Stabilization of a locally transmission problems of two strongly-weakly coupled wave systems

Abstract

In this paper, we embark on a captivating exploration of the stabilization of locally transmitted problems within the realm of two interconnected wave systems. To begin, we wield the formidable Arendt-Batty criteria (Trans. Am. Math. Soc. 306(2) (1988) 837–852) to affirm the resolute stability of our system. Then, with an artful fusion of a frequency domain approach and the multiplier method, we unveil the exquisite phenomenon of exponential stability, a phenomenon that manifests when the waves of the second system synchronize their propagation speeds. In cases where these speeds diverge, our investigation reveals a graceful decay of our system’s energy, elegantly characterized by a polynomial decline at a rate of $t^{- 1}$ .

Keywords

Transmission problems coupled wave equation strong stability exponential stability polynomial stability frequency domain approach

1. .Introduction

When a vibrating source disturbs the first particle of a medium, a wave is generated. This phenomenon propagates from particle to particle through the medium, typically modeled by a wave equation. To mitigate these vibrations, the most common approach is the addition of damping.

In recent years, the stabilization of wave systems, whether simple or coupled, with localized damping has garnered significant attention from researchers, particularly in the one-dimensional case (see [1,4–6,17,19,24,25]). Notably, in two papers, it was demonstrated that the smoothness of the damping and coupling coefficients plays a crucial role in the stability and regularity of the solutions of the studied systems. Specifically, in [2], Akil, Badawi, and Nicaise investigated the stability of locally coupled wave equations with local Kelvin–Voigt damping when the supports of the damping and coupling coefficients are disjoint. They showed that the energy of their system decays polynomially, with rates of $t^{- 1}$ and $t^{- 4}$ in different cases.

Additionally, there are numerous publications on the multi-dimensional setting (see [3,7–14,20,22,23,27–29,32,34]). Our objective in this work is to examine a more general problem. However, before presenting our main contributions, let us first recall some previous results for such systems. In 2020, S. Gerbi et al. [18] proved the exponential decay rate of solutions when the waves propagate with equal speeds, the coupling region is a subset of the damping region, and it satisfies the geometric control condition (GCC), with the damping and coupling coefficients in $W^{1, \infty} (Ω)$ . Similarly, Wehbe, Ibtissam, and Akil [33] recently showed that the energy of smooth solutions of the system decays polynomially, at a rate of $t^{- 1}$ , considering both the damping and coupling coefficients are non-smooth. They extended this work to the multidimensional case in [8], studying the stability of the system under various geometric control conditions. They established polynomial stability when there is an intersection between the damping and coupling regions, and also when the coupling region is a subset of the damping region under the GCC.

Within the intricate framework of this paper, our focus converges on an intriguing question: What specific qualities define the stability of our transmission problems (1.1)? Indeed, this problem involves two wave systems: one weakly coupled and the other strongly coupled with non-smooth coefficients. To the best of our knowledge, no results exist in the literature concerning our problem (1.1), especially in the one-dimensional case. The goal of this paper is to address this gap by studying the stability of the following locally transmitted problem:

\begin{aligned} {\begin{cases} u_{t t} - a_{1} u_{x x} + d_{1} (x) u_{t} + c_{1} (x) y = 0, & (x, t) \in (0, L_{0}) \times R_{+}^{*}, \\ y_{t t} - y_{x x} + c_{1} (x) u = 0, & (x, t) \in (0, L_{0}) \times R_{+}^{*}, \\ φ_{t t} - a_{2} φ_{x x} + d_{2} (x) φ_{t} + c_{2} (x) ψ_{t} = 0, & (x, t) \in (L_{0}, L) \times R_{+}^{*}, \\ ψ_{t t} - ψ_{x x} - c_{2} (x) φ_{t} = 0, & (x, t) \in (L_{0}, L) \times R_{+}^{*}, \end{cases} \end{aligned}

(1.1)

with fully Dirichlet boundary conditions,

\begin{aligned} u (0, t) = y (0, t) = φ (L, t) = ψ (L, t) = 0, t \in R_{+}^{*}, \end{aligned}

(1.2)

and the following transmission conditions,

\begin{aligned} {\begin{cases} u (L_{0}, t) = φ (L_{0}, t), & y (L_{0}, t) = ψ (L_{0}, t), t \in R_{+}^{*}, \\ a_{1} u_{x} (L_{0}, t) = a_{2} φ_{x} (L_{0}, t), & y_{x} (L_{0}, t) = ψ_{x} (L_{0}, t), t \in R_{+}^{*}, \end{cases} \end{aligned}

(1.3)

and with the following initial data

\begin{aligned} (u, y, φ, ψ, u_{t}, y_{t}, φ_{t}, ψ_{t}) (x, 0) = (u_{0}, y_{0}, φ_{0}, ψ_{0}, u_{1}, y_{1}, φ_{1}, ψ_{1}), \end{aligned}

(1.4)

where

\begin{aligned} d_{1} (x) = {\begin{cases} d_{1} & if x \in (α_{2}, α_{4}), \\ 0 & otherwise, \end{cases} c_{1} (x) = {\begin{cases} c_{1} & if x \in (α_{1}, α_{3}), \\ 0 & otherwise, \end{cases} \end{aligned}

(1.5)

\begin{aligned} d_{2} (x) = {\begin{cases} d_{2} & if x \in (β_{2}, β_{4}), \\ 0 & otherwise, \end{cases} and c_{2} (x) = {\begin{cases} c_{2} & if x \in (β_{1}, β_{3}), \\ 0 & otherwise, \end{cases} \end{aligned}

(1.6)

and

a_{1}

a_{2}

d_{1}

d_{2}

are strictly positives constants,

c_{1}, c_{2} \in R^{*}

, with

\begin{aligned} | c_{1} | < \frac{1}{C_{0}}, \end{aligned}

(1.7)

where

C_{0}

denotes the Poincaré constant. More precisely,

C_{0}

is the smallest positive constant such that

\begin{aligned} \int_{0}^{L_{0}} | f |^{2} d x ⩽ C_{0} \int_{0}^{L_{0}} | f_{x} |^{2} d x, \forall f \in H_{0}^{1} (0, L_{0}) . \end{aligned}

(1.8)

We consider

0 < α_{1} < α_{2} < α_{3} < α_{4} < L_{0} < β_{1} < β_{2} < β_{3} < β_{4} < L

. (See Fig. 1.)

The paper is structured as follows: First in Section 2, we prove the well-posedness of our system by using semigroup approach. Then in Section 2.1, following a general criteria of Arendt and Batty, we show the strong stability of our problem. Next, in Section 2.2, by using the frequency domain approach combining with a specific multiplier method, we establish exponential stability of the solution if and only if the waves of the second coupled equations have the same speed of propagation (i.e., $a_{2} = 1$ ). In the case when $a_{2} \neq 1$ , we prove that the energy of our problem decays polynomially with the rate $t^{- 1}$ .

Fig. 1.

Geometric description of the functions $d_{1}$ , $d_{2}$ , $c_{1}$ and $c_{2}$ .

2. .Well-posedness and stability results

Let’s start by rigorously establishing the well-posedness of the system (1.1)–(1.4) using a detailed semi-group approach.

Case1. If $d_{1} (x) = 0$ in $(0, L_{0})$ .

Let $(u, u_{t}, y, y_{t}, φ, φ_{t}, ψ, ψ_{t})$ be a regular solution of the system (1.1)–(1.4). The energy of the system is given by

\begin{aligned} E (t) & = \frac{1}{2} \int_{0}^{L_{0}} (| u_{t} |^{2} + a_{1} | u_{x} |^{2} + | y_{t} |^{2} + | y_{x} |^{2} + 2 ℜ (c_{1} (x) u \bar{y})) d x \\ + \frac{1}{2} \int_{L_{0}}^{L} (| φ_{t} |^{2} + a_{2} | φ_{x} |^{2} + | ψ_{t} |^{2} + | ψ_{x} |^{2}) d x . \end{aligned}

(2.1)

A straightforward computation gives

\begin{aligned} E^{'} (t) = - \int_{L_{0}}^{L} d_{2} (x) | φ_{t} |^{2} d x ⩽ 0. \end{aligned}

(2.2)

Thus, the system (1.1)–(1.4) is dissipative in the sense that its energy is a non increasing function with respect to the time variable t. We introduce the following Hilbert spaces

\begin{aligned} H_{L}^{1} (a, b) & = {f \in H^{1} (a, b); f (a) = 0}, \\ H_{R}^{1} (a, b) & = {f \in H^{1} (a, b); f (b) = 0}, \end{aligned}

for any real numbers a, b such that

a < b

. The energy space

H

is now defined by

\begin{aligned} H & = {[H_{L}^{1} (0, L_{0}) \times L^{2} (0, L_{0})]^{2} \times [H_{R}^{1} (L_{0}, L) \times L^{2} (L_{0}, L)]^{2}; u (L_{0}) = φ (L_{0}) \\ and~ y (L_{0}) = ψ (L_{0})} \end{aligned}

(2.3)

equipped with the following norm

\begin{aligned} ‖ U ‖_{H}^{2} & = a_{1} ‖ u_{x} ‖_{L^{2} (0, L_{0})}^{2} + ‖ v ‖_{L^{2} (0, L_{0})}^{2} + ‖ y_{x} ‖_{L^{2} (0, L_{0})}^{2} + ‖ z ‖_{L^{2} (0, L_{0})}^{2} + 2 ℜ \int_{0}^{L_{0}} c_{1} (x) u \bar{y} d x \\ + a_{2} ‖ φ_{x} ‖_{L^{2} (L_{0}, L)}^{2} + ‖ η ‖_{L^{2} (L_{0}, L)}^{2} + ‖ ψ_{x} ‖_{L^{2} (L_{0}, L)}^{2} + ‖ ξ ‖_{L^{2} (L_{0}, L)}^{2}, \end{aligned}

for all

U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in H

Let $U = (u, u_{t}, y, y_{t}, φ, φ_{t}, ψ, ψ_{t})^{T}$ , it’s easy to see that problem (1.1)–(1.4) is formally equivalent to the following abstract evolution equation in the Hilbert space $H$

\begin{aligned} {\begin{cases} U^{'} (t) = A U (t), t > 0, \\ U (0) = U_{0} = (u_{0}, u_{1}, y_{0}, y_{1}, φ_{0}, φ_{1}, ψ_{0}, ψ_{1})^{T}, \end{cases} \end{aligned}

