic behavior estimation

Ω |∇u n | 2 dx = o(1). (2.4.23)
Proof. Multiplying Equation (2.4.7) by u n and applying Green's formula, we obtain

iλ n Ω ρv n u n dx + Ω (a∇u n + b∇v n ) • ∇u n dx = λ -ℓ n Ω ρu n g n dx.
Taking into consideration (2.4.4), (2.4.9), (2.4.15) and the fact that g n converges to zero in L 2 (Ω), we deduce that

Ω a|∇u n | 2 dx = o(1)
which together with the fact that a(x) ≥ a 0 > 0 for almost every x ∈ Ω give the desired estimation (2.

Applications

Example 1. Consider System (2.1.1) with constant density and metric coefficients on a square domain with viscoelastic damping localized in a subdomain which contains a vertical strip. Thus, we construct the following geometry:

ω ⊃ ω 0 = {(x 1 , x 2 ) ∈ R 2 : ε 1 < x 1 < ε 2 , 0 < x 2 < L}. (LC1)
Setting a, ρ to be any strictly positive constants, it was proven that under conditions (LA) and (LC1), the energy of the wave equation (2.4.1) with local viscous damping decays polynomially as t -1 for smooth initial data (see Example 3 in [START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF]). Hence, taking β = 2 in Estimation (2.4.2) gives a polynomial decay rate of the energy of System (2.1.1) of type t -3 , i.e. there exists C > 0 such that

E(t) ≤ C t 1/3 ||U 0 || 2 D(A ) , ∀t > 0.
Example 2. Under the same conditions applied on System (2.1.1) in Example 1, we construct a different geometry:

ω ⊃ ω 0 = {(x 1 , x 2 ) ∈ R 2 : 0 < x 1 < ε 1 , 0 < x 2 < L}. (LC2)
Also, taking a, ρ to be any strictly positive constants in System (2.4.1), it was proven in [START_REF] Stahn | Optimal decay rate for the wave equation on a square with constant damping on a strip[END_REF] that the the energy of the wave equation (2.4.1) decays polynomially as t -4/3 for smooth initial data. Similarly, taking β = 3/2 in Estimation (2.4.2) improves the polynomial decay rate of the energy of System (2.1.1) to be of type t -2/5 i.e. there exists C > 0 such that

E(t) ≤ C t 2/5 ||U 0 || 2 D(A ) , ∀t > 0.

Stability of a string with localized Kelvin-Voigt damping

In this chapter, we consider the stabilization of a wave equation with localized internal viscoelastic damping of Kelvin-Voigt type in a bounded domain, in the 1-d case. Unlike [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF][START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF], we consider only one damping Kelvin-Voigt mechanism acting in the internal of the body. Using frequency domain arguments combined with the multiplier method, we establish an optimal energy decay rate of type t -4 .

Introduction

Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. In this chapter, we study the following elastic wave equation with local internal KelvinVoigt damping

           ρ U tt (x,t) -κ U x (x,t) + δ χ (α,β ) U xt (x,t) x = 0, (x,t) ∈ (0, L) × (0 + ∞), U(0,t) = U(L,t) = 0, t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), (3.1.1)
where

δ > 0, 0 < α < β < L, U = uχ (0,α) + vχ (α,β ) + wχ (β ,L) and χ (a,b) is the characteristic function of the interval (a, b).
Here ρ and κ are discontinuous positive functions, i.e., assume that there exist positive constant numbers ρ 1 , ρ 2 , ρ 3 , κ 1 , κ 2 and κ 3 such that

ρ =        ρ 1 , x ∈ (0, α), ρ 2 , x ∈ (α, β ), ρ 3 , x ∈ (β , L),
and κ =        κ 1 , x ∈ (0, α), κ 2 , x ∈ (α, β ), κ 3 , x ∈ (β , L).
In 1998, K. Liu and Z. Liu in [START_REF] Liu | Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping[END_REF] considered a wave equation with localized Kelvin-Voigt damping in the one-dimensional case. The dissipation is distributed on any subinterval of the region occupied by the beam and the damping coefficient is the characteristic function of the subinterval. They proved that the semigroup associated with the equation for the longitudinal motion of the beam is not exponentially stable. This result is due to the discontinuity of the viscoelastic materials. Later, in the one-dimensional case, it was found that the smoothness of the damping coefficient at the interface is an essential factor for the stability and the regularity of the solutions (see in [START_REF] Liu | Spectrum and Stability for Elastic Systems with Global or Local Kelvin-Voigt Damping[END_REF][START_REF] Liu | Exponential decay of energy of vibrating strings with local viscoelasticity[END_REF][START_REF] Liu | Eventual differentiability of a string with local Kelvin-Voigt damping[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF][START_REF] Liu | Stability of a String with Local Kelvin-Voigt Damping and Nonsmooth Coefficient at Interface[END_REF][START_REF] Zhang | Exponential stability of an elastic string with local Kelvin-Voigt damping[END_REF]).

Let us begin by recalling previous studies done on one-dimensional wave equations with Kelvin-Voigt damping. In 2011, Raposo et al. in [START_REF] Raposo | A Transmission Problem for Euler-Bernoulli beam with Kelvin-Voigt Damping[END_REF] considered the transmission problem for Euler-Bernoulli beam with one small part of Kelvin-Voigt damping material. They proved that the semigroup associated to the system is exponentially stable under the compatibility condition which states that u tx (L 0 ,t) = 0; t > 0, where L 0 is the point of separation between elastic and viscoelastic waves and u(x,t) denotes the longitudinal displacement of the beam. However, in 2013, Alves et al. in [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF] considered a model with only two parts where one part is elastic and the other is viscoelastic with Kelvin-Voigt constitutive relation. Under the assumption that the two materials are distributed equally, they established an optimal polynomial decay rate of the solution of type t -2 .

In 2014, Alves et al. in [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF] considered also the transmission problem of a material composed of three components; a Kelvin-Voigt viscoelastic material and two elastic materials where one of them is affected by frictional damping and the second is without any dissipation. They proved exponential stability of the system whenever the viscoelastic material is not in the middle of the material and they established a polynomial decay rate of the energy of type t -4 in the conjugate case. And hence, we are in the case of two controls distributed in a way that there is always damping at least on one part of the boundary. Then, in 2015, Hassine in [START_REF] Hassine | Stability of elastic transmission systems with a local Kelvin-Voigt damping[END_REF] examined the longitudinal and transversal vibrations of the Euler-Bernoulli beam with Kelvin-Voigt damping distributed locally in one side. He used the frequency domain approach to prove that the semigroup associated with the equation for transversal motion is exponentially stable, while that associated to the equation of longitudinal motion is polynomially stable. Also, in the same year, Hassine in [START_REF] Hassine | Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping[END_REF] studied the energy decay estimates of the beam equation coupled with a wave equation in an elastic beam. He established polynomial stability in both cases when the Kelvin-Voigt damping acts through the plate equation and when this dissipation acts through the wave equation. Later, in 2017, Oquendo in [START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF] considered two transmission problems with two damping mechanisms. He proved nonuniform stability and an optimal polynomial decay rate of the solution of type t -2 when both frictional and Kelvin-Voigt damping act on the same part of the body. On the contrary, he established a exponential stability whenever the dissipations act on complementary parts. Next, in 2018, Maryati et al. in [START_REF] Maryati | Stability of an N-component timoshenko beam with localized Kelvin-Voigt and frictional dissipation[END_REF] studied the stability of an N-component Timoshenko beam with localized Kelvin-Voigt and frictional dissipation. They proved that the type of stability depends on the position of each component where they showed that the model is exponentially stable if and only if any elastic component is connected with at least one component with frictional damping. Otherwise, the polynomial decay rate of type t -2 is obtained. Finally, in 2018, Rivera et al. in [START_REF] Rivera | Stability to localized viscoelastic transmission problem[END_REF] studied the wave propagation over materials consisting of three components; two of them formed by viscoelastic materials of Kelvin-Voigt type and the third is purely elastic. They attained exponential stability of the system in the case where the discontinuous viscoelastic component is not in the middle of the bar. Otherwise, they established polynomial decay rate of the solution of type t -2 . Thus, it seems to us that there is no result in the literature concerning the study of the the optimal energy decay rate of the transmission problem of wave equations with locally internal Kelvin-Voigt type damping, i.e. when the damping is in the middle of the material without any dissipation in the other parts. So, we are interested in studying the transmission problem of the one-dimensional wave equation with internal localized Kelvin-Voigt damping.

In [START_REF] Muní | Transmission Problems in (Thermo)Viscoelasticity with Kelvin-Voigt Damping: Nonexponential, Strong, and Polynomial Stability[END_REF], Rivera and Racke considered System (3.1.1) in n-dimensional case, where n ≥ 1. First, they showed that the system is strongly stable. Next, they showed that the system is not uniformly stable. Hence, they established a polynomial energy decay rate of type t -2 3 . But this result is not optimal. Later, in 2019, Ammari et al. in [START_REF] Ammari | Stability of the wave equations on a tree with local Kelvin-Voigt damping[END_REF] studied the stability of wave equations on a tree with local Kelvin-Voigt damping on some of the edges. In fact, they introduced a planar tree-shaped network; say T , of N elastic strings; N ≥ 2 and they considered the following initial and boundary value problem:

                             ∂ 2 u ᾱ ∂ 2 t (x,t) -∂ ∂ x ∂ u ᾱ ∂ x + a ᾱ (x) ∂ 2 u ᾱ ∂ x∂t (x,t) = 0, 0 < x < ℓ ᾱ , t > 0, ᾱ ∈ I := I M ∪ I S , u(0,t) = 0, u ᾱ (ℓ ᾱ ,t) = 0, ᾱ ∈ I S , t > 0, u ᾱ•β (0,t) = u ᾱ (ℓ ᾱ ,t), t > 0, β = 1, 2, . . . , m ᾱ , ᾱ ∈ I M , m ᾱ ∑ β =1 ∂ u ᾱ•β ∂ x (0,t) + a ᾱ•β (0) ∂ 2 u ᾱ•β ∂ x∂t (0,t) = ∂ u ᾱ ∂ x (ℓ ᾱ ,t) + a ᾱ (ℓ ᾱ ) ∂ 2 u ᾱ ∂ x∂t (ℓ ᾱ ,t), t > 0, ᾱ ∈ I M , u ᾱ (x, 0) = u 0 ᾱ (x), ∂ u ᾱ ∂t (x, 0) = u 1 ᾱ (x), 0 < x < ℓ ᾱ , ᾱ ∈ I,
where ᾱ = (α 1 , . . . , α k ), M is the set of interior vertices of T , S is the set of exterior vertices of T , I M is the index set of M, I S is the index set of S, e ᾱ•β ; β = 1, . . . , m ᾱ , denotes the edges different from e ᾱ and branching out from O ᾱ which represents the vertex of T with index ᾱ, u ᾱ : [0, ℓ ᾱ ] × (0, ∞) → R, ᾱ ∈ I be the transverse displacement with index ᾱ, a ᾱ ∈ L ∞ (0, ℓ ᾱ ) and there exists a subinterval ω ᾱ of (0, ℓ ᾱ ) such that a ᾱ (x) > 0 a.e. on ω ᾱ . In this case, e ᾱ is called a Kelvin-Voigt edge. However, if a ᾱ is zero, then e ᾱ is a purely elastic edge.

Concerning their result, under the following property

∀ ᾱ ∈ I, a ′ ᾱ , a ′′ ᾱ ∈ L ∞ (0, ℓ ᾱ ) and ∀ ᾱ ∈ I M , a ᾱ (ℓ ᾱ ) - m ᾱ ∑ β =1 a ′ ᾱ•β (0) ≤ 0, (P)
they established a polynomial energy decay rate of type t -4 whenever the damping coefficient is discontinuous at least at an inner node of T . They obtained their result by using frequency domain approach combined with the multiplier method, where they solved the main conflict using Gagliardo-Nirenberg inequality [START_REF] Liu | Semigroups associated with dissipative systems[END_REF]. However, it was not proven that this decay rate is optimal.

In this chapter, we restrict [START_REF] Ammari | Stability of the wave equations on a tree with local Kelvin-Voigt damping[END_REF] on a one elastic wave equation with discontinuous localized viscoelastic material. Thus, we study the transmission problem of the one-dimensional wave equation with internal localized Kelvin-Voigt damping (see System (3.1.1)) as in Figure 3.1. We divide the bar into 3 pieces; two pieces are purely elastic and one piece is of viscoelastic material. Using frequency domain approach combined with a different multiplier method, we acquire the asymptotic behavior estimations of the solution at the points of interface α and β , and thus we show that the energy of the System (3.1.1) has a polynomial decay rate of type t -4 . In addition, unlike [START_REF] Ammari | Stability of the wave equations on a tree with local Kelvin-Voigt damping[END_REF], in this chapter, we prove that this decay rate is optimal. Note that the technique used here was very helpful in proving Theorems 4.2.7 and 4.2.14 in Chapter 4. Last but not least, in addition to the previously cited papers, we briefly recall some previous studies done on N-dimensional wave equation with Kelvin-Voigt damping. In 1988, Huang in [START_REF] Huang | On the Mathematical Model for Linear Elastic Systems with Analytic Damping[END_REF] considered N-dimensional wave equation with global Kelvin-Voigt damping and they proved that the system is analytic. However, when the damping region is a neighborhood of the whole or part of the boundary and the damping coefficient is regular enough, in [START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF][START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF] it was proved that the energy of the system goes exponentially to zero as t goes to infinity for all usual initial data. In 2018, Ammari et al. in [START_REF] Ammari | Stabilization for the wave equation with singular kelvinâ Ȃ Şvoigt damping[END_REF] considered the case where the Kelvin-Voigt damping is localized in a subdomain ω faraway from the boundary without geometric conditions. They established a logarithmic energy decay rate for smooth initial data. Finally, in 2020, Wehbe et al. in [START_REF] Wehbe | Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions[END_REF] studied the stabilization of the wave equation with KelvinVoigt damping in a bounded domain with non-smooth damping coefficient. A polynomial energy decay estimate was established provided that the dissipation is internal and satisfies or does not satisfy any geometric conditions.

• • α β v(x) u(x) w(x) 0 L Elastic part Viscoelastic part Elastic part
This chapter is organized as follows: In Section 3.2, we set the framework of System (3.1.1). In Section 3.3, we estimate the upper bound of the polynomial decay rate. In Section 3.4, we establish an optimal polynomial energy decay rate of type t -4 (see Theorem 3.4.1).

Preliminaries and Well-Posedness

We start this section by formulating System (3.1.1) as an abstract Cauchy problem. For this aim, we denote the longitudinal displacement by U and this displacement is divided into three parts

U(x,t) =        u(x,t), (x,t) ∈ (0, α) × (0, +∞), v(x,t), (x,t) ∈ (α, β ) × (0, +∞), w(x,t), (x,t) ∈ (β , L) × (0, +∞).
In this case, System (3.1.1) is equivalent to the following system ρ 1 u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞), (3.2.1)

ρ 2 v tt -(κ 2 v x + δ v xt ) x = 0, (x,t) ∈ (α, β ) × (0, +∞), (3.2.2) ρ 3 w tt -κ 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞), (3.2.3)
with the Dirichlet boundary conditions

u(0,t) = w(L,t) = 0, t ∈ (0, +∞). (3.2.4)
The transmission conditions are given by

       u(α,t) = v(α,t), v(β ,t) = w(β ,t), t ∈ (0, +∞), κ 2 v x (α,t) + δ v xt (α,t) = κ 1 u x (α,t), t ∈ (0, +∞), κ 2 v x (β ,t) + δ v xt (β ,t) = κ 3 w x (β ,t), t ∈ (0, +∞). (3.2.5) System (3.2.1)-(3.2.5) is subjected to the following initial conditions        (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, α), (v(x, 0), v t (x, 0)) = (v 0 (x), v 1 (x)), x ∈ (α, β ), (w(x, 0), w t (x, 0)) = (w 0 (x), w 1 (x)), x ∈ (β , L). (3.2.6)
The energy of solutions of the System (3.2.1)-(3.2.6) is defined by:

E(t) = κ 1 2 α 0 |u x | 2 dx + ρ 1 2 α 0 |u t | 2 dx + κ 2 2 β α |v x | 2 dx + ρ 2 2 β α |v t | 2 dx + κ 3 2 L β |w x | 2 dx + ρ 3 2 L β |w t | 2 dx.
H m = H m (0, α) × H m (α, β ) × H m (β , L), m = 1, 2, L 2 = L 2 (0, α) × L 2 (α, β ) × L 2 (β , L), H 1 L = {(u, v, w) ∈ H 1 | u(0) = w(L) = 0, u(α) = v(α), v(β ) = w(β )}.
The Hilbert energy space is given by

H = H 1 L × L 2
and is equipped with the following inner product

U, Ũ H = κ 1 α 0 u x ũx dx + κ 2 β α v x ṽx dx + κ 3 L β w x wx dx + ρ 1 α 0 y ỹdx + ρ 2 β α zzdx + ρ 3 L β φ φ dx,
where U = (u, v, w, y, z, φ ) ∈ H and Ũ = ( ũ, ṽ, w, ỹ, z, φ ) ∈ H . We use U H to denote the corresponding norm. We define the linear unbounded operator A : D(A ) ⊂ H -→ H by:

D(A ) = {(u, v, w, y, z, φ ) ∈ H | (y, z, φ ) ∈ H 1 L , (u, κ 2 v + δ z, w) ∈ H 2 , κ 2 v x (α) + δ z x (α) = κ 1 u x (α), κ 2 v x (β ) + δ z x (β ) = κ 3 w x (β )}
and for all U = (u, v, w, y, z, φ ) ∈ D(A ),

A U = y, z, φ , κ 1 ρ 1 u xx , 1 ρ 2 (κ 2 v x + δ z x ) x , κ 3 ρ 3 w xx . If U = (u, v, w, u t , v t , w t ) is a regular solution of System (3.2.1)-(3.2.6
), then we transform this system into the following initial value problem

U t = A U, U(0) = U 0 , (3.2.7)
where

U 0 = (u 0 , v 0 , w 0 , u 1 , v 1 ,
U ∈ C 0 (R + , H ). Moreover, if U 0 ∈ D(A ), then U ∈ C 1 (R + , H ) ∩C 0 (R + , D(A )).
Now, for proving the polynomial stability of the C 0 -semigroup e tA t≥0 , we will rely on the frequency domain approach method that has been obtained by Borichev and Tomilov in [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Bátkai | Polynomial stability of operator semigroups[END_REF][START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF][START_REF] Liu | Characterization of polynomial decay rate for the solution of linear evolution equation[END_REF]).

Upper bound estimation of the polynomial decay rate

In this section, we estimate the upper bound of the polynomial decay rate. Our main result in this part is the following theorem. Theorem 3.3.1. For ε > 0 (small enough), we cannot expect the energy decay rate t -4-ε for all initial data U 0 ∈ D (A ) and for all t > 0.

Proof. According to the result of Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], it suffices to show the existence of sequences

(λ n ) n ⊂ R * + with λ n → +∞, (U n ) n ⊂ D (A ), and (F n ) n ⊂ H such that (iλ n I -A )U n = F n is bounded in H and lim n→∞ λ -1 2 +ε n U n H = ∞. (3.3.1) Set λ n = 1 α 2 πn + 1 √ n κ 1 ρ 1 , n ∈ N * (3.3.2)
and

F n (x) = 0, 0, 0, - 2κ 1 ρ 1 sin(η n x), 0, 0 , U n = (u n , v n , w n , iλ n u n , iλ n v n , iλ n w n ), such that                        u n (x) = sin (η n x) sin (η n α) A 1,n + α cos (η n α) η n - x cos (η n x) η n , x ∈ (0, α), v n (x) = sinh (ξ n (β -x)) A 1,n + sinh (ξ n (α -x)) A 2,n sinh (ξ n (β -α)) , x ∈ (α, β ), w n (x) = sin (γ n (L -x)) sin (γ n (L -β )) A 2,n , x ∈ (β , L), (3.3.3)
where

             η n = λ n ρ 1 κ 1 , γ n = λ n ρ 3 κ 3 , ξ n = ρ 2 λ n 2 δ 2 + κ 2 2 λ -2 n δ 2 + κ 2 2 λ -2 n -κ 2 λ -1 n + i δ 2 + κ 2 2 λ -2 n + κ 2 λ -1 n , (3.3.4) and                                    A 1,n = α sin (η n α) - cos (η n α) η n ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α)) + γ n κ 3 cot (γ n (L -β )) κ 3 Det(M n ) , A 2,n = (κ 2 + iδ λ n ) α sin (η n α) - cos (η n α) η n κ 3 Det(M n ) sinh (ξ n (β -α))
,

Det(M n ) = -κ -1 1 γ n (κ 1 η n cot (η n α) + ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α))) cot (γ n (L -β )) -κ -1 3 η n ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α)) cot (η n α) -κ -1 1 κ -1 3 ξ 2 n (κ 2 + iδ λ n ) 2 . ( 3
         δ 2 + κ 2 2 λ -2 n -κ 2 λ -1 n > 0, ξ 2 n = - ρ 2 λ 2 n (κ 2 -iδ λ n ) κ 2 2 + δ 2 λ 2 n = - ρ 2 λ 2 n κ 2 + iδ λ n , sin (η n α) = sin 1 √ n = 0, sin (γ n (L -β )) = 0, sinh (ξ n (β -α)) = 0.
The proof is divided into three steps.

Step 1. In this step, let us show that

(iλ n I -A )U n = F n .
Detailing iλ n U n -A U n , we get

iλ n U n -A U n = 0, 0, 0, - κ 1 ρ 1 (u n ) xx + η 2 n u n , - κ 2 + iδ λ n ρ 2 (v n ) xx -ξ 2 n v n , - κ 3 ρ 3 (w n ) xx + γ 2 n w n .
Inserting (3.3.3) in the above equation, gives after direct computation that

iλ n U n (x) -A U n (x) = 0, 0, 0, - 2κ 1 ρ 1 sin(η n x), 0, 0 = F n (x).
Step 2. In this step, we show that

U n ∈ D (A ) . Clear that (u n , (κ 2 + iδ λ n ) v n , w n ) ∈ H 2 , u n (0) = w n (L) = 0, u n (α) = v n (α) = A 1,n and w n (β ) = v n (β ) = A 2,n , so to get U n ∈ D (A ), we need to show (u n ) x (α) - 1 κ 1 (κ 2 + iδ λ n ) (v n ) x (α) = 0 and (w n ) x (β ) - 1 κ 3 (κ 2 (β ) + iδ λ n ) (v n ) x (β ) = 0. (3.3.6)
For this aim, detailing

(u n ) x (α) - 1 κ 1 (κ 2 + iδ λ n ) (v n ) x (α) and (w n ) x (β ) - 1 κ 3 (κ 2 (β ) + iδ λ n ) (v n ) x (β ) with u n ,
v n and w n are given in (3.3.3), we get the following system

M n A 1,n A 2,n =   cos(η n α) η n -α sin(η n α) 0   , (3.3.7)
where

M n =    η n cot (η n α) + ξ n (κ 2 +iδ λ n ) coth(ξ n (β -α)) κ 1 -ξ n (κ 2 +iδ λ n ) κ 1 sinh(ξ n (β -α)) ξ n (κ 2 +iδ λ n ) κ 3 sinh(ξ n (β -α)) -γ n cot (γ n (L -β )) -(κ 2 +iδ λ n )ξ n coth(ξ n (β -α)) κ 3    .
Denoting the determinant of a matrix M n by Det(M n ), one gets that

Det(M n ) = -κ -1 1 γ n (κ 1 η n cot (η n α) + ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α))) cot (γ n (L -β )) -κ -1 3 η n ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α)) cot (η n α) -κ -1 1 κ -1 3 ξ 2 n (κ 2 + iδ λ n ) 2 .
Then, solving System (3.3.7), we get A 1,n and A 2,n as given in (3.3.5). Hence, we get (3.3.6) and consequently, we get U n ∈ D (A ).

Step 3. In this step, we prove (3.3.1). First, from (3.3.3), we get

η n u n (x) = sin (η n x) η n A 1,n sin (η n α) + α cot (η n α) -x cos (η n x) . (3.3.8)
From (3.3.5), we get

η n A 1,n sin (η n α) = η n α 1 + cot 2 (η n α) -cot (η n α) ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α)) + γ n κ 3 cot (γ n (L -β )) κ 3 Det(M n ) .
Consequently, from the above equation and (3.3.5), we obtain

η n A 1,n sin (η n α) + α cot (η n α) = A 3,n (A 4,n + 1) + α cot (η n α) , (3.3.9)
where

A 3,n = - κ 1 η n α 1 + cot 2 (η n α) -cot (η n α) κ 1 η n cot (η n α) + ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α)) (3.3.10)
and

A 4,n = A 5,n coth 2 (ξ n (β -α)) -1 A 6,n + cot (γ n (L -β )) , (3.3.11) such that          A 5,n = ξ 2 n (κ 2 + iδ λ n ) 2 κ 3 γ n (κ 1 η n cot (η n α) + ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α))) , A 6,n = κ 1 η n ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α)) cot (η n α) + ξ 2 n (κ 2 + iδ λ n ) 2 κ 3 γ n (κ 1 η n cot (η n α) + ξ n (κ 2 + iδ λ n ) coth (ξ n (β -α))) . (3.3.12)
Now, inserting (3.3.2) in (3.3.4), then using asymptotic expansion, we get

         η n = 1 α 2 πn + 1 √ n , γ n = 1 α 2 πn + 1 √ n κ 1 ρ 3 ρ 1 κ 3 , ξ n = πρ 2 α δ 1 2 κ 1 ρ 1 1 4 (1 + i) √ n + O n -1 2 . (3.3.13)
Using asymptotic expansion, we obtain

cot (η n α) = √ n + O n -1 2 , coth (ξ n (β -α)) = 1 + o(1
A 3,n = √ n (c 1 + o(1)) , A 5,n = √ n (c 2 + o(1)) , A 6,n = √ n (c 3 + o(1)) , (3.3.15)
where c 1 , c 1 , c 3 ∈ C * and given by

                       c 1 = α κ 1 (1 -i) δ κ 1 ρ 1 3 4 πρ 2 αδ 1 2 -κ 1 , c 2 = 2iδ κ 1 ρ 2 π αρ 1 √ κ 3 ρ 3 (1 -i) δ κ 1 ρ 1 3 4 πρ 2 αδ 1 2 -κ 1 , c 3 = δ κ 1 (1 -i) αρ 3 4 1 κ 1 4 1 πρ 2 αδ 1 2 + 2iπρ 2 αρ 1 √ κ 3 ρ 3 (1 -i) δ κ 1 ρ 1 3 4 πρ 2 αδ 1 2 -κ 1
.

Substituting
A 4,n = (c 2 + o(1)) coth 2 ρ 2 π α δ 1 2 κ 1 ρ 1 1 4 (1 + i) √ n + O n -1 2 (β -α) -1 c 3 + o(1) + cot κ 4 2 πn+ 1 √ n √ n , (3.3.16)
where

κ 4 = L -β α κ 1 ρ 3 ρ 1 κ 3 .
We have the following two cases: Case 1, if κ 4 is a rational number, then there exists a subsequence of {λ n }, still denoted by λ n , such that sin (4 κ 4 πn) = 0. In this case, we have

1 √ n cot κ 4 2 πn + 1 √ n = 1 √ n cos (4 κ 4 πn) + cos 2κ 4 √ n sin (4 κ 4 πn) + sin 2κ 4 √ n = κ -1 4 + O n -1 .
Inserting the above estimation in (3.3.16), then using the fact that κ -1 4 + c 3 = 0 and c 2 = 0, we obtain

A 4,n → 0 as n → +∞.
Case 2, if κ 4 is an irrational number, then there exists a subsequence of {λ n }, still denoted by λ n , such that sin (4 κ 4 πn) ≥ c > 0. In this case, we have

1 √ n cot κ 4 2 πn + 1 √ n = 1 √ n cos (4 κ 4 πn) + cos 2κ 4 √ n sin (4 κ 4 πn) + sin 2κ 4 √ n → 0 as n → +∞.
Consequently, inserting the above estimation in (3.3.16), then using the fact that c 3 = 0 and c 2 = 0, we get A 4,n → 0 as n → +∞. Therefore, in both cases, we have

A 4,n → 0 as n → +∞. ( 3
η n A 1,n sin (η n α) + α cot (η n α) = √ n (c 4 + o(1)) , ( 3
c 4 = c 1 + α = (1 -i) αδ κ 1 ρ 1 3 4 πρ 2 αδ 1 2 (1 -i) δ κ 1 ρ 1 3 4 πρ 2 αδ 1 2 -κ 1 . Substituting (3.3.18) in (3.3.8), we get η n u n (x) = √ n (c 4 + o(1)) sin (η n x) -x cos (η n x) , x ∈ (0, α).
Consequently,

α 0 |η n u n (x)| 2 dx ∼ α n 2 |c 4 | 2 + o(1) . Since U n H ≥ ρ 1 α 0 |η n u n (x)| 2 dx ∼ √ n |c 4 | √ 2αρ 1 2 + o(1) ∼ λ n ,
then for all ε > 0 (small enough), we have λ

-1 2 +ε n U n H ∼ n ε → +∞,
hence, we get (3.3.1). The result follows from Borichev and Tomilov theorem (see Theorem 1.3.8). Note that estimation (3.3.1) also implies that our system is non-uniformly stable.

Polynomial Stability

In this section, we will prove the polynomial stability of System (3.2.1)-(3.2.6). Our main result in this part is the following theorem. Theorem 3.4.1. For all initial data U 0 ∈ D(A ), there exists a constant C > 0 independent of U 0 such that the energy of System (3.2.1)-(3.2.6) has the optimal polynomial decay rate

E(t) ≤ C t 4 ||U 0 || 2 D(A ) , ∀t > 0. (3.4.1)
Similar to [START_REF] Muní | Transmission Problems in (Thermo)Viscoelasticity with Kelvin-Voigt Damping: Nonexponential, Strong, and Polynomial Stability[END_REF] we can prove that iR ⊂ ρ(A ), then for the proof of Theorem 3.4.1, according to Theorem 1.3.8, we need to prove that sup

λ ∈R (iλ I -A ) -1 L (H ) = O |λ | 1 2 . (3.4.2)
We will argue by contradiction. Indeed, suppose there exists

{(λ n , U n := (u n , v n , w n , y n , z n , φ n ))} n≥1 ⊂ R * + × D (A ), such that λ n → +∞, U n H = 1 (3.4.3)
and there exists sequence

F n := ( f 1,n , f 2,n , f 3,n , f 4,n , f 5,n , f 6,n ) ∈ H , such that λ 1 2 n (iλ n I -A )U n = F n → 0 in H . (3.4.4)
We will check condition (3.4.2) by finding a contradiction with U n H = 1 such as U n H = o(1). From now on, for simplicity, we drop the index n. By detailing Equation (3.4.4), we get the following system

iλ u -y = λ -1 2 f 1 in H 1 (0, α), (3.4.5) iλ v -z = λ -1 2 f 2 in H 1 (α, β ), (3.4.6) iλ w -φ = λ -1 2 f 3 in H 1 (β , L), (3.4.7) iλ ρ 1 y -κ 1 u xx = ρ 1 λ -1 2 f 4 in L 2 (0, α), (3.4.8) iλ ρ 2 z -(κ 2 v x + δ z x ) x = ρ 2 λ -1 2 f 5 in L 2 (α, β ), (3.4.9) iλ ρ 3 φ -κ 3 w xx = ρ 3 λ -1 2 f 6 in L 2 (β , L). (3.4.10)
Remark that, since U = (u, v, w, y, z, φ ) ∈ D(A ), we have the following boundary conditions

|u x (α)| = κ -1 1 |κ 2 v x (α) + δ z x (α)|, |y(α)| = |z(α)| , |w x (β )| = κ -1 3 |κ 2 v x (β ) + δ z x (β )|, |z(β )| = |φ (β )| . (3.4.11)
Taking the inner product of (3.4.4) with U in H , then using the fact that U is uniformly bounded in H , we get

β α |z x | 2 dx = -δ -1 ℜ (iλ I -A )U, U H = o λ -1 2 . (3.4.12)
Next, from (3.4.6), (3.4.12) and the fact that ( f 2 ) x → 0 in L 2 (α, β ), we get

β α |v x | 2 dx = o λ -5 2 . ( 3
|z(α)| 2 = o λ -1 , |z(β )| 2 = o λ -1 , (3.4.15) |κ 2 v x (α) + δ z x (α)| = o(1), |κ 2 v x (β ) + δ z x (β )| = o(1). (3.4.16) Proof. Let g ∈ C 1 ([α, β ]) such that g(β ) = -g(α) = 1, max x∈[α,β ] |g(x)| = c g and max x∈[α,β ] |g ′ (x)| = c g ′ ,
where c g and c g ′ are strictly positive constant numbers independent from λ . The proof is divided into three steps.

Step 1. In this step, we prove the following asymptotic behavior estimate

|z(β )| 2 + |z(α)| 2 ≤ ρ 2 λ 1 2 2 + 2c g ′ β α |z| 2 dx + o λ -1 . (3.4.17)
First, from (3.4.6), we have

z x = iλ v x -λ -1 2 ( f 2 ) x in L 2 (α, β ). (3.4.18)
Multiplying (3.4.18) by 2 gz and integrating over (α, β ), then taking the real part, we get

β α g(x) (|z| 2 ) x dx = ℜ 2iλ β α g(x) v x zdx -ℜ 2λ -1 2 β α g(x) ( f 2 ) x zdx .
Using by parts integration in the left hand side of above equation, we get

g(x) |z| 2 β α = β α g ′ (x) |z| 2 dx + ℜ 2iλ β α g(x) v x zdx -ℜ 2λ -1 2 β α g(x) ( f 2 ) x zdx , consequently, |z(β )| 2 + |z(α)| 2 ≤ c g ′ β α |z| 2 dx + 2λ c g β α |v x | |z| dx + 2λ -1 2 c g β α |( f 2 ) x | |z| dx. (3.4.19)
On the other hand, we have

2λ c g |v x ||z| ≤ ρ 2 λ 1 2 |z| 2 2 + 2λ 3 2 c 2 g ρ 2 |v x | 2 and 2λ -1 2 |( f 2 ) x ||z| ≤ c g ′ |z| 2 + c 2 g λ -1 c g ′ |( f 2 ) x | 2 .
Inserting the above equation in (3.4.19), then using (3.4.13) and the fact that

( f 2 ) x → 0 in L 2 (α, β ), we get |z(β )| 2 + |z(α)| 2 ≤ ρ 2 λ 1 2 2 + 2 c g ′ β α |z| 2 dx + o λ -1 ,
hence, we get (3.4.17).

Step 2. In this step, we prove the following asymptotic behavior estimate

|κ 2 v x (α) + δ z x (α)| 2 + |κ 2 v x (β ) + δ z x (β )| 2 ≤ ρ 2 λ 3 2 2 β α |z| 2 dx + o (1) . (3.4.20)
First, multiplying (3.4.9) by -2 g (κ 2 v x + δ z x ) and integrating over (α, β ), then taking the real part, we get

β α g(x) |κ 2 v x + δ z x | 2 x dx = 2ℜ iλ ρ 2 β α g(x)z (κ 2 v x + δ z x ) dx -2ρ 2 λ -1 2 ℜ β α g(x) f 5 (κ 2 v x + δ z x ) dx .
Using by parts integration in the left hand side of above equation, we get

g(x) |κ 2 v x + δ z x | 2 β α = β α g ′ (x) |κ 2 v x + δ z x | 2 dx + 2ℜ iλ ρ 2 β α g(x) z (κ 2 v x + δ z x ) dx -2ρ 2 λ -1 2 ℜ β α g(x) f 5 (κ 2 v x + δ z x ) dx , consequently, |κ 2 v x (β ) + δ z x (β )| 2 + |κ 2 v x (α) + δ z x (α)| 2 ≤ c g ′ β α |κ 2 v x + δ z x | 2 dx + 2λ ρ 2 c g β α |z| |κ 2 v x + δ z x | dx +2ρ 2 λ -1 2 c g β α | f 5 | |κ 2 v x + δ z x | dx.
|κ 2 v x (β ) + δ z x (β )| 2 + |κ 2 v x (α) + δ z x (α)| 2 ≤ 2λ ρ 2 c g β α |z| |κ 2 v x + δ z x | dx + o λ -1 2 . (3.4.21)
On the other hand, we have

2λ ρ 2 c g |z| |κ 2 v x + δ z x | ≤ ρ 2 λ 3 2 2 |z| 2 + 2ρ 2 λ 1 2 c 2 g |κ 2 v x + δ z x | 2 .
Inserting the above equation in (3.4.21), then using (3.4.12)-(3.4.13), we get

|κ 2 v x (α) + δ z x (α)| 2 + |κ 2 v x (β ) + δ z x (β )| 2 ≤ ρ 2 λ 3 2 2 β α |z| 2 dx + o (1) ,
hence, we get (3.4.20).