(2.4)

and the unbounded operator

A

is defined by

\begin{aligned} A U = (\begin{matrix} v \\ a_{1} u_{x x} - c_{1} (x) y \\ z \\ y_{x x} - c_{1} (x) u \\ η \\ a_{2} φ_{x x} - c_{2} (x) ξ - d_{2} (x) η \\ ξ \\ ψ_{x x} + c_{2} (x) η \end{matrix}), \end{aligned}

(2.5)

for all

U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in D (A)

, with domain

\begin{aligned} D (A) & = {U \in H; v, z \in H_{L}^{1} (0, L_{0}), u, y \in H^{2} (0, L_{0}) \cap H_{L}^{1} (0, L_{0}), η, ξ \in H_{R}^{1} (L_{0}, L), \\ φ, ψ \in H^{2} (L_{0}, L) \cap H_{R}^{1} (L_{0}, L) ~and~ u (L_{0}) = φ (L_{0}), y (L_{0}) = ψ (L_{0})} . \end{aligned}

Proposition 2.1.

The unbounded linear operator $A$ generates a $C_{0}$ -semigroup of contractions on $H$ .

Proof.

Using Lumer–Phillips theorem (see [30]), it is sufficient to prove that $A$ is a maximal dissipative operator so that $A$ generates a C₀-semigroup of contractions on $H$ . First, let $U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in D (A)$ . Then, integrating by parts we have

\begin{aligned} ℜ ⟨ A U, U ⟩_{H} = - \int_{L_{0}}^{L} d_{2} (x) | η |^{2} d x ⩽ 0. \end{aligned}

(2.6)

This 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}, f_{8})^{T} \in H

, we look for

U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in D (A)

solution of

\begin{aligned} - A U = F . \end{aligned}

(2.7)

Equivalently, we have the following system

\begin{aligned} - v = f_{1}, \end{aligned}

(2.8)

\begin{aligned} - a_{1} u_{x x} + c_{1} (x) y = f_{2}, \end{aligned}

(2.9)

\begin{aligned} - z = f_{3}, \end{aligned}

(2.10)

\begin{aligned} - y_{x x} + c_{1} (x) u = f_{4}, \end{aligned}

(2.11)

\begin{aligned} - η = f_{5}, \end{aligned}

(2.12)

\begin{aligned} - a_{2} φ_{x x} + d_{2} (x) η + c_{2} (x) ξ = f_{6}, \end{aligned}

(2.13)

\begin{aligned} - ξ = f_{7}, \end{aligned}

(2.14)

\begin{aligned} - ψ_{x x} - c_{2} (x) η = f_{8} . \end{aligned}

(2.15)

Inserting (2.12) and (2.14) into (2.13) and (2.15), we get

\begin{aligned} - a_{2} φ_{x x} = f_{6} + c_{2} (x) f_{7} + d_{2} (x) f_{5}, \end{aligned}

(2.16)

and

\begin{aligned} - ψ_{x x} = f_{8} - c_{2} (x) f_{5} . \end{aligned}

(2.17)

Multiplying (2.9), (2.16) by

\bar{ϕ_{1}} \in H_{0}^{1} (0, L)

and (2.11), (2.17) by

\bar{ϕ_{2}} \in H_{0}^{1} (0, L)

, integrating over

(0, L)

, we get

\begin{aligned} - a_{1} \bar{ϕ_{1}} (L_{0}) u_{x} (L_{0}) + a_{1} \int_{0}^{L_{0}} u_{x} (\bar{ϕ_{1}})_{x} d x + \int_{0}^{L_{0}} c_{1} (x) y \bar{ϕ_{1}} d x = \int_{0}^{L_{0}} f_{2} \bar{ϕ_{1}} d x, \end{aligned}

(2.18)

\begin{aligned} - \bar{ϕ_{2}} (L_{0}) y_{x} (L_{0}) + \int_{0}^{L_{0}} y_{x} (\bar{ϕ_{2}})_{x} d x + \int_{0}^{L_{0}} c_{1} (x) u \bar{ϕ_{2}} d x = \int_{0}^{L_{0}} f_{4} \bar{ϕ_{2}} d x, \end{aligned}

(2.19)

\begin{aligned} a_{2} \bar{ϕ_{1}} (L_{0}) φ_{x} (L_{0}) + a_{2} \int_{L_{0}}^{L} φ_{x} (\bar{ϕ_{1}})_{x} d x = \int_{L_{0}}^{L} (f_{6} + c_{2} (x) f_{7} + d_{2} (x) f_{5}) \bar{ϕ_{1}} d x, \end{aligned}

(2.20)

and

\begin{aligned} \bar{ϕ_{2}} (L_{0}) ψ_{x} (L_{0}) + \int_{L_{0}}^{L} ψ_{x} (\bar{ϕ_{2}})_{x} d x = \int_{L_{0}}^{L} (f_{8} - c_{2} (x) f_{5}) \bar{ϕ_{2}} d x . \end{aligned}

(2.21)

Adding the above equations, we obtain the following variational problem:

\begin{aligned} Λ (((u, φ), (y, ψ)), (ϕ_{1}, ϕ_{2})) = l (ϕ_{1}, ϕ_{2}), \forall (ϕ_{1}, ϕ_{2}) \in H_{0}^{1} (0, L) \times H_{0}^{1} (0, L), \end{aligned}

(2.22)

where

\begin{aligned} Λ (((u, φ), (y, ψ)), (ϕ_{1}, ϕ_{2})) = ϑ_{1} ((u, y), (ϕ_{1}, ϕ_{2})) + ϑ_{2} ((φ, ψ), (ϕ_{1}, ϕ_{2})), \end{aligned}

with

\begin{aligned} {\begin{cases} ϑ_{1} ((u, y), (ϕ_{1}, ϕ_{2})) = \int_{0}^{L_{0}} (a_{1} u_{x} (\bar{ϕ_{1}})_{x} + y_{x} (\bar{ϕ_{2}})_{x} + c_{1} (x) y \bar{ϕ_{1}} + c_{1} (x) u \bar{ϕ_{2}}) d x, \\ ϑ_{2} ((φ, ψ), (ϕ_{1}, ϕ_{2})) = a_{2} \int_{L_{0}}^{L} φ_{x} (\bar{ϕ_{1}})_{x} d x + \int_{L_{0}}^{L} ψ_{x} (\bar{ϕ_{2}})_{x} d x, \end{cases} \end{aligned}

and

\begin{aligned} l (ϕ_{1}, ϕ_{2}) & = \int_{0}^{L_{0}} (f_{2} \bar{ϕ_{1}} + f_{4} \bar{ϕ_{2}}) d x + \int_{L_{0}}^{L} (f_{6} + c_{2} (x) f_{7} + d_{2} (x) f_{5}) \bar{ϕ_{1}} d x \\ + \int_{L_{0}}^{L} (f_{8} - c_{2} (x) f_{5}) \bar{ϕ_{2}} d x . \end{aligned}

First, thanks to (1.7) we have that

ϑ_{1}

is a bilinear, continuous and coercive form on

(H_{L}^{1} (0, L_{0}) \times H_{L}^{1} (0, L_{0}))^{2}

. Second it’s easy to see that

ϑ_{2}

is a bilinear, continuous and coercive form on

(H_{R}^{1} (L_{0}, L) \times H_{R}^{1} (L_{0}, L))^{2}

and l is linear continuous form on

H_{0}^{1} (L_{0}, L) \times H_{0}^{1} (L_{0}, L)

. Then, using Lax–Milgram theorem, we deduce that there exists

((u, φ), (y, ψ)) \in H_{0}^{1} (0, L) \times H_{0}^{1} (0, L)

unique solution of the variational problem (2.22). By using the classical elliptic regularity we deduce that

u, y \in H^{2} (0, L_{0}) \cap H_{L}^{1} (0, L_{0})

and

φ, ψ \in H^{2} (L_{0}, L) \cap H_{R}^{1} (L_{0}, L)

. Next by setting

v = - f_{1}

z = - f_{3}

η = - f_{5}

and

ξ = - f_{7}

, we deduce that

U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in D (A)

is solution of (2.7). To conclude, we need to show the uniqueness of such a solution. So, let

U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in D (A)

be a solution of (2.7) with

\begin{aligned} f_{1} = f_{2} = f_{3} = f_{4} = f_{5} = f_{6} = f_{7} = f_{8} = 0. \end{aligned}

Then we directly deduce that

v = z = η = ξ = 0

and therefore

(u, y) \in [H_{L}^{1} (0, L_{0})]^{2}

satisfies (2.22) with

l (ϕ_{1}, ϕ_{2}) = 0

and

(φ, ψ) \in [H_{R}^{1} (L_{0}, L)]^{2}

satisfies (2.22) with

l (ϕ_{1}, ϕ_{2}) = 0

. As

ϑ_{1}

ϑ_{2}

are two sesquilinears, continuous coercive forms, we deduce that

\begin{aligned} u = y = φ = ψ = 0. \end{aligned}

In other words,

\ker (A) = {0}

. Consequently, we get

U = (u, - f_{1}, y, - f_{3}, φ, - f_{5}, ψ, - f_{7})^{T} \in D (A)

is a unique solution of (2.7).

Since $0 \in ρ (A)$ the resolvent set of $A$ we easily get $R (λ I - A) = H$ for a sufficiently small $λ > 0$ (see [26, Thm. 1.2.4]). This, together with the dissipativeness of $A$ , imply that $D (A)$ is dense in $H$ (see [30, Thm. 4.6]). Then $A$ is m-dissipative in $H$ .

As $A$ generates a $C_{0}$ -semigroup of contractions $(e^{t A})_{t ⩾ 0}$ , we have the following result: Theorem 2.2 (Existence and uniqueness of the solution)

(1)
If $U_{0} \in D (A)$ , then problem (2.4) admits a unique strong solution U satisfying:
$\begin{aligned} U \in C^{1} (R_{+}, H) \cap C^{0} (R_{+}, D (A)) . \end{aligned}$

(2)
If $U_{0} \in H$ , then problem (2.4) admits a unique weak solution U satisfying:
$\begin{aligned} U \in C^{0} (R_{+}, H) . \end{aligned}$

2.1. Strong stability

Now the following result is about the strong stability of system (1.1)–(1.4)

Theorem 2.3.
The $C_{0}$ -semigroup of contractions $(e^{t A})_{t ⩾ 0}$ is strongly stable in the energy space $H$ in the sense that
$\begin{aligned} lim_{t \to + \infty} ‖ e^{t A} U_{0} ‖_{H} = 0, \forall U_{0} \in H . \end{aligned}$

Proof.
Since the resolvent of $A$ is compact in $H$ , it follows from the Arendt–Batty’s theorem (see [15]) that the system (1.1)–(1.4) is strongly stable if and only if $A$ does not have pure imaginary eigenvalues, i.e. $σ (A) \cap i R = \emptyset$ . From Proposition 2.1, we have that $0 \in ρ (A)$ . Therefore, only $σ (A) \cap i R^{*} = \emptyset$ must be proved. For this purpose, suppose that there exists a real number $λ \neq 0$ and $U = (u, v, y, z, φ, η, ψ, ξ)^{T} \in D (A)$ such that
$\begin{aligned} A U = i λ U . \end{aligned}$
(2.23)
From (2.6) and (2.23), we have
$\begin{aligned} 0 = ℜ (i λ ‖ U ‖_{H}^{2}) = ℜ (⟨ A U, U ⟩_{H}) = - \int_{L_{0}}^{L} d_{2} (x) | η |^{2} d x . \end{aligned}$
(2.24)
Condition (1.6) implies that
$\begin{aligned} \sqrt{d_{2}} η = 0 in (L_{0}, L) and η = 0 in (β_{2}, β_{4}) . \end{aligned}$
(2.25)
Detailing (2.23) and using (2.25), we get the following equations
$\begin{aligned} \begin{array}{lll} v = i λ u in (0, L_{0}), & z = i λ y in (0, L_{0}), \\ η = i λ φ in (L_{0}, L), & ξ = i λ ψ in (L_{0}, L), \end{array} \end{aligned}$
(2.26)
and
$\begin{aligned} {\begin{cases} λ^{2} u + a_{1} u_{x x} - c_{1} (x) y = 0, & x \in (0, L_{0}), \\ λ^{2} y + y_{x x} - c_{1} (x) u = 0, & x \in (0, L_{0}), \\ λ^{2} φ + a_{2} φ_{x x} - i λ c_{2} (x) ψ = 0, & x \in (L_{0}, L), \\ λ^{2} ψ + ψ_{x x} + i λ c_{2} (x) φ = 0, & x \in (L_{0}, L) . \end{cases} \end{aligned}$
(2.27)
Our goal is to prove that $u = y = 0$ in $(0, L_{0})$ and $φ = ψ = 0$ in $(L_{0}, L)$ . For simplicity, we divide the proof into two steps.

Step 1. The aim of this step is to show that $φ = ψ = 0$ in $(L_{0}, L)$ . So, using (2.25) and the third equation in (2.26), we have
$\begin{aligned} φ = 0 in (β_{2}, β_{4}) . \end{aligned}$
(2.28)
From (2.27)₃, (1.6) and the above equation, we get
$\begin{aligned} ψ = 0 in (β_{2}, β_{3}) . \end{aligned}$
(2.29)
Using the above result and (2.28), we have $φ = ψ = 0$ in $(β_{2}, β_{3})$ . Since $φ, ψ \in H^{2} (L_{0}, L) \subset C^{1} ([β_{2}, β_{3}])$ , then
$\begin{aligned} φ (ζ) = φ_{x} (ζ) = ψ (ζ) = ψ_{x} (ζ) = 0, \forall ζ \in {β_{2}, β_{3}} . \end{aligned}$
(2.30)
Let $V_{1} = (φ, φ_{x}, ψ, ψ_{x})^{T}$ , the following system
$\begin{aligned} {\begin{cases} λ^{2} φ + a_{2} φ_{x x} - i λ c_{2} (x) ψ = 0, & x \in (L_{0}, β_{2}), λ^{2} \\ ψ + ψ_{x x} + i λ c_{2} (x) φ = 0, & x \in (L_{0}, β_{2}), \\ φ (β_{2}) = φ_{x} (β_{2}) = ψ (β_{2}) = ψ_{x} (β_{2}) = 0, \end{cases} \end{aligned}$
(2.31)
can be written as
$\begin{aligned} {\begin{cases} (V_{1})_{x} = B_{1} V_{1} in (L_{0}, β_{2}), \\ V_{1} (β_{2}) = 0, \end{cases} \end{aligned}$
(2.32)
where
$\begin{aligned} B_{1} = (\begin{array}{cccc} 0 & 1 & 0 & 0 \\ \frac{- λ^{2}}{a_{2}} & 0 & \frac{i λ}{a_{2}} c_{2} 0 \\ 0 & 0 & 0 & 1 \\ - i λ c_{2} & 0 & - λ^{2} & 0 \end{array}) . \end{aligned}$
The solution of the differential equation (2.32) is given by
$\begin{aligned} V_{1} (x) = e^{B_{1} (x - β_{2})} V_{1} (β_{2}) = 0 in (L_{0}, β_{2}) . \end{aligned}$
(2.33)
Then,
$\begin{aligned} φ = ψ = 0 in (L_{0}, β_{2}) . \end{aligned}$
(2.34)
Now, we still need to prove that $φ = 0$ in $(β_{4}, L)$ and $ψ = 0$ in $(β_{3}, L)$ . Using (2.28), (2.29), the fact that $φ \in C^{1} ([β_{2}, β_{4}])$ , $ψ \in C^{1} ([β_{2}, β_{3}])$ and the equations (2.27)₃ and (2.27)₄, we get the following systems:
$\begin{aligned} {\begin{cases} λ^{2} φ + a_{2} φ_{x x} = 0, x \in (β_{4}, L), \\ φ (β_{4}) = φ_{x} (β_{4}) = φ (L) = 0, \end{cases} \end{aligned}$
(2.35)
and
$\begin{aligned} {\begin{cases} λ^{2} ψ + ψ_{x x} = 0, x \in (β_{3}, L), \\ ψ (β_{3}) = ψ_{x} (β_{3}) = ψ (L) = 0. \end{cases} \end{aligned}$
(2.36)
Then, using Holmgren uniqueness theorem, we get
$\begin{aligned} φ = 0 in (β_{4}, L) and ψ = 0 in (β_{3}, L) . \end{aligned}$
(2.37)
Hence from (2.28), (2.29), (2.34) and the above result, we obtain
$\begin{aligned} φ = ψ = 0 in (L_{0}, L) . \end{aligned}$
(2.38)
Step2. The aim of this step is to show that $u = y = 0$ in $(0, L_{0})$ . From (2.38) and the fact that φ, $ψ \in C^{1} ([L_{0}, L])$ , we have the following boundary condition
$\begin{aligned} φ (L_{0}) = φ_{x} (L_{0}) = ψ (L_{0}) = ψ_{x} (L_{0}) = 0. \end{aligned}$
(2.39)
So, from (1.3) and the above equation, we obtain
$\begin{aligned} u (L_{0}) = u_{x} (L_{0}) = y (L_{0}) = y_{x} (L_{0}) = 0. \end{aligned}$
(2.40)
From equations (2.27)₁ and (2.27)₂ and the fact that $c (x) = 0$ in $(α_{3}, L_{0})$ , we obtain the following system:
$\begin{aligned} {\begin{cases} λ^{2} u + a_{1} u_{x x} = 0, & x \in (α_{3}, L_{0}), \\ λ^{2} y + y_{x x} = 0, & x \in (α_{3}, L_{0}) . \end{cases} \end{aligned}$
(2.41)
It is easy to see that system (2.41) admits only a trivial solution on $(α_{3}, L_{0})$ under the boundary condition (2.40). Then
$\begin{aligned} u = y = 0 in (α_{3}, L_{0}) . \end{aligned}$
(2.42)
Using the above equation and the fact that φ, $ψ \in C^{1} ([L_{0}, L])$ , we have
$\begin{aligned} u (α_{3}) = u_{x} (α_{3}) = y (α_{3}) = y_{x} (α_{3}) = 0. \end{aligned}$
(2.43)
Next, using equations (2.27)₁ and (2.27)₂, we get
$\begin{aligned} {\begin{cases} λ^{2} u + a_{1} u_{x x} - c_{1} (x) y = 0, & x \in (α_{1}, α_{3}), \\ λ^{2} y + y_{x x} - c_{1} (x) u = 0, & x \in (α_{1}, α_{3}) . \end{cases} \end{aligned}$
(2.44)
Let $V_{2} = (u, u_{x}, y, y_{x})^{T}$ . From (2.43), $V_{2} (α_{3}) = 0$ . The system (2.44) can be written as the following equation
$\begin{aligned} (V_{2})_{x} = B_{2} V_{2} in (α_{1}, α_{3}), \end{aligned}$
(2.45)
where
$\begin{aligned} B_{2} = (\begin{array}{cccc} 0 & 1 & 0 & 0 \\ \frac{- λ^{2}}{a_{1}} & 0 & \frac{c_{1}}{a_{1}} & 0 \\ 0 & 0 & 0 & 1 \\ c_{1} & 0 & - λ^{2} & 0 \end{array}) . \end{aligned}$
The solution of the differential equation (2.45) is given by
$\begin{aligned} V_{2} (x) = e^{B_{2} (x - α_{3})} V_{2} (α_{3}) = 0 in (α_{1}, α_{3}) . \end{aligned}$
(2.46)
Then,
$\begin{aligned} u = y = 0 in (α_{1}, α_{3}) . \end{aligned}$
(2.47)
Now, we still need to show that $u = y = 0$ in $(0, α_{1})$ . Using the above result, the fact that $u, y \in C^{1} ([α_{1}, L_{0}])$ and the equations (2.27)₁ and (2.27)₂, we get:
$\begin{aligned} {\begin{cases} λ^{2} u + a_{1} u_{x x} = 0, & x \in (0, α_{1}), \\ u (α_{1}) = u_{x} (α_{1}) = u (0) = 0, \end{cases} \end{aligned}$
(2.48)
and
$\begin{aligned} {\begin{cases} λ^{2} y + y_{x x} = 0, x \in (0, α_{1}), \\ y (α_{1}) = y_{x} (α_{1}) = y (0) = 0. \end{cases} \end{aligned}$
(2.49)
Again, using Holmgren uniqueness theorem, we have
$\begin{aligned} u = y = 0 in (0, α_{1}) . \end{aligned}$
(2.50)
Finally, by using (2.26), (2.38), (2.42), (2.47) and (2.50) we deduce that $U = 0$ in $(0, L)$ and we reached our disered result.