Step 3. In this step, we prove the asymptotic behavior estimations of (3.4.14)- (3.4.16). First, multiplying (3.4.9) by -iλ -1 ρ -1 2 z and integrating over (α, β ), then taking the real part, we get

β α |z| 2 dx = -ℜ iρ -1 2 λ -1 β α (κ 2 v x + δ z x ) x z dx -ℜ i λ -3 2 β α f 5 z dx , consequently, β α |z| 2 dx ≤ ρ -1 2 λ -1 β α (κ 2 v x + δ z x ) x z dx + λ -3 2 β α | f 5 | |z| dx. (3.4.22)
From the fact that z is uniformly bounded in L 2 (α, β ) and

f 5 → 0 in L 2 (α, β ), we get λ -3 2 β α | f 5 | |z| dx = o λ -3 2 . (3.4.23)
On the other hand, using by parts integration and (3.4.12)-(3.4.13), we get

β α (κ 2 v x + δ z x ) x z dx = [(κ 2 v x + δ z x ) z] β α - β α (κ 2 v x + δ z x ) z x dx ≤ |κ 2 v x (β ) + δ z x (β )| |z(β )| + |κ 2 v x (α) + δ z x (α)| |z(α)| + β α |κ 2 v x + δ z x | |z x | dx ≤ |κ 2 v x (β ) + δ z x (β )| |z(β )| + |κ 2 v x (α) + δ z(α)| |z(α)| + o λ -1 2 . ( 3
β α |z| 2 dx ≤ ρ -1 2 λ -1 |κ 2 v x (β ) + δ z x (β )| |z(β )| + ρ -1 2 λ -1 |κ 2 v x (α) + δ z x (α)| |z(α)| + o λ -3 2 . (3.4.25) Now, for ζ = β or ζ = α, we have ρ -1 2 λ -1 |κ 2 v x (ζ ) + δ z x (ζ )| |z(ζ )| ≤ λ -1 2 2ρ 2 |z(ζ )| 2 + λ -3 2 2ρ 2 |κ 2 v x (ζ ) + δ z x (ζ )| 2 .
Rayan Nasser Page 65 of 161 rayan.nasser94@hotmail.com

β α |z| 2 dx ≤ λ -1 2 2ρ 2 |z(α)| 2 + |z(β )| 2 + λ -3 2 2ρ 2 |κ 2 v x (α) + δ z x (α)| 2 + |κ 2 v x (β ) + δ z x (β )| 2 + o λ -3 2 .
Next, inserting (3.4.17) and (3.4.20) in the above inequality, we obtain ). Thus, the proof of the Lemma is complete.

β α |z| 2 dx ≤ 1 2 + c g ′ ρ 2 λ 1 2 β α |z| 2 dx + o λ -3 2 , consequently, 1 2 - c g ′ ρ 2 λ 1 2 β α |z| 2 dx ≤ o λ -3 2 . Since λ → +∞, by choosing λ > 4ρ -2 2 c 2 g ′ , we get 0 < 1 2 - c g ′ ρ 2 λ 1 2 β α |z| 2 dx ≤ o λ -
Remark 3.4.3. In Lemma 3.4.2, the choice of g is not unique. For example, we can take From (3.4.5), we deduce that

g(x) = - 2 (β -x) β -α + 1 to get g(β ) = -g(α) = 1, g ∈ C 1 ([α, β ]), max x∈[α,β ] |g(x)| = 1, max x∈[α,β ] |g ′ (x)| = 2 β -α Also, we can take g(x) = cos (β -x)π β -α to get g(β ) = -g(α) = 1, g ∈ C 1 ([α, β ]), max x∈[α,β ] |g(x)| = 1, max x∈[α,β ] |g ′ (x)| = π β -α .
iλ ρ 1 u x = -ρ 1 y x -ρ 1 λ -1 2 ( f 1 ) x .
Inserting the above result in (3.4.28), then using the fact that u x is uniformly bounded in L 2 (0, α),

( f 1 ) x → 0 in L 2 (0, α) and f 4 → 0 in L 2 (0, α) gives -ρ 1 α 0 xyy x -κ 1 α 0 xu xx u x dx = o λ -1 2 .
Taking the real part in the above equation, then using by parts integration, we get Remark 3.4.5. Taking into consideration [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF], the result of Theorem 3.4.1 is surprising. In fact, in [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF] and [START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF], the authors proved that the energy corresponding to the wave equation with frictional damping and Kelvin-Voigt damping decays polynomially with optimal order t -4 . However, in our case, we get the same order and we prove that the decay rate is optimal in the absence of frictional damping where the Kelvin-Voigt damping is internal. Indeed, it seems that although the frictional damping facilitates the calculation, it does not affect the order of the energy decay rate.

ρ 1 2 α 0 |y| 2 dx + κ 1 2 α 0 |u x | 2 dx - α 2 κ 1 |u x (α)| 2 + ρ 1 |y(α)| 2 = o λ -1 2 . ( 3
α 0 |y| 2 dx + κ 1 2 α 0 |u x | 2 dx - α 2 κ -1 1 |κ 2 v x (α) + δ z x (α)| 2 + ρ 1 |z(α)| 2 = o λ -

Elastic part

Viscoelastic part & delay feedback

Wave equation with localized Kelvin-Voigt damping and internal/boundary time delay

In this chapter, we study the stabilization of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin-Voigt type and with boundary or localized internal delay feedback. Our main interest here is that the Kelvin-Voigt and the delay damping are both localized via non smooth coefficients. Under sufficient assumptions, in the case that the Kelvin-Voigt damping is localized faraway from the tip and the wave is subjected to a boundary delay feedback, we prove that the energy of the system decays polynomially of type t -4 . However, when the Kelvin-Voigt damping is localized near a part of the boundary and a time delay damping acts on the second boundary, we establish an exponential decay of the energy of the system. While, when the Kelvin-Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary, under sufficient assumptions, using frequency domain arguments combined with piecewise multiplier techniques, we prove that the energy of the system decays polynomially of type t -4 . Otherwise, if the above assumptions are not true, we establish instability results.

Introduction

Viscoelastic materials feature intermediate characteristics between purely elastic and purely viscous behaviors, i.e., they display both behaviors when undergoing deformation. In wave equations, when the viscoelastic controlling parameter is null, the viscous property vanishes and the wave equation becomes a pure elastic wave equation. However, time delays arise in many applications and practical problems like physical, chemical, biological, thermal and economic phenomena, where an arbitrary small delay may destroy the well-posedness of the problem and destabilize it. Actually, it is well-known that the simplest delay equations of parabolic type, u t (x,t) = ∆u(x,t -τ), or hyperbolic type u tt (x,t) = ∆u(x,t -τ), with a delay parameter τ > 0, are not well-posed. Their instability is due to the existence of a sequence of initial data remaining bounded, while the corresponding solutions go to infinity in an exponential manner at a fixed time (see [START_REF] Dreher | Ill-posed problems in thermomechanics[END_REF][START_REF] Jordan | A note on the delayed heat equation: Instability with respect to initial data[END_REF]).

The stabilization of a wave equation with Kelvin-Voigt type damping and internal or boundary time delay has attracted the attention of many authors in the last five years. Indeed, in 2016 Messaoudi et al. studied the stabilization of a wave equation with global Kelvin-Voigt damping and internal time delay in the multidimensional case (see [START_REF] Messaoudi | Well posedness and exponential stability in a wave equation with a strong damping and a strong delay[END_REF]), and they obtained an exponential stability result. In the same year, Nicaise et al. in [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF] considered the multidimensional wave equation with localized Kelvin-Voigt damping and mixed boundary condition with time delay. They obtained an exponential decay of the energy regarding that the damping is acting on a neighborhood of part of the boundary via a smooth coefficient. Also, in 2018, Anikushyn et al. in [START_REF] Demchenko | On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay[END_REF] considered the stabilization of a wave equation with global viscoelastic material subjected to an internal strong time delay where a global exponential decay rate was obtained. Thus, it seems to us that there are no previous results concerning the case of wave equations with internal localized Kelvin-Voigt type damping and boundary or internal time delay, especially in the absence of smoothness of the damping coefficient even in the one-dimensional case. So, we are interested in studying the stability of elastic wave equation with local KelvinVoigt damping and with boundary or internal time delay (see Systems (4.1.1) and (4.1.2)).

This chapter investigates the study of the stability of a string with Kelvin-Voigt type damping localized via a non-smooth coefficient and subjected to a localized internal or boundary time delay. Indeed, in the first part of this chapter, we study the stability of elastic wave equation with local Kelvin-Voigt damping, boundary feedback and time delay term at the boundary, i.e., we consider the following system

                   U tt (x,t) -κ U x (x,t) + δ 1 χ (α,β ) U xt (x,t) x = 0, (x,t) ∈ (0, L) × (0, +∞), U(0,t) = 0, t ∈ (0, +∞), U x (L,t) = -δ 3 U t (L,t) -δ 2 U t (L,t -τ), t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), U t (L,t) = f 0 (L,t), t ∈ (-τ, 0), (4.1.1)
where L, τ, δ 1 and δ 3 are strictly positive constant numbers, δ 2 is a non zero real number and the initial data (U 0 ,U 1 , f 0 ) belongs to a suitable space. Here 0 ≤ α < β < L and U = uχ (0,α) + vχ (α,β ) + wχ (β ,L) , where χ (a,b) is the characteristic function of the interval (a, b). We assume that there exist strictly positive constant numbers κ 1 , κ 2 , κ 3 , such that κ = κ 1 χ (0,α) + κ 2 χ (α,β ) + κ 3 χ (β ,L) . In fact, here we will consider two cases. In the first case, we divide the bar into 3 pieces; the first piece is an elastic part, the second piece is the viscoelastic part and in the third piece, the time delay feedback is effective at the ending point of the piece, i.e., we consider the case α > 0 (see Figure 4.1). While, in the second case, we divide the bar into 2 pieces; the first piece is the viscoelastic part and in the second piece the time delay feedback is effective at the ending point of the piece, i.e., we consider the case α = 0 (see Figure 4.2). Remark, here, in both cases, the KelvinVoigt damping is effective on a part of the piece and the time delay is effective at L. Regarding System (4.1.1), when δ 3 = 0, α = 0 and β = L = 1, in the case where the delay is only effective at 1, Datko in [START_REF] Datko | Two questions concerning the boundary control of certain elastic systems[END_REF] (see Example 3.5) proved that System (4.1.1) is unstable for an arbitrary small value of τ. Later, when δ 3 = 0 and in the absence of delay (i.e., δ 2 = 0), they proved in [START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Ammari | Stability of the wave equations on a tree with local Kelvin-Voigt damping[END_REF] that the energy of System (4.1.1) is polynomially stable. Indeed, when δ 2 = 0 and δ 3 = 0, the stability of System (4.1.1) is still an open problem. In this part, we assume that

δ 3 > 1 2κ 3 and |δ 2 | < 1 κ 3 2κ 3 δ 3 -1. (H)
We consider two cases. Case one, if α > 0 (see Figure 4.1), then using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov, we show that the energy of System (4.1.1) has a polynomial decay rate of type t -4 . Case two, if α = 0 (see Figure 4.2), then using the semigroup theory of linear operators and a result obtained by Huang and Prüss, we prove an exponential decay of the energy of System (4.1.1). In the second part of this work, we study the stability of the elastic wave equation with local KelvinVoigt damping and local internal time delay. This system takes the following form

             U tt (x,t) -[κ U x (x,t) + a(x) (δ 1 U xt (x,t) + δ 2 U xt (x,t -τ))] x = 0, (x,t) ∈ (0, L) × (0, +∞), U(0,t) = U(L,t) = 0, t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), U t (x,t) = f 0 (x,t), (x,t) ∈ (0, L) × (-τ, 0), (4.1.2)
where L, τ and δ 1 are strictly positive constant numbers, δ 2 is a non zero real number and the initial data (U 0 ,U 1 , f 0 ) belongs to a suitable space. Here 0 < α < L, a(x) = χ (α,L) and U = uχ (0,α) + vχ (α,L) , where χ (a,b) is the characteristic function of the interval (a, b). We assume that there exist strictly positive constant numbers κ 1 , κ 2 , such that κ = κ 1 χ (0,α) + κ 2 χ (α,L) . In fact, here we will divide the bar into 2 pieces; the first piece is an elastic part, while in the second piece the KelvinVoigt damping and the time delay are effective (see Figure 4.3). So, the KelvinVoigt damping and the time delay are effective on (α, L). Regarding System (4.1.2), in the absence of delay (i.e., δ 2 = 0), they proved in [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF][START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF] that the energy of System (4.1.2) is polynomially stable. Indeed, when δ 2 = 0, the stability of System (4.1.2) is still an open problem. In this part, we assume that

|δ 2 | < δ 1 . (H1)
By using the semigroup theory of linear operators and a result obtained by Borichev and Tomilov, we show that the energy of the System (4.1.2) has a polynomial decay rate of type t -4 . Moreover, when a(x) = 1, ∀x ∈ (0, L), if |δ 2 | ≥ δ 1 , we show that there exists a sequence of arbitrary small (and large) delays such that instabilities occur.

• α L v(x) u(x)
       u tt (x,t) -a 1 u xx (x,t) = 0, (x,t) ∈ (0, L) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (L,t) = a 2 u t (L,t -τ), t ∈ (0, +∞). (4.1.3)
For the stability of System (4.1.

3), what are the conditions on a 1 , a 2 , and τ? In 1985, Datko et al. in [START_REF] Datko | An example of the effect of time delays in boundary feedback stabilization of wave equations[END_REF] considered System (4.1.3) with L = a 1 = 1 and a 2 = -κ, where they added an internal term a 2 u(x,t) and the internal damping 2au t (x,t). The system is given by the following:

       u tt (x,t) -u xx (x,t) + 2au t (x,t) + a 2 u(x,t) = 0, (x,t) ∈ (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (1,t) = -κu t (1,t -τ), t ∈ (0, +∞), (4.1.4)
where the delay parameter τ is strictly positive, a > 0 and κ > 0. So, the above system models a string having a boundary feedback with delay at the free end. They showed that if κ e 2a + 1 < e 2a -1, then System (4.1.4) is strongly stable for all small enough delays. However, if κ e 2a + 1 > e 2a -1, then there exists an open set D dense in (0, +∞), such that for all τ in D, System (4.1.4) admits exponentially unstable solutions. Moreover, in the absence of delay in System (4.1.4) (i.e., τ = 0) and a ≥ 0, κ ≥ 0, its energy decays exponentially to zero under the condition a 2 + κ 2 > 0 (see [19]). In 1990, Datko in [START_REF] Datko | Two questions concerning the boundary control of certain elastic systems[END_REF] (see Example 3.5) considered System (4.1.3) with L = a 1 = 1 and a 2 = -κ, also he added the the strong internal damping -δ u xxt . The system is given by the following:

      
u tt (x,t) -u xx (x,t) -δ u xxt (x,t) = 0, (x,t) ∈ (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞),

u x (1,t) = -κu t (1,t -τ), t ∈ (0, +∞), (4.1.5)
where τ, κ and δ are strictly positive constant numbers. Even in the presence of the strong damping -δ u xxt , without any other damping, He proved that System (4.1.5) is unstable for an arbitrary small value of τ. In 2006, Xu et al. in [START_REF] Xu | Stabilization of wave systems with input delay in the boundary control[END_REF] considered System (4.1.3) with L = a 1 = 1 and a 2 = -κ(1 -µ), where they added the boundary term -κu t (1,t). The system is given by the following: In 2008, Guo and Xu in [START_REF] Guo | Boundary Output Feedback Stabilization of A One-Dimensional Wave Equation System With Time Delay[END_REF] studied the stabilization of a wave equation in the 1 -D case where it is affected by a boundary control and output observation suffering from time delay. The system is given by the following:

                   u tt (x,t) -u xx (x,t) = 0, (x,t) ∈ (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (1,t) = -κu t (1,t) -κ(1 -µ)u t (1,t -τ), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ (0, 1), u t (1,t) = f 0 (1,t), t ∈ (-τ, 0), ( 4
                   u tt (x,t) -u xx (x,t) = 0, (x,t) ∈ (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (1,t) = w(t), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ (0, 1), y(t) = u t (1,t -τ), t ∈ (0, +∞),
where w is the control and y is the output observation. Using the separation principle, the authors proved that the above delayed system is exponentially stable. In 2010, Gugat in [START_REF] Gugat | Boundary feedback stabilization by time delay for one-dimensional wave equations[END_REF] considered System (4.1.3) with

a 1 = 4L 2 τ -2 , a 2 = 0 if t ∈ (0, τ) and a 2 = 2 -1 L -1 τ λ if t ∈ (τ, ∞).
The problem is described by the following system

                   u tt (x,t) -4L 2 τ -2 u xx (x,t) = 0, (x,t) ∈ (0, L) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (L,t) = 0, t ∈ (0, τ), u x (L,t) = τ λ 2L u t (L,t -τ) , t ∈ (τ, +∞), (u(x, 0), u t (x, 0), u(0, 0)) = (u 0 (x), u 1 (x), 0) , x ∈ (0, L),
where τ and L are strictly positive constant numbers, while λ is a real number. When τ = 2Lc -1 with c > 0 and λ ∈ [3 -2 √ 2, 1), Gugat proved that the above system is exponentially stable. The result in [START_REF] Gugat | Boundary feedback stabilization by time delay for one-dimensional wave equations[END_REF] has been improved by J. Wang et al. in [START_REF] Wang | Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time[END_REF]. Indeed, in 2011, J. Wang et al. in [START_REF] Wang | Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time[END_REF] considered System (4.1.3) with L = a 1 = 1, a 2 = κ. The system is given by the following:

             u tt (x,t) -u xx (x,t) = 0, (x,t) ∈ (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (1,t) = κu t (1,t -τ), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ (0, 1),
where τ is a strictly positive constant number and κ is a real number. They showed that if the delay is equal to even multiples of the wave propagation time, then the above closed loop system is exponentially stable under sufficient and necessary conditions for κ. Else, if the delay is an odd multiple of the wave propagation time, thus the closed loop system is unstable. In 2013, H. Wang et al. in [START_REF] Wang | Exponential stabilization of 1-d wave equation with input delay[END_REF], studied System (4.1.6) under the feedback control law u t (1,t) = w(t) provided that the weight of the feedback with delay is a real β and that of the feedback without delay is a real α. They found a feedback control law that stabilizes exponentially the system for any |α| = |β |, by modifying the velocity feedback into the form u(t) = β w t (1,t) + α f (w(.,t), w t (.,t)), where f is a linear functional. Finally, in 2017, Xu et al. in [START_REF] Xie | Exponential stability of 1-d wave equation with the boundary time delay based on the interior control[END_REF], considered System (4.1.3) with L = a 1 = 1, a 2 = κ, also they added the internal term 2αu t (x,t). The system is given by the following:

                   u tt (x,t) -u xx (x,t) + 2αu t (x,t) = 0, (x,t) ∈ (0, 1) × (0, +∞), u(0,t) = 0, t ∈ (0, +∞), u x (1,t) = κu t (1,t -τ), t ∈ (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ (0, 1), u t (1,t) = f 0 (1,t), t ∈ (-τ, 0),
where τ > 0, α > 0 and κ is real number. Based on the idea of Lyapunov functional, they proved an exponential stability result of the above system under a certain relationship between α and κ.

Going to the multidimensional case, the stability of the wave equation with time delay has been studied in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF][START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF][START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF][START_REF] Messaoudi | Well posedness and exponential stability in a wave equation with a strong damping and a strong delay[END_REF][START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF][START_REF] Demchenko | On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay[END_REF][START_REF] Ammari | Asymptotic behavior of a delayed wave equation without displacement term[END_REF][START_REF] Ammari | On the exponential and polynomial convergence for a delayed wave equation without displacement[END_REF]. In 2006, Nicaise and Pignotti in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF] studied the multidimensional wave equation considering two cases. The first case concerns a wave equation with boundary feedback and a delay term at the boundary

                   u tt (x,t) -∆u(x,t) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ D × (0, +∞), ∂ u ∂ ν (x,t) = -µ 1 u t (x,t) -µ 2 u t (x,t -τ), (x,t) ∈ Γ N × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t) = f 0 (x,t), (x,t) ∈ Γ N × (-τ, 0). (4.1.7)
The second case concerns a wave equation with internal feedback and a delayed velocity term (i.e., an internal delay) and a mixed Dirichlet-Neumann boundary condition

                   u tt (x,t) -∆u(x,t) + µ 1 u t (x,t) + µ 2 u t (x,t -τ) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ D × (0, +∞), ∂ u ∂ ν (x,t) = 0, (x,t) ∈ Γ N × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t) = f 0 (x,t), (x,t) ∈ Ω × (-τ, 0). (4.1.8)
In both systems, Ω ⊂ R N be is considered to be an open bounded set with a boundary Γ of class C 2 , Γ is divided into two parts Γ D and Γ N , i.e., Γ = Γ D ∪ Γ N , such that Γ D ∩ Γ N = / 0 and Γ D = / 0. Moreover, ν(x) denotes the outer unit normal vector to the point x ∈ Γ N , ∂ u/∂ ν is the partial derivative, τ > 0 is the time delay, µ 1 and µ 2 are strictly positive constant numbers. Under the assumption that the weight of the feedback with delay is smaller than that without delay (µ 2 < µ 1 ), they obtained an exponential decay of the energy of both Systems (4.1.7) and (4.1.8). On the contrary, if the previous assumption does not hold (i.e., µ 2 ≥ µ 1 ), they found a sequence of delays for which the energy of some solutions does not tend to zero (see [START_REF] Benhassi | Feedback stabilization of a class of evolution equations with delay[END_REF] for the treatment of Problem (4.1.8) in more general abstract form). In 2009, Nicaise et al. in [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF] studied System (4.1.7) in the onedimensional case where the delay time τ is a function depending on time and they established an exponential stability result under the condition that the derivative of the decay function is upper bounded by a constant d < 1 and assuming that µ 2 < √ 1d µ 1 . In 2010, Ammari et al. in [START_REF] Ammari | Feedback boundary stabilization of wave equations with interior delay[END_REF] studied the wave equation with interior delay damping and dissipative undelayed boundary condition. The system is given by the following:

                   u tt (x,t) -∆u(x,t) + au t (x,t -τ) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ 0 × (0, +∞), ∂ u ∂ ν (x,t) = -κu t (x,t), (x,t) ∈ Γ 1 × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t) = f 0 (x,t), (x,t) ∈ Ω × (-τ, 0), (4.1.9)
where Ω ⊂ R N ; N ≥ 2, is an open bounded set with a boundary Γ, Γ is divided into two closed parts Γ 0 and Γ 1 , i.e., Γ = Γ 0 ∪ Γ 1 , such that Γ 0 ∩ Γ 1 = / 0 and Γ 0 = / 0. Moreover, τ > 0, a > 0 and κ > 0. Under the condition that Γ 1 satisfies the Γ-condition introduced in [START_REF] Lions | Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués[END_REF], they proved that System (4.1.9) is uniformly asymptotically stable whenever the delay coefficient is sufficiently small. In 2012, Pignotti in [START_REF] Pignotti | A note on stabilization of locally damped wave equations with time delay[END_REF] considered the wave equation with internal distributed time delay and local damping in a bounded and smooth domain Ω ⊂ R N ; N ≥ 1, with a boundary Γ. The system is given by the following:

             u tt (x,t) -∆u(x,t) + aχ ω u t (x,t) + κu t (x,t -τ) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t) = f (x,t), (x,t) ∈ Ω × (-τ, 0), ( 4
             u tt (x,t) -∆u(x,t) -µ 1 ∆u t (x,t) -µ 2 ∆u t (x,t -τ) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t) = f 0 (x,t), (x,t) ∈ Ω × (-τ, 0),
where Ω is a bounded and regular domain of R N ; N ≥ 1, with a boundary Γ, τ > 0 represents the time delay, µ 1 > 0 and µ 2 are real numbers such that |µ 2 | < µ 1 . They obtained an exponential stability result. In addition, in the same year, Nicaise et al. in [START_REF] Nicaise | Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback[END_REF] studied the multidimensional wave equation with localized Kelvin-Voigt damping and mixed boundary condition with time delay

                   u tt (x,t) -∆u(x,t) -div(a(x)∇u t ) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ 0 × (0, +∞), ∂ u ∂ ν (x,t) = -a(x) ∂ u t ∂ ν (x,t) -κu t (x,t -τ), (x,t) ∈ Γ 1 × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t) = f 0 (x,t), (x,t) ∈ Γ 1 × (-τ, 0), (4.1.11)
where

Ω ⊂ R N is an open bounded set with a boundary Γ of class C 2 , Γ is divided into two open parts Γ 0 and Γ 1 , i.e., Γ = Γ 0 ∪ Γ 1 , such that Γ 0 ∩ Γ 1 = / 0 and Γ i = / 0, i = 0, 1.
Moreover, ν(x) denotes the outer unit normal vector to the point x ∈ Γ 1 , ∂ u/∂ ν is the partial derivative, τ > 0 is the time delay, κ is a real number, a(x) ∈ L ∞ (Ω) and a(x) ≥ a 0 > 0 on ω such that ω ⊂ Ω is an open neighborhood of Γ 1 . Under an appropriate geometric condition on Γ 1 and assuming that a ∈ C 1,1 (Ω), ∆a ∈ L ∞ (Ω), they proved an exponential decay of the energy of System (4.1.11). In 2017, Ammari and Gerbi in [START_REF] Ammari | Interior Feedback Stabilization of Wave Equations with Dynamic Boundary Delay[END_REF] considered an interior stabilization problem for the wave equation with dynamic boundary delay. The system is given by the following:

                   u tt (x,t) -∆u + au t = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ 0 × (0, +∞), u tt (x,t) = -∂ u ∂ ν (x,t) -µu t (x,t -τ), (x,t) ∈ Γ 1 × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u t (x,t -τ) = f 0 (x,t -τ), (x,t) ∈ Ω × (0, τ),
where Ω ⊂ R N ; N ≥ 2 is an open bounded set with a boundary Γ, Γ is divided into two closed parts Γ 0 and Γ 1 , i.e., Γ = Γ 0 ∪ Γ 1 , such that Γ 0 ∩ Γ 1 = / 0 and Γ 0 = / 0. Moreover, ν(x) denotes the outer unit normal vector to the point x ∈ Γ 1 , ∂ u/∂ ν is the partial derivative, τ > 0 is the time delay, a and µ are positive numbers. They proved some stability results under the choice of the damping operator. The proof of the main result is based on a frequency domain method and combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent. Finally, in 2018, Anikushyn et al. in [START_REF] Demchenko | On a Kelvin-Voigt Viscoelastic Wave Equation with Strong Delay[END_REF] considered an initial boundary value problem for a viscoelastic wave equation subjected to a strong time localized delay in a Kelvin-Voigt type. The system is given by the following:

                   u tt (x,t) -c 1 ∆u(x,t) -c 2 ∆u(x,t -τ) -d 1 ∆u t (x,t) -d 2 ∆u t (x,t -τ) = 0, (x,t) ∈ Ω × (0, +∞), u(x,t) = 0, (x,t) ∈ Γ 0 × (0, +∞), ∂ u ∂ ν (x,t) = 0, (x,t) ∈ Γ 1 × (0, +∞), (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)) , x ∈ Ω, u(x,t) = f 0 (x,t), (x,t) ∈ Ω × (-τ, 0),
where

Ω ⊂ R N is an open bounded Lipschitz domain set with a boundary Γ of class C 2 , Γ is divided into two open parts Γ 0 and Γ 1 , i.e., Γ = Γ 0 ∪ Γ 1 , such that Γ 0 ∩ Γ 1 = / 0 and Γ 0 = / 0.
Moreover, ν(x) denotes the outer unit normal vector to the point x ∈ Γ 1 , ∂ u/∂ ν is the partial derivative, τ > 0 is the time delay, and c 1 , c 2 , d 1 , d 2 are positive real numbers. Under appropriate conditions on the coefficients, a global exponential decay rate is obtained. We can also mention that Ammari et al. in [START_REF] Ammari | Stability of an abstract-wave equation with delay and a Kelvin-Voigt damping[END_REF] considered the stabilization problem for an abstract equation with delay and a Kelvin-Voigt damping in 2015. The system is given by the following:

       u tt (t) + aBB * u t (t) + BB * u(t -τ), t ∈ (0, +∞), (u(0), u t (0)) = (u 0 , u 1 ) , B * u(t) = f 0 (t), t ∈ (-τ, 0),
for an appropriate class of operator B and a > 0. Using the frequency domain approach, they obtained an exponential stability result. Finally, the transmission problem of a wave equation with global or local Kelvin-Voigt damping and without any time delay was studied by many authors in the one-dimensional case (see [START_REF] Liu | Exponential Decay of Energy of the Euler-Bernoulli Beam with Locally Distributed Kelvin-Voigt Damping[END_REF][START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF][START_REF] Huang | On the Mathematical Model for Linear Elastic Systems with Analytic Damping[END_REF][START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Hassine | Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin-Voigt damping[END_REF][START_REF] Hassine | Stability of elastic transmission systems with a local Kelvin-Voigt damping[END_REF][START_REF] Oquendo | Frictional versus Kelvin-Voigt damping in a transmission problem[END_REF][START_REF] Rivera | Stability to localized viscoelastic transmission problem[END_REF][START_REF] Liu | Stability of a String with Local Kelvin-Voigt Damping and Nonsmooth Coefficient at Interface[END_REF][START_REF] Ghader | A transmission problem for the timoshenko system with one local kelvinvoigt damping and non-smooth coefficient at the interface[END_REF][START_REF] Wehbe | Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients[END_REF][START_REF] Hassine | Stability for coupled waves with locally disturbed Kelvinâ Ȃ ŞVoigt damping[END_REF]) and in the multidimensional case (see [START_REF] Wehbe | Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt damping under different types of geometric conditions[END_REF][START_REF] Zhang | Polynomial decay of an elastic/viscoelastic waves interaction system[END_REF][START_REF] Tebou | A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping[END_REF][START_REF] Liu | Exponential stability for the wave equations with local Kelvin-Voigt damping[END_REF][START_REF] Ammari | Stabilization for the wave equation with singular kelvinâ Ȃ Şvoigt damping[END_REF][START_REF] Hayek | A transmission problem of a system of weakly coupled wave equations with Kelvinâ Ȃ ŞVoigt dampings and non-smooth coefficient at the interface[END_REF]) and polynomial and exponential stability results were obtained. In addition, the stability of wave equations on a tree with local Kelvin-Voigt damping has been studied in [START_REF] Ammari | Stability of the wave equations on a tree with local Kelvin-Voigt damping[END_REF]. Thus, as we confirmed in the beginning, the case of wave equations with localized Kelvin-Voigt type damping and boundary or internal time delay; as in our Systems (4.

Wave equation with local Kelvin-Voigt damping and with boundary delay feedback

This section is devoted to our first aim, which is to study the stability of a wave equation with localized Kelvin-Voigt damping and boundary delay feedback (see System (4.1.1)). For this aim, like in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the auxiliary unknown η(L, ρ,t) = U t (L,t -ρ τ), ρ ∈ (0, 1), t > 0.

Thus, Problem (4.1.1) is equivalent to

                         U tt (x,t) -κ U x (x,t) + δ 1 χ (α,β ) U xt (x,t) x = 0, (x,t) ∈ (0, L) × (0, +∞), τη t (L, ρ,t) + η ρ (L, ρ,t) = 0, (ρ,t) ∈ (0, 1) × (0, +∞), U(0,t) = 0, t ∈ (0, +∞), U x (L,t) = -δ 3 U t (L,t) -δ 2 η(L, 1,t), t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), η(L, ρ, 0) = f 0 (L, -ρ τ), ρ ∈ (0, 1). (4.2.1)
Under assumption (H), let us define the energy of a solution of System (4.2.1) as

E(t) = 1 2 L 0 |U t | 2 + κ|U x | 2 dx + τ 2 1 0 |η| 2 dρ. (4.2.2)
Multiplying the first equation of (4.2.1) by U t , integrating over (0, L) with respect to x, then using integration by parts and the boundary conditions in (4.2.1) at x = 0 and at x = L, we get 1 2

d dt L 0 |U t | 2 + κ|U x | 2 dx = -δ 1 β α |U xt | 2 dx -κ 3 δ 3 |U t (L,t)| 2 -κ 3 δ 2 η(L, 1,t)U t (L,t). (4.2.3)
Multiplying the second equation of (4.2.1) by η, integrating over (0, 1) with respect to ρ, then using the fact that η(L, 0,t) = U t (L,t), we get

τ 2 d dt 1 0 |η| 2 dρ = - 1 2 |η(L, 1,t)| 2 + 1 2 |U t (L,t)| 2 . ( 4
d E(t) dt = -δ 1 β α |U xt | 2 dx -κ 3 δ 3 - 1 2 |U t (L,t)| 2 -κ 3 δ 2 η(L, 1,t)U t (L,t) - 1 2 |η(L, 1,t)| 2 . (4.2.5)
For all p > 0, we have

-κ 3 δ 2 η(L, 1,t)U t (L,t) ≤ κ 3 |δ 2 | |η(L, 1,t)| 2 2p + κ 3 |δ 2 | p |U t (L,t)| 2 2 .
Inserting the above equation in (4.2.5), we get

d E(t) dt ≤ -δ 1 β α |U xt | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1,t)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |U t (L,t)| 2 . (4.2.6)
Under assumption (H), we can easily check that there exists a strictly positive number p satisfying

κ 3 |δ 2 | < p < 2 κ 3 |δ 2 | κ 3 δ 3 - 1 2 , (4.2.7) such that 1 2 - κ 3 |δ 2 | 2p > 0 and κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 > 0,
so that the energies of the strong solutions satisfy E ′ (t) ≤ 0. Hence, System (4.2.1) is dissipative in the sense that its energy is non increasing with respect to the time t.

For studying the stability of System (4.2.1), we consider two cases. In Subsection 4.2.1, we consider the first case, when the Kelvin-Voigt damping is localized in the internal of the body, i.e., α > 0. While, in Subsection 4.2.2, we consider the second case, when the Kelvin-Voigt damping is localized near the boundary of the body, i.e., α = 0.

Wave equation with local Kelvin-Voigt damping far from the boundary and with boundary delay feedback

In this subsection, we assume that there exist α and β such that 0 < α < β < L, in this case, the Kelvin-Voigt damping is localized in the internal of the body (see Figure 4.1). For this aim, we denote the longitudinal displacement by U and this displacement is divided into three parts

U(x,t) =        u(x,t), (x,t) ∈ (0, α) × (0, +∞), v(x,t), (x,t) ∈ (α, β ) × (0, +∞), w(x,t), (x,t) ∈ (β , L) × (0, +∞).
In this case, System (4.2.1) is equivalent to the following system

             u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞), v tt -(κ 2 v x + δ 1 v xt ) x = 0, (x,t) ∈ (α, β ) × (0, +∞), w tt -κ 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞), τη t (L, ρ,t) + η ρ (L, ρ,t) = 0, (0, 1) × (0, +∞), (4.2.8)
with the following boundary and transmission conditions

                   u(0,t) = 0, t ∈ (0, +∞), u(α,t) = v(α,t), v(β ,t) = w(β ,t), t ∈ (0, +∞), w x (L,t) = -δ 3 w t (L,t) -δ 2 η(L, 1,t), t ∈ (0, +∞), κ 2 v x (α,t) + δ 1 v xt (α,t) = κ 1 u x (α,t), t ∈ (0, +∞), κ 2 v x (β ,t) + δ 1 v xt (β ,t) = κ 3 w x (β ,t), t ∈ (0, +∞), (4.2.9)
and with the following initial conditions

             (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, α), (v(x, 0), v t (x, 0)) = (v 0 (x), v 1 (x)), x ∈ (α, β ), (w(x, 0), w t (x, 0)) = (w 0 (x), w 1 (x)), x ∈ (β , L), η(L, ρ, 0) = f 0 (L, -ρ τ), ρ ∈ (0, 1), (4.2.10)
where the initial data (u 0 , u 1 , v 0 , v 1 , w 0 , w 1 , f 0 ) belongs to a suitable Hilbert space. So, using (4.2.2), the energy of System (4.2.8)-(4.2.10) is given by

E(t) = 1 2 α 0 |u t | 2 + κ 1 |u x | 2 dx + 1 2 β α |v t | 2 + κ 2 |v x | 2 dx + 1 2 L β |w t | 2 + κ 3 |w x | 2 dx + τ 2 1 0 |η| 2 dρ.
Similar to (4.2.5) and (4.2.6), we get

d E(t) dt = -δ 1 β α |v xt | 2 dx - 1 2 |η(L, 1,t)| 2 -κ 3 δ 2 η(L, 1,t)w t (L,t) -κ 3 δ 3 - 1 2 |w t (L,t)| 2 , ≤ -δ 1 β α |v xt | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1,t)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |w t (L,t)| 2 ,
where p is defined in (4.2.7). Thus, under hypothesis (H), the System (4.2.8)-(4.2.10) is dissipative in the sense that its energy is non increasing with respect to the time t. Now, we are in position to prove the existence and uniqueness of the solution of our system.

Well-posedness of the problem

We start this part by formulating System (4.2.8)-(4.2.10) as an abstract Cauchy problem. For this aim, let us define

H m = H m (0, α) × H m (α, β ) × H m (β , L), m = 1, 2, L 2 = L 2 (0, α) × L 2 (α, β ) × L 2 (β , L), H 1 L = {(u, v, w) ∈ H 1 | u(0) = 0, u(α) = v(α), v(β ) = w(β )}.
Remark 4.2.1. The Hilbert space L 2 is equipped with the norm:

(u, v, w) 2 L 2 = α 0 |u| 2 dx + β α |v| 2 dx + L β |w| 2 dx.
Also, it is easy to check that the space H 1 L is Hilbert space over C equipped with the norm:

(u, v, w) 2 H 1 L = κ 1 α 0 |u x | 2 dx + κ 2 β α |v x | 2 dx + κ 3 L β |w x | 2 dx.
Moreover, by Poincaré inequality, we can easily verify that there exists C > 0 depending on κ 1 , κ 2 , κ 3 , α, β and L, such that

(u, v, w) L 2 ≤ C (u, v, w) H 1 L , ∀(u, v, w) ∈ H 1 L .

We now define the Hilbert energy space H by

H = H 1 L × L 2 × L 2 (0, 1) equipped with the following inner product U, Ũ H = κ 1 α 0 u x ũx dx + κ 2 β α v x ṽx dx + κ 3 L β w x wx dx + α 0 y ỹdx + β α zzdx + L β φ φ dx + τ 1 0 η(L, ρ) η(L, ρ)dρ,
where U = (u, v, w, y, z, φ , η(L, •)) ∈ H and Ũ = ( ũ, ṽ, w, ỹ, z, φ , η(L, •)) ∈ H . We use U H to denote the corresponding norm. We define the linear unbounded operator A : D(A ) ⊂ H -→ H by:

D(A ) = (u, v, w, y, z, φ , η(L, •)) ∈ H 1 L × H 1 L × H 1 (0, 1) | (u, κ 2 v + δ 1 z, w) ∈ H 2 , φ (L) = η(L, 0) κ 2 v x (α) + δ 1 z x (α) = κ 1 u x (α), κ 2 v x (β ) + δ 1 z x (β ) = κ 3 w x (β ),w x (L) = -δ 3 η(L, 0) -δ 2 η(L, 1)
and for all U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ),

A U = y, z, φ , κ 1 u xx , (κ 2 v x + δ 1 z x ) x , κ 3 w xx , -τ -1 η ρ (L, •) . If U = (u, v, w, u t , v t , w t , η(L, •)
) is a regular solution of System (4.2.8)-(4.2.10), then we transform this system into the following initial value problem

U t = A U, U(0) = U 0 , (4.2.11)
where

U 0 = (u 0 , v 0 , w 0 , u 1 , v 1 , w 1 , f 0 (L, -•τ)) ∈ H .
We now use semigroup approach to establish well-posedness result for the System (4.2.8)-(4.2.10). According to Lumer-Phillips theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we need to prove that the operator A is m-dissipative in H . Therefore, we prove the following proposition.

Proposition 4.2.2. Under hypothesis (H), the unbounded linear operator A is m-dissipative in the energy space H .

Proof. For all U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ), we have

ℜ (A U, U) H = κ 1 ℜ α 0 (y x u x + u xx y) dx + ℜ β α (κ 2 z x v x + (κ 2 v x + δ 1 z x ) x z) dx +κ 3 ℜ L β φ x w x + w xx φ dx -ℜ 1 0 η ρ (L, ρ) η(L, ρ)dρ.
Using integration by parts in the above equation, we get

ℜ (A U, U) H = -δ 1 β α |z x | 2 dx - 1 2 |η(L, 1)| 2 + 1 2 |η(L, 0)| 2 + κ 3 ℜ w x (L)φ (L)
-κ 1 ℜ (u x (0)y(0)) + ℜ (κ 1 u x (α)y(α) -κ 2 v x (α)z(α) -δ 1 z x (α)z(α)) +ℜ κ 2 v x (β )z(β ) + δ 1 z x (β )z(β ) -κ 3 w x (β )φ (β ) .

(4.2.12)

On the other hand, since U ∈ D(A ), we have

y(0) = 0, y(α) = z(α), z(β ) = φ (β ), w x (L) = -δ 3 η(L, 0) -δ 2 η(L, 1), κ 1 u x (α) -κ 2 v x (α) -δ 1 z x (α) = 0, κ 2 v x (β ) + δ 1 z x (β ) -κ 3 w x (β ) = 0, φ (L) = η(L, 0).
Inserting the above equation in (4.2.12), we get

ℜ (A U, U) H = -δ 1 β α |z x | 2 dx - 1 2 |η(L, 1)| 2 -κ 3 δ 3 - 1 2 |η(L, 0)| 2 -κ 3 δ 2 ℜ (η(L, 0)η(L, 1)) . (4.2.13)
Under hypothesis (H), we easily check that there exists p > 0 such that

κ 3 |δ 2 | < p < 2 κ 3 |δ 2 | κ 3 δ 3 - 1 2 .
By Young's inequality, we get

-κ 3 δ 2 ℜ (η(L, 0)η(L, 1)) ≤ κ 3 |δ 2 | |η(L, 1)| 2 2p + κ 3 |δ 2 | p |η(L, 0)| 2 2 .
Inserting the above inequality in (4.2.13), we get

ℜ (A U, U) H ≤ -δ 1 β α |z x | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |η(L, 0)| 2 . (4.2.14)
From the construction of p, we have

1 2 - κ 3 |δ 2 | 2p > 0 and κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 > 0.
Therefore, from (4.2.14), we get

ℜ (A U, U) H ≤ 0,
which implies that A is dissipative. Now, let us go on with maximality.

Let F = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 (L, •)) ∈ H we look for U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) solution of the equation -A U = F. (4.2.15)
Equivalently, we consider the following system

-y = f 1 , (4.2.16)
-z = f 2 , (4.2.17)

-φ = f 3 , (4.2.18) -κ 1 u xx = f 4 , (4.2.19) -(κ 2 v x + δ 1 z x ) x = f 5 , (4.2.20)
-κ 3 w xx = f 6 , (4.2.21)
η ρ (L, ρ) = τ f 7 (ρ). (4.2.22)
In addition, we consider the following boundary conditions

u(0) = 0, u(α) = v(α), v(β ) = w(β ), (4.2.23) κ 2 v x (α) + δ 1 z x (α) = κ 1 u x (α), κ 2 v x (β ) + δ 1 z x (β ) = κ 3 w x (β ), (4.2
.24)

w x (L) = -δ 3 η(L, 0) -δ 2 η(L,
f 3 ∈ H 1 (β , L), we get η(L, 0) = φ (L) = -f 3 (L).
From the above equation and Equation (4.2.22), we can determine

η(L, ρ) = τ ρ 0 f 7 (ξ ) dξ -f 3 (L). It is clear that η(L, •) ∈ H 1 (0, 1) and η(L, 0) = φ (L) = -f 3 (L).
= -f 1 , z = -f 2 , φ = -f 3 , η(L, ρ) = τ ρ 0 f 7 (ξ ) dξ -f 3 (L), (4.2.27) -κ 1 u xx = f 4 , (4.2.28) -(κ 2 v x + δ 1 z x ) x = f 5 , (4.2.29)
-κ 3 w xx = f 6 , (4.2.30)

u(0) = 0, u(α) = v(α), v(β ) = w(β ), (4.2.31) κ 2 v x (α) + δ 1 z x (α) = κ 1 u x (α), κ 2 v x (β ) + δ 1 z x (β ) = κ 3 w x (β ), (4.2
.32) ) by ϕ, ψ, θ , integrating over (0, α), (α, β ) and (β , L) respectively, taking the sum, then using integration by parts, we get From the fact that (ϕ, ψ, θ ) ∈ H 1 L , we have

w x (L) = (δ 3 + δ 2 ) f 3 (L) -τδ 2 1 0 f 7 (ξ ) dξ . ( 4
κ 1 α 0 u x ϕ x dx + β α (κ 2 v x + δ 1 z x )ψ x dx + κ 3 L β w x θ x dx + κ 1 u x (0)ϕ(0) -κ 1 u x (α)ϕ(α) + (κ 2 v x (α) + δ 1 z x (α))ψ(α) -(κ 2 v x (β ) + δ 1 z x (β ))ψ(β ) + κ 3 w x (β )θ (β ) = α 0 f 4 ϕdx + β α f 5 ψdx + L β f 6 θ dx + κ 3 w x (L)θ (L).
ϕ(0) = 0, ϕ(α) = ψ(α), θ (β ) = ψ(β ).
Inserting

κ 1 α 0 u x ϕ x dx + κ 2 β α v x ψ x dx + κ 3 L β w x θ x dx = α 0 f 4 ϕdx + β α (δ 1 ( f 2 ) x ψ x + f 5 ψ) dx + L β f 6 θ dx + κ 3 (δ 3 + δ 2 ) f 3 (L) -τδ 2 1 0 f 7 (ξ ) dξ θ (L). (4.2.35)
We can easily verify that the left-hand side of (4.2.35) is a bilinear continuous coercive form on H 1 L × H 1 L , and the right-hand side of (4.2.35) is a linear continuous form on H 1 L . Then, using Lax-Milgram theorem, we deduce that there exists (u, v, w) ∈ H 1 L unique solution of the variational Problem (4.2.35). Using standard arguments, we can show that (u, κ 2 v + δ 1 z, w) ∈ H 2 . Finally, by setting y = -f 1 , z = -f 2 , φ =f 3 and η(L, ρ) = τ ρ 0 f 7 (ξ ) dξf 3 (L) and by applying the classical elliptic regularity we deduce that U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) is the solution of Equation (4.2.15). To conclude, we need to show the uniqueness of U. So, let U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) be a solution of (4.2.15) with F = 0, then we directly deduce that y = z = φ = η(L, ρ) = 0 and that (u, v, w) ∈ H 1 L satisfies Problem (4.2.35) with zero in the right hand side. This implies that u = v = w = 0, in other words, ker A = {0} and 0 belongs to the resolvent set ρ(A ) of A . Then, by contraction principle, we easily deduce that R(λ I -A ) = H for sufficiently small λ > 0. This, together with the dissipativeness of A , imply that D (A ) is dense in H and that A is m-dissipative in H (see Theorems 4.5, 4.6 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). The proof is thus complete.

Thanks to Lumer-Philips theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that A generates a C 0 -semigroup of contractions e tA in H and therefore Problem (4.2.8)-(4.2.10) is well-posed. Then we have the following result: Theorem 4.2.3. Under hypothesis (H), for any U 0 ∈ H , Problem (4.2.11) admits a unique weak solution,

U(x, ρ,t) = e tA U 0 (x, ρ), such that U ∈ C 0 (R + , H ). Moreover, if U 0 ∈ D(A ), then U ∈ C 1 (R + , H )∩C 0 (R + , D(A )).

Strong Stability

Our main result in this part is the following theorem. Theorem 4.2.4. Under hypothesis (H), the C 0 -semigroup of contractions e tA is strongly stable on the Hilbert space H in the sense that

lim t→+∞ ||e tA U 0 || H = 0, ∀ U 0 ∈ H .
For the proof of Theorem 4.2.4, using a general criteria of Arendt-Batty [START_REF] Arendt | Tauberian theorems and stability of one-parameter semigroups[END_REF], we need to prove that operator A has no pure imaginary eigenvalues and σ (A ) ∩ iR contains only a countable number of continuous spectrum of A . The argument for Theorem 4.2.4 relies on the subsequent Lemmas.

Lemma 4.2.5. Under hypothesis (H), for λ ∈ R, we have iλ I -A is injective, i.e., ker(iλ

I -A ) = {0}, ∀λ ∈ R.
Proof. From Proposition 4.2.2, we have 0 ∈ ρ(A ). We still need to show the result for λ ∈ R * . Suppose that there exists a real number λ = 0 and U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) such that

A U = iλ U. (4.2.36)
First, similar to Equation (4.2.14), we have

0 = ℜ (A U, U) H ≤ -δ 1 β α |z x | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |η(L, 0)| 2 ≤ 0.
Thus,

z x = 0 in (α, β ) and η(L, 1) = η(L, 0) = 0. ( 4
y = iλ u, x ∈ (0, α), (4.2.38)
z = iλ v, x ∈ (α, β ), (4.2.39)
w = iλ φ , x ∈ (β , L), (4.2.40)
κ 1 u xx = iλ y, x ∈ (0, α), (4.2.41)
(κ 2 v x + δ 1 z x ) x = iλ z, x ∈ (α, β ), ( 4
v x = z x = 0 in (α, β ). (4.2.46)
Thus,

v xx = z xx = 0 in (α, β ).
Inserting the above result in (4.2.42), then taking into consideration (4.2.39), we obtain

v = z = 0 in (α, β ). (4.2.47)
From the definition of D(A ) and using (4.2.45)-(4.2.47), we get

       u(α) = v(α) = 0, w(β ) = v(β ) = 0, y(α) = z(α) = 0, φ (β ) = z(β ) = 0, κ 1 u x (α) = κ 2 v x (α) + δ 1 z x (α) = 0, κ 3 w x (β ) = k 2 v x (β ) + δ 1 z x (β ) = 0, w(L) = iλ φ (L) = iλ η(L, 0) = 0, w x (L) = -δ 3 η(L, 0) -δ 2 η(L,
     u xx + λ 2 κ 1 u = 0, x ∈ (0, α), u(0) = u(α) = u x (α) = 0,      w xx + λ 2 κ 3 w = 0, x ∈ (β , L), w(β ) = w(L) = w x (β ) = w x (L) = 0.
Thus,

u(x) = 0 ∀ x ∈ (0, α) and w(x) = 0 ∀ x ∈ (β , L). ( 4
y(x) = 0 ∀ x ∈ (0, α) and φ (x) = 0 ∀ x ∈ (β , L).
Finally, from the above result, (4.2.45), (4.2.47) and (4.2.48), we get that U = 0. The proof is thus complete.

Lemma 4.2.6. Under hypothesis (H), for λ ∈ R, we have iλ I -A is surjective, i.e., R(iλ

I -A ) = H , ∀ λ ∈ R.
Proof. Since 0 ∈ ρ(A ), we still need to show the result for λ ∈ R * .

For any

F = ( f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , f 7 (L, •)
) ∈ H and λ ∈ R * , we prove the existence of U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) solution for the following equation

(iλ I -A )U = F.
Equivalently, we consider the following problem

y = iλ u -f 1 in H 1 (0, α), (4.2.49)
z = iλ v -f 2 in H 1 (α, β ), (4.2.50)
φ = iλ w -f 3 in H 1 (β , L), (4.2.51) iλ y -κ 1 u xx = f 4 in L 2 (0, α), (4.2.52) iλ z -(κ 2 v x + δ 1 z x ) x = f 5 in L 2 (α, β ), (4.2.53) iλ φ -κ 3 w xx = f 6 in L 2 (β , L), (4.2.54) η ρ (L, •) + iτλ η(L, •) = τ f 7 (L, •) in L 2 (0, 1), (4.2.55)
with the following boundary conditions

u(0) = 0, u(α) = v(α), v(β ) = w(β ), (4.2.56) κ 2 v x (α) + δ 1 z x (α) = κ 1 u x (α), κ 2 v x (β ) + δ 1 z x (β ) = κ 3 w x (β ), (4.2
.57)

w x (L) = -δ 3 η(L, 0) -δ 2 η(L,
η(L, ρ) = (iλ w(L) -f 3 (L)) e -iτλ ρ + τ ρ 0 e iτλ (ξ -ρ) f 7 (L, ξ )dξ . ( 4
-λ 2 u -κ 1 u xx = iλ f 1 + f 4 , (4.2.61) -λ 2 v -(κ 2 v x + δ 1 z x ) x = iλ f 2 + f 5 , (4.2.62)
-λ 2 w -κ 3 w xx = iλ f 3 + f 6 , (4.2.63)

z x = iλ v x -( f 2 ) x , (4.2.64)
u(0) = 0, u(α) = v(α), κ 1 u x (α) = κ 2 v x (α) + δ 1 z x (α), (4.2.65)
w(β ) = v(β ), κ 3 w x (β ) = κ 2 v x (β ) + δ 1 z x (β ), (4.2
.66) (α,β ) and (β , L) respectively, taking the sum, then using integration by parts, we get

w x (L) = -iλ δ 3 + δ 2 e -iτλ w(L) + δ 3 + δ 2 e -iτλ f 3 (L) -τδ 2 1 0 e iτλ (ξ -1) f 7 (L, ξ )dξ . ( 4
κ 1 α 0 u x ϕ x dx + β α (κ 2 v x + δ 1 z x )ψ x dx + κ 3 L β w x θ x dx + κ 1 u x (0)ϕ(0) -κ 1 u x (α)ϕ(α) + (κ 2 v x (α) + δ 1 z x (α))ψ(α) -(κ 2 v x (β ) + δ 1 z x (β ))ψ(β ) + κ 3 w x (β )θ (β ) -λ 2 α 0 uϕdx -λ 2 β α vψdx -λ 2 L β wθ dx -κ 3 w x (L)θ (L) = α 0 (iλ f 1 + f 4 ) ϕdx + β α (iλ f 2 + f 5 ) ψdx + L β (iλ f 3 + f 6 ) θ dx. (4.2.68)
From the fact that (ϕ, ψ, θ ) ∈ H 1 L , we have

ϕ(0) = 0, ϕ(α) = ψ(α), θ (β ) = ψ(β ).
Inserting the above equation in (4.2.68), then using (4.2.64)-(4.2.67), we get

a ((u, v, w) , (ϕ, ψ, θ )) = F (ϕ, ψ, θ ) , ∀ (ϕ, ψ, θ ) ∈ H 1 L , (4.2.69)
where

F (ϕ, ψ, θ ) = α 0 (iλ f 1 + f 4 ) ϕdx + β α (iλ f 2 + f 5 ) ψdx + δ 1 β α ( f 2 ) x ψ x dx + L β (iλ f 3 + f 6 ) θ dx + κ 3 δ 3 + δ 2 e -iτλ f 3 (L) -τδ 2 1 0 e iτλ (ξ -1) f 7 (L, ξ )dξ θ (L) and a ((u, v, w) , (ϕ, ψ, θ )) = a 1 ((u, v, w) , (ϕ, ψ, θ )) + a 2 ((u, v, w) , (ϕ, ψ, θ )) , such that          a 1 ((u, v, w) , (ϕ, ψ, θ )) = κ 1 α 0 u x ϕ x dx + (κ 2 + iδ 1 λ ) β α v x ψ x dx + κ 3 L β w x θ x dx, a 2 ((u, v, w) , (ϕ, ψ, θ )) = -λ 2 α 0 uϕdx -λ 2 β α vψdx -λ 2 L β wθ dx + iκ 3 λ δ 3 + δ 2 e -iτλ w(L)θ (L).

DAMPING AND WITH AN INTERNAL OR BOUNDARY TIME DELAY

Let H 1 L ′ be the dual space of H 1 L . We define the operators A, A 1 and A 2 by A :

H 1 L → H 1 L ′ (u, v, w) → A(u, v, w) A 1 : H 1 L → H 1 L ′ (u, v, w) → A 1 (u, v, w) A 2 : H 1 L → H 1 L ′ (u, v, w) → A 2 (u, v, w) such that        (A(u, v, w)) (ϕ, ψ, θ ) = a ((u, v, w) , (ϕ, ψ, θ )) , ∀ (ϕ, ψ, θ ) ∈ H 1 L , (A 1 (u, v, w)) (ϕ, ψ, θ ) = a 1 ((u, v, w) , (ϕ, ψ, θ )) , ∀ (ϕ, ψ, θ ) ∈ H 1 L , (A 2 (u, v, w)) (ϕ, ψ, θ ) = a 2 ((u, v, w) , (ϕ, ψ, θ )) , ∀ (ϕ, ψ, θ ) ∈ H 1 L . (4.2.70)
Our aim is to prove that the operator A is an isomorphism. For this aim, we proceed the proof in three steps.

Step 1. In this step, we prove that the operator A 1 is an isomorphism. For this aim, according to (4.2.70), we have

a 1 ((u, v, w) , (ϕ, ψ, θ )) = κ 1 α 0 u x ϕ x dx + (κ 2 + iδ 1 λ ) β α v x ψ x dx + κ 3 L β w x θ x dx.
We can easily verify that a 1 is a bilinear continuous coercive form on H 1 L × H 1 L . Then, by Lax-Milgram Lemma, the operator A 1 is an isomorphism.

Step 2. In this step, we prove that the operator A 2 is compact. First, for 1 2 < r < 1, we introduce the Hilbert space H r L by

H r L = {(ϕ, ψ, θ ) ∈ H r (0, α) × H r (α, β ) × H r (β , L) | ϕ(0) = 0, ϕ(α) = ψ(α), ψ(β ) = θ (β )}.
Thus by trace theorem, there exists C > 0, such that

|θ (L)| ≤ C (ϕ, ψ, θ ) H r L . (4.2.71)
Now, according to (4.2.70), we have

a 2 ((u, v, w) , (ϕ, ψ, θ )) = -λ 2 α 0 uϕdx -λ 2 β α vψdx -λ 2 L β wθ dx + iκ 3 λ δ 3 + δ 2 e -iτλ w(L)θ (L).
Then, by using (4.2.71), we get

|a 2 ((u, v, w) , (ϕ, ψ, θ ))| ≤ C 1 (u, v, w) H 1 L (ϕ, ψ, θ ) L 2 +C 1 (u, v, w) H 1 L (ϕ, ψ, θ ) H r L ,
where C 1 > 0. Therefore, for all r ∈ ( 1 2 , 1) there exists C 2 > 0, such that

|a 2 ((u, v, w) , (ϕ, ψ, θ ))| ≤ C 2 (u, v, w) H 1 L (ϕ, ψ, θ ) H r L , which implies that A 2 ∈ L H 1 L , (H r L ) ′ .
Finally, using the compactness embedding from (H r L ) ′ into H 1 L ′ we deduce that A 2 is compact.

From steps 1 and 2, we get that the operator A = A 1 + A 2 is a Fredholm operator of index zero 0. Consequently, by Fredholm alternative, proving the operator A is an isomorphism reduces to proving ker(A) = {0}.

Step 3. In this step, we prove that the ker(A) = {0}. For this aim, let ( ũ, ṽ, w) ∈ ker(A), i.e., a (( ũ, ṽ, w) , (ϕ, ψ, θ

)) = 0, ∀ (ϕ, ψ, θ ) ∈ H 1 L .
Equivalently,

α 0 κ 1 ũx ϕ x -λ 2 ũϕ dx + β α (κ 2 + iδ 1 λ ) ṽx ψ x -λ 2 ṽψ dx + L β κ 3 wx θ x -λ 2 wθ dx +iκ 3 λ δ 3 + δ 2 e -iτλ w(L)θ (L) = 0, ∀ (ϕ, ψ, θ ) ∈ H 1 L .
Then, we find that

         -λ 2 ũ -κ 1 ũxx = 0, -λ 2 ṽ -(κ 2 + iδ 1 λ ) ṽxx = 0, -λ 2 w -κ 3 wxx = 0, ũ(0) = 0, ũ(α) = ṽ(α), κ 1 ũx (α) = (κ 2 + iδ 1 λ ) ṽx (α), w(β ) = ṽ(β ), κ 3 wx (β ) = (κ 2 + iδ 1 λ ) ṽx (β ), wx (L) = -iλ δ 3 + δ 2 e -iτλ w(L).
Therefore, the vector Ṽ define by Ṽ = ũ, ṽ, w, iλ ũ, iλ ṽ, iλ w, iλ w(L)e -iτλ • belongs to D(A ) and we have iλ Ṽ -A Ṽ = 0. Thus, Ṽ ∈ ker(iλ I -A ), therefore by Lemma 4.2.5 , we get Ṽ = 0, this implies that ũ = 0, ṽ = 0 and w = 0, so ker(A) = {0}.

Therefore, from step 3 and Fredholm alternative, we get that the operator A is an isomorphism. It easy to see that the operator F is continuous form on H 1 L . Consequently, Equation (4.2.69) admits a unique solution (u, v, w) ∈ H 1 L . Thus, using (4.2.49)-(4.2.51), (4.2.60) and a classical regularity arguments, we conclude that (iλ I -A )U = F admits a unique solution U ∈ D (A ). The proof is thus complete.

Proof of Theorem 4.2.4. From Lemma 4.2.5, we have that the operator A has no pure imaginary eigenvalues and by Lemma 4.2.6, R(iλ I -A ) = H for all λ ∈ R. Therefore, the closed graph theorem implies that σ (A ) ∩ iR = / 0. Thus, we get the conclusion by applying Theorem 1.3.3 of Arendt and Batty. The proof is thus complete.

Polynomial Stability

In this part, we will prove the polynomial stability of System (4.2.8)-(4.2.10). Our main result in this part is the following theorem. Theorem 4.2.7. Under hypothesis (H), for all initial data U 0 ∈ D(A ), there exists a constant C > 0 independent of U 0 such that the energy of System (4.2.8)-(4.2.10) satisfies the following estimation

E(t) ≤ C t 4 U 0 2 D(A ) , ∀t > 0. (4.2.72)
From Lemma 4.2.5 and Lemma 4.2.6, we have seen that iR ⊂ ρ(A ), then for the proof of Theorem 4.2.7, according to the result of Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] (see also [START_REF] Batty | Non-uniform stability for bounded semi-groups on Banach spaces[END_REF]), we need to prove that sup

λ ∈R (iλ I -A ) -1 L (H ) = O |λ | 1 2 . ( 4

.2.73)

We will argue by contradiction. Indeed, suppose there exists

{(λ n , U n := (u n , v n , w n , y n , z n , φ n , η n (L, •)))} n≥1 ⊂ R * + × D (A ) , such that λ n → +∞, U n H = 1 (4.2.74)
and there exists sequence

F n := ( f 1,n , f 2,n , f 3,n , f 4,n , f 5,n , f 6,n , f 7,n (L, •)) ∈ H , such that λ ℓ n (iλ n I -A )U n = F n → 0 in H . ( 4
iλ u -y = λ -ℓ f 1 in H 1 (0, α), (4.2.76) iλ v -z = λ -ℓ f 2 in H 1 (α, β ), (4.2.77) iλ w -φ = λ -ℓ f 3 in H 1 (β , L), (4.2.78) iλ y -κ 1 u xx = λ -ℓ f 4 in L 2 (0, α), (4.2.79) iλ z -(κ 2 v x + δ 1 z x ) x = λ -ℓ f 5 in L 2 (α, β ), (4.2.80) iλ φ -κ 3 w xx = λ -ℓ f 6 in L 2 (β , L), (4.2.81) η ρ (L, •) + iτλ η(L, •) = τλ -ℓ f 7 (L, •) in L 2 (0, 1). (4.2.82)
Remark that, since U = (u, v, w, y, z, φ , η(L, •)) ∈ D(A ), we have the following boundary conditions

|u x (α)| = κ -1 1 |κ 2 v x (α) + δ 1 z x (α)|, |y(α)| = |z(α)| , |w x (β )| = κ -1 3 |κ 2 v x (β ) + δ 1 z x (β )|, |z(β )| = |φ (β )| (4.2.83)
and

w x (L) = -δ 3 η(L, 0) -δ 2 η(L, 1), φ (L) = η(L, 0). ( 4
|z x | 2 = o λ -ℓ , (4.2.85) |φ (L)| 2 = |η(L, 0)| 2 = o λ -ℓ , |η(L, 1)| 2 = o λ -ℓ , (4.2.86) β α |v x | 2 dx = o λ -ℓ-2 , (4.2.87) |w x (L)| 2 = o λ -ℓ . (4.2.88)
Proof. Taking the inner product of (4.2.75) with U in H , then using the fact that U is uniformly bounded in H , we get

-ℜ A U, U H = ℜ (iλ I -A )U, U H = o λ -ℓ .
Now, under hypothesis (H), similar to Equation (4.2.14), we get

0 ≤ δ 1 β α |z x | 2 dx +C 1 |η(L, 1)| 2 +C 2 |η(L, 0)| 2 ≤ -ℜ (A U, U) H = o λ -ℓ , (4.2.89)
where

C 1 = 1 2 - κ 3 |δ 2 | 2p > 0 and C 2 = κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 > 0.
Therefore

η(L, ρ) = η(L, 0)e -iτλ ρ + τλ -ℓ ρ 0 e iτλ (ξ -ρ) f 7 (L, ξ )dξ ∀ ρ ∈ (0, 1).
By using Cauchy Schwarz inequality, we get

|η(L, ρ)| 2 ≤ 2|η(L, 0)| 2 +2τ 2 λ -2ℓ 1 0 | f 7 (L, ξ )|dξ 2 ≤ 2|η(L, 0)| 2 +2τ 2 λ -2ℓ 1 0 | f 7 (L, ξ )| 2 dξ ∀ ρ ∈ (0, 1).
Integrating over (0, 1) with respect to ρ, then using (4.2.86) and the fact that f 7 (L, •) → 0 in L 2 (0, 1), we get

1 0 |η(L, ρ)| 2 dρ ≤ 2|η(L, 0)| 2 + 2τ 2 λ -2ℓ 1 0 | f 7 (L, ξ )| 2 dξ = o λ -ℓ ,
hence, we get (4.2.90). Thus, the proof of the Lemma is complete.

Lemma 4.2.10. Under hypothesis (H), for all ℓ ≥ 0, the solution (u, v, w, y, z, φ , η(L,

β |φ | 2 dx = o λ -ℓ , L β |w x | 2 dx = o λ -ℓ , (4.2.91) |w x (β )| 2 = o λ -ℓ , |φ (β )| 2 = o λ -ℓ , (4.2.92) |κ 2 v x (β ) + δ 1 z x (β )| 2 = o λ -ℓ , |z(β )| 2 = o λ -ℓ . ( 4
iλ w x = -φ x -λ -ℓ ( f 3 ) x .
Inserting the above result in (4.2.94), then using the fact that φ , w x are uniformly bounded in L 2 (β , L) and

( f 3 ) x , f 6 converge to zero in L 2 (β , L) gives - L β x φ φ x dx -κ 3 L β xw xx w x dx = o λ -ℓ .
Taking the real part in the above equation, then using integration by parts, we get

1 2 L β |φ | 2 dx + κ 3 2 L β |w x | 2 dx + β 2 κ 3 |w x (β )| 2 + |φ (β )| 2 = L 2 κ 3 |w x (L)| 2 + |φ (L)| 2 + o λ -ℓ .
Inserting (4.2.86) and (4.2.88) in the above equation, we get

1 2 L β |φ | 2 dx + κ 3 2 L β |w x | 2 dx + β 2 κ 3 |w x (β )| 2 + |φ (β )| 2 = o λ -ℓ ,
|z(α)| 2 = o λ -min(2ℓ,ℓ+ 1 2 ) , |z(β )| 2 = o λ -min(2ℓ,ℓ+ 1 2 ) , (4.2.96) |κ 2 v x (α) + δ 1 z x (α)| 2 = o λ -min(2ℓ-1,ℓ-1 2 ) . (4.2.97) Proof. Let g ∈ C 1 ([α, β ]) such that g(β ) = -g(α) = 1, max x∈[α,β ] |g(x)| = c g and max x∈[α,β ] |g ′ (x)| = c g ′ ,
where c g and c g ′ are strictly positive constant numbers independent from λ . The proof is divided into three steps.

Step 1. In this step, we prove the following asymptotic behavior estimate

|z(β )| 2 + |z(α)| 2 ≤ λ 1 2 2 + 2c g ′ β α |z| 2 dx + o λ -min(2ℓ,ℓ+ 1 2 ) . (4.2.98)
First, from (4.2.77), we have

z x = iλ v x -λ -ℓ ( f 2 ) x in L 2 (α, β ).
Multiplying the above equation by 2 gz and integrating over (α, β ), then taking the real part, we get

β α g(x) (|z| 2 ) x dx = ℜ 2iλ β α g(x) v x zdx -ℜ 2λ -ℓ β α g(x) ( f 2 ) x zdx ,
using integration by parts in the left hand side of above equation, we get

g(x) |z| 2 β α = β α g ′ (x) |z| 2 dx + ℜ 2iλ β α g(x) v x zdx -ℜ 2λ -ℓ β α g(x) ( f 2 ) x zdx .
Consequently, we obtain

|z(β )| 2 + |z(α)| 2 ≤ c g ′ β α |z| 2 dx + 2λ c g β α |v x | |z| dx + 2λ -ℓ c g β α |( f 2 ) x | |z| dx. (4.2.99)
On the other hand, we have

2λ c g |v x ||z| ≤ λ 1 2 |z| 2 2 + 2λ 3 2 c 2 g |v x | 2 and 2λ -ℓ c g |( f 2 ) x ||z| ≤ c g ′ |z| 2 + c 2 g λ -2ℓ c g ′ |( f 2 ) x | 2 .
Inserting the above equation in (4.2.99), then using (4.2.87) and the fact that ( f 2 ) x → 0 in L 2 (α, β ), we get

|z(β )| 2 + |z(α)| 2 ≤ λ 1 2 2 + 2 c g ′ β α |z| 2 dx + o λ -min(2ℓ,ℓ+ 1 2 ) ,
hence, we get (4.2.98).

Step 2. In this step, we prove the following asymptotic behavior estimate

|κ 2 v x (α) + δ 1 z x (α)| 2 + |κ 2 v x (β ) + δ 1 z x (β )| 2 ≤ λ 3 2 2 β α |z| 2 dx + o λ -ℓ+ 1 2 . (4.2.100)
First, multiplying (4.2.80) by -2 g (κ 2 v x + δ 1 z x ) and integrating over (α, β ), then taking the real part, we get

β α g(x) |κ 2 v x + δ 1 z x | 2 x dx = 2ℜ iλ β α g(x)z (κ 2 v x + δ 1 z x ) dx -2λ -ℓ ℜ β α g(x) f 5 (κ 2 v x + δ 1 z x ) dx ,
using integration by parts in the left-hand side of the above equation, we get

g(x) |κ 2 v x + δ 1 z x | 2 β α = β α g ′ (x) |κ 2 v x + δ 1 z x | 2 dx + 2ℜ iλ β α g(x) z (κ 2 v x + δ 1 z x ) dx -2λ -ℓ ℜ β α g(x) f 5 (κ 2 v x + δ 1 z x ) dx .
Consequently, we obtain

|κ 2 v x (β ) + δ 1 z x (β )| 2 + |κ 2 v x (α) + δ 1 z x (α)| 2 ≤ c g ′ β α |κ 2 v x + δ 1 z x | 2 dx + 2λ c g β α |z| |κ 2 v x + δ 1 z x | dx +2λ -ℓ c g β α | f 5 | |κ 2 v x + δ 1 z x | dx.
Now, using Cauchy Schwarz inequality, Equations (4.2.85), (4.2.87) and the fact that f 5 → 0 in L 2 (α, β ) in the right hand side of above equation, we get

|κ 2 v x (β ) + δ 1 z x (β )| 2 + |κ 2 v x (α) + δ 1 z x (α)| 2 ≤ 2λ c g β α |z| |κ 2 v x + δ 1 z x | dx + o λ -ℓ . (4.2.101)
On the other hand, we have

2λ c g |z| |κ 2 v x + δ 1 z x | ≤ λ 3 2 2 |z| 2 + 2λ 1 2 c 2 g |κ 2 v x + δ 1 z x | 2 .
Inserting the above equation in (4.2.101), then using Equations (4.2.85) and (4.2.87), we get

|κ 2 v x (α) + δ 1 z x (α)| 2 + |κ 2 v x (β ) + δ 1 z x (β )| 2 ≤ λ 3 2 2 β α |z| 2 dx + o λ -ℓ+ 1 2
, hence, we get (4.2.100).