2.2. Exponential and polynomial stability

In this section, we will study the exponential and polynomial stabilities of the system (1.1)–(1.4). Our main result in this part is the following theorems.

Theorem 2.4.
If $a_{2} = 1$ , then the C ₀ -semigroup $(e^{t A})_{t ⩾ 0}$ is exponentially stable; i.e., there exists constants $M ⩾ 1$ and $ϵ > 0$ independent of $U_{0}$ such that
$\begin{aligned} ‖ e^{t A} U_{0} ‖_{H} ⩽ M e^{- ϵ t} ‖ U_{0} ‖_{H} . \end{aligned}$

Theorem 2.5.
If $a_{2} \neq 1$ , then there exists $C > 0$ such that for every $U_{0} \in D (A)$ , we have
$\begin{aligned} E (t) ⩽ \frac{C}{t} ‖ U_{0} ‖_{D (A)}^{2}, \forall t > 0. \end{aligned}$

Since $i R \subset ρ (A)$ (see Section 2.1), according to Huang [21], Prüss [31], Borichev and Tomilov [16], to proof Theorems 2.4 and 2.5, we still need to check if the following condition hold:
$\begin{aligned} sup_{λ \in R} \frac{1}{| λ |^{ℓ}} ‖ (i λ I - A)^{- 1} ‖_{L (H)} < \infty with~ ℓ = {\begin{cases} 0 & for Theorem 2.4, \\ 1 & for Theorem 2.5. \end{cases} \end{aligned}$
(2.51)
We will prove condition (2.51) by an argument of contradiction. For this purpose, suppose that (2.51) is false, then there exists ${(λ_{n}, U_{n} := (u_{n}, v_{n}, y_{n}, z_{n}, φ_{n}, η_{n}, ψ_{n}, ξ_{n})^{T})} \subset R^{} \times D (A)$ with
$\begin{aligned} | λ_{n} | ⟶ + \infty and ‖ U_{n} ‖_{H} = 1, \end{aligned}$
(2.52)
such that
$\begin{aligned} (λ_{n})^{ℓ} (i λ_{n} I - A) U_{n} = F_{n} := (f_{1, n}, f_{2, n}, f_{3, n}, f_{4, n}, f_{5, n}, f_{6, n}, f_{7, n}, f_{8, n})^{T} ⟶ 0 in H . \end{aligned}$
(2.53)
For simplicity, we drop the index n. Equivalently, from (2.53), we have
$\begin{aligned} i λ u - v = λ^{- ℓ} f_{1} \to 0 in H_{L}^{1} (0, L_{0}), \end{aligned}$
(2.54)

$\begin{aligned} i λ v - a_{1} u_{x x} + c_{1} (x) y = λ^{- ℓ} f_{2} \to 0 in L^{2} (0, L_{0}), \end{aligned}$
(2.55)

$\begin{aligned} i λ y - z = λ^{- ℓ} f_{3} \to 0 in H_{L}^{1} (0, L_{0}), \end{aligned}$
(2.56)

$\begin{aligned} i λ z - y_{x x} + c_{1} (x) u = λ^{- ℓ} f_{4} \to 0 in L^{2} (0, L_{0}), \end{aligned}$
(2.57)

$\begin{aligned} i λ φ - η = λ^{- ℓ} f_{5} \to 0 in H_{R}^{1} (L_{0}, L), \end{aligned}$
(2.58)

$\begin{aligned} i λ η - a_{2} φ_{x x} + c_{2} (x) ξ + d_{2} (x) η = λ^{- ℓ} f_{6} \to 0 in L^{2} (L_{0}, L), \end{aligned}$
(2.59)

$\begin{aligned} i λ ψ - ξ = λ^{- ℓ} f_{7} \to 0 in H_{R}^{1} (L_{0}, L), \end{aligned}$
(2.60)

$\begin{aligned} i λ ξ - ψ_{x x} - c_{2} (x) η = λ^{- ℓ} f_{8} \to 0 in L^{2} (L_{0}, L) . \end{aligned}$
(2.61)
Here we will check the condition (2.51) by finding a contradiction with (2.52) by showing $‖ U ‖_{H} = o (1)$ . From (2.52), (2.54), (2.56), (2.58) and (2.60), we obtain
$\begin{aligned} ‖ λ u ‖_{L^{2} (0, L_{0})} = O (1), ‖ λ y ‖_{L^{2} (0, L_{0})} = O (1), \end{aligned}$
(2.62)
and
$\begin{aligned} ‖ λ φ ‖_{L^{2} (L_{0}, L)} = O (1), ‖ λ ψ ‖_{L^{2} (L_{0}, L)} = O (1) . \end{aligned}$
(2.63)
For clarity, we will divide the proof into several lemmas.
Lemma 2.6.
The solution* $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimation
$\begin{aligned} \int_{β_{2}}^{β_{4}} | η |^{2} d x = o (λ^{- ℓ}) a n d \int_{β_{2}}^{β_{4}} | λ φ |^{2} d x = o (λ^{- ℓ}) . \end{aligned}$
(2.64)

Proof.
Taking the inner product of (2.53) with U in $H$ and using (2.6), we get
$\begin{aligned} \int_{β_{2}}^{β_{4}} | η |^{2} d x = - ℜ (⟨ A U, U ⟩_{H}) = ℜ (⟨ (i λ I - A) U, U ⟩_{H}) = λ^{- ℓ} ℜ ⟨ F, U ⟩_{H} ⩽ | λ |^{- ℓ} ‖ F ‖_{H} ‖ U ‖_{H} . \end{aligned}$
Thus, from the above equation, the fact that $‖ F ‖_{H} = o (1)$ and $‖ U ‖_{H} = 1$ , we obtain the first estimation in (2.64). By using (2.58) and the first estimation in (2.64), we get the last estimation.

Inserting (2.58) and (2.60) into (2.59) and (2.61), we get the following system
$\begin{aligned} λ^{2} φ + a_{2} φ_{x x} - i λ d_{2} (x) φ - i λ c_{2} (x) ψ = F_{1}, \end{aligned}$
(2.65)

$\begin{aligned} λ^{2} ψ + ψ_{x x} + i λ c_{2} (x) φ = F_{2}, \end{aligned}$
(2.66)
where
$\begin{aligned} {\begin{cases} F_{1} = - i λ^{- ℓ + 1} f_{5} - d_{2} (x) λ^{- ℓ}, f_{5} - λ^{- ℓ} f_{6} - λ^{- ℓ} c_{2} (x) f_{7}, \\ F_{2} = λ^{- ℓ} c_{2} (x) f_{5} - i λ^{- ℓ + 1} f_{7} - λ^{- ℓ} f_{8} . \end{cases} \end{aligned}$

Lemma 2.7.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimation
$\begin{aligned} \int_{β_{2} + δ}^{β_{4} - δ} | φ_{x} |^{2} d x = o (λ^{- min (ℓ, \frac{ℓ}{2} + 1)}) . \end{aligned}$
(2.67)