Step 3. In this step, we prove the asymptotic behavior estimations of (4.2.95)-(4.2.97). First, multiplying (4.2.80) by -iλ -1 z and integrating over (α, β ), then taking the real part, we get

β α |z| 2 dx = -ℜ iλ -1 β α (κ 2 v x + δ 1 z x ) x z dx -ℜ i λ -ℓ-1 β α f 5 z dx , consequently, β α |z| 2 dx ≤ λ -1 β α (κ 2 v x + δ 1 z x ) x z dx + λ -ℓ-1 β α | f 5 | |z| dx. (4.2.102)
From the fact that z is uniformly bounded in L 2 (α, β ) and f 5 → 0 in L 2 (α, β ), we get

λ -ℓ-1 β α | f 5 | |z| dx = o λ -ℓ-1 . (4.2.103)
On the other hand, using integration by parts and (4.2.85), (4.2.87), we get

β α (κ 2 v + δ 1 z) xx z dx = [(κ 2 v x + δ 1 z x ) z] β α - β α (κ 2 v x + δ 1 z x ) z x dx ≤ |κ 2 v x (β ) + δ 1 z x (β )| |z(β )| + |κ 2 v x (α) + δ 1 z x (α)| |z(α)| + β α |κ 2 v x + δ 1 z x | |z x | dx ≤ |κ 2 v x (β ) + δ 1 z x (β )| |z(β )| + |κ 2 v x (α) + δ 1 z(α)| |z(α)| + o λ -ℓ .
Inserting the above equation and Equation (4.2.103) in (4.2.102), we get

β α |z| 2 dx ≤ λ -1 |κ 2 v x (β ) + δ 1 z x (β )| |z(β )| + λ -1 |κ 2 v x (α) + δ 1 z x (α)| |z(α)| + o λ -ℓ-1 . (4.2.104) Now, for ζ = β or ζ = α, we have λ -1 |κ 2 v x (ζ ) + δ 1 z x (ζ )| |z(ζ )| ≤ λ -1 2 2 |z(ζ )| 2 + λ -3 2 2 |κ 2 v x (ζ ) + δ 1 z x (ζ )| 2 .
Substituting the above equation in (4.2.104), we get

β α |z| 2 dx ≤ λ -1 2 2 |z(α)| 2 + |z(β )| 2 + λ -3 2 2 |κ 2 v x (α) + δ 1 z x (α)| 2 + |κ 2 v x (β ) + δ 1 z x (β )| 2 + o λ -ℓ-1 .
Next, inserting Equations (4.2.98) and (4.2.100) in the above inequality, we obtain

β α |z| 2 dx ≤ 1 2 + c g ′ λ 1 2 β α |z| 2 dx + o λ -min(2ℓ+ 1 2 ,ℓ+1) , consequently, 1 2 - c g ′ λ 1 2 β α |z| 2 dx ≤ o λ -min(2ℓ+ 1 2 ,ℓ+1) .
Since λ → +∞, by choosing λ > 4 c 2 g ′ , we get

0 < 1 2 - c g ′ λ 1 2 β α |z| 2 dx ≤ o λ -min(2ℓ+
(x) = cos (β -x)π β -α to get g(β ) = -g(α) = 1, g ∈ C 1 ([α, β ]), max x∈[α,β ] |g(x)| = 1, max x∈[α,β ] |g ′ (x)| = π β -α .
Also, we can take

g(x) = β -x β -α 2 -3 β -x β -α + 1.
Lemma 4.2.13. Under hypothesis (H), for all ℓ ≥ 0, the solution (u, v, w, y, z, φ , η(L,

α 0 |y| 2 dx = o λ -min(2ℓ-1,ℓ-1 2
) and

α 0 |u x | 2 dx = o λ -min(2ℓ-1,ℓ-1 2 ) . ( 4
iλ u x = -y x -λ -ℓ ( f 1 ) x .
Inserting the above result in (4.2.106), then using the fact that u x , y are uniformly bounded in L 2 (0, α) and ( f 1 ) x , f 4 converge to zero in L 2 (0, α) gives

- α 0 xyy x dx -κ 1 α 0 xu xx u x dx = o λ -ℓ .
Taking the real part in the above equation, then using integration by parts, we get 1 2

α 0 |y| 2 dx + κ 1 2 α 0 |u x | 2 dx - α 2 κ 1 |u x (α)| 2 + |y(α)| 2 = o λ -ℓ .
Inserting the boundary conditions (4.2.83) at x = α in the above equation to get

1 2 α 0 |y| 2 dx + κ 1 2 α 0 |u x | 2 dx = α 2 κ -1 1 |κ 2 v x (α) + δ 1 z x (α)| 2 + |z(α)| 2 + o λ -ℓ .
Finally, substituting (4.

U 2 H = κ 1 α 0 |u x | 2 dx + κ 2 β α |v x | 2 dx + κ 3 L β |w x | 2 dx + α 0 |y| 2 dx + β α |z| 2 dx + L β |φ | 2 dx + τ 1 0 |η(L, ρ)| 2 dρ = o λ -min(2ℓ-1,ℓ-1 2 ) .
To obtain U H = o(1), we need min 2ℓ -1, ℓ -1 2 ≥ 0, so we choose ℓ = 1 2 as the optimal value. Hence, we obtain that U H = o(1) which contradicts (4.2.74). Therefore, the energy of System (4.2.8)-(4.2.10) satisfies estimation (4.2.72) for all initial data U 0 ∈ D(A ). The proof is thus complete.

Wave equation with local Kelvin-Voigt damping near the boundary and boundary delay feedback

In this subsection, we study the stability of System (4.2.1), but in the case that the Kelvin-Voigt damping is near the boundary, i.e., α = 0 and 0 < β < L (see Figure 4.2). For this aim, we denote the longitudinal displacement by U and this displacement is divided into two parts

U(x,t) = v(x,t), (x,t) ∈ (α, β ) × (0, +∞), w(x,t), (x,t) ∈ (β , L) × (0, +∞).
In this case, System (4.2.1) is equivalent to the following system

                                                 v tt -(κ 2 v x + δ 1 v xt ) x = 0, (x,t) ∈ (0, β ) × (0, +∞), w tt -κ 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞), τη t (L, ρ,t) + η ρ (L, ρ,t) = 0, (ρ,t) ∈ (0, 1) × (0, +∞), v(0,t) = 0, t ∈ (0, +∞), w x (L,t) = -δ 3 w t (L,t) -δ 2 η(L, 1,t), t ∈ (0, +∞), v(β ,t) = w(β ,t), t ∈ (0, +∞), κ 2 v x (β ,t) + δ 1 v xt (β ,t) = κ 3 w x (β ,t), t ∈ (0, +∞), (v(x, 0), v t (x, 0)) = (v 0 (x), v 1 (x)) , x ∈ (α, β ), (w(x, 0), w t (x, 0)) = (w 0 (x), w 1 (x)) , x ∈ (β , L), η(L, ρ, 0) = f 0 (L, -ρ τ), ρ ∈ (0, 1). ( 4
X m = H m (0, β ) × H m (β , L), m = 1, 2, X 0 = L 2 (0, β ) × L 2 (β , L), X 1 L = {(v, w) ∈ X 1 | v(0) = 0, v(β ) = w(β )},
where the Hilbert space X 0 is equipped with the norm:

(v, w) 2 X 0 = β 0 |v| 2 dx + L β |w| 2 dx.
Moreover, it is easy to check that the space X 1 L is Hilbert space over C equipped with the norm:

(v, w) 2 X 1 L = κ 2 β 0 |v x | 2 dx + κ 3 L β |w x | 2 dx.
In addition, by Poincaré inequality, we can easily verify that there exists C > 0 depending on κ 2 , κ 3 , β and L, such that

(v, w) X 0 ≤ C (v, w) X 1 L , ∀(v, w) ∈ X 1 L .
We now define the Hilbert energy space by

H 1 = X 1 L × X 0 × L 2 (0, 1)
equipped with the following inner product

(U, Ũ) H 1 = κ 2 β 0 v x ṽx dx + κ 3 L β w x wx dx + β 0 zzdx + L β φ φ dx + τ 1 0 η(L, ρ) η(L, ρ)dρ,
where U = (v, w, z, φ , η(L, •)) ∈ H 1 and Ũ = ( ṽ, w, z, φ , η(L, •)) ∈ H 1 . We use U H 1 to denote the corresponding norm. We define the linear unbounded operator A 1 : D(A 1 ) ⊂ H 1 -→ H 1 by:

D(A 1 ) = U = (v, w, z, φ , η(L, •)) ∈ X 1 L × X 1 L × H 1 (0, 1) | (κ 2 v + δ 1 z, w) ∈ X 2 , κ 2 v x (β ) + δ 1 z x (β ) = κ 3 w x (β ), w x (L) = -δ 3 η(L, 0) -δ 2 η(L, 1), φ (L) = η(L, 0)
and for all U = (v, w, z, φ , η(L, •)) ∈ D(A 1 ),

A 1 U = z, φ , (κ 2 v x + δ 1 z x ) x , κ 3 w xx , -τ -1 η ρ (L, •) . If U = (v, w, v t , w t , η(L, •)
) is a regular solution of System (4.2.107), then we transform this system into the following initial value problem

U t = A 1 U, U(0) = U 0 , (4.2.108)
where

U 0 = (v 0 , w 0 , v 1 , w 1 , f 0 (L, -• τ)) ∈ H 1 . Note that D(A 1
) is dense in H 1 and that for all U ∈ D(A 1 ), we have

ℜ (A 1 U, U) H 1 ≤ -δ 1 β 0 |z x | 2 dx - 1 2 - κ 3 |δ 2 | 2p |η(L, 1)| 2 -κ 3 δ 3 - 1 2 - κ 3 |δ 2 | p 2 |η(L, 0)|
(iλ I -A 1 ) -1 L (H 1 ) = O (1) . ( 4
{(λ n , U n := (v n , w n , z n , φ n , η n (L, •)))} n≥1 ⊂ R * + × D (A 1 ) , such that λ n → +∞, U n H 1 = 1 (4.2.112)
and there exists sequence G n := (g 1,n , g 2,n , g 3,n , g 4,n , g 5,n (L, •)) ∈ H 1 , such that

(iλ n I -A 1 )U n = G n → 0 in H 1 . ( 4
iλ v -z = g 1 in H 1 (0, β ), (4.2.114) iλ w -φ = g 2 in H 1 (β , L), (4.2.115) iλ z -(κ 2 v x + δ 1 z x ) x = g 3 in L 2 (0, β ), (4.2.116) iλ φ -κ 3 w xx = g 4 in L 2 (β , L), (4.2.117) η ρ (L, •) + iτλ η(L, •) = τg 5 (L, •) in L 2 (0, 1). (4.2.118) Remark that, since U = (v, w, z, φ , η(L, •)) ∈ D(A 1 )
, we have the following boundary conditions

|w x (β )| = κ -1 3 |κ 2 v x (β ) + δ 1 z x (β )| , |z(β )| = |φ (β )| , (4.2.119) w x (L) = -δ 3 η(L, 0) -δ 2 η(L, 1), φ (L) = η(L, 0). (4.2.120)
Taking the inner product of (4.2.113) with U in H 1 , then using (4.2.109), hypothesis (H) and the fact that U is uniformly bounded in H 1 , we obtain

β 0 |z x | 2 = o (1) , |φ (L)| 2 = |η(L, 0)| 2 = o (1) , |η(L, 1)| 2 = o (1) . (4.2.121)
Taking (4.2.114), then using the first asymptotic estimate of (4.2.121) and the fact that (g 1 ) x → 0 in L 2 (0, β ), we get

β 0 |v x | 2 dx = o λ -2 . (4.2.122)
Taking the first asymptotic estimate of (4.2.120), then using the second and the third asymptotic estimates of (4.2.121), we obtain

|w x (L)| 2 = o (1) . ( 4

.2.123)

Similarly as in Lemma 4.2.9, with ℓ = 0, taking (4.2.118), then using the second and the third asymptotic estimates of (4.2.121), we obtain

1 0 |η(L, ρ)| 2 dρ = o (1) . (4.2.124)
Similarly as in Lemma 4.2.10, with ℓ = 0, multiplying Equation (4.2.117) by xw x and integrating over (β , L), after that using the fact that iλ w x = -φ x -(g 2 ) x , then using the fact that φ , w x are uniformly bounded in L 2 (β , L) and (g 2 ) x , g 4 converge to zero in L 2 (β , L) gives

- L β x φ φ x dx -κ 3 L β xw xx w x dx = o (1) .
Taking the real part in the above equation, then using integration by parts, Equation (4.2.123) and the second asymptotic estimate of (4.2.121), we obtain

1 2 L β |φ | 2 dx + κ 3 2 L β |w x | 2 dx + β 2 κ 3 |w x (β )| 2 + |φ (β )| 2 = o (1) , hence, we get L β |φ | 2 dx = o(1), L β |w x | 2 dx = o(1), |w x (β )| 2 = o(1), |φ (β )| 2 = o (1) . (4.2.125)
Inserting the third and the fourth asymptotic estimates of (4.2.125) in (4.2.119), we get

|κ 2 v x (β ) + δ 1 z x (β )| = o(1), |z(β )| = o(1). (4.2.126)
Similar to step 3 of Lemma 4.2.11, with α = 0 and ℓ = 0, multiplying (4.2.116) by -iλ -1 z and integrating over (0, β ), taking the real part, then using the fact that z is uniformly bounded in L 2 (0, β ) and g 3 → 0 in L 2 (0, β ), we get

β 0 |z| 2 dx ≤ λ -1 β 0 (κ 2 v x + δ 1 z x ) x z dx + o λ -1 . (4.2.127)
On the other hand, using integration by parts, the fact that z(0) = 0, and Equations ( 4

β 0 (κ 2 v x + δ 1 z x ) x z dx = [(κ 2 v x + δ 1 z x ) z] β 0 - β 0 (κ 2 v x + δ 1 z x ) z x dx ≤ |κ 2 v x (β ) + δ 1 z x (β )| |z(β )| + β 0 |κ 2 v x + δ 1 z x | |z x | dx = o(1

Wave equation with local internal Kelvin-Voigt damping and local internal delay feedback

In this section, we study the stability of System (4.1.2). We assume that there exist α such that 0 < α < L, in this case, the Kelvin-Voigt damping and the time delay feedback are locally distributed near the boundary (see Figure 4.3). For this aim, we denote the longitudinal displacement by U and this displacement is divided into two parts

U(x,t) =    u(x,t), (x,t) ∈ (0, α) × (0, +∞), v(x,t), (x,t) ∈ (α, L) × (0, +∞).
Furthermore, like in [START_REF] Nicaise | Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks[END_REF], we introduce the auxiliary unknown η(x, ρ,t) = v t (x,t -ρ τ), x ∈ (α, L), ρ ∈ (0, 1), t > 0.

In this case, System (4.1.2) is equivalent to the following system

u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞), (4.3.1) v tt -(κ 2 v x + δ 1 v xt (x,t) + δ 2 η x (x, 1,t)) x = 0, (x,t) ∈ (α, L) × (0, +∞), (4.3.2)
τη t (x, ρ,t) + η ρ (x, ρ,t) = 0, (x, ρ,t) ∈ (α, L) × (0, 1) × (0, +∞), (4. 3.3) with the Dirichlet boundary conditions

u(0,t) = v(L,t) = η(L, ρ,t) = 0, t ∈ (0, +∞), ρ ∈ (0, 1), (4.3.4)
with the following transmission conditions u(α,t) = v(α,t), t ∈ (0, +∞),

κ 1 u x (α,t) = κ 2 v x (α,t) + δ 1 v xt (α,t) + δ 2 η x (α, 1,t), t ∈ (0, +∞), (4.3.5)
and with the following initial conditions

       (u(x, 0), u t (x, 0)) = (u 0 (x), u 1 (x)), x ∈ (0, α), (v(x, 0), v t (x, 0)) = (v 0 (x), v 1 (x)), x ∈ (α, L), η(x, ρ, 0) = f 0 (x, -ρτ), (x, ρ) ∈ (α, L) × (0, 1). (4.3.6)
Under assumption (H1), let us define the energy of a solution of System (4.3.1)-(4.3.6) as

E(t) = 1 2 α 0 |u t (x,t)| 2 + κ 1 |u x (x,t)| 2 dx + 1 2 L α |v t (x,t)| 2 + κ 2 |v x (x,t)| 2 dx + τ|δ 2 | 2 L α 1 0 |η x (x, ρ,t)| 2 dρ dx.
Multiplying (4.3.1), (4.3.2) and (4. 3.3) x by u t , y t and |δ 2 |η x , integrating over (0, α), (α, L) and (α, L) × (0, 1) respectively, taking the sum, then using integration by parts and the boundary conditions in (4.3.4)-(4.3.5), we get

E ′ (t) = -δ 1 + |δ 2 | 2 L α |v xt (x,t)| 2 dx - |δ 2 | 2 L α |η x (x, 1,t)| 2 dx -δ 2 L α v xt (x,t) η x (x, 1,t)dx,
Using Young's inequality for the third term in the right, we get

E ′ (t) ≤ (-δ 1 + |δ 2 |) L α |v xt (x,t)| 2 dx.
Under assumption (H1), the energies of the strong solutions satisfy E ′ (t) ≤ 0. Hence, the System (4.3.1)-(4.3.6) is dissipative in the sense that its energy is non increasing with respect to the time t.

Well-posedness of the problem

We start this part by formulating System (4.3.1)-(4.3.6) as an abstract Cauchy problem. For this aim, let us define

H 1 R (α, L) = v ∈ H 1 (α, L) | v(L) = 0 , L 2 = L 2 (0, α) × L 2 (α, L), H 2 = H 2 (0, α) × H 2 (α, L), H 1 = {(u, v) ∈ H 1 (0, α) × H 1 (α, L) | u(0) = 0, u(α) = v(α), v(L) = 0}.
The spaces L 2 , H 1 and H 1 R (α, L) are obviously a Hilbert spaces over C equipped respectively with the norms

(u, v) 2 L 2 = α 0 |u| 2 dx + L α |v| 2 dx, (u, v) 2 H 1 = κ 1 α 0 |u x | 2 dx + κ 2 L α |v x | 2 dx and v 2 H 1 R (α,L) = L α |v x | 2 dx.
In addition by Poincaré inequality, we can easily verify that there exist C 1 > 0 and C 2 > 0 depending on κ 1 , κ 2 , α and L, such that

(u, v) L 2 ≤ C 1 (u, v) H 1 and L α |w| 2 dx ≤ C 2 w 2 H 1 R (α,L) , ∀(u, v) ∈ H 1 , w ∈ H 1 R (α, L).
Let us define the energy Hilbert space H 2 by

H 2 = H 1 × L 2 × L 2 (0, 1), H 1 R (α, L) equipped with the following inner product U, Ũ H 2 = κ 1 α 0 u x ũx dx + κ 2 L α v x ṽx dx + α 0 y ỹdx + L α zzdx + τ|δ 2 | L α 1 0 η x (x, ρ) ηx (x, ρ)dρ dx, where U = (u, v, y, z, η(•, •)) ∈ H 2 and Ũ = ( ũ, ṽ, ỹ, z, η(•, •)) ∈ H 2 .
We use U H 2 to denote the corresponding norm. We define the linear unbounded operator A 2 : D(A 2 ) ⊂ H 2 -→ H 2 by:

D(A 2 ) = (u, v, y, z, η(•, •)) ∈ H 2 | (y, z) H 1 , (u, κ 2 v + δ 1 z + δ 2 η(•, 1)) ∈ H 2 , κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1) = κ 1 u x (α), η, η ρ ∈ L 2 (0, 1), H 1 R (α, L) , η(•, 0) = z(•)
and for all

U = (u, v, y, z, η(•, •)) ∈ D(A 2 ), A 2 U = y, z, κ 1 u xx , (κ 2 v x + δ 1 z x + δ 2 η x (•, 1)) x , -τ -1 η ρ (•, •) . If U = (u, v, u t , v t , η(•, •)
) is a regular solution of System (4.3.1)-(4.3.6), then we transform this system into the following initial value problem

U t = A 2 U, U(0) = U 0 , (4.3.7)
where

U 0 = (u 0 , v 0 , u 1 , v 1 , f 0 (•, -• τ)) ∈ H 2 .
We now use semigroup approach to establish well-posedness result for the System (4.3.1)-(4.3.6). We prove the following proposition.

Proposition 4.3.1. Under hypothesis (H1), the unbounded linear operator A 2 is m-dissipative in the energy space H 2 .

Proof. For all U = (u, v, y, z, η(•, •)) ∈ D(A 2 ), we have

ℜ (A 2 U,U) H 2 = κ 1 ℜ α 0 (y x u x + u xx y) dx + ℜ L α (κ 2 z x v x + (κ 2 v x + δ 1 z x + δ 2 η x (x, 1)) x z) dx -|δ 2 |ℜ L α 1 0 η xρ (x, ρ) η x (x, ρ)dρ dx.
Using integration by parts in the above equation, we get

ℜ A 2 U,U 2 H 2 = -δ 1 L α |z x | 2 dx -δ 2 ℜ L α η x (•, 1)z x dx + |δ 2 | 2 L α |η x (x, 0)| 2 dx - |δ 2 | 2 L α |η x (x, 1)| 2 dx -κ 1 ℜ (u x (0)y(0)) +ℜ (κ 1 u x (α)y(α) -κ 2 v x (α)z(α) -δ 1 z x (α)z(α) -δ 2 η x (α, 1)z(α)) +ℜ (κ 2 v x (L)z(L) + δ 1 z x (L)z(L) + δ 2 η x (L, 1)z(L)) . (4.3.8)
Since U ∈ D(A 2 ), we have

y(0) = z(L) = 0, y(α) = z(α), z(x) = η(x, 0), κ 1 u x (α) -κ 2 v x (α) -δ 1 z x (α) -δ 2 η x (α, 1) = 0.
Substituting the above boundary conditions in (4.3.8), then using Young's inequality, we get

ℜ (A 2 U,U) H 2 = -δ 1 + |δ 2 | 2 L α |z x | 2 dx - |δ 2 | 2 L α |η x (x, 1)| 2 dx -δ 2 ℜ L α η x (•, 1)z x dx ≤ (-δ 1 + |δ 2 |) L α |z x | 2 dx, (4.3.9)
hence under hypothesis (H1), we get

ℜ (A 2 U,U) H 2 ≤ 0, which implies that A 2 is dissipative. We next prove that A 2 is m-dissipative. Let F = ( f 1 , f 2 , g 1 , g 2 , h(•, •)) ∈ H 2 .
We should prove that there exists a unique solution U = (u, v, y, z, η(•, •)) ∈ D(A 2 ) of the equation

-A 2 U = F.
Equivalently, we consider the following system

-y = f 1 , (4.3.10) -z = f 2 , (4.3.11) -κ 1 u xx = g 1 , (4.3.12) -(κ 2 v x + δ 1 z x + δ 2 η x (•, 1)) x = g 2 , (4.3.13) η ρ (x, ρ) = τh(x, ρ). (4.3.14)
In addition, we consider the following boundary conditions

u(0) = v(L) = 0, u(α) = v(α), (4.3.15) κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1) = κ 1 u x (α), (4.3.16) η(•, 0) = z(•). ( 4
( f 1 , f 2 ) ∈ H 1 , we get η(•, 0) = z(•) = -f 2 (•) ∈ H 1 (α, L).
From the above equation and Equation (4.3.14), we can determine

η(x, ρ) = τ ρ 0 h(x, ξ ) dξ -f 2 (x). (4.3.18) Since f 2 ∈ H 1 (α, L), f 2 (L) = 0 and h ∈ L 2 (0, 1), H 1 R (α, L) , then it is clear that η, η ρ ∈ L 2 ((0, 1), H 1 R (0, L)).
(u, v) ∈ H 1 , in addition (u, κ 2 v + δ 1 z + δ 2 η(•, 1)) ∈ H 2 .
For this aim, let (ϕ, ψ) ∈ H 1 . Multiplying Equations (4.3.12) and (4.3.13) by ϕ and ψ, integrating over (0, α) and (α, L) respectively, taking the sum, then using integration by parts, we get From the fact that (ϕ, ψ) ∈ H 1 , we have ϕ(0) = ψ(L) = 0 and ϕ(α) = ψ(α).

κ 1 α 0 u x ϕ x dx + L α (κ 2 v x + δ 1 z x + δ 2 η x (•, 1))ψ x dx + κ 1 u x (0)ϕ(0) -κ 1 u x (α)ϕ(α) + (κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1))ψ(α) -(κ 2 v x (L) + δ 1 z x (L) + δ 2 η x (L, 1))ψ(L) = α 0 g 1 ϕdx + L α g 2 ψdx.
Inserting the above equation in (4.3.19), then using (4.3.11), (4.3.16) and (4.3.18), we get

κ 1 α 0 u x ϕ x dx + κ 2 L α v x ψ x dx = α 0 g 1 ϕdx + L α g 2 ψdx + (δ 1 + δ 2 ) L α ( f 2 ) x ψ x dx -δ 2 τ L α 1 0 h x (•, ξ ) ψ x dξ dx. (4.3.20)
We can easily verify that the left-hand side of (4.3.20) is a bilinear continuous coercive form on H 1 × H 1 and the right-hand side of (4.3.20) is a linear continuous form on H 1 . Then, using Lax-Milgram theorem, we deduce that there exists (u, v) ∈ H 1 unique solution of the variational Problem (4.3.20). Using (4.3.11), (4.3.18) and (4.3.20) once we obtain

κ 1 α 0 u x ϕ x dx + L α (κ 2 v + δ 1 z + δ 2 η(•, ρ)) x ψ x dx = α 0 g 1 ϕdx + L α g 2 ψdx, ∀ (ϕ, ψ) ∈ H 1 . (4.3.21) Since (g 1 , g 2 ) ∈ L 2 , z ∈ H 1 (α, L), η ∈ L 2 ((0, 1), H 1 R (0, L)
) and (u, v) ∈ H 1 is a unique solution of (4.3.21), then by classical elliptic regularity we deduce that the unique solution (u, v) ∈ H 1 of (4.3.20) satisfies (u, κ 2 v + δ 1 z + δ 2 η(•, 1)) ∈ H 2 . On the other hand, using (4.3.21) once more we obtain (4.3.22), there remains

α 0 (κ 1 u xx + g 1 ) ϕ dx + L α [(κ 2 v + δ 1 z + δ 2 η(•, ρ)) xx + g 2 ] ψ dx + (κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, ρ) -κ 1 u x (α)) ψ(α) = 0, ∀ (ϕ, ψ) ∈ H 1 . (4.3.22) In (4.3.22) by choosing (ϕ, ψ) ∈ H 1 0 (0, α) × H 1 0 (α, L), we get κ 1 u xx + g 1 = (κ 2 v + δ 1 z + δ 2 η(•, ρ)) xx + g 2 = 0 a.e. Returning
(κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, ρ) -κ 1 u x (α)) ψ(α) = 0, ∀ (ϕ, ψ) ∈ H 1 .
Since ψ(α) is arbitrary, we deduce that κ 2 v x (α)+δ 1 z x (α)+δ 2 η x (α, ρ)-κ 1 u x (α) = 0. Then, System (4.3.12)-(4.3.13) with boundary conditions (4.3.15)-(4.3.16) has a unique solution (u, v) ∈ H 1 and (u, κ 2 v + δ 1 z + δ 2 η(•, 1)) ∈ H 2 . Thus, we deduce that there exists a unique solution U = (u, v, y, z, η(•, •)) ∈ D(A 2 ) of the equation -A 2 U = F. Thus, 0 belongs to the resolvent set ρ(A 2 ) of A 2 . Then, by contraction principle, we easily deduce that R(λ I -A 2 ) = H 2 for sufficiently small λ > 0. This, together with the dissipativeness of A 2 , imply that D (A 2 ) is dense in H 2 and that A 2 is m-dissipative in H 2 (see Theorems 4.5, 4.6 in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). The proof is thus complete. Thanks to Lumer-Philips theorem (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), we deduce that A 2 generates a C 0 -semigroup of contractions e tA 2 in H 2 and therefore Problem (4.3.1)-(4.3.6) is well-posed.

Polynomial Stability

The main result in this subsection is the following theorem.

Theorem 4.3.2. Under hypothesis (H1), for all initial data U 0 ∈ D(A 2 ), there exists a constant C > 0 independent of U 0 such that the energy of System (4.3.1)-(4.3.6) satisfies the following estimation

E(t) ≤ C t 4 U 0 2 D(A 2 )
, ∀t > 0. According to the result of Borichev and Tomilov [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF], we have to check if the following conditions hold:

iR ⊆ ρ (A 2 ) (4.3.23) and sup λ ∈R (iλ I -A 2 ) -1 L (H 2 ) = O |λ | 1 2 . (4.3.24)
The next proposition is a technical result to be used in the proof of Theorem 4.3.2 given below.

Proposition 4.3.3. Under hypothesis (H1), let (λ ,U := (u, v, y, z, η(•, •))) ∈ R * × D (A 2 ) , such that (iλ I -A 2 )U = F := ( f 1 , f 2 , f 3 , f 4 , g(•, •)) ∈ H 2 . (4.3.25) That is iλ u -y = f 1 in H 1 (0, α), (4.3.26)
iλ v -z = f 2 in H 1 (α, L), (4.3.27)
iλ y -κ 1 u xx = f 3 in L 2 (0, α), (4.3.28
)

iλ z -(κ 2 v x + δ 1 z x + δ 2 η x (•, 1)) x = f 4 in L 2 (α, L), (4.3.29)
η ρ (•, •) + iτλ η(•, •) = τg(•, •) in L 2 (0, 1), H 1 R (α, L) . (4.3.30)
Then, we have the following inequality

U 2 H 2 ≤ K 1 λ -4 (|λ | + 1) 6 F H 2 U H 2 + F 2 H 2 . (4.3.31)
In addition, if |λ | ≥ M > 0, then we have

U 2 H 2 ≤ K 2 √ M + 1 √ M 2 |λ | 1 2 1 + |λ | -1 2 8 F H 2 U H 2 + F 2 H 2 . (4.3.32)
Here and below we denote by K j a positive constant number independent of λ .

For the proof of Proposition 4.3.3, we need the following Lemmas.

L α |z x | 2 dx ≤ K 3 F H 2 U H 2 , (4.3.33)
L α |v x | 2 dx ≤ K 4 λ -2 F H 2 U H 2 + F 2 H 2 , (4.3.34)
L α 1 0 |η x (x, ρ)| 2 dρ dx ≤ K 5 F H 2 U H 2 + F 2 H 2 , (4.3.35)
L α |κ 2 v x + δ 1 z x + δ 2 η x (x, 1)| 2 dx ≤ K 6 1 + λ -2 F H 2 U H 2 + F 2 H 2 , (4.3.36)
where

K 3 = (δ 1 -|δ 2 |) -1 , K 4 = 2 max K 3 , κ -1 2 , K 5 = 2 max K 3 , τ|δ 2 | -1 , K 6 = 3 max κ 2 2 K 4 , δ 2 1 K 3 + δ 2 2 K 5 .
Proof. First, taking the inner product of (4.3.25) with U in H 2 , then using hypothesis (H1), arguing in the same way as (4.3.9), we obtain

L α |z x | 2 dx ≤ - 1 δ 1 -|δ 2 | ℜ A 2 U,U H 2 = 1 δ 1 -|δ 2 | ℜ F,U H 2 ≤ 1 δ 1 -|δ 2 | F H 2 U H 2 ,
hence we get (4.3.33). Next, from (4.3.27), (4.3.33) and the fact that κ 2

L α |( f 2 ) x | 2 dx ≤ F 2 H 2 , we obtain L α |v x | 2 dx ≤ 2λ -2 L α |z x | 2 dx + 2λ -2 L α |( f 2 ) x | 2 dx ≤ 2λ -2 K 3 F H 2 U H 2 + κ -1 2 F 2 H 2 ≤ 2λ -2 max K 3 , κ -1 2 F H 2 U H 2 + F 2 H 2 ,
therefore we get (4.3.34). Now, from (4.3.30) and using the fact that

U ∈ D (A 2 ) (i.e.,η(•, 0) = z(•)), we obtain η(x, ρ) = z(x)e -iτλ ρ + τ ρ 0 e iτλ (ξ -ρ) g(x, ξ )dξ (x, ρ) ∈ (α, L) × (0, 1), (4.3.37)
consequently, we obtain

L α 1 0 |η x (x, ρ)| 2 dρ dx ≤ 2 L α |z x | 2 dx + 2τ 2 L α 1 0 |g x (x, ξ )| 2 dξ dx.
Inserting (4.3.33) in the above equation, then using the fact that τ|δ (4.3.35). On the other hand, from (4.3.37), we get

2 | L α 1 0 |g x (x, ξ )| 2 dξ dx ≤ F 2 H 2 , we obtain L α 1 0 |η x (x, ρ)| 2 dρ dx ≤ 2K 3 F H 2 U H 2 + 2τ|δ 2 | -1 F 2 H 2 ≤ 2 max K 3 , τ|δ 2 | -1 F H 2 U H 2 + F 2 H 2 , hence we get
η x (x, 1) = z x (x)e -iτλ + τ 1 0 e iτλ (ξ -1) g x (x, ξ )dξ x ∈ (α, L).
From the above equation and (4.3.33), we obtain

L α |η x (x, 1)| 2 dx ≤ 2 L α |z x | 2 dx + 2τ 2 L α 1 0 |g x (x, ξ )| 2 dξ dx ≤ 2K 3 F H 2 U H 2 + 2τ|δ 2 | -1 F 2 H 2 ≤ 2 max K 3 , τ|δ 2 | -1 F H 2 U H 2 + F 2 H 2 .
Finally, from (4.3.33), (4.3.34) and the above inequality, we get

L α |κ 2 v x + δ 1 z x + δ 2 η x (x, 1)| 2 dx ≤ 3κ 2 2 L α |v x | 2 dx + 3δ 2 1 L α |z x | 2 dx + 3δ 2 2 L α |η x (x, 1)| 2 dx ≤ 3 κ 2 2 K 4 λ -2 + δ 2 1 K 3 + δ 2 2 K 5 F H 2 U H 2 + F 2 H 2 ≤ 3 max κ 2 2 K 4 , δ 2 1 K 3 + δ 2 2 K 5 1 + λ -2 F H 2 U H 2 + F 2 H 2 , hence we get (4.3.36
). The proof is thus complete.

|z(α)| 2 ≤ 1 + |λ | 1 2 -s 1 r 1 L α |z| 2 dx + K 7 r 1 |λ | -1 2 +s 1 F H 2 U H 2 + F 2 H 2 (4.3.38)
and

|κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 ≤ |λ | 3 2 -s 2 r 2 L α |z| 2 dx + K 8 1 + r 2 |λ | 1 2 +s 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 , (4.3.39)
where

K 7 = 2 (L -α + 1) 2 K 4 + κ -1 2 and K 8 = (L -α) -1 + 1 (K 6 + 1) .
Proof. First, from Equation (4.3.27), we have

-z x = ( f 2 ) x -iλ v x .
Multiplying the above equation by 2(x + 1 -α)z, integrating over (α, L) and taking the real parts, then using integration by parts and the fact that z(L) = 0, we get

|z(α)| 2 = - L α |z| 2 dx -2ℜ L α (x + 1 -α) (iλ v x -( f 2 ) x ) zdx ≤ L α |z| 2 dx + 2 (L -α + 1) |λ | L α |v x ||z|dx + 2 (L -α + 1) L α |( f 2 ) x ||z|dx. (4.3.40)
On the other hand, for all s 1 ∈ R and

r 1 ∈ R * + , we have      2 (L -α + 1) |λ ||v x ||z| ≤ |λ | 1 2 -s 1 |z| 2 2r 1 + 2 (L -α + 1) 2 r 1 |λ | 3 2 +s 1 |v x | 2 , 2 (L -α + 1) |( f 2 ) x ||z| ≤ |λ | 1 2 -s 1 |z| 2 2r 1 + 2 (L -α + 1) 2 r 1 |λ | -1 2 +s 1 |( f 2 ) x | 2 .
Inserting the above equation in (4.3.40), then using (4.3.34) and the fact that

L α |( f 2 ) x | 2 dx ≤ κ -1 2 F 2 H 2 , we get |z(α)| 2 ≤ 1 + |λ | 1 2 -s 1 r 1 L α |z| 2 dx + 2r 1 (L -α + 1) 2 |λ | s 1 |λ | 3 2 L α |v x | 2 dx + |λ | -1 2 L α |( f 2 ) x | 2 dx ≤ 1 + |λ | 1 2 -s 1 r 1 L α |z| 2 dx + 2r 1 (L -α + 1) 2 |λ | s 1 -1 2 K 4 F H 2 U H 2 + F 2 H 2 + κ -1 2 F 2 H 2 ≤ 1 + |λ | 1 2 -s 1 r 1 L α |z| 2 dx + 2r 1 (L -α + 1) 2 |λ | s 1 -1 2 K 4 + κ -1 2 F H 2 U H 2 + F 2 H 2 ,
hence we get (4.3.38). Next, multiplying Equation (4.3.29) by 2

(L -x) (L -α) -1 (κ 2 v x + δ 1 z x + δ 2 η x (•, 1
)), integrating over (α, L) and taking the real parts, then using integration by parts, we get

|κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 = (L -α) -1 L α |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| 2 dx +2 (L -α) -1 ℜ L α (L -x) ( f 4 -iλ z) (κ 2 v x + δ 1 z x + δ 2 η x (•, 1)) dx.
Consequently, we have

|κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 ≤ (L -α) -1 L α |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| 2 dx +2 L α | f 4 | |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| dx + 2|λ | L α |z| |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| dx. (4.3.41)
On the other hand, for all s 2 ∈ R and

r 2 ∈ R * + , we have    2| f 4 || |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| ≤ | f 4 | 2 + |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| 2 , 2|λ ||z|| |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| ≤ |λ | 3 2 -s 2 r 2 |z| 2 + r 2 |λ | 1 2 +s 2 | |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| 2 .
Inserting the above equation in (4.3.41), we get

|κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 ≤ |λ | 3 2 -s 2 r 2 L α |z| 2 dx + (L -α) -1 + 1 + r 2 |λ | 1 2 +s 2 L α | |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| 2 dx + L α | f 4 | 2 dx .
Substituting (4.3.36) in the above equation, then using the fact that

L α | f 4 | 2 dx ≤ F 2 H 2 ≤ 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
we get

|κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 ≤ |λ | 3 2 -s 2 r 2 L α |z| 2 dx + (L -α) -1 + 1 (K 6 + 1) 1 + r 2 |λ | 1 2 +s 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
hence we get (4.3.39). The proof is thus complete.