Proof.
First, we define the cut-off function $θ_{1} \in C^{1} (L_{0}, L)$ by
$\begin{aligned} 0 ⩽ θ_{1} ⩽ 1, θ_{1} = 1 on (β_{2} + δ, β_{4} - δ) and θ_{1} = 0 on (L_{0}, β_{2}) \cup (β_{4}, L) . \end{aligned}$
(2.68)
Multiplying (2.65) by $θ_{1} \bar{φ}$ , integrating over $(L_{0}, L)$ and taking the real part, we get
$\begin{aligned} \int_{L_{0}}^{L} θ_{1} | λ φ |^{2} d x - a_{2} \int_{L_{0}}^{L} θ_{1} | φ_{x} |^{2} d x - ℜ {a_{2} \int_{L_{0}}^{L} θ_{1}^{^{'}} φ_{x} \bar{φ} d x} \\ - ℜ {i λ c_{2} \int_{β_{2}}^{β_{3}} θ_{1} ψ \bar{φ} d x} = ℜ {\int_{L_{0}}^{L} F_{1} θ_{1} \bar{φ} d x} . \end{aligned}$
(2.69)
Using the fact that $‖ F ‖ \to 0$ in $H$ and $λ φ$ is uniformly bounded in $L^{2} (L_{0}, L)$ , we obtain
$\begin{aligned} ℜ {\int_{L_{0}}^{L} F_{1} θ_{1} \bar{φ} d x} & = ℜ {\int_{L_{0}}^{L} θ_{1} (- i λ^{- ℓ + 1} f_{5} - d_{2} (x) λ^{- ℓ} f_{5} - λ^{- ℓ} f_{6} - λ^{- ℓ} c_{2} (x) f_{7}) \bar{φ} d x} \\ = o (λ^{- ℓ}) . \end{aligned}$
(2.70)
On the other hand, using Lemma 2.6, the fact that $λ ψ$ and $φ_{x}$ are uniformly bounded in $L^{2} (L_{0}, L)$ and the definition of $θ_{1}$ , we get
$\begin{aligned} ℜ {i λ c_{2} \int_{β_{2}}^{β_{3}} θ_{1} ψ \bar{φ} d x} = o (λ^{- (\frac{ℓ}{2} + 1)}), \end{aligned}$
(2.71)
and
$\begin{aligned} ℜ {a_{2} \int_{L_{0}}^{L} θ_{1}^{^{'}} φ_{x} \bar{φ} d x} = o (λ^{- (\frac{ℓ}{2} + 1)}) . \end{aligned}$
(2.72)
Furthermore, using Lemma 2.6 and the definition of the function $θ_{1}$ in (2.68), we get
$\begin{aligned} \int_{L_{0}}^{L} θ_{1} | λ φ |^{2} d x = o (λ^{- ℓ}) . \end{aligned}$
(2.73)
Inserting (2.70)–(2.73) in (2.69), we get (2.67).
Lemma 2.8.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimation
$\begin{aligned} \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x ⩽ \frac{| a_{2} - 1 | | λ |}{| c_{2} |} o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1) . \end{aligned}$
(2.74)

Proof.
First, we fix a cut-off function $θ_{2} \in C^{1} (L_{0}, L)$ such that $0 ⩽ θ_{2} (x) ⩽ 1$ , for all $x \in [L_{0}, L]$ and
$\begin{aligned} θ_{2} (x) = {\begin{cases} 1 & if x \in [β_{2} + 2 δ, β_{3} - δ], \\ 0 & if x \in [L_{0}, β_{2} + δ] \cup [β_{3}, L] . \end{cases} \end{aligned}$
From (2.61), $i λ^{- 1} θ_{2} \bar{ψ_{x x}}$ is uniformly bounded in $L^{2} (L_{0}, L)$ . Multiplying (2.59) by $i λ^{- 1} θ_{2} \bar{ψ_{x x}}$ , using integration by parts over $(L_{0}, L)$ and using the fact that $‖ f_{6} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} \int_{L_{0}}^{L} θ_{2}^{^{'}} η \bar{ψ_{x}} d x + \int_{L_{0}}^{L} θ_{2} η_{x} \bar{ψ_{x}} d x - \frac{i}{λ} a_{2} \int_{L_{0}}^{L} θ_{2} φ_{x x} \bar{ψ_{x x}} d x \\ - \frac{i}{λ} c_{2} \int_{β_{2} + δ}^{β_{3}} (θ_{2}^{^{'}} ξ + θ_{2} ξ_{x}) \bar{ψ_{x}} d x - \frac{i}{λ} d_{2} \int_{β_{2} + δ}^{β_{3}} (θ_{2}^{^{'}} η + θ_{2} η_{x}) \bar{ψ_{x}} d x = o (λ^{- ℓ}) . \end{aligned}$
(2.75)
From (2.58) and (2.60), we obtain
$\begin{aligned} η_{x} = i λ φ_{x} - λ^{- ℓ} (f_{5})_{x} and - \frac{i}{λ} ξ_{x} = ψ_{x} + i λ^{- (ℓ + 1)} (f_{7})_{x} . \end{aligned}$
Inserting the above equations in (2.75) and taking the real part, we get
$\begin{aligned} ℜ {i λ \int_{L_{0}}^{L} θ_{2} φ_{x} \bar{ψ_{x}} d x} - ℜ {\frac{i}{λ} a_{2} \int_{L_{0}}^{L} θ_{2} φ_{x x} \bar{ψ_{x x}} d x} + c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} | ψ_{x} |^{2} d x \\ = - ℜ {\int_{L_{0}}^{L} θ_{2}^{^{'}} η \bar{ψ_{x}} d x} + ℜ {\frac{1}{λ^{ℓ}} \int_{L_{0}}^{L} θ_{2} (f_{5})_{x} \bar{ψ_{x}} d x} + ℜ {\frac{i}{λ} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} ξ \bar{ψ_{x}} d x} \\ - ℜ {\frac{i}{λ^{(ℓ + 1)}} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} (f_{7})_{x} \bar{ψ_{x}} d x} + ℜ {\frac{i}{λ} d_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} η \bar{ψ_{x}} d x} \\ - ℜ {d_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} φ_{x} \bar{ψ_{x}} d x} - ℜ {\frac{i}{λ^{(ℓ + 1)}} d_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} (f_{5})_{x} \bar{ψ_{x}} d x} + o (λ^{- ℓ}) . \end{aligned}$
(2.76)
Using the fact that $ψ_{x}$ is uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ F ‖_{H} = o (1)$ , we get
$\begin{aligned} ℜ {\frac{1}{λ^{ℓ}} \int_{L_{0}}^{L} θ_{2} (f_{5})_{x} \bar{ψ_{x}} d x} = o (λ^{- ℓ}), \\ - ℜ {\frac{i}{λ^{(ℓ + 1)}} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} (f_{7})_{x} \bar{ψ_{x}} d x} = o (λ^{- (ℓ + 1)}) and \\ - ℜ {\frac{i}{λ^{(ℓ + 1)}} d_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} (f_{5})_{x} \bar{ψ_{x}} d x} = o (λ^{- (ℓ + 1)}) . \end{aligned}$
(2.77)
Next, using the fact that $ψ_{x}$ , η and ξ are uniformly bounded in $L^{2} (L_{0}, L)$ , we have
$\begin{aligned} ℜ {\frac{i}{λ} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} ξ \bar{ψ_{x}} d x} = O (λ^{- 1}) = o (1) and \\ ℜ {\frac{i}{λ} d_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} η \bar{ψ_{x}} d x} = O (λ^{- 1}) = o (1) . \end{aligned}$
(2.78)
Furthermore, using the estimations (2.64), (2.67) and the fact that $ψ_{x}$ is uniformly bounded in $L^{2} (L_{0}, L)$ , we get
$\begin{aligned} - ℜ {\int_{L_{0}}^{L} θ_{2}^{^{'}} η \bar{ψ_{x}} d x} = o (λ^{- \frac{ℓ}{2}}) and - ℜ {d_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} φ_{x} \bar{ψ_{x}} d x} = o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) . \end{aligned}$
(2.79)
Inserting (2.77)–(2.79) in (2.76) and using the definition of $(θ_{2})$ , we get
$\begin{aligned} c_{2} \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x + ℜ {i λ \int_{β_{2} + δ}^{β_{3}} θ_{2} φ_{x} \bar{ψ_{x}} d x} - ℜ {\frac{i}{λ} a_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} φ_{x x} \bar{ψ_{x x}} d x} = o (1) . \end{aligned}$
(2.80)
From (2.59), $i λ^{- 1} a_{2} θ_{2} \bar{φ_{x x}}$ is uniformly bounded in $L^{2} (L_{0}, L)$ . Multiplying (2.61) by $i λ^{- 1} a_{2} θ_{2} \bar{φ_{x x}}$ , using integration by parts over $(L_{0}, L)$ and the fact that $‖ f_{8} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} a_{2} \int_{L_{0}}^{L} θ_{2}^{^{'}} ξ \bar{φ_{x}} d x + a_{2} \int_{L_{0}}^{L} θ_{2} ξ_{x} \bar{φ_{x}} d x - \frac{i}{λ} a_{2} \int_{L_{0}}^{L} θ_{2} ψ_{x x} \bar{φ_{x x}} d x + \frac{i}{λ} a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} η \bar{φ_{x}} d x \\ + \frac{i}{λ} a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} η_{x} \bar{φ_{x}} d x = o (λ^{- ℓ}) . \end{aligned}$
(2.81)
From (2.58) and (2.60), we obtain
$\begin{aligned} η_{x} = i λ φ_{x} - λ^{- ℓ} (f_{5})_{x} and ξ_{x} = i λ ψ_{x} - λ^{- ℓ} (f_{7})_{x} . \end{aligned}$
Inserting the above equations in (2.81) and taking the real part, we get
$\begin{aligned} ℜ {i λ a_{2} \int_{L_{0}}^{L} θ_{2} ψ_{x} \bar{φ_{x}} d x} - ℜ {\frac{i}{λ} a_{2} \int_{L_{0}}^{L} θ_{2} ψ_{x x} \bar{φ_{x x}} d x} \\ = - ℜ {a_{2} \int_{L_{0}}^{L} θ_{2}^{^{'}} ξ \bar{φ_{x}} d x} + ℜ {\frac{a_{2}}{λ^{ℓ}} \int_{L_{0}}^{L} θ_{2} (f_{7})_{x} \bar{φ_{x}} d x} - ℜ {\frac{i}{λ} a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} η \bar{φ_{x}} d x} \\ + ℜ {\frac{i}{λ^{(ℓ + 1)}} a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} (f_{5})_{x} \bar{φ_{x}} d x} + a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} | φ_{x} |^{2} d x + o (λ^{- ℓ}) . \end{aligned}$
(2.82)
First, using the fact that $φ_{x}$ is uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ F ‖_{H} = o (1)$ , we get
$\begin{aligned} ℜ {\frac{a_{2}}{λ^{ℓ}} \int_{L_{0}}^{L} θ_{2} (f_{7})_{x} \bar{φ_{x}} d x} = o (λ^{- ℓ}) and \\ ℜ {\frac{i}{λ^{(ℓ + 1)}} a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} (f_{5})_{x} \bar{φ_{x}} d x} = o (λ^{- (ℓ + 1)}) . \end{aligned}$
(2.83)
Next, by the fact that $φ_{x}$ and η are uniformly bounded in $L^{2} (L_{0}, L)$ , we have
$\begin{aligned} - ℜ {\frac{i}{λ} a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2}^{^{'}} η \bar{φ_{x}} d x} = O (λ^{- 1}) = o (1) . \end{aligned}$
(2.84)
On the other hand, using the estimation (2.67), the fact that ξ is uniformly bounded in $L^{2} (L_{0}, L)$ , we get
$\begin{aligned} - ℜ {a_{2} \int_{L_{0}}^{L} θ_{2}^{^{'}} ξ \bar{φ_{x}} d x} = o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) and a_{2} c_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} | φ_{x} |^{2} d x = o (λ^{- min (ℓ, \frac{ℓ}{2} + 1)}) . \end{aligned}$
(2.85)
Inserting (2.83)–(2.85) in (2.82) and using the definition of $(θ_{2})$ , we get
$\begin{aligned} ℜ {i λ a_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} ψ_{x} \bar{φ_{x}} d x} - ℜ {\frac{i}{λ} a_{2} \int_{β_{2} + δ}^{β_{3}} θ_{2} ψ_{x x} \bar{φ_{x x}} d x} = o (1) . \end{aligned}$
(2.86)
Now, adding (2.80) and (2.86), we obtain
$\begin{aligned} \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x ⩽ \frac{| a_{2} - 1 | | λ |}{| c_{2} |} \int_{β_{2} + δ}^{β_{3}} | φ_{x} | | ψ_{x} | d x + o (1) . \end{aligned}$
(2.87)
Finally, using Lemma 2.7 and the fact that $ψ_{x}$ is uniformly bounded in $L^{2} (L_{0}, L)$ , we get our estimation (2.74).
Lemma 2.9.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimation
$\begin{aligned} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | ξ |^{2} d x ⩽ \frac{3 | a_{2} - 1 | | λ |}{| c_{2} |} o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1) . \end{aligned}$
(2.88)