Lemma 4.3.6. Under hypothesis (H1), for all s 1 , s 2 ∈ R, the solution (u, v, y, z, η(•, •)) ∈ D(A 2 ) of Equation (4.3.25) satisfies the following estimate

α 0 |y| 2 dx + κ 1 α 0 |u x | 2 dx ≤ K 9 1 + |λ | 1 2 -s 1 + |λ | 3 2 -s 2 L α |z| 2 dx +K 10 1 + |λ | -1 2 +s 1 + |λ | 1 2 +s 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 , (4.3.42)
where

K 9 = α max 1, κ -1 1 and K 10 = 2α max 4κ -1 2 1 , κ -1 1 K 8 , K 7 .
Proof. First, multiplying Equation (4.3.28) by 2xu x , integrating over (0, α) and taking the real parts, then using integration by parts, we get 2ℜ iλ

α 0 xyu x dx + κ 1 α 0 |u x | 2 dx = κ 1 α|u x (α)| 2 + 2ℜ α 0 x f 3 u x dx . (4.3.43)
From (4.3.26), we deduce that

iλ u x = -y x -( f 1 ) x .
Inserting the above result in (4.3.43), then using integration by parts, we get

α 0 |y| 2 dx + κ 1 α 0 |u x | 2 dx = κ 1 α|u x (α)| 2 + α|y(α)| 2 + 2ℜ α 0 x f 3 u x dx + 2ℜ α 0 x y ( f 1 ) x dx , consequently, we get α 0 |y| 2 dx + κ 1 α 0 |u x | 2 dx ≤ κ 1 α|u x (α)| 2 + α|y(α)| 2 + 2α α 0 |u x || f 3 |dx + α 0 |y||( f 1 ) x |dx . (4.3.44)
Using Cauchy Schwarz inequality, we get

α 0 |u x || f 3 |dx + α 0 |y||( f 1 ) x |dx ≤ 2κ -1 2 1 F H 2 U H 2 . (4.3.45)
On the other hand, since U ∈ D (A 2 ), we have

κ 1 |u x (α)| = |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)|, |y(α)| = |z(α)|. ( 4
α 0 |y| 2 dx + κ 1 α 0 |u x | 2 dx ≤ ακ -1 1 |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 + α|z(α)| 2 + 4ακ -1 2 1 F H 2 U H 2 .
Inserting (4.3.38) and (4.3.39) with r 1 = r 2 = 1 in the above estimation, we get

α 0 |y| 2 dx + κ 1 α 0 |u x | 2 dx ≤ α max 1, κ -1 1 1 + |λ | 1 2 -s 1 + |λ | 3 2 -s 2 L α |z| 2 dx +αK 7 |λ | -1 2 +s 1 F H 2 U H 2 + F 2 H 2 + 4ακ -1 2 1 F H 2 U H 2 +ακ -1 1 K 8 1 + |λ | 1 2 +s 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 .
In the above equation, using the fact that

     F H 2 U H 2 ≤ 1 + λ -2 F H 2 U H 2 + F 2 H 2 , F H 2 U H 2 + F 2 H 2 ≤ 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
we get

α 0 |y| 2 dx + κ 1 α 0 |u x | 2 dx ≤ α max 1, κ -1 1 1 + |λ | 1 2 -s 1 + |λ | 3 2 -s 2 L α |z| 2 dx +α max 4κ -1 2 1 , κ -1 1 K 8 , K 7 2 + |λ | -1 2 +s 1 + |λ | 1 2 +s 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
hence, we get (4.3.42). The proof is thus complete.

Lemma 4.3.7. Under hypothesis (H1), for all s 1 , s 2 , s 3 ∈ R and r 1 , r 2 ,

r 3 ∈ R * + , the solution (u, v, y, z, η(•, •)) ∈ D(A 2 ) of Equation (4.3.25) satisfies the following estimates U 2 H 2 ≤ K 11 1 + |λ | 1 2 -s 1 + |λ | 3 2 -s 2 L α |z| 2 dx +K 12 1 + |λ | 1 2 +s 2 + |λ | -1 2 +s 1 1 + λ -2 F H 2 U H 2 + F 2 H 2 (4.3.47) and R 1,λ L α |z| 2 dx ≤ K 13 R 2,λ 1 + λ -2 F H 2 U H 2 + F 2 H 2 , (4.3.48
)

such that    R 1,λ = 1 -1 2 |λ | -s 3 -s 2 r 2 r 3 + r 3 |λ | -s 1 +s 3 r 1 + r 3 |λ | s 3 -1 2 , R 2,λ = |λ | -1 r 1 r 3 |λ | s 1 +s 3 + r 2 r -1 3 |λ | s 2 -s 3 + 1 + r -1 3 |λ | -s 3 -3 2 ,
where

K 11 = K 9 + 1, K 12 = K 10 + max (κ 2 K 4 , τ|δ 2 |K 5 ) , K 13 = max (K 3 + K 6 + 2, max(K 7 , K 8 )) 2 .
Proof. First, from (4.3.34), (4.3.35) and (4.3.42), we get

U 2 H 2 ≤ K 9 + 1 + K 9 |λ | 1 2 -s 1 + |λ | 3 2 -s 2 L α |z| 2 dx +K 10 1 + |λ | 1 2 +s 2 + |λ | -1 2 +s 1 1 + λ -2 F H 2 U H 2 + F 2 H 2 + max (κ 2 K 4 , τ|δ 2 |K 5 ) 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
hence we get (4.3.47). Next, multiplying (4.3.29) by -iλ -1 z and integrating over (α, L), then taking the real part, then using integration by parts and the fact that z(L) = 0, we get

L α |z| 2 dx = -ℜ i λ -1 L α f 4 z dx + ℜ iλ -1 L α (κ 2 v + δ 1 z + δ 2 η(•, 1)) x z x dx +ℜ iλ -1 (κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)) z(α) , consequently, L α |z| 2 dx ≤ |λ | -1 L α | f 4 ||z| dx + |λ | -1 L α |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| |z x | dx +|λ | -1 |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| |z(α)|. (4.3.49)
Using Cauchy Schwarz inequality, we have

|λ | -1 L α | f 4 ||z| dx ≤ |λ | -1 F H 2 U H 2 ≤ |λ | -1 1 + λ -2 F H 2 U H 2 + F 2 H 2 . ( 4
L α |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| |z x | dx ≤ 1 2 L α |z x | 2 dx + 1 2 L α |κ 2 v x + δ 1 z x + δ 2 η x (•, 1)| 2 dx ≤ K 3 2 F H 2 U H 2 + K 6 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 ≤ K 3 +K 6 2 1 + λ -2 F H 2 U H 2 + F 2 H 2 .
Inserting (4.3.50) and the above estimation in (4.3.49), we get

L α |z| 2 dx ≤ K 3 + K 6 + 2 2 |λ | -1 1 + λ -2 F H 2 U H 2 + F 2 H 2 +|λ | -1 |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| |z(α)|. (4.3.51)
Now, for all

s 3 ∈ R, r 3 ∈ R * + , we get |λ | -1 |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| |z(α)| ≤ r 3 |λ | s 3 -1 2 2 |z(α)| 2 + |λ | -s 3 -3 2 2r 3 |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| 2 .
Substituting (4.3.38) and (4.3.39) in the above estimation, we obtain

|λ | -1 |κ 2 v x (α) + δ 1 z x (α) + δ 2 η x (α, 1)| |z(α)| ≤ 1 2 |λ | -s 3 -s 2 r 2 r 3 + r 3 |λ | -s 1 +s 3 r 1 + r 3 |λ | s 3 -1 2 L α |z| 2 dx + max(K 7 ,K 8 ) 2 R 3,λ 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
where

R 3,λ = |λ | -1 (r 1 r 3 |λ | s 1 +s 3 + r 2 r -1 3 |λ | s 2 -s 3 ) + r -1 3 |λ | -s 3 -3 2 .
Finally, inserting the above equation in (4.3.51), we get

1 -1 2 |λ | -s 3 -s 2 r 2 r 3 + r 3 |λ | -s 1 +s 3 r 1 + r 3 |λ | s 3 -1 2 L α |z| 2 dx ≤ max(K 3 +K 6 +2,max(K 7 ,K 8 )) 2 R 3,λ + |λ | -1 1 + λ -2 F H 2 U H 2 + F 2 H 2 ,
hence we get (4.3.48). The proof is thus complete.

Proof of Proposition 4.3.3. We now divide the proof into two steps:

Step 1. In this step, we prove the asymptotic behavior estimate (4.3.31). Taking

s 3 = s 1 = -s 2 = 1 2 , r 1 = 1, r 2 = 9 and r 3 = 1 3 in Lemma 4.3.7, we get      1 2 L α |z| 2 dx ≤ K 13 λ -4 1 3 λ 2 + |λ | + 30 λ 2 + 1 F H 2 U H 2 + F 2 H 2 , U 2 H 2 ≤ 2K 11 λ 2 + 1 L α |z| 2 dx + 3K 12 λ -2 λ 2 + 1 F H 2 U H 2 + F 2 H 2 .
In the above equation, using the fact that

1 3 λ 2 + |λ | + 30 ≤ 30 λ 2 + |λ | + 1 ≤ 30 (|λ | + 1) 2 and λ 2 + 1 ≤ (|λ | + 1) 2 , we get L α |z| 2 dx ≤ 30K 13 λ -4 (|λ | + 1) 4 F H 2 U H 2 + F 2 H 2 (4.3.52)
and

U 2 H 2 ≤ 2K 11 (|λ | + 1) 2 L α |z| 2 dx + 3K 12 λ -2 (|λ | + 1) 2 F H 2 U H 2 + F 2 H 2 . ( 4
U 2 H 2 ≤ 60K 11 K 13 (|λ | + 1) 4 + 3K 12 λ 2 λ -4 (|λ | + 1) 2 F H 2 U H 2 + F 2 H 2 , ≤ (60K 11 K 13 + 3K 12 ) λ -4 (|λ | + 1) 6 F H 2 U H 2 + F 2 H 2 ,
hence we get (4.3.31).

Step 2. In this step, we prove the asymptotic behavior estimate (4.3.32). Let M ∈ R * such that |λ | ≥ M > 0.

In this case, taking

s 1 = s 2 = s 3 = 0, r 1 = 3 √ M 2 , r 2 = 3 √ M and r 3 = √ M 2 in Lemma 4.3.7, we get U 2 H 2 ≤ K 11 |λ | 3 2 1 + |λ | -1 + |λ | -3 2 L α |z| 2 dx +K 12 |λ | 1 2 1 + |λ | -1 2 + |λ | -1 1 + λ -2 F H 2 U H 2 + F 2 H 2 (4.3.54) and 1 2 1 - √ M 2|λ | 1 2 L α |z| 2 dx ≤ K 13 |λ | -1 1 + 3M 4 + 6 M + 2|λ | -1 2 √ M 1 + λ -2 F H 2 U H 2 + F 2 H 2 . (4.3.55)
From the fact that |λ | ≥ M, we get

1 2 1 - √ M 2|λ | 1 2 ≥ 1 4 > 0.
Therefore, from the above inequality and (4.3.55), we get

L α |z| 2 dx ≤ 24K 13 |λ | -1 1 + M + 1 M + |λ | -1 2 √ M 1 + λ -2 F H 2 U H 2 + F 2 H 2 . (4.3.56)
In Estimation (4.3.56), using the fact that

1 + M + 1 M ≤ √ M + 1 √ M 2 , 1 √ M ≤ √ M + 1 √ M 2 , 1 + λ -2 ≤ 1 + |λ | -1 2 4 , we get L α |z| 2 dx ≤ 24K 13 |λ | -1 √ M + 1 √ M 2 1 + |λ | -1 2 5 F H 2 U H 2 + F 2 H 2 . (4.3.57)
Inserting (4.3.57) in (4.3.54), then using the fact that

1 + |λ | -1 + |λ | -3 2 ≤ 1 + |λ | -1 2 3 , 1 + |λ | -1 2 + |λ | -1 1 + λ -2 ≤ 1 + |λ | -1 2 6 ≤ 1 + |λ | -1 2 8
, we get

U 2 H 2 ≤ max (24K 11 K 13 , K 12 ) |λ | 1 2 1 + |λ | -1 2 8 √ M + 1 √ M 2 + 1 F H 2 U H 2 + F 2 H 2 ≤ 2 max (24K 11 K 13 , K 12 ) |λ | 1 2 1 + |λ | -1 2 8 √ M + 1 √ M 2 F H 2 U H 2 + F 2 H 2 ,
hence we get estimate (4.3.32). The proof is thus complete.

Proof of Theorem 4.3.2. First, we will prove (4.3.23). Remark that it has been proved in Proposition 4.3.1 that 0 ∈ ρ(A 2 ). Now, suppose (4.3.23) is not true, then there exists ω ∈ R * such that iω ∈ ρ(A 2 ). According to [57, page 25] (see Lemma 1.3.5 in Chapter 1), there exists

{(λ n ,U n := (u n , v n , y n , z n , η n (•, •)))} n≥1 ⊂ R * × D (A 2 ) , with λ n → ω as n → ∞, |λ n | < |ω| and U n H 2 = 1, such that (iλ n I -A 2 )U n = F n := ( f 1,n , f 2,n , f 3,n , f 4,n , f 5,n (•, •)) → 0 in H 2 ,
as n → ∞.

We will check (4.3.23) by finding a contradiction with U n H 2 = 1 such as U n H 2 → 0. According to Equation (4.3.31) in Proposition 4.3.3 with U = U n , F = F n and λ = λ n , we obtain

0 ≤ U n 2 H 2 ≤ K 1 |λ n | -4 (|λ n | + 1) 6 F n H 2 U n H 2 + F n 2 H 2 ,
as n → ∞, we get U n 2 H 2 → 0, which contradicts U n H 2 = 1. Thus, condition (4.3.23) holds true. Next, we will prove (4.3.24) by a contradiction argument. Suppose there exists

{(λ n ,U n := (u n , v n , y n , z n , η n (•, •)))} n≥1 ⊂ R * × D (A 2 ) ,
with |λ n | ≥ 1 without affecting the result, such that |λ n | → +∞, and U n H 2 = 1 and there exists a sequence

G n := (g 1,n , g 2,n , g 3,n , g 4,n , g 5,n (•, •)) ∈ H 2 , such that (iλ n I -A 2 )U n = λ -1 2 n G n → 0 in H 2 .
We will check (4.3.24) by finding a contradiction with U n H 2 = 1 such as U n H 2 = o(1). According to Equation (4.3.32) in Proposition 4.3.3 with U = U n , F = λ -1 2 G n , λ = λ n and M = 1, we get

U n 2 H 2 ≤ 4K 2 1 + |λ n | -1 2 8 G n H 2 U n H 2 + |λ n | -1 2 G n 2 H 2 , as |λ n | → ∞, we get U n 2 H 2 = o(1)
, which contradicts U n H 2 = 1. Thus, condition (4.3.24) holds true and the proof is thus complete.

Instability

In this part, we restrict our analysis to the case a

(x) = 1, ∀x ∈ (0, L). Then, System (4.1.2) becomes              U tt (x,t) -[κ U x (x,t) + δ 1 U xt (x,t) + δ 2 U xt (x,t -τ)] x = 0, (x,t) ∈ (0, L) × (0, +∞), U(0,t) = U(L,t) = 0, t ∈ (0, +∞), (U(x, 0),U t (x, 0)) = (U 0 (x),U 1 (x)) , x ∈ (0, L), U t (x,t) = f 0 (x,t), (x,t) ∈ (0, L) × (-τ, 0), (4.3.58)
where κ, L, τ and δ 1 are strictly positive constant numbers, δ 2 is a non zero real number and the initial data (U 0 ,U 1 , f 0 ) belongs to a suitable space. Let us define the energy of a solution of System (4.3.58) as

E(U,t) = 1 2 L 0 |U t (x,t)| 2 + κ |U x (x,t)| 2 dx + τ|δ 2 | 2 L 0 1 0 |U xt (x,t -ρ τ)| 2 dρ dx.
The main result in this subsection is the following theorem.

Theorem 4.3.8. If (H1) does not hold (i.e., |δ 2 | ≥ δ 1 ), then there exist a sequence of arbitrary small (or large) delays, and solutions of Problem (4.3.58) corresponding to these delays, such that their standard energy does not tend to 0.

Proof. We seek a solution of (4.3.58) in the form

U(x,t) = sin mπx L e iλt , m ∈ N * , λ ∈ R * + . (4.3.59)
Substituting (4.3.59) in (4.3.58), we obtain

-sin mπx L λ 2 - i δ 1 + δ 2 e -iλ τ π 2 m 2 λ L 2 - κπ 2 m 2 L 2 e iλt = 0, ∀ x ∈ (0, L).
Consequently,

λ 2 - (δ 2 sin (λ τ) + i (δ 1 + δ 2 cos (λ τ))) π 2 m 2 λ L 2 - κπ 2 m 2 L 2 = 0. (4.3.60) Assume that cos (λ τ) = - δ 1 δ 2 and sin (λ τ) = 1 δ 2 δ 2 2 -δ 2 1 . (4.3.61)
Remark that, since we are considering the case |δ 2 | ≥ δ 1 , then there exist λ , τ such that (

, we get

λ 2 - δ 2 2 -δ 2 1 π 2 m 2 L 2 λ - κπ 2 m 2 L 2 = 0. (4.3.62)
We distinguish two cases.

Case 1. If |δ 2 | = δ 1 , then we take λ m = √ κ πm L and τ m,n = (2n + 1) L √ κ m , m, n ∈ N * . (4.3.63)
We can easily check that λ m and τ m,n are solutions of (4. 3.61)

U m (x,t) = sin mπx L e i √ κ πmt L , m ∈ N * ,
with τ m,n is a set of time delays that become arbitrarily small (or large) for suitable choices of the indices n, m ∈ N * . Furthermore, we have

E(U m ,t) = κ |δ 2 | π 2 m 2 τ m,n + 2 L m 2 π 2 2L 2 .
Thus, the energy is constant and strictly positive. Therefore, we have instability phenomena for a sequence of arbitrarily small or large time delays.

Case 2. If |δ 2 | > δ 1 , then we take λm = δ 2 2 -δ 2 1 2L 2 1 + 1 + 4κL 2 δ 2 2 -δ 2 1 π 2 m 2 π 2 m 2 , τm,n = (2n + 1) π -δ 2 |δ 2 | arccos δ 1 δ 2 λm , m, n ∈ N * .
We can easily check that λm and τm,n are solutions of (4.3.61) and (4.3.62). Inserting the above equation in (4.3.59), we get that (4.3.58) admits solutions in the form

Ũm (x,t) = sin mπx L e i λm t , m ∈ N * ,
with τm,n is a set of time delays that become arbitrarily small (or large) for suitable choices of the indices n, m ∈ N * . On the other hand, we have

E( Ũm ,t) = m 2 2L |δ 2 | π 2 τm,n + L 2 m 2 λm + κπ 2 .
Therefore, the energy is constant and strictly positive. Thus, we have instability phenomena for a sequence of arbitrarily small or large time delays. The proof is thus complete.

Numerical study of a wave equation with internal localized Kelvin-Voigt damping

In this chapter, we study the numerical solution of an elastic/viscoelastic wave equation with non smooth localized distributed Kelvin-Voigt damping acting faraway from the boundary. Our method is based on the Finite Volume Method (FVM) and we are interested in deriving the stability estimates and the convergence of the numerical solution to the continuous one.

Introduction

Numerical simulation is a computation that implements a mathematical model for a physical system. It is required to study the behavior of systems representing complicated mathematical models in order to provide good approximations to the analytical solutions. Numerical modeling uses mathematical models to describe the physical conditions necessary especially for engineers. Still, some of their equations, mostly partial differential equations, are somehow impossible to solve directly. With numerical models, the approximate solutions can be obtained; for example, by Finite Difference Method (FDM), Finite Element Method (FEM) or Finite Volume Method (FVM), with each method having its own merits and limitations. Nevertheless, several results are interpreted within the framework of an engineering process concerning the numerical experiments carried out in these models. However, as mathematicians, numerical studies help us not only in confirming the theoretical results, but also in predicting the solutions of some conjectures. In fact, there are many numerical studies in the literature related to FDM, FEM and FVM. For example, Larsson et al. applied FEM on a strongly damped wave equation (see [START_REF] Larsson | Finite-Element Methods for a Strongly Damped Wave Equation[END_REF]). In [START_REF] Tebou | Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity[END_REF], Zuazua et al. considered a problem that models the damped vibrations of a string with fixed ends. Assuming that the damping is localized on a subinterval, FDM with artificial viscosity was applied to show the exponential decay of the discrete energy and to obtain the convergence of the scheme. However, the error estimates were obtained later on by Rincon et al. after applying a spatial FEM (see [START_REF] Rincon | Numerical analysis for a locally damped wave equation[END_REF]). Moreover, Gao et al., in [START_REF] Gao | Unconditionally stable difference schemes for a one-space-dimensional linear hyperbolic equation[END_REF], presented an unconditionally stable finite difference scheme to find the solution of a one-dimensional linear hyperbolic equation with global damping. Also, Wang et al. used high order FDM to solve the wave equation in the second order form in two space dimensions (see [START_REF] Wang | High order finite difference methods for the wave equation with nonconforming grid interfaces[END_REF]). Furthermore, Xu et al., in [START_REF] Yu | A Numerical Method of the Euler-Bernoulli Beam with Optimal Local Kelvin-Voigt Damping[END_REF], considered the Euler-Bernoulli beam equation with local Kelvin-Voigt damping acing via nonsmooth coefficient. Using FEM followed by the control parameterization method, they aim to design a control input numerically distributed locally on a subinterval such that the total energy of the beam and the control on a given time period is minimal. As to FVM, which is based on an integral formulation, it is very popular in solving linear hyperbolic equations. For instance, LeVeque, in [START_REF] Leveque | Finite-Volume Methods for Hyperbolic Problems[END_REF], used FVM for hyperbolic equations. Also, considering that wave equations are important hyperbolic equations that arise in acoustics, numerical studies were carried on wave-based modeling of room acoustics using FDM and FVM (see [START_REF] Hamilton | Finite Difference and Finite Volume Methods for Wave-based Modelling of Room Acoustics[END_REF]). In addition, in

• For i = 0, . . . , N α , x i+ 1 2 = ih α , which yields, x 1 2 = 0 and x N α + 1 2 = α. • For i = 1, . . . , N α , x i = i - 1 2 h α , which yields, x 1 = h α 2 and x N α = α - h α 2 .
We discretize [α, β ] such that:

• For i = N α + 1, . . . , N α + N, x i+ 1 2 = α + (i -N α )h, which yields, x N α + 3 2 = α + h and x N α +N+ 1 2 = β . • For i = N α + 1, . . . , N α + N, x i = α + i -N α - 1 2 h, which yields, x N α +1 = α + h 2 and x N α +N = β - h 2 .
We discretize [β , L] such that: The interval (0, L) is discretized using the admissible mesh provided above. The time discretization is performed with a constant time step ∆t = T N with t n+1t n = ∆t for all n = 0, 1, • • • , N . Also, we denote the first and second discrete time derivatives following a backward difference and a second-order central difference at time t n :

• For i = N α + N + 1, . . . , N max , x i+ 1 2 = β + (i -N α -N)h β , which yields, x N α +N+ 3 2 = β + h β and x N max + 1 2 = L. • For i = N α + N + 1, . . . , N max , x i = β + i -N α -N - 1 2 h β , which yields, x N α +N+1 = β + h β 2 and x N max = L - h β 2 . Now, we set • h i = x i+ 1 2 -x i-1 2 ; i = 1, . . . , N max , • h i+ 1 2 = x i+1 -x i ; i = 0, . . . , N max , • h i-1 2 = x i -x i-1 ; i = 1, . . . , N max + 1, • h + i = x i+ 1 2 -x i ; i = 1, . . . , N max , • h - i = x i -x i-1 2 ; i = 1, . . . , N max , • size(T ) = max{h i , i = 1, . . . , N max }. h α h α h h h β h β 0 α β L x i-1 x i-1 2 x i x i+ 1 2 x i+1 x i-1 x i-1 2 x i x i+ 1 2 x i+1 x i-1 2 x i-1 x i x i+ 1 2 x i+1 i = 2, • • • , N α -1 i = N α + 2, • • • , N α + N -1 i = N α + N + 2, • • • , N max -1
∂ 1 v n = v n -v n-1 ∆t and ∂ 2 v n = v n+1 -2v n + v n-1 (∆t) 2 .
Also, the centered time discretization is denoted by:

∂ 1/2 v n = v n+1 -v n-1 2∆t .
Now, for the time step ∆t, we set the following condition There exists τ 0 ∈ [0, T ] such that ∆t ≤ τ 0 .

(TA)

We designate the discrete unknowns by {u n over the control volumes K i respectively. However, this data is used to construct the approximations of u, v, w within the cells. In fact, we have:

i ; i = 1, • • • , N α , n ∈ N}, {v n i ; i = N α + 1, • • • , N α + N, n ∈ N}, {w n i ; i = N α + N + 1, • • • , N max ,
u i (t) = 1 h i x i + 1 2 x i -1 2 u(x,t)dx, i = 1, . . . , N α , v i (t) = 1 h i x i + 1 2 x i -1 2 v(x,t)dx, i = N α + 1, . . . , N α + N, w i (t) = 1 h i x i + 1 2 x i -1 2 w(x,t)dx, i = N α + N + 1, . . . , N max .
Define

U i =        u i , i = 1, . . . , N α , v i , i = N α + 1, . . . , N α + N, w i , i = N α + N + 1, . . . , N max .
We start the solution with Φ i = (ϕ i , η i , γ i ) regarding that

u 0 i = ϕ i where ϕ i = 1 h i K i ϕ(x)dx, for i = 1, • • • , N α , v 0 i = η i where η i = 1 h i K i η(x)dx, for i = N α + 1, • • • , N α + N, w 0 i = γ i where ϕ i = 1 h i K i γ(x)dx, for i = N α + N + 1, • • • , N max . (5.2.1)
For the solution of the first time step; i.e. for n = 0, we use the imaginary time level -1 and we symbolize the values of the approximation functions at this step by:

u -1 i for i = 1, • • • , N α , v -1 i for i = N α + 1, • • • , N α + N, w -1 i for i = N α + N + 1 • • • , N max . Also, we set Ψ i = (ψ i , ζ i , θ i ) such that ψ i = 1 h i K i ψ(x)dx, for i = 1, • • • , N α , ζ i = 1 h i K i ζ (x)dx, for i = N α + 1, • • • , N α + N, θ i = 1 h i K i θ (x)dx, for i = N α + N + 1, • • • , N max .
Therefore, using the above notations, we can approximate the initial conditions of u t , v t and w t using centered time approximation to get

u -1 i = u 1 i -2∆tψ i , for i = 1, • • • , N α , v -1 i = v 1 i -2∆tζ i , for i = N α + 1, • • • , N α + N, w -1 i = w 1 i -2∆tθ i , for i = N α + N + 1, • • • , N max . Consequently, U -1 i = U 1 i -2∆tΨ i , for i = 1, • • • , N max . (5.2.2)
Finally, we set u 0 = 0 and w N max +1 = 0.

The spatial discretization is based on FVM where the key argument of this work refers to [27, Chap.2] which deals with partial differential equations of second order of elliptic type. In our case, the system is elliptic in space. Now, the speed c 2 defined by

c =        c 1 , x ∈ (0, α), c 2 , x ∈ (α, β ), c 3 , x ∈ (β , L)
is discontinuous at α and β . Due to this discontinuity and following the discussion in [START_REF] Eymard | Handbook of numerical analysis[END_REF], we set

c i = 1
h i K i c(x)dx
where we have c

∈ C 1 (K i ) for i = 1, • • • , N max .
For the scheme to be conservative, the discretization of the flux at x N α + 1 2 and x N α +N+ 1 2 should have the same value over K i and K i+1 ; i = N α , N α + N.

Now, we will construct the numerical schemes.

Explicit Discretization

In this part, we will construct the numerical scheme of System (5.1.4)-(5.1.9) using FVM in space. We also approximate the time derivatives using FDM.

Discretization of Equation (5.1.4). Before we start, note that a reasonable choice for the approximation of -c 2 i U x (x i+ 1 2 ,t n ) is the differential quotient

F i+ 1 2 =                -c 2 1 u n 1 h 1 2 , i = 0, -c 2 i U n i+1 -U n i h i+ 1 2 , i = 1, . . . , N max -1, c 2 3 w n Nmax h Nmax+ 1 2 , i = N max . Denote ℓ i+ 1 2 = c 2 i h i+ 1 2
and proceed as the following.

For i = 1, • • • , N α -1: K i u tt dx -c 2 1 K i u xx dx = 0, x i+ 1 2 x i-1 2 u tt dx -c 2 1 x i+ 1 2 x i-1 2 u xx dx = 0.
To begin with, remark that

1 h i x i+ 1 2 x i-1 2 u tt (x,t)dx = 1 h i x i+ 1 2 x i-1 2 u(x,t)dx tt = [u i (t)] tt .
A direct calculation gives Using second-order central time discretization at t n and a forward difference method for the space derivative at position x i , we obtain

h i [u i (t)] tt -c 2 1 [u x (x i+ 1 2 ,t n ) -u x (x i-1 2 ,t n )] = 0.
h α u n+1 i -2u n i + u n-1 i (∆t) 2 -ℓ i+ 1 2 (u n i+1 -u n i ) -ℓ i-1 2 (u n i -u n i-1 ) = 0.
(5.2.3)

For i = N α :

x Nα + 1 2 x Nα -1 2 u tt dx -c 2 1 x Nα + 1 2 x Nα -1 2 u xx dx = 0, h N α [u N α (t)] tt -c 2 1 [u x (x N α + 1 2 ,t n ) -u x (x N α -1 2 ,t n )] = 0. Let H i+ 1 2 represent the approximation of c 2 i u x (x i+ 1 2 ,t n ) for i = N α .
Using the finite difference principle, we have

H i+ 1 2 = c 2 i u n i+ 1 2 -u n i h + i over K i ; i = N α , H i+ 1 2 = c 2 i+1 u n i+1 -u n i+ 1 2 h - i+1 over K i+1 ; i = N α .
Since x i is the center of K i for all i, then substituting i = N α gives

H i+ 1 2 = c 2 i u n i+ 1 2 -u n i hα 2 over K i ; i = N α , H i+ 1 2 = c 2 i+1 u n i+1 -u n i+ 1 2 h 2 over K i+1 ; i = N α
The above approximations should be equal to guarantee the conservation of the flux (see [START_REF] Eymard | Handbook of numerical analysis[END_REF]), and thus we get by a direct calculation

H i+ 1 2 = c 2 i c 2 i+1 h 2 c 2 i + h α 2 c 2 i+1 (u n i+1 -u n i ) for i = N α . (5.2.4)
Given that for i = N α , c i+1 = c 2 and c i = c 1 , we obtain

c 2 1 u x (x N α + 1 2 ,t n ) = ℓ N α + 1 2 (v n N α +1 -u n N α ); with ℓ N α + 1 2 = 2c 2 1 c 2 2 c 2 1 h + c 2 2 h α taking into consideration that u N α +1 is actually v N α +1
due to the continuity of the functions at the interface. Now, using the second-order central difference in time and forward difference discretization in space, we obtain

h α u n+1 N α -2u n N α + u n-1 N α (∆t) 2 -ℓ N α + 1 2 (v n N α +1 -u n N α ) -ℓ N α -1 2 (u n N α -u n N α -1 ) = 0.
Discretization of Equation (5.1.5).

For i = N α + 1:

x Nα + 3 2 x Nα + 1 2 v tt dx -c 2 2 x Nα + 3 2 x Nα + 1 2 v xx dx -δ x Nα + 3 2 x Nα + 1 2 v xxt dx = 0, h N α +1 [v N α +1 (t)] tt -c 2 2 [v x (x N α + 3 2 ,t n ) -v x (x N α + 1 2 ,t n )] -δ [v xt (x N α + 3 2 ,t n ) -v xt (x N α + 1 2 ,t n )] = 0. ,t n ) = 0.
For the first term of the above equation we apply the second-order central difference in time at t n and for the third and fourth terms we use spatial forward difference, where the central difference in time is applied only to the fourth term. However, the second term is treated for i = N α similarly as (5.2.4). Therefore, we obtain

h v n+1 N α +1 -2v n N α +1 + v n-1 N α +1 (∆t) 2 -ℓ N α + 3 2 (v n N α +2 -v n N α +1 ) -ℓ N α + 1 2 (v n N α +1 -u n N α ) (5.2.5) = δ 2∆t v n+1 N α +2 -v n+1 N α +1 h - v n-1 N α +2 -v n-1 N α +1
h .