Proof.
First, we fix a cut-off function $θ_{3} \in C^{1} (L_{0}, L)$ such that $0 ⩽ θ_{3} (x) ⩽ 1$ , for all $x \in [L_{0}, L]$ and
$\begin{aligned} θ_{3} (x) = {\begin{cases} 1 & if x \in [β_{2} + 3 δ, β_{3} - 2 δ], \\ 0 & if x \in [L_{0}, β_{2} + 2 δ] \cup [β_{3} - δ, L] . \end{cases} \end{aligned}$
Multiplying (2.61) by $- i λ^{- 1} θ_{3} \bar{ξ}$ , using integration by parts over $(L_{0}, L)$ , the fact that ξ is uniformly bounded in $L^{2} (0, L)$ and $‖ f_{8} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} \int_{L_{0}}^{L} θ_{3} | ξ |^{2} d x - \frac{i}{λ} \int_{L_{0}}^{L} θ_{3}^{^{'}} ψ_{x} \bar{ξ} d x - \frac{i}{λ} \int_{L_{0}}^{L} θ_{3} ψ_{x} \bar{ξ_{x}} d x + \frac{i}{λ} c_{2} \int_{β_{2} + 2 δ}^{β_{3} - δ} θ_{3} η \bar{ξ} d x = o (λ^{- (ℓ + 1)}) . \end{aligned}$
(2.89)
From (2.60), we obtain
$\begin{aligned} - \frac{i}{λ} \bar{ξ_{x}} = - \bar{ψ_{x}} + i λ^{- (ℓ + 1)} \bar{(f_{7})_{x}} . \end{aligned}$
Inserting the above equation in (2.89), we get
$\begin{aligned} \int_{L_{0}}^{L} θ_{3} | ξ |^{2} d x & = \int_{L_{0}}^{L} θ_{3} | ψ_{x} |^{2} d x - i λ^{- (ℓ + 1)} \int_{L_{0}}^{L} θ_{3} \bar{(f_{7})_{x}} ψ_{x} d x \\ + \frac{i}{λ} \int_{L_{0}}^{L} θ_{3}^{^{'}} ψ_{x} \bar{ξ} d x - \frac{i}{λ} c_{2} \int_{β_{2} + 2 δ}^{β_{3} - δ} θ_{3} η \bar{ξ} d x + o (λ^{- (ℓ + 1)}) . \end{aligned}$
(2.90)
Using the fact that $ψ_{x}$ , η and ξ are uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ (f_{7})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} | i λ^{- (ℓ + 1)} \int_{L_{0}}^{L} θ_{3} \bar{(f_{7})_{x}} ψ_{x} d x | = o (λ^{- (ℓ + 1)}), | \frac{i}{λ} \int_{L_{0}}^{L} θ_{3}^{^{'}} ψ_{x} \bar{ξ} d x | = o (1) and \\ | \frac{i}{λ} c_{2} \int_{β_{2} + 2 δ}^{β_{3} - δ} θ_{3} η \bar{ξ} d x | = o (1) . \end{aligned}$
(2.91)
Using (2.74) and the definition of $θ_{3}$ , we get
$\begin{aligned} \int_{L_{0}}^{L} θ_{3} | ψ_{x} |^{2} d x & ⩽ 3 \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x \\ ⩽ \frac{3 | a_{2} - 1 | | λ |}{| c_{2} |} o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1) . \end{aligned}$
(2.92)
Inserting (2.91) and (2.92) in (2.90) and using the definition of $θ_{3}$ , we get the desired estimation (2.88).

Now, we fix a function $g \in C^{1} ([β_{1}, β_{2} + 3 δ])$ such that
$\begin{aligned} g (β_{1}) = - g (β_{2} + 3 δ) = 1, max_{x \in [β_{1}, β_{2} + 3 δ]} | g (x) | = M_{g} and max_{x \in [β_{1}, β_{2} + 3 δ]} | g^{'} (x) | = M_{g^{'}}, \end{aligned}$
(2.93)
where $M_{g}$ and $M_{g^{'}}$ are strictly positive constant numbers.
Remark 2.10.
It is easy to see the existence of $g (x)$ . For example, we can take
$\begin{aligned} g (x) = \frac{1}{(β_{2} + 3 δ - β_{1})^{2}} (- 2 x^{2} + 4 β_{1} x + (β_{2} + 3 δ)^{2} - β_{1}^{2} - 2 (β_{2} + 3 δ) β_{1}), \end{aligned}$
(2.94)
to get
$\begin{aligned} g (β_{1}) = - g (β_{2} + 3 δ) = 1, g \in C^{1} ([β_{1}, β_{2} + 3 δ]), \end{aligned}$
and
$\begin{aligned} max_{x \in [β_{1}, β_{2} + 3 δ]} | g (x) | = 1 and max_{x \in [β_{1}, β_{2} + 3 δ]} | g^{'} (x) | = \frac{4}{β_{2} + 3 δ - β_{1}} . \end{aligned}$

Lemma 2.11.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimations
$\begin{aligned} | η (β_{2} + 3 δ) |^{2} + | η (β_{1}) |^{2} = O (λ), \end{aligned}$
(2.95)
and
$\begin{aligned} | ξ (β_{2} + 3 δ) |^{2} + | ξ (β_{1}) |^{2} + | ψ_{x} (β_{2} + 3 δ) |^{2} + {| ψ_{x} (β_{1}) |}^{2} = O (1) . \end{aligned}$
(2.96)

Proof.
First, deriving (2.58) with respect to x, we have
$\begin{aligned} i λ φ_{x} - η_{x} = λ^{- ℓ} (f_{5})_{x} . \end{aligned}$
(2.97)
Multiplying the above equation by $2 g \bar{η}$ , integrating over $(β_{1}, β_{2} + 3 δ)$ , then taking the real part, we get
$\begin{aligned} ℜ {2 i λ \int_{β_{1}}^{β_{2} + 3 δ} g φ_{x} \bar{η} d x} - \int_{β_{1}}^{β_{2} + 3 δ} g {(| η |^{2})}_{x} d x \\ = ℜ {2 λ^{- ℓ} \int_{β_{1}}^{β_{2} + 3 δ} g (f_{5})_{x} \bar{η} d x} . \end{aligned}$
(2.98)
Using integration by parts in (2.98), we get
$\begin{aligned} {[- g | η |^{2}]}_{β_{1}}^{β_{2} + 3 δ} & = - \int_{β_{1}}^{β_{2} + 3 δ} g^{'} | η |^{2} d x - ℜ {2 i λ \int_{β_{1}}^{β_{2} + 3 δ} g φ_{x} \bar{η} d x} \\ + ℜ {2 λ^{- ℓ} \int_{β_{1}}^{β_{2} + 3 δ} g (f_{5})_{x} \bar{η} d x} . \end{aligned}$
(2.99)
Using the definition of g and Cauchy–Schwarz inequality in (2.99), we obtain
$\begin{aligned} | η (β_{2} + 3 δ) |^{2} + | η (β_{1}) |^{2} & ⩽ M_{g^{'}} \int_{β_{1}}^{β_{2} + 3 δ} | η |^{2} d x \\ + 2 | λ | M_{g} {(\int_{β_{1}}^{β_{2} + 3 δ} | φ_{x} |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{1}}^{β_{2} + 3 δ} | η |^{2} d x)}^{\frac{1}{2}} \\ + 2 | λ |^{- ℓ} M_{g} {(\int_{β_{1}}^{β_{2} + 3 δ} | (f_{5})_{x} |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{1}}^{β_{2} + 3 δ} | η |^{2} d x)}^{\frac{1}{2}} . \end{aligned}$
(2.100)
Thus, from (2.100) and the fact that $φ_{x}$ , η are uniformly bounded in $L^{2} (L_{0}, L)$ in particular in $L^{2} (β_{1}, β_{2} + 3 δ)$ and $‖ (f_{5})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we have the first estimation (2.95).