For i = N α + 2, • • • , N α + N -1: x i + 1 2 x i -1 2 v tt dx -c 2 2 x i + 1 2 x i -1 2 v xx dx -δ x i + 1 2 x i -1 2 v xxt dx = 0, h i [v i (t)] tt -c 2 2 [v x (x i+ 1 2 ,t n ) -v x (x i-1 2 ,t n )] -δ [v xt (x i+ 1 2 ,t n ) -v xt (x i-1 2 ,t n )] = 0.
Similarly, applying a second-order central difference to the second time derivative and a forward difference in space to the other terms combined with centered time discretization for the last two terms, we obtain

h v n+1 i -2v n i +v n-1 i (∆t) 2 -ℓ i+ 1 2 (v n i+1 -v n i ) -ℓ i-1 2 (v n i -v n i-1 ) = δ 2∆t v n+1 i+1 -v n+1 i h - v n-1 i+1 -v n-1 i h - v n+1 i -v n+1 i-1 h + v n-1 i -v n-1 i-1 h . For i = N α + N: x Nα +N+ 1 2 x Nα +N-1 2 v tt dx -c 2 2 x Nα +N+ 1 2 x Nα +N-1 2 v xx dx -δ x Nα +N+ 1 2 x Nα +N-1 2 v xxt dx = 0, h N α +N [v N α +N (t)] tt -c 2 2 [v x (x N α +N+ 1 2 ,t n ) -v x (x N α +N-1 2 ,t n )] -δ [v xt (x N α +N+ 1 2 ,t n ) -v xt (x N α +N-1 2 ,t n )] = 0.
Noting that the position x N α +N+ 1 2 represents the point β , then we use the transmission condition (5.1.9) at point β and thus

-c 2 2 v x (x N α +N+ 1 2 ,t n ) -δ v xt (x N α +N+ 1 2 ,t n ) = -c 2 3 w x (x N α +N+ 1 2 ,t n ).
Consequently, we obtain

h[v N α +N (t)] tt -c 2 3 w x (x N α +N+ 1 2 ,t n ) + c 2 2 v x (x N α +N-1 2 ,t n ) + δ v xt (x N α +N-1 2 ,t n ) = 0.
Similar to the way used for i = N α + 1 (see (5.2.5)), we apply the second-order central difference in time for the first term of the above equation. With respect to the third and fourth terms, we use spatial forward difference, where the central difference in time is applied only to the fourth term. Concerning the second term, we denote for i = N α + N. We argue in the same way as in [START_REF] Eymard | Handbook of numerical analysis[END_REF] to obtain J i+ 1 2 as well as we obtained H i+ 1 2 (see (5.2.4)). Given that for i = N α + N, κ i+1 = c 2 3 and κ i = c 2 2 , we obtain

c 2 3 w x (x N α +N+ 1 2 ,t n ) = ℓ N α +N+ 1 2 (w n N α +N+1 -v n N α +N ); with ℓ N α +N+ 1 2 = 2c 2 2 c 2 3 c 2 3 h + c 2 2 h β (5.2.6)
where w n N α +N is v n N α +N due to the continuity of the functions at the interface. Therefore, we obtain

h v n+1 N α +N -2v n N α +N + v n-1 N α +N (∆t) 2 -ℓ N α +N+ 1 2 (w n N α +N+1 -v n N α +N ) -ℓ N α +N-1 2 (v n N α +N -v n N α +N-1 ) (5.2.7) = - δ 2∆t v n+1 N α +N -v n+1 N α +N-1 h - v n-1 N α +N -v n-1 N α +N-1 h .
Discretization of Equation (5.1.6).

For i = N α + N + 1:

x Nα +N+ 1 2

x Nα +N+ 3 2 w tt dxc 2 3

x Nα +N+ 3 2

x Nα +N+ 1 2

w xx dx = 0, h N α +N+1 [w N α +N+1 (t)] tt -c 2 3 [w x (x N α +N+ 3 2 ,t n ) -w x (x N α +N+ 1 2 ,t n )] = 0.
Note that the last term of the above equation is treated in the previous part (for i = N α + N). Again, using the second-order central difference in time and forward difference discretization in space, we obtain

h β w n+1 N α +N+1 -2w n N α +N+1 + w n-1 N α +N+1 (∆t) 2 -ℓ N α +N+ 3 2 (w n N α +N+2 -w n N α +N+1 ) -ℓ N α +N+ 1 2 (w n N α +N+1 -v n N α +N ) = 0. For i = N α + N + 2, • • • , N max :
Similar to the treatment of the first equation (for i = 1, • • • , N α -1, see (5.2.3)), we do the same thing to obtain

h β w n+1 i -2w n i + w n-1 i (∆t) 2 -ℓ i+ 1 2 (w n i+1 -w n i ) -ℓ i-1 2 (w n i -w n i-1 ) = 0.
Now, the discrete problem of System (5.1.4)-(5.1.9) can be written as

h i U n+1 i -2U n i +U n-1 i (∆t) 2 -[ℓ i+ 1 2 (U n i+1 -U n i ) -ℓ i-1 2 (U n i -U n i-1 )] =                                0, i = 1, . . . , N α , δ 2∆t
U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h , i = N α + 1, δ 2∆t U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h - U n+1 i -U n+1 i-1 h + U n-1 i -U n-1 i-1 h , i = N α + 2, . . . , N α + N -1, -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h , i = N α + N, 0, i = N α + N + 1, . . . , N max .
(5.2.8)

It follows that

[h i U n+1 i ] | i=1,••• ,Nmax -δ ∆t 2h (U n+1 N α +2 -U n+1 N α +1 ) -δ ∆t 2h (U n+1 i+1 -2U n+1 i +U n+1 i-1 ) | i=Nα +2,••• ,Nα +N-1 + δ ∆t 2h (U n+1 N α +N -U n+1 N α +N-1 ) = [(2h i -(∆t) 2 ℓ i+ 1 2 -(∆t) 2 ℓ i-1 2 )U n i ] | i=1,••• ,Nmax + [(∆t) 2 ℓ i+ 1 2 U n i+1 ] | i=1,••• ,Nmax + [(∆t) 2 ℓ i-1 2 U n i-1 ] | i=1,••• ,Nmax -δ ∆t 2h (U n-1 N α +2 -U n-1 N α +1 ) -δ ∆t 2h (U n-1 i+1 -2U n-1 i +U n-1 i-1 ) | i=Nα +2,••• ,Nα +N-1 + δ ∆t 2h (U n-1 N α +N -U n-1 N α +N-1 ).
Consequently, the discrete problem of System (5.1.4)-(5.1.9) can be written as

M + δ ∆t h A U n+1 = 2M + (∆t) 2 B U n -M - δ ∆t h A U n-1 , (5.2.9)
where

U =   u i=1,••• ,N α v i=N α +1,•••N α +N w i=N α +N+1,••• ,N max   , A =   0 0 0 0 A 22 0 0 0 0   , B =   B 11 B 12 0 B 21 B 22 B 23 0 B 32 B 33   and M is a diagonal matrix of size N max such that M 11 = diag(h i ) i=1,...,N α , M 22 = diag(h i ) i=N α +1,...,N α +N and M 33 = diag(h i ) i=N α +N+1,...,N max . B 11 =      -ℓ 3/2 -ℓ 1/2 ℓ 3/2 0 • • • • • • 0 ℓ 3/2 -ℓ 5/2 -ℓ 3/2 ℓ 5/2 0 • • • 0 0 . . . • • • . . . 0 • • • • • • • • • ℓ N α -1/2 -ℓ N α +1/2 -ℓ N α -1/2      , B 22 =      -ℓ N α +3/2 -ℓ N α +1/2 ℓ N α +3/2 0 • • • 0 ℓ N α +3/2 -ℓ N α +5/2 -ℓ N α +3/2 ℓ N α +5/2 0 • • • 0 . . . • • • . . . 0 • • • • • • ℓ N α +N-1/2 -ℓ N α +N+1/2 -ℓ N α +N-1/2      , B 33 =      -ℓ N α +N+3/2 -ℓ N α +N+1/2 ℓ N α +N+3/2 0 • • • • • • 0 ℓ N α +N+3/2 -ℓ N α +N+5/2 -ℓ N α +N+3/2 ℓ N α +N+5/2 0 • • • 0 0 . . . • • • . . . 0 • • • • • • • • • ℓ N max -1/2 -ℓ N max +1/2 -ℓ N max -1/2      , B 21 =     0 • • • 0 ℓ N α +1/2 . . . • • • . . . 0 . . . • • • . . . . . .     , B 12 =     0 • • • 0 • • • . . . • • • . . . . . . ℓ N α +1/2 • • • . . . . . .     , B 32 =     0 • • • 0 ℓ N α +N+1/2 . . . • • • . . . 0 . . . • • • . . . . . .     , B 23 =     0 • • • 0 • • • . . . • • • . . . . . . ℓ N α +N+1/2 • • • . . . . . .     , A 22 =       1/2 -1/2 0 • • • 0 -1/2 1 -1/2 0 • • • 0 . . . . . . . . . 0 • • • . . . -1/2 1/2       .
The explicit numerical scheme of System (5.1.4)-(5.1.9) is stated by (5.2.1) and (5.2.9), where U -1 i = (u -1 i , v -1 i , w -1 i ); defined by (5.2.2), is used for the solution at n = 0. By discrete Fourier analysis, the numerical scheme (5.2.9) is stable if and only if the following Courant-Friedrichs-Lewy; i.e. CFL condition holds:

c 2 i (∆t) 2 ≤ (h i ) 2 which is equivalent to ∆t ≤ min(c -1 i )∆x; with ∆x = min{h i ; i = 1, . . . , N max }. (CFL condition)
The number min(c -1 i ) stands for the CFL number.

Estimation of the discrete energy for explicit form

In this subsection, we plan to design a discrete energy that is conserved when δ = 0 and dissipates when δ > 0.

For this aim, we define:

• the discrete kinetic energy for U as:

E k (U n ) = 1 2 N max ∑ i=1 h i U n+1 i -U n i ∆t 2
• the discrete potential energy for U as:

E p (U n ) = 1 2 N max ∑ i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(U n i+1 -U n i )
The total discrete energy is then defined as

E n = E k (U n ) + E p (U n ).
(5.2.10)

Now, we want to prove that the above stated goals for the discrete energy (5.2.10) are fulfilled. For this purpose, we aim to prove the following theorem:

Theorem 5.2.1. The total discrete energy defined by (5.2.10) of the explicit numerical scheme (5.2.8) satisfies the following estimation

E n -E n-1 = -δ ∆t N α +N-1 ∑ i=N α +1 U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i 2∆t h 2 .
The proof of Theorem 5.2.1 is similar to the continuous case where we multiply the discrete problem by the approximation of U t . Proof. To obtain the energy estimates, we multiply the left hand side and right hand side of (5.2.8) by (U n+1 i -U n-1 i ) and we sum over i = 1, . . . , N max .

Estimation of the left hand side of (5.2.8): First, we have

∑ N max i=1 h i U n+1 i -2U n i +U n-1 i (∆t) 2 (U n+1 i -U n-1 i ) -∑ N max i=1 [ℓ i+ 1 2 (U n i+1 -U n i ) -ℓ i-1 2 (U n i -U n i-1 )](U n+1 i -U n-1 i
).

(5.2.11)

Estimation of the first term of (5.2.11):

∑ N max i=1 h i (U n+1 i -U n i )-(U n i -U n-1 i ) (∆t) 2 [(U n+1 i -U n i ) + (U n i -U n-1 i )] = ∑ N max i=1 h i U n+1 i -U n i ∆t 2 -∑ N max i=1 h i U n i -U n-1 i ∆t 2 = 2(E k (U n ) -E k (U n-1 )).
(5.2.12)

Estimation of the second term of (5.2.11):

-

∑ N max i=1 [ℓ i+ 1 2 (U n i+1 -U n i ) -ℓ i-1 2 (U n i -U n i-1 )](U n+1 i -U n-1 i ) = -∑ N max i=1 ℓ i+ 1 2 (U n i+1 -U n i )(U n+1 i -U n-1 i ) + ∑ N max i=1 ℓ i-1 2 (U n i -U n i-1 )(U n+1 i -U n-1 i ).
By translation of index i of the second term of the right hand side of the above equation, we obtain:

-∑ N max i=1 [ℓ i+ 1 2 (U n i+1 -U n i ) -ℓ i-1 2 (U n i -U n i-1 )](U n+1 i -U n-1 i ) = -∑ N max i=1 ℓ i+ 1 2 (U n i+1 -U n i )(U n+1 i -U n-1 i ) + ∑ N max -1 i=0 ℓ i+ 1 2 (U n i+1 -U n i )(U n+1 i+1 -U n-1 i+1
). Taking into consideration that U 0 = U N max +1 = 0, we obtain: ) stands for the discrete time derivative of the total discrete energy defined by (5.2.10). It is left to show that this energy provides the needed properties; i.e. the energy is preserved with δ = 0 and dissipative whenever δ > 0. To this end, we study the right hand side of Equation ( 5

-∑ N max i=1 [ℓ i+ 1 2 (U n i+1 -U n i ) -ℓ i-1 2 (U n i -U n i-1 )](U n+1 i -U n-1 i ) = ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )[(U n+1 i+1 -U n+1 i ) -(U n-1 i+1 -U n-1 i )] = ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(U n+1 i+1 -U n+1 i ) -∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(U n-1 i+1 -U n-1 i ) = 2(E p (U n ) -E p (U n-1 )).
.2.8)×(U n+1 i -U n-1 i ).
Estimation of the right hand side of (5.2.8): First of all, we have

δ 2∆t ∑ N α +N-1 i=N α +2 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h - U n+1 i -U n+1 i-1 h + U n-1 i -U n-1 i-1 h (U n+1 i -U n-1 i ) + δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ) = δ 2∆t ∑ N α +N-1 i=N α +2 U n+1 i+1 -U n+1 i h - U n+1 i -U n+1 i-1 h (U n+1 i -U n-1 i ) -δ 2∆t ∑ N α +N-1 i=N α +2 U n-1 i+1 -U n-1 i h - U n-1 i -U n-1 i-1 h (U n+1 i -U n-1 i ) + δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ).
Rayan Nasser Page 124 of 161 rayan.nasser94@hotmail.com

δ 2∆t ∑ N α +N-1 i=N α +2 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h - U n+1 i -U n+1 i-1 h + U n-1 i -U n-1 i-1 h (U n+1 i -U n-1 i ) + δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ) = δ 2∆t ∑ N α +N-1 i=N α +2 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h (U n+1 i -U n-1 i ) -δ 2∆t ∑ N α +N-2 i=N α +1 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h (U n+1 i+1 -U n-1 i+1 ) + δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ) = δ 2∆t ∑ N α +N-1 i=N α +1 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h (U n+1 i -U n-1 i ) -δ 2∆t ∑ N α +N-1 i=N α +1 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h (U n+1 i+1 -U n-1 i+1 ) -δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) + δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ) + δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ).
It follows that

δ 2∆t ∑ N α +N-1 i=N α +2 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h - U n+1 i -U n+1 i-1 h + U n-1 i -U n-1 i-1 h (U n+1 i -U n-1 i ) + δ 2∆t U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h (U n+1 N α +1 -U n-1 N α +1 ) -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h (U n+1 N α +N -U n-1 N α +N ) = -δ 2∆t ∑ N α +N-1 i=N α +1 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h 2 .
(5.2.13) Thus, setting

E n = E k (U n ) + E p (U n ), we get E n -E n-1 = -δ ∆t N α +N-1 ∑ i=N α +1 U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i 2∆t h 2 .
Therefore, the proof of Theorem 5.2.1 is complete and the discrete total energy (5.2.10) is non-increasing over time.

,t n )] = 0.

So, using second-order central time discretization at t n and a forward difference method for the space derivative at position x i together with the average of diffusion terms in n + 1 and n -1 time steps, we obtain

h α u n+1 i -2u n i + u n-1 i (∆t) 2 - 1 2 ℓ i+ 1 2 (u n+1 i+1 -u n+1 i )-ℓ i-1 2 (u n+1 i -u n+1 i-1 ) - 1 2 ℓ i+ 1 2 (u n-1 i+1 -u n-1 i )-ℓ i-1 2 (u n-1 i -u n-1 i-1 ) = 0. (5.2.

14)

For i = N α :

x Nα + 1 2 x Nα -1 2 u tt dx -c 2 1 x Nα + 1 2 x Nα -1 2 u xx dx = 0, h N α [u N α (t)] tt -c 2 1 [u x (x N α + 1 2 ,t n ) -u x (x N α -1 2 ,t n )] = 0.
Applying the second-order central difference in time, a forward difference discretization in space at position x N α -1 2 and using (5.2.4); together with an average of n + 1 and n -1 time step diffusion, we obtain

h α u n+1 Nα -2u n Nα +u n-1 Nα (∆t) 2 -1 2 ℓ N α + 1 2 (v n+1 N α +1 -u n+1 N α ) -ℓ N α -1 2 (u n+1 N α -u n+1 N α -1 ) -1 2 ℓ N α + 1 2 (v n-1 N α +1 -u n-1 N α ) -ℓ N α -1 2 (u n-1 N α -u n-1 N α -1 ) = 0.
For the first term of the above equation we apply the second-order central time discretization at t n . However, for the third term, we use spatial forward difference combined with an average of n + 1 and n -1 time step diffusion. Concerning the fourth term, a forward difference in space with central difference in time is applied. Note that the second term is treated previously (for i = N α ). Therefore, we obtain

h v n+1 Nα +1 -2v n Nα +1 +v n-1 Nα +1 (∆t) 2 -1 2 ℓ N α + 3 2 (v n+1 N α +2 -v n+1 N α +1 ) -ℓ N α + 1 2 (v n+1 N α +1 -u n+1 N α ) -1 2 ℓ N α + 3 2 (v n-1 N α +2 -v n-1 N α +1 ) -ℓ N α + 1 2 (v n-1 N α +1 -u n-1 N α ) = δ 2∆t v n+1 Nα +2 -v n+1 Nα +1 h - v n-1 Nα +2 -v n-1 Nα +1 h (5.2.15)
For i = N α + 2, • • • , N α + N -1:

x i + 1 2 x i -1 2 v tt dx -c 2 2 x i + 1 2 x i -1 2 v xx dx -δ x i + 1 2 x i -1 2 v xxt dx = 0, h i [v i (t)] tt -c 2 2 [v x (x i+ 1 2 ,t n ) -v x (x i-1 2 ,t n )] -δ [v xt (x i+ 1 2 ,t n ) -v xt (x i-1 2 ,t n )] = 0.
We use a second-order central difference on the double time derivative. For the second and third terms we apply a forward difference in space combined with an average of n + 1 and n -1 time step diffusion. However, we use a forward difference together with centered time discretization for the last two terms and we obtain

h v n+1 i -2v n i +v n-1 i (∆t) 2 -1 2 ℓ i+ 1 2 (v n+1 i+1 -v n+1 i ) -ℓ i-1 2 (v n+1 i -v n+1 i-1 ) -1 2 ℓ i+ 1 2 (v n-1 i+1 -v n-1 i ) -ℓ i-1 2 (v n-1 i -v n-1 i-1 ) = δ 2∆t v n+1 i+1 -v n+1 i h - v n-1 i+1 -v n-1 i h - v n+1 i -v n+1 i-1 h + v n-1 i -v n-1 i-1 h For i = N α + N: x Nα +N+ 1 2 x Nα +N-1 2 v tt dx -c 2 2 x Nα +N+ 1 2 x Nα +N-1 2 v xx dx -δ x Nα +N+ 1 2 x Nα +N-1 2 v xxt dx = 0, h N α +N [v N α +N (t)] tt -c 2 2 [v x (x N α +N+ 1 2 ,t n ) -v x (x N α +N-1 2 ,t n )] -δ [v xt (x N α +N+ 1 2 ,t n ) -v xt (x N α +N-1 2 ,t n )] = 0.
Similarly to (5.2.7), as the position x N α +N+ 1 2 represents the point β , then we use the transmission condition (5.1.9) at point β and thus

-c 2 2 v x (x N α +N+ 1 2 ,t n ) -δ v xt (x N α +N+ 1 2 ,t n ) = -c 2 3 w x (x N α +N+ 1 2 ,t n ).
[v N α +N (t)] tt -c 2 3 w x (x N α +N+ 1 2 ,t n ) + c 2 2 v x (x N α +N-1 2 ,t n ) + δ v xt (x N α +N-1 2 ,t n ) = 0.
Similar to the way used for i = N α + 1 (see (5.2.15)), we apply the second-order central difference approximation in time for the second time derivative. For the third term of the above equation, we apply a forward difference in space at position x N α +N+ 1 2 combined with an average of n + 1 and n -1 time step diffusion. A central difference in time and a spatial forward difference in space is applied on the fourth term. However, we treat the second term exactly as we treated it in the explicit scheme (see Subsection 5.2.1; (5.2.6)), using the average of diffusion terms in n + 1 and n -1 time steps. Therefore, we obtain

h v n+1 Nα +N -2v n Nα +N +v n-1 Nα +N (∆t) 2 -1 2 ℓ N α +N+ 1 2 (w n+1 N α +N+1 -v n+1 N α +N ) -ℓ N α +N-1 2 (v n+1 N α +N -v n+1 N α +N-1 ) -1 2 ℓ N α +N+ 1 2 (w n-1 N α +N+1 -v n-1 N α +N ) -ℓ N α +N-1 2 (v n-1 N α +N -v n-1 N α +N-1 ) = -δ 2∆t v n+1 Nα +N -v n+1 Nα +N-1 h - v n-1 Nα +N -v n-1 Nα +N-1 h .
Discretization of Equation (5.1.6). For i = N α + N + 1:

x Nα +N+ 1 2 x Nα +N+ 3 2 w tt dx -c 2 3 x Nα +N+ 3 2 x Nα +N+ 1 2 w xx dx = 0, h N α +N+1 [w N α +N+1 (t)] tt -c 2 3 [w x (x N α +N+ 3 2 ,t n ) -w x (x N α +N+ 1 2 ,t n )] = 0.
The last term of the above equation is treated in the previous part (for i = N α +N). Also, using the second-order central difference in time and forward difference discretization in space with an average of n + 1 and n -1 time step diffusion, we obtain

h β w n+1 Nα +N+1 -2w n Nα +N+1 +w n-1 Nα +N+1 (∆t) 2 -1 2 ℓ N α +N+ 3 2 (w n+1 N α +N+2 -w n+1 N α +N+1 ) -ℓ N α +N+ 1 2 (w n+1 N α +N+1 -v n+1 N α +N ) -1 2 ℓ N α +N+ 3 2 (w n-1 N α +N+2 -w n-1 N α +N+1 ) -ℓ N α +N+ 1 2 (w n-1 N α +N+1 -v n-1 N α +N ) = 0. For i = N α + N + 2, • • • , N max :
Just like the treatment of the first equation (for i = 1, • • • , N α -1, see (5.2.14)), we do the same thing to obtain

h β w n+1 i -2w n i + w n-1 i (∆t) 2 - 1 2 ℓ i+ 1 2 (w n+1 i+1 -w n+1 i )-ℓ i-1 2 (w n+1 i -w n+1 i-1 ) - 1 2 ℓ i+ 1 2 (w n-1 i+1 -w n-1 i )-ℓ i-1 2 (w n-1 i -w n-1 i-1 ) = 0.
Now, the implicit discrete problem with an average of n + 1 and n -1 time step diffusion of System (5.1.4) -(5.1.9) can be written as

h i U n+1 i -2U n i +U n-1 i (∆t) 2 -1 2 [ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) -ℓ i-1 2 (U n+1 i -U n+1 i-1 )] -1 2 [ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) -ℓ i-1 2 (U n-1 i -U n-1 i-1 )] =                                0, i = 1, . . . , N α , δ 2∆t
U n+1 Nα +2 -U n+1 Nα +1 h - U n-1 Nα +2 -U n-1 Nα +1 h , i = N α + 1, δ 2∆t U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h - U n+1 i -U n+1 i-1 h + U n-1 i -U n-1 i-1 h , i = N α + 2, . . . , N α + N -1, -δ 2∆t U n+1 Nα +N -U n+1 Nα +N-1 h - U n-1 Nα +N -U n-1 Nα +N-1 h , i = N α + N, 0, i = N α + N + 1, . . . , N max .
(5.2.16)

It follows that

h i + (∆t) 2 2 ℓ i+ 1 2 + (∆t) 2 2 ℓ i-1 2 U n+1 i -(∆t) 2 2 ℓ i+ 1 2 U n+1 i+1 -(∆t) 2 2 ℓ i-1 2 U n+1 i-1 | i=1,••• ,Nmax -δ ∆t 2h (U n+1 N α +2 -U n+1 N α +1 ) -δ ∆t 2h (U n+1 i+1 -2U n+1 i +U n+1 i-1 ) | i=Nα +2,••• ,Nα +N-1 + δ ∆t 2h (U n+1 N α +N -U n+1 N α +N-1 ) = 2[h i U n i ] | i=1,••• ,Nmax -h i + (∆t) 2 2 ℓ i+ 1 2 + (∆t) 2 2 ℓ i-1 2 U n-1 i -(∆t) 2 2 ℓ i+ 1 2 U n-1 i+1 -(∆t) 2 2 ℓ i-1 2 U n-1 i-1 | i=1,••• ,Nmax -δ ∆t 2h (U n-1 N α +2 -U n-1 N α +1 ) -δ ∆t 2h (U n-1 i+1 -2U n-1 i +U n-1 i-1 ) | i=Nα +2,••• ,Nα +N-1 + δ ∆t 2h (U n-1 N α +N -U n-1 N α +N-1 ).
Now, the semi-implicit discrete problem with an average of n + 1 and n -1 time step diffusion of System (5.1.4) -(5.1.9) can be written as

M + (∆t) 2 2 B + δ ∆t h A U n+1 = 2M U n -M + (∆t) 2 2 B - δ ∆t h A U n-1
(5.2.17)

concerning that vector U and all matrices are defined in (5.2.9). Eventually, the numerical scheme of System (5.1.4)-(5.1.9) is stated by (5.2.1) and (5.2.17), where

U -1 i = (u -1 i , v -1 i , w -1 i ); defined by (5.2.
2), is used for the solution at n = 0. However, System (5.2.17) is not fully implicit.

Estimation of the discrete energy for implicit form

In this subsection, we also seek to design a discrete energy that is preserved when δ = 0, whereas when δ > 0, we aim to have a dissipation in the discrete energy. To this end, we define:

• the discrete kinetic energy for U as:

E k (U n ) = 1 2 N max ∑ i=1 h i U n+1 i -U n i ∆t 2
• the discrete potential energy for u as:

E p (U n ) = 1 4 N max ∑ i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) 2 + 1 4 N max ∑ i=0 ℓ i+ 1 2 (U n i+1 -U n i ) 2
The total discrete energy is then defined as

E n = E k (U n ) + E p (U n ).
(5.2.18) Like Subsection 5.2.1, we want to prove that the discrete energy defined by (5.2.18) fulfills the properties mentioned above. To this end, we prove the following theorem:

Theorem 5.2.2. The total discrete energy defined by (5.2.18) of the semi-implicit numerical scheme (5.2.16) satisfies the following estimation

E n -E n-1 = -δ ∆t N α +N-1 ∑ i=N α +1 U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i 2∆t h 2 .
Also, the proof of Theorem 5.2.2 is similar to the continuous case where we multiply the discrete problem by the approximation of U t . Proof. Similar to the technique used in Subsection 5.2.1, to obtain the energy estimates, we multiply the left and right hand sides of (5.2.16) by (U n+1 i -U n-1 i ) and we sum over i = 1, . . . , N max .

Estimation of the left hand side of (5.2.16):

Concerning the first term, we get exactly the same result as (5.2.12).

Estimation of the second term of the left hand side of (5.2.16): First of all, we have

-1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) -ℓ i-1 2 (U n+1 i -U n+1 i-1 )](U n+1 i -U n-1 i ) -1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) -ℓ i-1 2 (U n-1 i -U n-1 i-1 )](U n+1 i -U n-1 i ) = -1 2 ∑ N max i=1 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(U n+1 i -U n-1 i ) + 1 2 ∑ N max i=1 ℓ i-1 2 (U n+1 i -U n+1 i-1 )(U n+1 i -U n-1 i ) -1 2 ∑ N max i=1 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(U n+1 i -U n-1 i ) + 1 2 ∑ N max i=1 ℓ i-1 2 (U n-1 i -U n-1 i-1 )(U n+1 i -U n-1 i
).

(5.2.19)

By translation of index i for the second and fourth term of the right hand side of the above equation, we obtain

-1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) -ℓ i-1 2 (U n+1 i -U n+1 i-1 )](U n+1 i -U n-1 i ) -1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) -ℓ i-1 2 (U n-1 i -U n-1 i-1 )](U n+1 i -U n-1 i ) = -1 2 ∑ N max i=1 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(U n+1 i -U n-1 i ) + 1 2 ∑ N max -1 i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(U n+1 i+1 -U n-1 i+1 ) -1 2 ∑ N max i=1 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(U n+1 i -U n-1 i ) + 1 2 ∑ N max -1 i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(U n+1 i+1 -U n-1 i+1 ).
Taking into consideration that U 0 = U N max +1 = 0, it follows

-1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) -ℓ i-1 2 (U n+1 i -U n+1 i-1 )](U n+1 i -U n-1 i ) -1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) -ℓ i-1 2 (U n-1 i -U n-1 i-1 )](U n+1 i -U n-1 i ) = 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i ) + 1 2 ∑ N max i=1 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i ).
A direct calculation gives

-1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) -ℓ i-1 2 (U n+1 i -U n+1 i-1 )](U n+1 i -U n-1 i ) -1 2 ∑ N max i=1 [ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) -ℓ i-1 2 (U n-1 i -U n-1 i-1 )](U n+1 i -U n-1 i ) = 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i +U n-1 i+1 -U n-1 i )(U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i ) = 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i ) 2 -1 2 ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) 2 -1 2 ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i ) 2 = 2(E p (U n ) -E p (U n-1 )).
The left hand side of (5.2.16)×(U n+1 i -U n-1 i

) represents the discrete time derivative of the total discrete energy defined by (5.2.18). It remains to show that this energy is conservative when δ = 0 and dissipative when δ > 0. Therefore, we should study the right hand side of Equation (5.2.16)×(U n+1 i -U n-1 i ). In fact, we get exactly the same result as in (5.2.13).

Thus, setting E

n = E k (U n ) + E p (U n ), we get E n -E n-1 = -δ ∆t N α +N-1 ∑ i=N α +1 U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i 2∆t h 2 .
Consequently, the proof of Theorem 5.2.2 is complete and the discrete total energy (5.2.18) is non-increasing over time.

Before we start figuring out the stability estimates, we will write Equation (5.2.16) for n = 0.

SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

Step 1. Multiply the left and right hand side of Equation (5.2.20) by 2(∆t) 2 U 1 i and sum over i to obtain

4 ∑ N max i=1 h i (U 1 i ) 2 -4 ∑ N max i=1 h i U 0 i U 1 i -4∆t ∑ N max i=1 h i Ψ i U 1 i -2(∆t) 2 ∑ N max i=1 ℓ i+ 1 2 (U 1 i+1 -U 1 i )U 1 i + 2(∆t) 2 ∑ N max i=1 ℓ i-1 2 (U 1 i -U 1 i-1 )U 1 i +2(∆t) 3 ∑ N max i=1 ℓ i+ 1 2 (Ψ i+1 -Ψ i )U 1 i -2(∆t) 3 ∑ N max i=1 ℓ i-1 2 (Ψ i -Ψ i-1 )U 1 i =                        0, i = 1, . . . , N α , 2(∆t) 2 δ (Ψ Nα +2 -Ψ Nα +1 )U 1 Nα +1 h , i = N α + 1, 2(∆t) 2 δ ∑ N α +N-1 i=N α +2 (Ψ i+1 -Ψ i )U 1 i h -2(∆t) 2 δ ∑ N α +N-1 i=N α +2 (Ψ i -Ψ i-1 )U 1 i h , i = N α + 2, . . . , N α + N -1, -2(∆t) 2 δ (Ψ Nα +N -Ψ Nα +N-1 )U 1 Nα +N h , i = N α + N, 0, i = N α + N + 1, . . . , N max .
Substituting (5.2.1) in the left hand side of the above equation and applying the translation of index i for the fifth and seventh term of the left hand side and for the second term | i=Nα +2,...,Nα +N-1 of the right hand side, we obtain

4 ∑ N max i=1 h i (U 1 i ) 2 -4 ∑ N max i=1 h i Φ i U 1 i -4∆t ∑ N max i=1 h i Ψ i U 1 i -2(∆t) 2 ∑ N max i=1 ℓ i+ 1 2 (U 1 i+1 -U 1 i )U 1 i + 2(∆t) 2 ∑ N max -1 i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )U 1 i+1 +2(∆t) 3 ∑ N max i=1 ℓ i+ 1 2 (Ψ i+1 -Ψ i )U 1 i -2(∆t) 3 ∑ N max -1 i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )U 1 i+1 = 2(∆t) 2 δ (Ψ Nα +2 -Ψ Nα +1 )U 1 Nα +1 h + 2(∆t) 2 δ ∑ N α +N-1 i=N α +2 (Ψ i+1 -Ψ i )U 1 i h -2(∆t) 2 δ ∑ N α +N-2 i=N α +1 (Ψ i+1 -Ψ i )U 1 i+1 h -2(∆t) 2 δ (Ψ Nα +N -Ψ Nα +N-1 )U 1 Nα +N h .
Using the fact that U 0 = U N max +1 = 0, we get

4 ∑ N max i=1 h i (U 1 i ) 2 = 4 ∑ N max i=1 h i Φ i U 1 i + 4∆t ∑ N max i=1 h i Ψ i U 1 i -2(∆t) 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 + 2(∆t) 3 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(U 1 i+1 -U 1 i ) -2(∆t) 2 δ ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(U 1 i+1 -U 1 i ) h (5.3.2)
Step 2. Estimation of the terms of the right hand side of (5.3.2):

Estimation of the first term. Applying Young's inequality, we obtain

4 ∑ N max i=1 h i Φ i U 1 i ≤ 8 ∑ N max i=1 h i (Φ i ) 2 + 1 2 ∑ N max i=1 h i (U 1 i ) 2 .
Estimation of the second term. We apply similarly Young's inequality to get 4∆t ∑

N max i=1 h i Ψ i U 1 i ≤ 8(∆t) 2 ∑ N max i=1 h i (Ψ i ) 2 + 1 2 ∑ N max i=1 h i (U 1 i ) 2 .
Estimation of the fourth and fifth terms. Like above, we apply Young's inequality to get

2(∆t) 3 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(U 1 i+1 -U 1 i ) -2(∆t) 2 δ ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(U 1 i+1 -U 1 i ) h ≤ (∆t) 4 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 + (∆t) 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 + δ 2 (∆t) 2 c 2 2 ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i ) 2 h + (∆t) 2 c 2 2 ∑ N α +N-1 i=N α +1 (U 1 i+1 -U 1 i ) 2 h . Taking into consideration that 1 h = ℓ i+ 1 2 c 2 2 ; i = N α + 1, . . . , N α + N -1, we obtain 2(∆t) 3 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(U 1 i+1 -U 1 i ) -2(∆t) 2 δ ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(U 1 i+1 -U 1 i ) h ≤ (∆t) 2 (∆t) 2 + δ 2 c 4 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 + 2(∆t) 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 .
Rayan Nasser Page 132 of 161 rayan.nasser94@hotmail.com

∑ N max i=1 h i (U 1 i ) 2 ≤ 8 ∑ N max i=1 h i (Φ i ) 2 + 8(∆t) 2 ∑ N max i=1 h i (Ψ i ) 2 + (∆t) 2 (∆t) 2 + δ 2 c 4 2 ∑ N max i=0 (Ψ i+1 -Ψ i ) 2 h i+ 1 2 . (5.3.3)
Hence, we deduce the following estimation

||U 1 T || 2 ≤ 8 3 ||Φ T || 2 + 8(∆t) 2 3 ||Ψ T || 2 + (∆t) 2 3 (∆t) 2 + δ 2 c 4 2 ||Ψ T || 2 1,T .
Due to Hypothesis (H ′ ) and time assumption (TA), we get

||U 1 T || 2 ≤ C, (5.3.4)
where C depends only on the initial data of the problem and on the constant τ 0 (C is independent of the discretization parameters). From estimation (5.3.4), we conclude that U Proof. The proof of Lemma 5.3.3 is divided into two steps.

Step 1. Multiply Equation (5.2.20) by (U 1 i -U 0 i ) and sum over i to obtain

2 ∑ N max i=1 h i (U 1 i -U 0 i ) 2 (∆t) 2 -2 ∆t ∑ N max i=1 h i Ψ i (U 1 i -U 0 i ) -∑ N max i=1 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(U 1 i -U 0 i ) + ∑ N max i=1 ℓ i-1 2 (U 1 i -U 1 i-1 )(U 1 i -U 0 i ) +∆t ∑ N max i=1 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(U 1 i -U 0 i ) -∆t ∑ N max i=1 ℓ i-1 2 (Ψ i -Ψ i-1 )(U 1 i -U 0 i ) = δ (Ψ Nα +2 -Ψ Nα +1 )(U 1 Nα +1 -U 0 Nα +1 ) h + δ ∑ N α +N-1 i=N α +2 (Ψ i+1 -Ψ i )(U 1 i -U 0 i ) h -δ ∑ N α +N-1 i=N α +2 (Ψ i -Ψ i-1 )(U 1 i -U 0 i ) h -δ (Ψ Nα +N -Ψ Nα +N-1 )(U 1 Nα +N -U 0 Nα +N ) h .
By translation of index i for the fourth and sixth terms of the left hand side of the above equation and the third term of the right hand side of the above equation, we get

2 ∑ N max i=1 h i (U 1 i -U 0 i ) 2 (∆t) 2 -2 ∆t ∑ N max i=1 h i Ψ i (U 1 i -U 0 i ) -∑ N max i=1 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(U 1 i -U 0 i ) + ∑ N max -1 i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(U 1 i+1 -U 0 i+1 ) +∆t ∑ N max i=1 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(U 1 i -U 0 i ) -∆t ∑ N max -1 i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(U 1 i+1 -U 0 i+1 ) = δ (Ψ Nα +2 -Ψ Nα +1 )(U 1 Nα +1 -U 0 Nα +1 ) h + δ ∑ N α +N-1 i=N α +2 (Ψ i+1 -Ψ i )(U 1 i -U 0 i ) h -δ ∑ N α +N-2 i=N α +1 (Ψ i+1 -Ψ i )(U 1 i+1 -U 0 i+1 ) h -δ (Ψ Nα +N -Ψ Nα +N-1 )(U 1 Nα +N -U 0 Nα +N ) h . Using the fact that U 0 = U N max +1 = 0, it follows 2 ∑ N max i=1 h i (U 1 i -U 0 i ) 2 (∆t) 2 -2 ∆t ∑ N max i=1 h i Ψ i (U 1 i -U 0 i ) + ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )[(U 1 i+1 -U 1 i ) -(U 0 i+1 -U 0 i )] -∆t ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )[(U 1 i+1 -U 1 i ) -(U 0 i+1 -U 0 i )] = -δ ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )[(U 1 i+1 -U 1 i )-(U 0 i+1 -U 0 i )] h , Rayan
h i (U 1 i -U 0 i ) 2 (∆t) 2 + ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 = 2 ∆t ∑ N max i=1 h i Ψ i (U 1 i -U 0 i ) + ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(U 0 i+1 -U 0 i ) + ∆t ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )[(U 1 i+1 -U 1 i ) -(U 0 i+1 -U 0 i )] -δ ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )[(U 1 i+1 -U 1 i )-(U 0 i+1 -U 0 i )] h .
(5.3.5)

Step 2. Estimation of the terms of the right hand side of (5.3.5):

Estimation of the first term. Apply Young's inequality to obtain

2 ∆t ∑ N max i=1 h i Ψ i (U 1 i -U 0 i ) ≤ ∑ N max i=1 h i (U 1 i -U 0 i ) 2 (∆t) 2 + ∑ N max i=1 h i (Ψ i ) 2 .
Estimation of the second term. Also, apply Young's inequality to get

∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(U 0 i+1 -U 0 i ) ≤ 1 6 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 + 3 2 ∑ N max i=0 ℓ i+ 1 2 (U 0 i+1 -U 0 i ) 2 .
Estimation of the last two terms. Setting ε > 0, we apply Young's inequality as well to obtain ∆t ∑

N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )[(U 1 i+1 -U 1 i ) -(U 0 i+1 -U 0 i )] -δ ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )[(U 1 i+1 -U 1 i )-(U 0 i+1 -U 0 i )] h ≤ (∆t) 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U 0 i+1 -U 0 i ) 2 + δ 2 ε c 4 2 ∑ N α +N-1 i=N α +1 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 + 1 2ε ∑ N α +N-1 i=N α +1 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 + 1 2ε ∑ N α +N-1 i=N α +1 ℓ i+ 1 2 (U 0 i+1 -U 0 i ) 2 ≤ (∆t) 2 + δ 2 ε c 4 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 + 1 2 (1 + ε -1 ) ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 + 1 2 (1 + ε -1 ) ∑ N max i=0 ℓ i+ 1 2 (U 0 i+1 -U 0 i ) 2 , concerning that 1 h = ℓ i+ 1 2 c 2 2 .
⋄ Inserting the above estimations in (5.3.5), we deduce

∑ N max i=1 h i (U 1 i -U 0 i ) 2 (∆t) 2 + 1 3 -1 2ε ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i ) 2 ≤ ∑ N max i=1 h i (Ψ i ) 2 + 2 + 1 2ε ∑ N max i=0 ℓ i+ 1 2 (U 0 i+1 -U 0 i ) 2 + (∆t) 2 + δ 2 ε c 4 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 .
(5.3.6)

Setting ε = 2, we deduce that

||∂ 1 U 1 T || 2 + ||U 1 T || 2 1,T ≤ 12||Ψ T || 2 + 27||Φ T || 2 1,T + 12 (∆t) 2 + 2δ 2 c 4 2 ||Ψ T || 2 1,T ≤ C, (5.3.7)
where C depends only on the initial data of the problem and on τ 0 (C is independent of the discretization parameters under Hypothesis (H ′ ) and assumption (TA)). Therefore, the proof of Lemma 5.

-U n-1 i for i = 1, • • • , N max .
Applying exactly the same steps in calculating the discrete energy in Subsection 5.2.4; i.e. the steps before writing the terms as discrete derivatives with respect to time, we can use the results obtained to get directly

∑ N max i=1 h i U n+1 i -U n i ∆t 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) 2 = ∑ N max i=1 h i U n i -U n-1 i ∆t 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) 2 -δ 2∆t ∑ N α +N-1 i=N α +1 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h 2 .
Using the fact thatδ 2∆t

N α +N-1 ∑ i=N α +1 U n+1 i+1 -U n+1 i h - U n-1 i+1 -U n-1 i h 2 ≤ 0, we obtain directly ∑ N max i=1 h i U n+1 i -U n i ∆t 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i ) 2 ≤ ∑ N max i=1 h i U n i -U n-1 i ∆t 2 + 1 2 ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i ) 2 .
(

For n = 1, we get from (5.3.8)

||∂ 1 U 2 T || 2 + ||U 2 T || 2 1,T ≤ 2||∂ 1 U 1 T || 2 + ||Φ T || 2 1,T .
(5.3.9)

From the Discrete Poincare inequality [27, Lemma 9.1], we have

||U 2 T || 2 ≤ c 0 ||U 2 T || 2
1,T , with c 0 a strictly positive constant. Thus, we obtain from (5.3.9)

||U 2 T || 2 ≤ 2c 0 ||∂ 1 U 1 T || 2 + c 0 ||Φ T || 2 1,T .
Thus,

||U 2 T || 2 + ||∂ 1 U 2 T || 2 + ||U 2 T || 2 1,T ≤ C
where C depends only on the initial data of the problem and τ 0 (see Estimation (5.3.7)) and is independent of the discretization parameters due to Hypothesis (H ′ ) and assumption (TA). However, we continue similarly for

n = 2, • • • , N -1, to obtain finally ||∂ 1 U n T || 2 + ||U n T || 2 + ||U n T || 2 1,T ≤ C; n = 1, • • • , N -1 (5.3.10)
with C being a strictly positive constant independent of the discretization parameters and depending only on τ 0 and the initial values of the problem. The proof of Theorem 5.3.4 is thus complete.

Convergence

In this section, we want to prove the convergence of the numerical solution of Problem (5.2.16). The strategy in this section is based on the stability estimates obtained in Section 5.3, and passing through the limit, we obtain the convergence to the weak solution. To this end, we will write first the continuous weak formulation problem (5.1.10) in its discrete form. As we will use a density argument, we set function P ∈ C ∞ ([0, T ];C ∞ 0 (0, L)) such that P(x, T ) = 0 and let size(T ) be small enough. Define P n i = P(x i ,t n ) for i = 0, • • • , N max . Also, let U T ,∆t be the notation of the numerical solution under the integral (0, L). Thus,

U T ,∆t (x,t) = U n i ; x ∈ [x i-1 2 , x i+
h i U 1 i -U 0 i (∆t) 2 P 0 i -∑ N max i=1 h i Ψ i P 0 i -∆t 2 ∑ N max i=1 ℓ i+ 1 2 (U 1 i+1 -U 1 i )P 0 i + ∆t 2 ∑ N max i=1 ℓ i-1 2 (U 1 i -U 1 i-1 )P 0 i + (∆t) 2 2 ∑ N max i=1 ℓ i+ 1 2 (Ψ i+1 -Ψ i )P 0 i -(∆t) 2 2 ∑ N max i=1 ℓ i-1 2 (Ψ i -Ψ i-1 )P 0 i = δ ∆t 2 Ψ Nα +2 -Ψ Nα +1 h P 0 N α +1 + δ ∆t 2 ∑ N α +N-1 i=N α +2 Ψ i+1 -Ψ i h P 0 i -δ ∆t 2 ∑ N α +N-1 i=N α +2 Ψ i -Ψ i-1 h P 0 i -δ ∆t 2 Ψ Nα +N -Ψ Nα +N-1 h P 0 N α +N .
By translation of index i for the fourth and sixth terms of the left hand side and the third term of the right hand side of the above equation, and using the fact that P 0 = P N max +1 = 0, a direct calculation gives

∆t ∑ N max i=1 h i U 1 i -U 0 i (∆t) 2 P 0 i -∑ N max i=1 h i Ψ i P 0 i -∆t 2 ∑ N max i=1 ℓ i+ 1 2 (U 1 i+1 -U 1 i )P 0 i + ∆t 2 ∑ N max -1 i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )P 0 i+1 + (∆t) 2 2 ∑ N max i=1 ℓ i+ 1 2 (Ψ i+1 -Ψ i )P 0 i -(∆t) 2 2 ∑ N max -1 i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )P 0 i+1 = δ ∆t 2 Ψ Nα +2 -Ψ Nα +1 h P 0 N α +1 + δ ∆t 2 ∑ N α +N-1 i=N α +2 Ψ i+1 -Ψ i h P 0 i -δ ∆t 2 ∑ N α +N-2 i=N α +1 Ψ i+1 -Ψ i h P 0 i+1 -δ ∆t 2 Ψ Nα +N -Ψ Nα +N-1 h P 0 N α +N . Consequently, ∆t ∑ N max i=1 h i U 1 i -U 0 i (∆t) 2 P 0 i -∑ N max i=1 h i Ψ i P 0 i + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(P 0 i+1 -P 0 i ) -(∆t) 2 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(P 0 i+1 -P 0 i ) = δ ∆t 2 ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(P 0 i -P 0 i+1 ) h
. Now, we modify the above equation as follows:

∆t ∑ N max i=1 h i U 1 i -U 0 i (∆t) 2 P 0 i -∑ N max i=1 h i Ψ i P 0 i + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(P 0 i+1 -P 0 i + P 1 i+1 -P 1 i+1 -P 1 i + P 1 i ) -(∆t) 2 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(P 0 i+1 -P 0 i ) = δ ∆t 2 ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(P 0 i -P 0 i+1 ) h , to get finally after small rearrangement ∆t ∑ N max i=1 h i U 1 i -U 0 i (∆t) 2 P 0 i -∑ N max i=1 h i Ψ i P 0 i + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(P 1 i+1 -P 1 i ) + δ ∆t 2 ∑ N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(P 0 i+1 -P 0 i ) h = (∆t) 2 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(P 0 i+1 -P 0 i ) + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U 1 i+1 -U 1 i )(P 1 i+1 -P 1 i -P 0 i+1 + P 0 i ) (5.4.1)
Step 2. Now, we continue for n ≥ 1 and we multiply (5.2.16) by ∆tP n i . Using the same techniques of translation of index i, we obtain as a result

∆t ∑ N max i=1 h i U n+1 i -U n i (∆t) 2 P n i -∆t ∑ N max i=1 U n i -U n-1 i (∆t) 2 P n i + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i ) + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(P n i+1 -P n i ) + δ 2 ∑ N α +N-1 i=N α +1 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i ) h -δ 2 ∑ N α +N-1 i=N α +1 (U n-1 i+1 -U n-1 i )(P n i+1 -P n i ) h = 0.
Step 3. Summing the above equation for n

= 1, • • • , N -1, we obtain ∑ N -1 n=1 ∆t ∑ N max i=1 h i U n+1 i -U n i (∆t) 2 P n i -∑ N -1 n=1 ∆t ∑ N max i=1 U n i -U n-1 i (∆t) 2 P n i + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(P n i+1 -P n i ) + δ ∑ N -1 n=1 ∆t ∑ N α +N-1 i=N α +1 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i ) 2h∆t -δ ∑ N -1 n=1 ∆t ∑ N α +N-1 i=N α +1 (U n-1 i+1 -U n-1 i
)(P n i+1 -P n i ) 2h∆t = 0. Working on the first two terms of Equation (5.4.2): Using the fact that U N i = 0, we translate the index n of the first term to get

∑ N -1 n=1 ∆t ∑ N max i=1 h i U n+1 i -U n i (∆t) 2 P n i -∑ N -1 n=1 ∆t ∑ N max i=1 U n i -U n-1 i (∆t) 2 P n i = ∑ N n=2 ∆t ∑ N max i=1 h i U n i -U n-1 i (∆t) 2 P n-1 i -∑ N n=2 ∆t ∑ N max i=1 h i U n i -U n-1 i (∆t) 2 P n i -∆t ∑ N max i=1 h i U 1 i -U 0 i (∆t) 2 P 1 i .
(5.4.3)

Working on the second two terms of Equation (5.4.2): We do the following modification

1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(P n i+1 -P n i ) = 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i
)(P n i+1 -P n i + P n+1 i+1 -P n+1 i+1 + P n+1 i

-P n+1 i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i
)(P n i+1 -P n i + P n-1 i+1 -P n-1 i+1 + P n-1 i

-P n-1 i ) = 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(P n+1 i+1 -P n+1 i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(P n-1 i+1 -P n-1 i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i -P n+1 i+1 + P n+1 i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i
)(P n i+1 -P n i -P n-1 i+1 + P n-1 i

).

By translation of index n for the first two terms of the right hand side of the above equation, we obtain )(P n i+1 -P n i -P n-1 i+1 + P n-1 i ).

1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i )(P n i+1 -P n i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i )(P n i+1 -P n i ) = 1 2 ∑ N n=2 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(P n i+1 -P n i ) + 1 2 ∑ N -2 n=0 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(P n i+1 -P n i ) + 1 2 ∑ N -1 n=1 ∆t ∑
(5.4.4) Substituting (5.4.3) and (5.4.4) in (5.4.2), we obtain

-∑ N n=2 ∆t ∑ N max i=1 h i (U n i -U n-1 i )(P n i -P n-1 i ) (∆t) 2 -∆t ∑ N max i=1 h i U 1 i -U 0 i (∆t) 2 P 1 i + ∑ N n=2 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(P n i+1 -P n i ) + 1 2 ∑ 1 n=0 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(P n i+1 -P n i ) +δ ∑ N -1 n=1 ∆t ∑ N α +N-1 i=N α +1 (U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i )(P n i+1 -P n i ) 2h∆t = R 1 ,
(5. 4.5) where

R 1 = ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U N-1 i+1 -U N-1 i )(P N-1 i+1 -P N-1 i ) + 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n+1 i+1 -U n+1 i
)(P n+1 i+1 -P n+1 i -P n i+1 + P n i )

+ 1 2 ∑ N -1 n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n-1 i+1 -U n-1 i
)(P n-1 i+1 -P n-1 i -P n i+1 + P n i ).

Combining (5.4.1) and (5.4.5) , we obtain (∆tΨ i+1 -∆tΨ i -U 0 i+1 +U 0 i )(P 0 i+1 -P 0 i ) -δ ∆t 2 ∑

-∑ N n=1 ∆t ∑ N max i=1 h i (U n i -U n-1 i )(P n i -P n-1 i ) (∆t) 2 -∑ N max i=1 h i Ψ i P 0 i + ∑ N n=1 ∆t ∑ N max i=0 ℓ i+ 1 2 (U n i+1 -U n i )(P n i+1 -P n i ) +δ ∑ N -1 n=1 ∆t ∑ N α +N-1 i=N α +1 (U n+1 i+1 -U n+1 i -U n-1 i+1 +U n-1 i )(P n i+1 -P n i ) 2h∆t = R,
N α +N-1 i=N α +1 (Ψ i+1 -Ψ i )(P 0 i+1 -P 0 i ) h . Thus, we deduce from (5.4.6) (w i+1,∆t -w i,∆t )(P i+1,∆t -P i,∆t )dt + δ T 0 ∑ ). Thanks to [START_REF] Eymard | Study of a finite volume scheme for the regularized mean curvature flow level set equation[END_REF], there exists (u, v, w) ∈ C 0 ([0, T ]; L 2 ) with (u(., 0), v(., 0), w(., 0)) = (ϕ(.), η(.), γ(.)) and a subsequence denoted again by (u T m ,∆t m , v T m ,∆t m , w T m ,∆t m ) converging to (u, v, w) in L ∞ ([0, T ]; L 2 ). Now, the proof of Theorem 5.4.1 is divided into steps.

-T 0 L 0 ∂ 1 U T ,
N α +N-1 i=N α +1 [∂ 1/2 (v i+1,∆t -v i,
Step 1. Before we study the convergence of the numerical solution to the weak one, we have to show that R → 0 as m → ∞. The first term of R can be estimated as the following using Holder's inequality

∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U N-1 i+1 -U N-1 i )(P N-1 i+1 -P N-1 i ) ≤ ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (u N-1 i+1 -u N-1 i ) 2 1/2 ∑ N max i=0 ℓ i+ 1 2 (P N-1 i+1 -P N-1 i ) 2 1/2 = ∆t 2 ||U N-1 T m || 1,T ||P N-1
T m || 1,T → 0 as m → ∞ (∆t → 0). Also, we consider the second term of R and we apply similarly Holder's inequality to obtain (∆tΨ i+1 -∆tΨ i -U 0 i+1 +U 0 i )(P 0 i+1 -

P 0 i ) = (∆t) 2 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i )(P 0 i+1 -P 0 i ) -∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U 0 i+1 -U 0 i )(P 0 i+1 -P 0 i ) ≤ (∆t) 2 2 ∑ N max i=0 ℓ i+ 1 2 (Ψ i+1 -Ψ i ) 2 1/2 ∑ N max i=0 ℓ i+ 1 2 (P 0 i+1 -P 0 i ) 2 1/2 + ∆t 2 ∑ N max i=0 ℓ i+ 1 2 (U 0 i+1 -U 0 i ) 2 1/2 ∑ N max i=0 ℓ i+ 1 2 (P 0 i+1 -P 0 i ) 2 1/2 = (∆t) 2 2 ||Ψ T m || 1,T ||P 0 T m || 1,T + ∆t 2 ||U 0 T m || 1,T ||P 0 T m || 1,T → 0 as m → ∞ (∆t → 0).
We do the same technique for the other terms and we deduce that R → 0 as m → ∞.

Step 2. Now, we move to show the convergence of the numerical solution to the weak solution of problem (5.1.10). Due to the regularity of the function P, ∂ 1 P T m ,∆t m → P t strongly in L 2 ([0, T ]; (0, α)). However, the stability estimate results show that ∂ 1 u T m ,∆t m is bounded in L ∞ ([0, T ]; L 2 (0, α)) and thus ∂ 1 u T m ,∆t m ⇀ ⋆ u t in L ∞ ([0, T ]; L 2 (0, α)). We obtain Also, using the fact that for all ψ ∈ H 1 (0, α), ψ T m → ψ ∈ L 2 (0, α), we deduce that Regarding that x N α +1 = α + h 2 , we get x N α +1 → α as h → 0. However, from the stability results, v T m is bounded in H 1 (α, β ) and hence, there exists a subsequence, denoted again by v T m converging to v in H 1 (α, β ) as m → ∞. Thus, due to the continuity of the trace operator, tr(v T m ) → tr(v) in H 1/2 ({α, β }), and therefore v N α +1 → v(α) = u(α). So, we can say that The continuity of the trace operator implies that |v N α +1,∆t m | ≤ c||v T m ,∆t m || where c is a positive constant. Concerning that u T m ,∆t m is bounded in L ∞ ([0, T ]; L 2 (0, α)), v T m ,∆t m is bounded in L ∞ ([0, T ]; L 2 (α, β )) and the function P is smooth enough, then T13 → 0 as m → ∞. Now, we can write T11 in the following form using the integral formulation:

T11 = -∑ N n=1 ∆t ∑ N α i=1 u n i K i c 2 i P xx (x,t n )dx = -∑ N n=1 ∆t ∑ N α -1 i=1 ℓ i+ 1 2 u n i (P n i+1 -P n i ) + ∑ N n=1 ∆t ∑ N α -1 i=1 ℓ i-1 2 u n i (P n i -P n i-1 ) -∑ N n=1 ∆t u n N α ℓ N α + 1 2 (P n N α +1 -P n N α ) -ℓ N α -1 2 (P n N α -P n N α -1
) . Using translation of index i for the second term of the above equation, we obtain Regarding that x N α +N = βh 2 , we get x N α +N → β as h → 0. However, from the stability results, v T m is bounded in H 1 (α, β ) and hence, there exists a subsequence, denoted again by v T m converging to v in H 1 (α, β ) as m → ∞. Thus, due to the continuity of the trace operator, tr(v T m ) → tr(v) in H 1/2 ({α, β }), and therefore v N α +N → v(β ) . So, we can say that v N α +N (t) → v(β ,t) almost everywhere in time as size(T ) → 0. Consequently, v N α +N,∆t m → v(β ,t) as m → ∞ for all t ∈ [0, T ] due to the continuity of the functions in time.

T11 = ∑ N n=1 ∆t ∑ N α -1 i=0 ℓ i+ 1 2 (u n i+1 -u n i )(P n i+1 -P n i ) -
As v T m ,∆t m → v in L ∞ ([0, T ]; L 2 (α, β )), we obtain Again, we reformulate T2 as the following: The continuity of the trace operator implies that |v N α +N,∆t m | ≤ c||v T m ,∆t m || where c is a positive constant. Concerning that v T m ,∆t m is bounded in L ∞ ([0, T ]; L 2 (α, β )) and the function P is smooth enough, then T23 → 0 as m → ∞. Now, we can write T21 in the following form using the integral formulation: Finally,

T2 = -∑ N n=1 ∆t ∑ N α +N N α +1 c 2 i v n i K i P xx (x,t n )dx + ∑ N n=1 ∆t c 2 N α +N v n N α +N P x (β ,t n ) -∑ N n=1 ∆t c 2 N α +1 v n N α +1 P x (α,t n ) -c 2
T21 = -∑ N n=1 ∆t ∑ N α +N N α +1 c 2 i v n i K i P xx (x,t n )dx = -∑ N n=1 ∆t ∑ N α +N-1 i=N α +1 ℓ
T22 = ∑ N n=1 ∆t c 2 N α +N v n N α +N P x (β ,t n ) -∑ N n=1 ∆t c 2 N α +1 v n N α +1 P x (α,t n ) = ∑ N n=1 ∆t ℓ N α +N+ 1 2 v n N α +N (P n N α +N+1 -P n N α +N ) -∑ N n=1 ∆t ℓ N α + 1 2 v n N α +1 (P n N α +1 -P n N α ).
Similarly, using the same argument, we introduce the auxiliary term Since w T m ,∆t m is bounded in L ∞ ([0, T ]; L 2 (β , L)) and the function P is smooth enough, and making the same discussion for the boundedness of the term v N α +N,∆t m , we deduce that T33 → 0 as m → ∞. Similarly, we obtain T31 in the following form:

T31 = ∑ N n=1 ∆t ∑ N max N α +N+1 ℓ i+ 1 2
(w n i+1w n i )(P n i+1 -P n i ) + Finally, Also, we reformulate T4 as the following:

T32 = -∑ N n=1 ∆t c 2 N α +N+1 v n N α +N P x (β ,t n ) = -∑ N n=1 ∆t ℓ N α +N+ 1 2 v n N α +N (P n N α +N+1 -P n N α +N ).
T4 = -δ ∑ N n=1 ∆t ∑ N α +N-1 i=N α +1 ∂ 1/2 v n i K i P xx (x,t n )dx -δ ∑ N n=1 ∆t∂ 1/2 v n N α +1 P x (α,t n ) + δ ∑ N n=1 ∆t∂ 1/2 v n N α +N P x (β ,t n ) +δ T 0 β α ∂ 1/2 v T m
,∆t m (P xx (x,t n ) -P xx (x,t))dxdt + δ T 0 ∂ 1/2 v N α +1,∆t m (P x (α,t n ) -P x (α,t))dt -δ T 0 ∂ 1/2 v N α +N,∆t m (P x (β ,t n ) -P x (β ,t))dt = T41 + T42 + T43 .

Since ∂ 1/2 v T m ,∆t m is bounded in L ∞ ([0, T ]; L 2 (α, β )), function P is smooth enough and using the fact that ∂ 1/2 v n N α +1 and ∂ 1/2 v n N α +N are bounded due to the continuity of the trace operator, then T43 → 0 as m → ∞ . Now, we can write T41 in the following form using the centered time discretization and the integral formulation:

T41 = -δ ∑ N n=1 ∆t ∑ N α +N-1 i=N α +1 ∂ 1/2 v n i K i P xx (x,t n )dx = -δ ∑ N n=1 ∆t ∑ N α +N-1 i=N α +1 v n+1 i -v n-1 i 2∆t P n i+1 -P n i h i+ 1 2 - P n i -P n i+1 h i-1 2 .
Using translation of index i for the second term of the above equation, we obtain Therefore, for the limit solution U = (u, v, w), the variational problem (5.1.10) holds for any arbitrary function P ∈ C ∞ ([0, T ];C ∞ 0 (0, L)) regarding that P(T, x) = 0. By density argument, Problem (5.1.10) holds for (p, q, z) ∈ L 2 ([0, T ]; H 1 L ) with (p t , q t , z t ) ∈ L 2 ([0, T ]; L 2 ) such that p(T, x) = q(T, x) = z(T, x) = 0. The proof of Theorem (5.4.1) is thus complete.

T41 = δ ∑ N n=1 ∆t ∑ N α +N-1 i=N α +1 (v n+1 i+1 -v n+1 i -v n-

Conclusion and open problems

In Chapter 2, we studied the stabilization of a wave equation with local internal Kelvin-Voigt damping under different geometries not studied in previous literature. We obtained polynomial decays of the energy of systems with different rates, regarding that the damping region satisfies or does not satisfy the GCC. However, the optimality of these decay rates is still an open problem where we conjecture t -2 to be the optimal one. In fact, in order to get the optimality, we suggest to take Ω to be a disk where the viscoelastic part is near the boundary. Also, in [START_REF] Ammari | Stabilization for the wave equation with singular kelvinâ Ȃ Şvoigt damping[END_REF], they studied problem (2.1.1) with constant coefficients and on C ∞ domains where the damping region doesn't obey any geometry. So, it would be interesting to study the stabilization of System (2.1.1) on bounded domains with boundary of class C 2 in R N , N ≥ 2, whenever the damping region is localized internally and faraway from the boundary without any geometric conditions. However, the coefficients of System (2.1.1) can be taken of class C 1 (Ω).

In the second part of Chapter 4, we studied the stability of a wave equation with locally distributed Kelvin-Voigt damping and internal time delay (see System (4.1.2)). Here, the Kelvin-Voigt and the internal delay damping are both localized via non smooth coefficients near the boundary. Thus, it is interesting to consider the stabilization of System (4.1.2), such that the Kelvin-Voigt and the internal delay damping are non smooth and both localized faraway from the boundary. Moreover, the treatment of Systems (4.1.1) and (4.1.2) is still an open problem in R N , N ≥ 2. However, in this case, we can have plenty of studies including several situations; whether the Kelvin-Voigt damping obeys or does not obey any geometric conditions.

In Chapter 5, we study the numerical solution of a one-dimensional elastic/viscoelastic wave equation with non smooth localized distributed Kelvin-Voigt damping acting faraway from the boundary, using the Finite Volume Method. Indeed, this case will be important to be investigated in the N-d case; N ≥ 2, on unstructured meshes.

κ 1 α 0 u x ũx dx + κ 2 β α v x ṽx dx + κ 3 L β w x wx dx + ρ 1 α 0 y ỹdx + ρ 2

 103102 2 and is equipped with the following inner product U, Ũ H =

H = κ 1 α0 u x ũx dx + κ 2 β α v x ṽx dx + κ 3 L

 123 β w x wx dx + α 0 y ỹdx+ β α zzdx + L β φ φ dx + τ 1 0 η(L, ρ) η(L, ρ)dρ,

Definition 1 . 1 . 3 .•Definition 1 . 1 . 5 .Definition 1 . 1 . 6 .

 113115116 Let T ∈ L (E, F). • We define the range of T by R (T ) = {Tu : u ∈ E} ⊂ F. We define the Kernel of T by ker (T ) = {u ∈ E : Tu = 0} ⊂ E. Theorem 1.1.4. (Fredholm alternative [16, Theorem 6.6]) If T ∈ K (E), then • dim(ker (I -T )) < ∞, where dim is the dimension of the space and I is the identity operator on E. • R (I -T ) is closed. • ker (I -T ) = 0 ⇔ R (I -T ) = E. An unbounded linear operator T from E into F is a pair (T, D (T )) presented by the following linear transformation, with D (T ) ⊂ E being a subspace called in particular the domain of T T : D (T ) ⊂ E -→ F. Let T : D (T ) ⊂ E -→ F be an unbounded linear operator. • We define the range of T by R (T ) = {Tu : u ∈ D (T )} ⊂ F. • We define the kernel of T by ker (T ) = {u ∈ D (T ) : Tu = 0} ⊂ E.

Definition 1 . 1 . 8 .

 118 The closedness of an unbounded linear operator T can be characterized as follows if u n ∈ D (T ) such that u n → u in E and Tu n → v in F, then u ∈ D (T ) and Tu = v. Let T : D (T ) ⊂ E -→ F be a closed unbounded linear operator.

Figure 1 .

 1 Figure 1. Comparison between different geometric conditions in a disk

  u D,τ (x,t) = u n+1 p , for a.e. x ∈ p, ∀t ∈]nτ, (n + 1)τ[, ∀p ∈ M, ∀n = 0, . . . , N T , and the function z D,τ by z D,τ (x,t) = u n+1 pu n p τ for a.e. x ∈ p, ∀t ∈]nτ, (n + 1)τ[, ∀p ∈ M, ∀n ∈ N.

CHAPTER 2 .

 2 STABILITY OF N-D TRANSMISSION PROBLEM IN VISCOELASTICITY WITH LOCALIZED KELVIN-VOIGT DAMPING UNDER DIFFERENT TYPES OF GEOMETRIC CONDITIONS

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :

 1234 Fig.1: Elastic/viscoelastic waves interaction models satisfying the assumption (A1)Rayan NasserPage 41 of 161 rayan.nasser94@hotmail.com

Lemma 2 . 3 . 7 .

 237 Assume that assumptions (LA) and (A2) hold. Then the solution (u n , v n ) ∈ D(A ) of Equations (2.3.4)-(2.3.5) satisfies the following asymptotic behavior estimation ω |u n | 2 = o λ -2 n .

Remark 2 . 3 .

 23 11. i) The main result of Section 2.3 covers the stabilization over several geometries not taken into consideration in previous works; for example, Fig.1-c, Fig.2, Fig.3, Fig.4and Fig.6.

CHAPTER 2 .

 2 STABILITY OF N-D TRANSMISSION PROBLEM IN VISCOELASTICITY WITH LOCALIZED KELVIN-VOIGT DAMPING UNDER DIFFERENT TYPES OF GEOMETRIC CONDITIONS viscoelsatic wave

Lemma 2 . 4 . 2 .

 242 Assume that conditions (LA) and (LE) hold. Then the solution (u n , v n ) ∈ D(A ) of Equations (2.4.6)-(2.4.7) satisfies the following asymptotic behavior estimations

Lemma 2 . 4 . 5 .

 245 Assume that conditions (LA) and (LE) hold. Then if ℓ ≥ 2β + 2, the solution (u n , v n ) ∈ D(A ) of (2.4.6)-(2.4.7) satisfies the following asymptotic behavior estimation Ω |λ n u n | 2 dx = o(1).

Figure 3 . 1 :

 31 Figure 3.1: Partial viscoelastic material

  .3.18) CHAPTER 3. OPTIMAL POLYNOMIAL STABILITY OF A STRING WITH LOCALLY DISTRIBUTED KELVIN-VOIGT DAMPING AND NONSMOOTH COEFFICIENT AT THE INTERFACE where c 4 ∈ C * and given by

2 ,

 2 .4.13) The proof of Theorem 3.4.1 is divided into several Lemmas. Lemma 3.4.2. The solution (u, v, w, y, z, φ ) ∈ D(A ) of Equations (3.4.5)-(3.4.10) satisfies the following asymptotic behavior estimations β α |z| 2 dx = o λ -3 (3.4.14)

CHAPTER 3 .

 3 OPTIMAL POLYNOMIAL STABILITY OF A STRING WITH LOCALLY DISTRIBUTED KELVIN-VOIGT DAMPING AND NONSMOOTH COEFFICIENT AT THE INTERFACE Inserting the above equation in (3.4.25), we get

3 2 ,

 2 hence, we get (3.4.14). Finally, inserting (3.4.14) in (3.4.17) and (3.4.20), we get (3.4.15) and (3.4.16

Lemma 3 . 4 . 4 . 1 α 0 xyu x dx -κ 1 α 0 xu xx u x dx = ρ 1 λ -1 2 α 0 x f 4

 3441010204 The solution (u, v, w, y, z, φ ) ∈ D(A ) of Equations (3.4.5)-(3.4.10) satisfies the following asymptotic behavior estimations α 0 |y| 2 dx = o (1) , α 0 |u x | 2 dx = o (1) (3.4.26) and L β |φ | 2 dx = o (1) , L β |w x | 2 dx = o (1) . (3.4.27) Proof. Multiplying (3.4.8) by xu x and integrating over (0, α), we get iλ ρ u x dx. (3.4.28)

  .4.29) Inserting the boundary conditions (3.4.11) at x = α in (3.4.29) gives ρ 1 2

1 2 .

 2 Inserting (3.4.15)-(3.4.16) in the above equation, we obtain the first and the second asymptotic estimates of (3.4.26). Similarly, we multiply equation (3.4.10) by (x -L)w x and integrate over (β , L), then we use (3.4.7). Arguing in the same way as (3.4.29), then using the boundary conditions (3.4.11) at x = β and (3.4.15)-(3.4.16), we get the first and the second asymptotic estimates of (3.4.27). The proof is thus complete. Proof of Theorem 3.4.1. Combining estimations (3.4.13), (3.4.14), (3.4.26) and (3.4.27), we obtain that U H = o(1) which contradicts(3.4.3). Therefore, the energy of System (3.2.1)-(3.2.6) satisfies estimation (3.4.1) for all initial data U 0 ∈ D(A ). Moreover, following Theorem 3.3.1 we cannot expect the energy decay rate t -4-ε for all initial data U 0 ∈ D (A ) and for all t > 0. Hence, the energy decay rate in (3.4.1) is optimal.

Figure 4 . 1 :Figure 4 . 2 :

 4142 Figure 4.1: K-V damping is acting localized in the internal of the body and time delay feedback is effective at L

Figure 4 . 3 :

 43 Figure 4.3: Local internal K-V damping and Local internal delay feedback

Lemma 4 . 3 . 4 .

 434 Under hypothesis (H1), the solution (u, v, y, z, η(•, •)) ∈ D(A 2 ) of Equation (4.3.25) satisfies the following estimates

Lemma 4 . 3 . 5 .

 435 Under hypothesis (H1), for all s 1 , s 2 ∈ R and r 1 , r 2 ∈ R * + , the solution (u, v, y, z, η(•, •)) ∈ D(A 2 ) of Equation (4.3.25) satisfies the following estimates

Figure 5 . 1 :

 51 Figure 5.1: A model representing the admissible one-dimensional mesh

CHAPTER 5 .

 5 NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING By translation of index of the second term of the right hand side of the above equation, we obtain

CHAPTER 5 .

 5 NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING ⋄ Applying the estimates to (5.3.2), we get 3

  161 rayan.nasser94@hotmail.com CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

  161 rayan.nasser94@hotmail.com CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING where

7 ) 5 . 4 . 1 .

 7541 ∆t )](P i+1,∆t -P i,∆t ) h dt α 0 ψ T P 0 T dx -β α ζ T P 0 T dx -L β θ T P 0 T dx = R.(5.4.Theorem Assume that Hypothesis (H ′ ) holds. For m ∈ N, let T m be the admissible mesh described in Section (5.2) and ∆t m be the time step satisfying condition (TA). Let (u T m ,∆t m , v T m ,∆t m , w T m ,∆t m ) be the discrete solution of (5.2.20) and (5.2.16). Assume that size(T m ) → 0 and ∆t m → 0 as m → ∞. Then there exists a subsequence denoted also by (u T m ,∆t m , v T m ,∆t m , w T m ,∆t m ) that converges weakly ⋆ to the weak solution (u, v, w) ∈ L ∞ ([0, T ]; (0, α)) × L ∞ ([0, T ]; (α, β )) × L ∞ ([0, T ]; (β , L)) of problem (5.1.10).Proof. From Theorems 5.3.1 and 5.3.4, we obtain(u T m ,∆t m , v T m ,∆t m , w T m ,∆t m ) bounded in L ∞ ([0, T ]; (0, α)) × L ∞ ([0, T ]; (α, β ))×L ∞ ([0, T ]; (β , L)).Then there exists a subsequence, still denoted by(u T m ,∆t m , v T m ,∆t m , w T m ,∆t m ), such that (u T m ,∆t m , v T m ,∆t m , w T m ,∆t m ) ⇀ ⋆ (u, v, w) in L ∞ ([0, T ]; (0, α)) × L ∞ ([0, T ]; (α, β )) × L ∞ ([0, T ]; (β , L)) as m → ∞.It is left to show that (u, v, w) is the weak solution of problem (5.1.10

1 2 ∑i=0 ℓ i+ 1 2 2 T0 2 ∑ N max i=0 ℓ i+ 1 2

 21222 N -1 n=0 ∆t ∑ N max ||U T m ,∆t m || 1,T ||∂ 1 P T m ,∆t m || 1,T dt → 0 as m → ∞ (∆t → 0).Rayan Nasser Page 138 of 161 rayan.nasser94@hotmail.com CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING We continue in a similar way ∆t

0 ∂ 1 u∂ 1 v∂ 1 w

 0111 T m ,∆t m ∂ 1 P T m ,∆t m dxdt ⇀ ⋆ - T m ,∆t m ∂ 1 P T m ,∆t m dxdt ⇀ ⋆ - T m ,∆t m ∂ 1 P T m ,∆t m dxdt ⇀ ⋆ - t dxdt as m → ∞.

1 = T 0 N α - 1 ∑ i=0 ℓ i+ 1 2 ( 2 = T 0 N α +N- 1 ∑ i=N α +1 ℓ i+ 1 2 ( 2 ( 0 N α +N- 1 ∑Firstu 1 T 0 v

 1012201220110 )P(x, 0)dx as m → ∞, for function P regular enough. Similarly,β α ζ T m P 0 T m dx → -β α ζ (x)P(x, 0)dx as m → ∞, -L β θ T m P 0 T m dx → -L β θ (x)P(x, 0)dx as m → ∞.Rayan Nasser Page 139 of 161 rayan.nasser94@hotmail.com CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPINGStep 2. Let T u i+1,∆tu i,∆t )(P i+1,∆t -P i,∆t )dt+ (v N α +1,∆tu N α ,∆t )(P N α +1,∆t -P N α ,∆t )dt, T v i+1,∆tv i,∆t )(P i+1,∆t -P i,∆t )dt + (w N α +N+1,∆tv N α +N,∆t )(P N α +N+1,∆t -P N α +N,∆t )dt, w i+1,∆tw i,∆t )(P i+1,∆t -P i,∆t )dt, T 4 = δ T i=N α +1 [∂ 1/2 (v i+1,∆t mv i,∆t m )](P i+1,∆t m -P i,∆t m ) h . T m ,∆t m P xx (x,t)dxdt + c 2 N α +1,∆t m P x (α)dt.Set the following trace operator tr :H 1 (α, β ) -→ L 2 ((α, β )),and define the following space H 1/2 ((α, β )) = {v ∈ L 2 ((α, β )) | ∃ ṽ ∈ H 1 (α, β ) : v = tr( ṽ)}.

  v N α +1 (t) → u(α,t)almost everywhere in time as size(T ) → 0. Consequently, v N α +1,∆t m → u(α,t) as m → ∞ for all t ∈ [0, T ] due to the continuity of the functions in time.Also, as uT m ,∆t m → u in L ∞ ([0, T ]; L 2 (0, α)), x dxdt.Now, we reformulate T1 as the following:T1 = -∑ N n=1 ∆t ∑ N α i=0 c 2 i u n i K i P xx (x,t n )dx + ∑ N n=1 ∆t c 2 N α v n N α +1 P x (α,t n ) -c 2 1 T 0 α 0 u T m ,∆t m (P xx (x,t) -P xx (x,t n ))dxdtc 2 1 T 0 v N α +1,∆t m (P x (α,t n ) -P x (α,t))dt = T11 + T12 + T13 .Rayan NasserPage 140 of 161 rayan.nasser94@hotmail.com CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

2 2v 2 T 0 v 2 T 0 v

 22020 h α ∑ N n=1 ∆t u n N α (P n N α +1 -P n N α ). Finally, T12 = ∑ N n=1 ∆t c 2 N α v n N α +1 P x (α,t n ) = ∑ N n=1 ∆t ℓ N α + 1 2 v n N α +1 (P n N α +1 -P n N α ).Similarly, consider the following auxiliary term T2 = T m ,∆t m P xx (x,t)dxdt + c 2 N α +N,∆t m P x (β )dtc 2 N α +1,∆t m P x (α)dt.

T2

2 T 2 T

 22 v T m ,∆t m (P xx (x,t) -P xx (x,t n ))dxdtc 2 0 v N α +N,∆t m (P x (β ,t n ) -P x (β ,t))dt +c 2 0 v N α +1,∆t m (P x (α,t n ) -P x (α,t))dt = T21 + T22 + T23 .

T3 = -c 2 3 T 0 L 3 T 0 v 3 T

 30303 β w T m ,∆t m P xx (x,t)dxdtc 2 N α +N,∆t m P x (β )dt.Applying the trace operator as previous and taking into consideration that w T m ,∆t m → w in L ∞ ([0, T ]; L 2 (β , L)), reformulate T3 as the following:T3 = -∑ N n=1 ∆t ∑ N max i=N α +N+1 c 2 i w n i K i P xx (x,t n )dx -∑ N n=1 ∆t c 2 N α +N+1 v n N α +N P x (β ,t n ) +c 2 0 v N α +N,∆t m (P x (β ,t n ) -P x (β ,t))dtc 2 3 T 0 Lβ w T m ,∆t m (P xx (x,t) -P xx (x,t n ))dxdt = T31 + T32 + + T33 .

  ∆t w n N α +N+1 (P n N α +N+1 -P n N α +N ).

∂ 1 / 2 v 0 ∂ 1 / 0 ∂ 1 /

 120101 + T31 = T 1 + T 2 + T 3 , and T12 + T22 + T32 = 0 and thereforeT 1 + T 2 + T 3 → c x dxdt as m → ∞.Finally, consider the following termT4 = -δ T m ,∆t m P xx (x,t)dxdt -δ T 2 v N α +1,∆t m P x (α) + δ T 2 v N α +N,∆t m P x (β ).Rayan NasserPage 142 of 161 rayan.nasser94@hotmail.com CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING Applying the trace operator and the fact that ∂ 1/2 v T m ,∆t m ⇀ * v t in L ∞ ([0, T ]; L 2 (α, β )) (due to the stability results)

  P x dxdt as m → ∞.

Figure 5 . 2 :Figure 5 . 4 :Figure 5 . 5 :Figure 5 . 6 :Figure 5 . 7 :Figure 5 . 8 :Figure 5 . 9 :

 52545556575859 Figure 5.2: Initial profile Figure 5.3: No damping/equal speed

  Wave equation with local Kelvin-Voigt damping far from the boundary and with boundary delay feedback 4.2.2 Wave equation with local Kelvin-Voigt damping near the boundary and boundary delay feedback 4.3 Wave equation with local internal Kelvin-Voigt damping and local internal delay feedback 4.3.1 Well-posedness of the problem 4.3.2 Polynomial Stability 4.3.3 Instability 5 Numerical analysis of a finite volume method for a 1-d wave equation with non smooth wave speed and internal

	List of symbols
	List of symbols
	R	The set of real numbers.
	R +
	localized Kevin-Voigt damping
	5.1 Introduction
	5.2 Construction of the discretization of the problem
	5.2.1 Explicit Discretization
	5.2.2 Estimation of the discrete energy for explicit form
	5.2.3 Implicit Discretization
	5.2.4 Estimation of the discrete energy for implicit form
	5.3 Stability estimates
	5.4 Convergence
	5.5 Numerical Experiments: Validation of the theoretical results
	5.5.1 Equal speed of propagation
	5.5.2 Different speed of propagation
	Conclusion and open problems
	Bibliography
			damping
	under different types of geometric conditions
	2.1 Introduction
	2.2 Well-Posedness and strong stability
	2.2.1 Well-Posedness of the problem
	2.2.2 Strong stability
	2.3 Polynomial stability
	2.4 Polynomial stability in the absence of GCC
	2.4.1 Applications
	3 Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and
	nonsmooth coefficient at the interface
	3.1 Introduction
	3.2 Preliminaries and Well-Posedness
	3.3 Upper bound estimation of the polynomial decay rate
	3.4 Polynomial Stability
	Rayan Nasser	Page 6 of 161	rayan.nasser94@hotmail.com

Introduction 1 Preliminaries 1.1 Bounded and Unbounded linear operators 1.2 Semigroups, Existence and uniqueness of solution 1.3 Stability of a semigroup 1.4 Geometric conditions 1.5 Numerical analysis results 2 Stability of N-D transmission problem in viscoelasticity with localized Kelvin-Voigt 4 Stability results for an elastic-viscoelastic wave equation with localized Kelvin-Voigt damping and with an internal or boundary time delay 4.1 Introduction 4.2 Wave equation with local Kelvin-Voigt damping and with boundary delay feedback Rayan Nasser Page 5 of 161 rayan.nasser94@hotmail.com Contents 4.2.1

  Thus, as b is nonnegative implies that System (2.1.1) is dissipative in the sense that its energy is decreasing with respect to time t. Assume that b ≥ b 0 > 0 in a nonempty open subset ω of Ω (see condition (LA) below), then the decay is strict and E(u,t) goes to zero as t goes to infinity (see Subsection 2.2.2). The question is then to know, under the localization condition (LA), what is the energy decay rate under geometric conditions satisfied by ω. First, we establish a polynomial energy decay estimate of type t -1 for smooth initial data provided that the damping region ω satisfies the Geometric Control Condition GCC and meas(ω ∪ Γ) > 0 (condition (A1) below) or ω satisfies Strictly Geometric Control Condition SGCC (condition (A2) below). Second, in the case where ω does not satisfy the GCC condition, i.e. in the presence of trapped rays that do not meet the damped region ω, we focus on the 2-dimensional square. We prove that the energy of smooth initial data

	decays polynomially like t -2/ℓ , where ℓ = 6 if condition (LC1) holds and ℓ = 5 if condition (LC2) holds (see
	below). The frequency domain approach and new multiplier technics are employed. The results of Theorem
	2.3.4 and Theorem 2.4.1 are new. Indeed, the geometric situations covered by these theorems are richer than
	that considered in all previous literature (see for instance

  We continue under conditions (LA) and (LE), following the same technique used to get Estimation (2.3.23) and taking into consideration Estimations (2.4.9) and (2.4.18) on the second term of the left hand side of Equation (2.4.17), we get

.

4.19)

  .4.20) So, adding (2.4.16) and (2.4.17) and using (2.4.19) and (2.4.20), we obtain

  OPTIMAL POLYNOMIAL STABILITY OF A STRING WITH LOCALLY DISTRIBUTED KELVIN-VOIGT DAMPING AND NONSMOOTH COEFFICIENT AT THE INTERFACE Let (u, v, w) be a regular solution for the System (3.2.1)-(3.2.6). Multiplying (3.2.1), (3.2.2) and (3.2.3) by u t , v t and w t , respectively, then using the boundary conditions in (3.2.4) and (3.2.5), we get

	d dt	E (t) = -δ	β α	|v xt | 2 dx ≤ 0.
	Thus System (3.2.1)-(3.2.6) is dissipative in the sense that its energy is non increasing with respect to the time
	t. For well-posedness, let us define

Rayan Nasser

Page 56 of 161 rayan.nasser94@hotmail.com CHAPTER 3.

  (3.3.13) and (3.3.15) in (3.3.11), we get

  OPTIMAL POLYNOMIAL STABILITY OF A STRING WITH LOCALLY DISTRIBUTED KELVIN-VOIGT DAMPING AND NONSMOOTH COEFFICIENT AT THE INTERFACE Now, using Cauchy Schwarz inequality, (3.4.12)-(3.4.13) and the fact that f 5 → 0 in L 2 (α, β ) in the right hand side of above equation, we get

Rayan Nasser

Page 64 of 161 rayan.nasser94@hotmail.com CHAPTER 3.

  .1.10) where κ real, τ > 0 and a > 0. System (4.1.10) shows that the damping is localized, indeed, it acts on a neighborhood of a part of the boundary of Ω. Under the assumption that |κ| < κ 0 < a, the author established an exponential decay rate. Later, in 2016, Messaoudi et al. in[START_REF] Messaoudi | Well posedness and exponential stability in a wave equation with a strong damping and a strong delay[END_REF] considered the stabilization of the following wave equation with strong time delay

  1.1) and (4.1.2), where the damping is acting in a non-smooth region is still an open problem. The aim of this chapter consists in studying the stability of the Systems (4.1.1) and (4.1.2).This chapter is organized as follows: In Section 4.2, we study the stability of System (4.1.1). Indeed, in Subsection 4.2.1, we consider the case α > 0. Under hypothesis (H), first, we prove the well-posedness of System (4.1.1). Next, we prove the strong stability of the system in the lack of the compactness of the

resolvent of the generator. Then, we establish a polynomial energy decay rate of type t -4 (see Theorem 4.2.7). In addition, in Subsection 4.2.2, we consider the case α = 0 and we prove the exponential stability of System (4.1.1) (see Theorem 4.2.14). In Section 4.3, we study the stability of System (4.1.2). Under hypothesis (H1), first, we prove the well-posedness of System (4.1.2). Next, we establish a polynomial energy decay rate of type t -4 (see Theorem 4.3.2). Finally, when a(x) = 1, ∀x ∈ (0, L), and |δ 2 | ≥ δ 1 , we show that System (4.2.1) is unstable (see Theorem 4.3.8).

  .2.33) Let (ϕ, ψ, θ ) ∈ H 1 L . Multiplying Equations (4.2.28), (4.2.29), (4.2.30

  , from (4.2.89), we get (4.2.85) and (4.2.86). Next, from (4.2.77), (4.2.85) and the fact that ( f 2 ) x → 0 in L 2 (α, β ), we get (4.2.87). Finally, from (4.2.84) and (4.2.86), we obtain (4.2.88). The proof is thus complete. Lemma 4.2.9. Under hypothesis (H), for all ℓ ≥ 0, the solution (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) of Equations (4.2.76)-(4.2.82) satisfies the following asymptotic behavior estimation

	0	1	|η(L, ρ)| 2 dρ = o λ -ℓ .	(4.2.90)
	Proof. It follows from (4.2.82) that

  .2.93)

	Proof. Multiplying Equation (4.2.81) by xw x and integrating over (β , L), we get
	iλ	L β	xφ w x dx -κ 3	L β	xw xx w x dx = λ -ℓ	L β	x f 6 w x dx.	(4.2.94)
	From (4.2.78), we deduce that

  Under hypothesis (H), for all ℓ ≥ 0, the solution (u, v, w, y, z, φ , η(L, •)) ∈ D(A ) of Equations (4.2.76)-(4.2.82) satisfies the following asymptotic behavior estimations

	hence, we get (4.2.91) and (4.2.92). Finally, from (4.2.83) and (4.2.92), we obtain (4.2.93). The proof is thus
	complete.
	Lemma 4.2.11. β α	|z| 2 dx = o λ -min(2ℓ+ 1 2 ,ℓ+1) ,	(4.2.95)

  Remark 4.2.12. An example about g, we can take g

1 2 ,ℓ+1) , hence, we get (4.2.95). Finally, inserting (4.2.95) in (4.2.98) and (4.2.100) and using the first asymptotic estimate of (4.2.93), we get (4.2.96) and (4.2.97). The proof is thus complete.

  .2.105) Proof. Multiplying Equation (4.2.79) by xu x and integrating over (0, α), we get

	iλ	0	α	xyu x dx -κ 1	0	α	xu xx u x dx = λ -ℓ	0	α	x f 4 u x dx.	(4.2.106)
	From (4.2.76), we deduce that

  2.96) and (4.2.97) in the above equation, we obtain (4.2.105). The proof is thus complete.

	Proof of Theorem 4.2.7. From Lemma 4.2.8, Lemma 4.2.9, Lemma 4.2.10, Lemma 4.2.11 and Lemma 4.2.13,
	we get

  .2.113) We will check condition (4.2.111) by finding a contradiction with U n H 1 = 1 such as U n H 1 = o(1). From now on, for simplicity, we drop the index n. By detailing Equation (4.2.113), we get the following system

  ).

	(4.2.128)
	Inserting (4.2.128) in (4.2.127), we get

β 0 |z| 2 dx = o λ -1

. (4.2.129) Finally, from (4.2.122), (4.2.124), (4.2.125) and (4.2.129), we get U H 1 = o(1), which contradicts (4.2.112). Therefore, (4.2.111) holds and the proof is thus complete.

  Our next aim is to prove that System (4.3.12)-(4.3.13) with boundary conditions (4.3.15)-(4.3.16) has a unique solution

  n ∈ N} where they stand for approximation of the mean values of u, v, w

	CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON
	SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING
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  1 T ∈ L 2 . The proof of Lemma 5.3.2 is thus complete. Lemma 5.3.3. Under Hypothesis (H ′ ) and Condition (TA), the numerical solution of the first time step U 1 T of the implicit numerical scheme (5.2.20) is bounded in H 1 L and the approximation of the first time derivation is bounded in L 2 .

  3.3 is thus complete. Proof of Theorem 5.3.1. Combining estimations (5.3.4) and (5.3.7) concludes the boundedness of the first time step U 1T in spaces H 1 L and L 2 . Also, its first time derivation approximation is bounded in L 2 . Now, we will move to demonstrate the stability estimates of the discretized numerical scheme (5.2.16) for n ≥ 1. To this end, we aim to prove the following theorem: Assume that Hypothesis (H ′ ) and Condition (TA) hold. Then, the numerical solution U T ,∆t of the numerical scheme (5.2.16) is bounded in L ∞ ([0, T ]; (0, L)) ∩ L ∞ ([0, T ]; H 1 L ) and its time derivative approximation is bounded in L ∞ ([0, T ]; L 2 ).

	CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON
	SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING
	Theorem 5.3.4. Proof. For the proof of Theorem 5.3.4, we multiply (5.2.16) by U n+1 i
	Rayan Nasser	Page 134 of 161	rayan.nasser94@hotmail.com

  NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING [t n ,t n+1 [. Now, to get the discrete variational problem, we will pass through several steps.

	CHAPTER 5. Step 1. Multiply (5.2.20) by	∆t 2	P 0 i and sum over i to obtain
	N max ∆t ∑ i=1
	Rayan Nasser		Page 135 of 161	rayan.nasser94@hotmail.com

1 2 [, t ∈

  ∆t ∂ 1 P T ,∆t dxdt + T 0 ∑ (U i+1,∆t -U i,∆t )(P i+1,∆t -P i,∆t )dt +δ T 0 ∑ N α +N-1 i=N α +1 [∂ 1/2 (U i+1,∆t -U i,∆t )](P i+1,∆t -P i,∆t ) h dt -L 0 Ψ T P 0 T dx = R.The above equation can be reformulated as the following∂ 1 u T ,∆t ∂ 1 P T ,∆t dxdt -T 0 β α ∂ 1 v T ,∆t ∂ 1 P T ,∆t dxdt -T 0 L β ∂ 1 w T ,∆t ∂ 1 P T ,∆t dxdt (u i+1,∆tu i,∆t )(P i+1,∆t -P i,∆t )dt + T (v N α +1,∆tu N α ,∆t )(P N α +1,∆t -P N α ,∆t )dt (v i+1,∆tv i,∆t )(P i+1,∆t -P i,∆t )dt + T (w N α +N+1,∆tv N α +N,∆t )(P N α +N+1,∆t -P N α +N,∆t )dt

		N max i=0 ℓ i+ 1 2
	-T 0 0 + T α 0 ∑ N α -1 i=0 ℓ i+ 1 2	0	2c 2 1 c 2 2 1 h+c 2 c 2 2 h α
	+ T 0 ∑ N α +N-1 i=N α +1 ℓ i+ 1 2		0	2c 2 3 c 2 2 3 h+c 2 c 2 2 h β
	+ T 0 ∑ N max i=N α +N+1 ℓ i+ 1 2

  CHAPTER 5. NUMERICAL ANALYSIS OF A FINITE VOLUME METHOD FOR A 1-D WAVE EQUATION WITH NON SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING Using translation of index i for the second term of the above equation, we obtain T21 = ∑ N n=1 ∆t ∑

	i+ 1 2 ℓ i+ 1 2 (v n v n i (P n i+1 -P n i ) + ∑ N n=1 ∆t ∑ N α +N-1 i=N α +1 ℓ i-1 2 i+1 -v n i )(P n i+1 -P n i ) + 2c 2 1 c 2 2 c 2 1 h+c 2 2 h α ∑ N n=1 ∆t v n v n i (P n i -P n i-1 ) N α +1 (P n N α +1 -P n N α ) n=1 ∆t v n n=1 ∆t v n -∑ N N α +1 N N α +N-1 -2c 2 2 c 2 3 c 2 3 h+c 2 2 h β ∑ N N α +N (P n N α +N+1 -P n N α +N ).

α +N ℓ N α +N+ 1 2 (P n N α +N+1 -P n N α +N ) -ℓ N α +N-1 2 (P n N α +N -P n N α +N-1 ) .

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  T42 = -δ ∑ N n=1 ∆t∂ 1/2 v n N α +1 P x (α,t n ) + δ ∑ N n=1 ∆t∂ 1/2 v n N α +N P x (β ,t n )

				1 i+1 +v n-1 i 2∆t h i+ 1 2	)(P n i+1 -P n i )
	+δ ∑ N n=1 ∆t	(v n+1 Nα +1 -v n-1 Nα +1 )(P n Nα +1 -P n Nα ) 2∆t h Nα + 1 2	-δ ∑ N n=1 ∆t	(v n+1 Nα +N -v n-1 Nα +N )(P n Nα +N -P n Nα +N-1 ) 2∆t h Nα +N+ 1 2
	= T 4 + δ ∑ N n=1 ∆t	(v n+1 Nα +1 -v n-1 Nα +1 )(P n Nα +1 -P n Nα ) 2∆t h Nα + 1 2	-δ ∑ N n=1 ∆t	(v n+1 Nα +N -v n-1 Nα +N )(P n Nα +N -P n Nα +N-1 ) 2 2∆t h Nα +N+ 1	.
	In addition,
	= -δ ∑ N n=1 ∆t	(v n+1 Nα +1 -v n-1 Nα +1 )(P n Nα +1 -P n Nα ) 2∆t h Nα + 1 2	+ δ ∑ N n=1 ∆t	(v n+1 Nα +N -v n-1 Nα +N )(P n Nα +N -P n Nα +N-1 ) 2∆t h Nα +N+ 1 2
	Consequently,
	T 4 ⇀

* δ

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SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

 [START_REF] Sato | Explicit numerical simulation of time-dependent viscoelastic flow problems by a finite element/finite volume method[END_REF], explicit numerical simulation has been developed for time dependent viscoelastic flow problems using a combination between FVM and FEM. Zhang et al.,[START_REF] Zhang | A new spectral finite volume method for elastic wave modelling on unstructured meshes[END_REF], illustrated a new spectral FVM for a 2-d and 3-d elastic wave equations with external sources on unstructured meshes. Zhang et al. also presented a new efficient FVM for 3-d elastic wave simulation on unstructured tetrahedral meshes (see [START_REF] Zhang | A new high-order finite volume method for 3D elastic wave simulation on unstructured meshes[END_REF]). We aslo mention Riečanová et al. [START_REF] Rieä | ą. Study of the numerical solution to the wave equation[END_REF], were the authors obtained the stability estimates and convergence of the numerical scheme of a wave equation with Dirichlet boundary condition on a rectangular domain. Their method was based on FVM in space together with the average of n + 1 and n -1 time step diffusion. However, in the one-dimensional case, some authors ensured the decay of energy and gave examples that verify its asymptotic behavior by implementing numerical schemes (see [START_REF] Alves | The asymptotic behavior of the linear transmission problem in viscoelasticity[END_REF][START_REF] Alves | The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin-Voigt Dissipation[END_REF][START_REF] Raposo | A Transmission Problem for Euler-Bernoulli beam with Kelvin-Voigt Damping[END_REF][START_REF] Maryati | Stability of an N-component timoshenko beam with localized Kelvin-Voigt and frictional dissipation[END_REF]). Thus, to our knowledge, it seems that there are no results in the literature on the convergence and stability estimates, concerning the transmission problem of elastic/viscoelastic systems where there is a discontinuity at the interface, based especially on FVM. The main objective of this chapter is to fill this gap. More precisely, we study the numerical solution of the transmission problem of a wave equation with localized Kelvin-Voigt damping acting faraway from the boundary via nonsmooth coefficient. In Section 5.2, we discretize our system using an admissible mesh and set the explicit numerical scheme in Subsection (5.2.1) and the semi-implicit numerical scheme with an average of n + 1 and n -1 time step diffusion in Subsection (5.2.3) using FVM in space. However, in Subsections (5.2.2) and (5.2.4), we design a discrete energy that dissipates when the control is acting and is conserved during its absence. Sections (5.3) and (5.4) are devoted to stability estimates and convergence of the discrete solution to the continuous one. Finally, in Section 5.5, we give some numerical examples which demonstrate the theoretical results obtained in [START_REF] Nasser | Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface[END_REF].

Consider the following transmission problem of a wave equation with localized Kelvin-Voigt type damping ρ 1 u tt -κ 1 u xx = 0, (x,t) ∈ (0, α) × (0, +∞),

(5.1.1)

ρ 3 w tt -κ 3 w xx = 0, (x,t) ∈ (β , L) × (0, +∞).

(5.1.3)

Here ρ 1 , ρ 2 , ρ 3 , κ 1 , κ 2 and κ 3 are strictly positive constant numbers representing the density and metric coefficients of Equations (5.1.1), (5.1.2) and (5.1.3) respectively and the damping constant ρ is strictly positive.

In fact, we divide Equations (5.1.1)-(5.1.3) by ρ 1 , ρ 2 and ρ 3 respectively to obtain and the following initial conditions

where

and c 3 = κ 3 ρ 3 denote the speed of the wave propagation of Equations (5.1.4)-(5.1.6) respectively. Also, δ = ρ ρ 2 .

SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

The transmission conditions are given by

(5.1.9)

In [START_REF] Nasser | Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface[END_REF], the authors studied the asymptotic behavior of the energy of the continuous case of System (5.1.4)-(5.1.9) and obtained an optimal energy decay rate of type t -4 . In this chapter, our objective is to study this system numerically. To this aim, we construct a numerical scheme based on FVM in space. In Subsection 5.2.1, we develop a conditionally stable numerical scheme using explicit discretization. However, in Subsection 5.2.3, the space discretization is combined with the average of n + 1 and n -1 time step diffusion and we obtain a semi-implicit numerical scheme.

Definition of the space. Let us define

We say that (u, v, w) is a weak solution of (5.1.4)- (5.1.8) if for all T > 0, the following conditions hold:

1.

) such that p(T, x) = 0, q(T, x) = 0 and z(T, x) = 0. We set Φ = (ϕ, η, γ) and Ψ = (ψ, ζ , θ ) and we study our problem under the following hypothesis (H ′ ):

Construction of the discretization of the problem

In this section, we will define the admissible mesh then we will construct an explicit scheme and finally a semi-implicit numerical scheme.

Admissible one-dimensional mesh. An admissible mesh T of the interval (0, L) is given by a family

) and a family

We discretize the intervals [0, α], [α, β ] and [β , L] into N α , N and N β internal points respectively such that N α , N, N β ∈ N * and N max = N α + N + N β . To be clear, let h α = α/N α , h = (β -α)/N and h β = (L -β )/N β and discretize as the following:

We discretize [0, α] such that:

SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

First of all, since the position x N α + 1 2 represents the point α, then we use the transmission condition (5.1.9) at point α and thus

Consequently, we obtain

SMOOTH WAVE SPEED AND INTERNAL LOCALIZED KEVIN-VOIGT DAMPING

After substituting (5.2.2), a direct calculation gives

(5.2.20)

We denote by U 1 T the numerical solution of the first time step of the numerical scheme (5.2.20); i.e., U

represents the solution of this scheme for n = 1. Now, denote the discrete norm in L 2 by

. Also, we denote the discrete norm in H 1 L by

, with U 0 = U N max +1 = 0.

Stability estimates

Stability usually refers to numerical schemes producing bounded solution errors based on the approximation scheme being used. In this section, we derive the desired stability estimates for the numerical solution of the implicit scheme (5.2.16).

Theorem 5.3.1. Assume that Hypothesis (H ′ ) and Condition (TA) hold. Then, the discrete L 2 and H 1 L norms of the numerical solution of the first time step U 1 T of the implicit numerical scheme (5.2.20) are bounded. Moreover, the approximation of the first time derivation is also bounded in L 2 . Therefore, U 1 T satisfies the following estimation

where C is a strictly positive constant independent of time and space discretization parameters.

The proof of Theorem 5.3.1 is divided into two Lemmas.

Lemma 5.3.2. Under Hypothesis (H ′ ) and Condition (TA), the numerical solution of the first time step U 1 T of the implicit numerical scheme (5.2.20) is bounded in L 2 .

Numerical Experiments: Validation of the theoretical results

In this section, we present some examples to illustrate graphically the theoretical results obtained in [START_REF] Nasser | Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface[END_REF]. In all experiments, we test the explicit scheme (5.2.8) and suppose L = 3, α = 1, β = 2 and the final time T = 10000. The discretization is given by N α = 20, N = 10, N β = 20 and N = 400000. Consequently, h α = 0.05, h = 0.1, h β = 0.05 and the time step is chosen as ∆t = T N = CFL × ∆x. We will study the asymptotic behavior for the following initial conditions of the form

Equal speed of propagation

Case 1. No damping: conservation of the total energy. When δ = 0, Figure 5.2 shows that the total energy is conserved along time. Thus, this numerical test shows that in the absence of the damping term, the total energy is completely conserved. Consequently, the numerical scheme (5.2.8) does not produce any numerical dissipation and therefore the numerical behavior observed is only due to the considered model.

Remark 5.5.1. Indeed, in the case of different propagation speed, numerical tests show also conservation of the energy in the absence of the damping, i.e., when δ = 0 (see for instance Figure 5.4).

Case 2. Exponential stability. When δ = 1, Figure 5.5-a shows that the total energy goes to zero as t → ∞, and thus we have dissipation. First, for a graph in ln-ln scale, there is a an energy decay of order t -α , whereas, for a graph of ln-time scale, we obtain an energy decay of order e -ωt . By simple linear regression (least squares), Figure 5.5-c shows that the polynomial decay of the energy is very fast (α ≃ 54.8760) and thus we deduce that the energy decays faster than polynomial. In fact, we can observe from Figure 5.5-b an exponential decay, as the graph ofln(E(t)) versus time t plots a straight line. By simple linear regression (least squares), we obtain the rate of decay numerically (ω ≃ 0.0008) and it is found to be very small. However, the final time profile confirms that U is small but it shows that high frequencies are not completely controlled (see Figure 5.5-d).

Different speed of propagation

In this part, we aim to verify the theoretical results obtained in [START_REF] Nasser | Optimal polynomial stability of a string with locally distributed Kelvin-Voigt damping and nonsmooth coefficient at the interface[END_REF]. To this end, we consider several cases and we obtain the decay rates numerically by simple linear regression (least squares).

Since the final time is large, we show two figures for the energy, the first one is for t = 0 to t = 100 and the second one represents the energy from T /2 till T . Indeed, Figure 5.6 shows that the total energy is decreasing and goes finally to zero, and thus we look for an exponential or polynomial decay. The graph ofln(E(t)) versus t plots a curve of the decay of the energy with a very small coefficient (ω = 0.0001) which shows that the energy decays slower than exponential (see Figure 5.6-c). However, Figure 5.6-d plots the graph ofln(E(t)) versus ln(t) which permits to show that E(t) tends to zero as 1/t α with α ≃ 4.17, since the curve is asymptotically a straight line. Finally, the final time profile confirms that U is small but also shows that high frequencies are not completely controlled (see Figure 5.6-e).

Case 2. (c

25 and δ = 1. Similarly, since the final time is large, we show two figures for the energy, the first one is for t = 0 to t = 100 and the second one represents the energy from T /2 till T . Under the same discussion, Figure 5.7 shows that E(t) tends to zero as 1/t α with α ≃ 4.4608. Also, the final time profile confirms that U is small, although the high frequencies are not completely controlled, especially at the beginning of the experiment.

= 6 and δ = 1. Figure 5.8 shows that the total energy goes finally to zero. Therefore, to explore the speed of convergence to zero, we plot similarly the graph ofln(E(t)) versus time and the graph ofln(E(t)) versus ln(t). Figure 5.8-c permits to show that the energy decays slower than exponential (ω = 0.0005), and hence we look for a polynomial decay. As a result, E(t) tends to zero as 1/t α with α ≃ 3.3990 (see Figure 5.8-d where the graph is asymptotically a straight line). Finally, the final time profile confirms that U is small but it shows that high frequencies are not completely controlled.

= 4 and δ = 1. Since Figure 5.9 shows that the total energy goes finally to zero, then we intent to study the speed of convergence to zero. Using the same argument, Figure 5.9-c permits to show that the energy decays slower than exponential (ω = 0.0006), while Figure 5.9-d states that E(t) tends to zero as 1/t α with α ≃ 4.4350. Finally, the final time profile confirms that although U is small, high frequencies are not completely controlled.

Remark 5.5.2. When the propagation speeds are not equal, we obtain a polynomial decay of the energy with slight different rates. However, in some cases, we get a numerical polynomial convergence better than t -4 , but it will probably be t -4 if we increase the time. In this test, we do not perform very long simulation to confirm, for reason of computation time. Abstract : My thesis is devoted to study the stabilization of the system of wave equations with locally distributed viscoelastic material of Kelvin-Voigt type damping. Also, we aim to study the numerical solution of these types of transmission problems where there is a discontinuity at the interface using the Finite volume method. First, we study the stability of a multidimensional wave equation on smooth and non smooth domains with localized internal Kelvin-Voigt damping in the presence or absence of geometric conditions; regarding that the damping is distributed faraway from the boundary or near a part of the boundary.

Second, we study the optimal stability of the previous system in the one-dimensional case where the Kelvin-Voigt damping is distributed randomly faraway from the boundary. Also, we investigate the stability of a one-dimensional wave equation with non smooth localized internal viscoelastic damping of Kelvin-Voigt type and with boundary or localized internal delay feedback. Finally, using the Finite volume criteria, we obtain the stability and convergence estimates of an elastic/viscoelastic transmission problem with localized internal Kelvin-Voigt damping acting via non smooth coefficient at the interface and we verify the theoretical results by numerical simulations.