Next, from (2.60) and (2.61), we have
$\begin{aligned} i λ ψ_{x} - ξ_{x} = λ^{- ℓ} (f_{7})_{x} and i λ ξ - ψ_{x x} - c_{2} (x) η = λ^{- ℓ} f_{8} . \end{aligned}$
(2.101)
Multiplying the above equations by $2 g \bar{ξ}$ and $2 g \bar{ψ_{x}}$ respectively, integrating over $(β_{1}, β_{2} + 3 δ)$ , taking the real part, then using the fact that $ψ_{x}$ , ξ are uniformly bounded in $L^{2} (L_{0}, L)$ in particular in $L^{2} (β_{1}, β_{2} + 3 δ)$ , $‖ f_{8} ‖_{L^{2} (L_{0}, L)} = o (1)$ and $‖ (f_{7})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we have
$\begin{aligned} ℜ {2 i λ \int_{β_{1}}^{β_{2} + 3 δ} g ψ_{x} \bar{ξ} d x} - \int_{β_{1}}^{β_{2} + 3 δ} g (| ξ |^{2})_{x} d x = o (λ^{- ℓ}), \end{aligned}$
(2.102)
and
$\begin{aligned} ℜ {2 i λ \int_{β_{1}}^{β_{2} + 3 δ} g ξ \bar{ψ_{x}} d x} - \int_{β_{1}}^{β_{2} + 3 δ} g (| ψ_{x} |^{2})_{x} d x - ℜ {2 \int_{β_{1}}^{β_{2} + 3 δ} c_{2} (x) g η \bar{ψ_{x}} d x} \\ = o (λ^{- ℓ}) . \end{aligned}$
(2.103)
Adding (2.102) and (2.103), then using integration by parts, we obtain
$\begin{aligned} {[- g (| ξ |^{2} + | ψ_{x} |^{2})]}_{β_{1}}^{β_{2} + 3 δ} & = - \int_{β_{1}}^{β_{2} + 3 δ} g^{'} (| ξ |^{2} + | ψ_{x} |^{2}) d x \\ + ℜ {2 \int_{β_{1}}^{β_{2} + 3 δ} c_{2} (x) g η \bar{ψ_{x}} d x} + o (λ^{- ℓ}) . \end{aligned}$
(2.104)
Using the definition of g and Cauchy–Schwarz inequality in the above equation, we obtain
$\begin{aligned} | ξ (β_{2} + 3 δ) |^{2} + | ξ (β_{1}) |^{2} + | ψ_{x} (β_{2} + 3 δ) |^{2} + | ψ_{x} (β_{1}) |^{2} \\ ⩽ M_{g^{'}} \int_{β_{1}}^{β_{2} + 3 δ} (| ξ |^{2} + | ψ_{x} |^{2}) d x \\ + 2 | c_{2} | M_{g} {(\int_{β_{1}}^{β_{2} + 3 δ} | η |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{1}}^{β_{2} + 3 δ} | ψ_{x} |^{2} d x)}^{\frac{1}{2}} + o (λ^{- ℓ}) . \end{aligned}$
(2.105)
Finally, from (2.105) and the fact that $ψ_{x}$ , ξ and η are uniformly bounded in $L^{2} (L_{0}, L)$ in particular in $L^{2} (β_{1}, β_{2} + 3 δ)$ , we have the second estimation (2.96).
Lemma 2.12.
Let $h \in C^{1} (L_{0}, L)$ be a function with $h (L_{0}) = h (L) = 0$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimation
$\begin{aligned} \int_{L_{0}}^{L} h^{'} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x + ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h ξ \bar{φ_{x}} d x} \\ + ℜ {2 \int_{L_{0}}^{L} d_{2} (x) h η \bar{φ_{x}} d x} - ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h η \bar{ψ_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.106)

Proof.
Multiplying (2.59) by $2 h \bar{φ_{x}}$ , integrating over $(L_{0}, L)$ , taking the real part, then using the fact that $φ_{x}$ is uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ f_{6} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} ℜ {2 i λ \int_{L_{0}}^{L} h η \bar{φ_{x}} d x} - a_{2} \int_{L_{0}}^{L} h (| φ_{x} |^{2})_{x} d x + ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h ξ \bar{φ_{x}} d x} \\ + ℜ {2 \int_{L_{0}}^{L} d_{2} (x) h η \bar{φ_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.107)
From (2.58), we deduce that
$\begin{aligned} i λ \bar{φ_{x}} = - \bar{η_{x}} - λ^{- ℓ} \bar{(f_{5})_{x}} . \end{aligned}$
Inserting the above equation in (2.107), then using the fact that η is uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ (f_{5})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} - \int_{L_{0}}^{L} h (| η |^{2} + a_{2} | φ_{x} |^{2})_{x} d x + ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h ξ \bar{φ_{x}} d x} \\ + ℜ {2 \int_{L_{0}}^{L} d_{2} (x) h η \bar{φ_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.108)
Using integration by parts in (2.108) and the fact that $h (L_{0}) = h (L) = 0$ , we obtain
$\begin{aligned} \int_{L_{0}}^{L} h^{'} (| η |^{2} + a_{2} | φ_{x} |^{2}) d x + ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h ξ \bar{φ_{x}} d x} \\ + ℜ {2 \int_{L_{0}}^{L} d_{2} (x) h η \bar{φ_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.109)
Next, multiplying (2.61) by $2 h \bar{ψ_{x}}$ , integrating over $(L_{0}, L)$ , taking the real part, then using the fact that $ψ_{x}$ is uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ f_{8} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we obtain
$\begin{aligned} ℜ {2 i λ \int_{L_{0}}^{L} h ξ \bar{ψ_{x}} d x} - \int_{L_{0}}^{L} h (| ψ_{x} |^{2})_{x} d x - ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h η \bar{ψ_{x}} d x} = o (λ^{- ℓ}), \end{aligned}$
(2.110)
from (2.60), we deduce that
$\begin{aligned} i λ \bar{ψ_{x}} = - \bar{ξ_{x}} - λ^{- ℓ} \bar{(f_{7})_{x}} . \end{aligned}$
Inserting the above equation in (2.110), then using the fact that ξ is uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ (f_{7})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} - \int_{L_{0}}^{L} h (| ξ |^{2} + | ψ_{x} |^{2})_{x} d x - ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h η \bar{ψ_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.111)
Using integration by parts in (2.111) and the fact that $h (L_{0}) = h (L) = 0$ , we obtain
$\begin{aligned} \int_{L_{0}}^{L} h^{'} (| ξ |^{2} + | ψ_{x} |^{2}) d x - ℜ {2 \int_{L_{0}}^{L} c_{2} (x) h η \bar{ψ_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.112)
Finally, adding (2.109) and (2.112), we obtain the desired estimation (2.106).

Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ , we fix cut-off functions $θ_{4}, θ_{5} \in C^{1} (L_{0}, L)$ such that $0 ⩽ θ_{4} (x) ⩽ 1$ , $0 ⩽ θ_{5} (x) ⩽ 1$ for all $x \in [L_{0}, L]$ and
$\begin{aligned} θ_{4} (x) = {\begin{cases} 1 & if x \in [L_{0}, β_{2} + 3 δ], \\ 0 & if x \in [β_{3} - 2 δ, L], \end{cases} \end{aligned}$
and
$\begin{aligned} θ_{5} (x) = {\begin{cases} 0 & if x \in [L_{0}, β_{2} + 3 δ], \\ 1 & if x \in [β_{3} - 2 δ, L], \end{cases} \end{aligned}$
and set $max_{[L_{0}, L]} | θ_{4}^{^{'}} (x) | = M_{θ_{4}^{^{'}}}$ and $max_{[L_{0}, L]} | θ_{5}^{^{'}} (x) | = M_{θ_{5}^{^{'}}}$ . Lemma 2.13.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following asymptotic behavior estimations
$\begin{aligned} \int_{L_{0}}^{β_{2} + 3 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x ⩽ K_{1} | a_{2} - 1 | | λ | o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1), \end{aligned}$
(2.113)
and
$\begin{aligned} \int_{β_{3} - 2 δ}^{L} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x ⩽ K_{2} | a_{2} - 1 | | λ | o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1), \end{aligned}$
(2.114)
where $K_{1} = \frac{4}{| c_{2} |} (1 + (β_{3} - 2 δ - L_{0}) M_{θ_{4}^{^{'}}})$ and $K_{2} = \frac{4}{| c_{2} |} (1 + (L - β_{2} - 3 δ) M_{θ_{5}^{^{'}}})$ .
Proof.
First, using the result of Lemma 2.12 with $h = (x - L_{0}) θ_{4}$ , we obtain
$\begin{aligned} \int_{L_{0}}^{β_{2} + 3 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ = - \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (θ_{4} + (x - L_{0}) θ_{4}^{^{'}}) (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ - ℜ {2 c_{2} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (x - L_{0}) θ_{4} ξ \bar{φ_{x}} d x} - ℜ {2 d_{2} \int_{β_{2}}^{β_{3} - 2 δ} (x - L_{0}) θ_{4} η \bar{φ_{x}} d x} \\ + ℜ {2 c_{2} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (x - L_{0}) θ_{4} η \bar{ψ_{x}} d x} - ℜ {2 c_{2} \int_{β_{1}}^{β_{2} + 3 δ} (x - L_{0}) ξ \bar{φ_{x}} d x} \\ + ℜ {2 c_{2} \int_{β_{1}}^{β_{2} + 3 δ} (x - L_{0}) η \bar{ψ_{x}} d x} + o (λ^{- ℓ}) . \end{aligned}$
(2.115)
Using Cauchy–Schwarz inequality, we have
$\begin{aligned} {- \int}_{β_{2} + 3 δ}^{β_{3} - 2 δ} (θ_{4} + (x - L_{0}) θ_{4}^{^{'}}) (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ - ℜ {2 c_{2} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (x - L_{0}) θ_{4} ξ \bar{φ_{x}} d x} - ℜ {2 d_{2} \int_{β_{2}}^{β_{3} - 2 δ} (x - L_{0}) θ_{4} η \bar{φ_{x}} d x} \\ + ℜ {2 c_{2} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (x - L_{0}) θ_{4} η \bar{ψ_{x}} d x} \\ ⩽ (1 + (β_{3} - 2 δ - L_{0}) M_{θ_{4}^{^{'}}}) \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ + 2 c_{2} (β_{3} - 2 δ - L_{0}) {(\int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | ξ |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | φ_{x} |^{2} d x)}^{\frac{1}{2}} \\ + 2 d_{2} (β_{3} - 2 δ - L_{0}) {(\int_{β_{2}}^{β_{3} - 2 δ} | η |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{2}}^{β_{3} - 2 δ} | φ_{x} |^{2} d x)}^{\frac{1}{2}} \\ + 2 c_{2} (β_{3} - 2 δ - L_{0}) {(\int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | η |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | ψ_{x} |^{2} d x)}^{\frac{1}{2}} . \end{aligned}$
(2.116)
Inserting (2.116) in (2.115), using Lemmas 2.6–2.9 and the fact that $ψ_{x}$ , ξ are uniformly bounded in $L^{2} (L_{0}, L)$ , we obtain
$\begin{aligned} \int_{L_{0}}^{β_{2} + 3 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ ⩽ \frac{4}{| c_{2} |} (1 + (β_{3} - 2 δ - L_{0}) M_{θ_{4}^{^{'}}}) | a_{2} - 1 | | λ | o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + I + o (1), \end{aligned}$
(2.117)
where
$\begin{aligned} I := ℜ {2 c_{2} \int_{β_{1}}^{β_{2} + 3 δ} (x - L_{0}) η \bar{ψ_{x}} d x} - ℜ {2 c_{2} \int_{β_{1}}^{β_{2} + 3 δ} (x - L_{0}) ξ \bar{φ_{x}} d x} . \end{aligned}$
(2.118)
From (2.58) and (2.60), we have
$\begin{aligned} \bar{φ_{x}} = i λ^{- 1} \bar{η_{x}} + i λ^{- (ℓ + 1)} \bar{(f_{5})_{x}} and \bar{ψ_{x}} = i λ^{- 1} \bar{ξ_{x}} + i λ^{- (ℓ + 1)} \bar{(f_{7})_{x}} . \end{aligned}$
Inserting the above equations in (2.118), then using the fact that η and ξ are uniformly bounded in $L^{2} (L_{0}, L)$ and $‖ (f_{5})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , $‖ (f_{7})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we obtain
$\begin{aligned} I & = ℜ {2 c_{2} i λ^{- 1} \int_{β_{1}}^{β_{2} + 3 δ} (x - L_{0}) η \bar{ξ_{x}} d x} - ℜ {2 c_{2} i λ^{- 1} \int_{β_{1}}^{β_{2} + 3 δ} (x - L_{0}) ξ \bar{η_{x}} d x} \\ + o (λ^{- (ℓ + 1)}) . \end{aligned}$
(2.119)
Using integration by parts to the second term in the above equation, we obtain
$\begin{aligned} I = ℜ {2 c_{2} i λ^{- 1} \int_{β_{1}}^{β_{2} + 3 δ} ξ \bar{η} d x} - ℜ {2 c_{2} i λ^{- 1} {[(x - L_{0}) ξ \bar{η}]}_{β_{1}}^{β_{2} + 3 δ}} . \end{aligned}$
(2.120)
From Lemma 2.11, we deduce that
$\begin{aligned} | η (β_{2} + 3 δ) | = O (λ^{\frac{1}{2}}), | η (β_{1}) | = O (λ^{\frac{1}{2}}), \end{aligned}$
(2.121)

$\begin{aligned} | ξ (β_{2} + 3 δ) | = O (1), | ξ (β_{1}) | = O (1) . \end{aligned}$
(2.122)
Using Cauchy–Schwarz inequality, (2.121), (2.122) and the fact that η, ξ are uniformly bounded in $L^{2} (L_{0}, L)$ , we obtain
$\begin{aligned} ℜ {2 c_{2} i λ^{- 1} \int_{β_{1}}^{β_{2} + 3 δ} ξ \bar{η} d x} = O (λ^{- 1}) = o (1), \end{aligned}$
(2.123)
and
$\begin{aligned} - ℜ {2 c_{2} i λ^{- 1} {[(x - L_{0}) ξ \bar{η}]}_{β_{1}}^{β_{2} + 3 δ}} = O (λ^{- \frac{1}{2}}) = o (1) . \end{aligned}$
(2.124)
Inserting the above estimations in (2.120), we get
$\begin{aligned} I = o (1) . \end{aligned}$
Finally, from the above estimation and (2.117), we obtain the desired estimation (2.113). Next, using the result of Lemma 2.12 with $h = (x - L) θ_{5}$ , we obtain
$\begin{aligned} \int_{β_{3} - 2 δ}^{L} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ = - ℜ {2 c_{2} \int_{β_{2} + 3 δ}^{β_{3}} (x - L) θ_{5} ξ \bar{φ_{x}} d x} \\ - \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (θ_{5} + (x - L) θ_{5}^{^{'}}) (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ - ℜ {2 d_{2} \int_{β_{2} + 3 δ}^{β_{4}} (x - L) θ_{5} η \bar{φ_{x}} d x} + ℜ {2 c_{2} \int_{β_{2} + 3 δ}^{β_{3}} (x - L) θ_{5} η \bar{ψ_{x}} d x} + o (λ^{- ℓ}) . \end{aligned}$
Using Cauchy–Schwarz inequality in the above equation, we get
$\begin{aligned} \int_{β_{3} - 2 δ}^{L} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ ⩽ (1 + (L - β_{2} - 3 δ) M_{θ_{5}^{^{'}}}) \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ + 2 c_{2} (L - β_{2} - 3 δ) {(\int_{β_{2} + 3 δ}^{β_{3}} | ξ |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{2} + 3 δ}^{β_{3}} | φ_{x} |^{2} d x)}^{\frac{1}{2}} \\ + 2 d_{2} (L - β_{2} - 3 δ) {(\int_{β_{2} + 3 δ}^{β_{4}} | η |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{2} + 3 δ}^{β_{4}} | φ_{x} |^{2} d x)}^{\frac{1}{2}} \\ + 2 c_{2} (L - β_{2} - 3 δ) {(\int_{β_{2} + 3 δ}^{β_{3}} | η |^{2} d x)}^{\frac{1}{2}} {(\int_{β_{2} + 3 δ}^{β_{3}} | ψ_{x} |^{2} d x)}^{\frac{1}{2}} + o (λ^{- ℓ}) . \end{aligned}$
(2.125)
Thus, from the above inequality, Lemmas 2.6–2.9 and the fact that $φ_{x}$ , $ψ_{x}$ , ξ are uniformly bounded in $L^{2} (L_{0}, L)$ , we get (2.114).
Lemma 2.14.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following estimate
$\begin{aligned} | v (L_{0}) |^{2} + a_{1} | u_{x} (L_{0}) |^{2} + | z (L_{0}) |^{2} + | y_{x} (L_{0}) |^{2} \\ ⩽ K_{3} | a_{2} - 1 | | λ | o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1), \end{aligned}$
(2.126)
where $K_{3} = \frac{K_{1}}{(β_{1} - L_{0})}$ .
Proof.
The proof is split into two steps.

Step 1. Letting $h_{1} \in C^{1} ([L_{0}, β_{1}])$ , the following estimate is targeted to prove
$\begin{aligned} - \int_{L_{0}}^{β_{1}} h_{1}^{^{'}} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x \\ + h_{1} (β_{1}) (| η (β_{1}) |^{2} + a_{2} | φ_{x} (β_{1}) |^{2} + | ξ (β_{1}) |^{2} + | ψ_{x} (β_{1}) |^{2}) \\ - h_{1} (L_{0}) (| η (L_{0}) |^{2} + a_{2} | φ_{x} (L_{0}) |^{2} + | ξ (L_{0}) |^{2} + | ψ_{x} (L_{0}) |^{2}) = o (λ^{- ℓ}) . \end{aligned}$
(2.127)
First, multiplying (2.59) and (2.61) by $2 h_{1} \bar{φ_{x}}$ and $2 h_{1} \bar{ψ_{x}}$ respectively, integrating over $(L_{0}, β_{1})$ , taking the real part, then using the fact that $φ_{x}$ and $ψ_{x}$ are uniformly bounded in $L^{2} (L_{0}, L)$ in particular in $L^{2} (L_{0}, β_{1})$ and $‖ f_{6} ‖_{L^{2} (L_{0}, L)} = o (1)$ , $‖ f_{8} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we obtain
$\begin{aligned} ℜ {2 i λ \int_{L_{0}}^{β_{1}} h_{1} η \bar{φ_{x}} d x} - a_{2} \int_{L_{0}}^{β_{1}} h_{1} (| φ_{x} |^{2})_{x} d x = o (λ^{- ℓ}), \end{aligned}$
(2.128)
and
$\begin{aligned} ℜ {2 i λ \int_{L_{0}}^{β_{1}} h_{1} ξ \bar{ψ_{x}} d x} - \int_{L_{0}}^{β_{1}} h_{1} (| ψ_{x} |^{2})_{x} d x = o (λ^{- ℓ}), \end{aligned}$
(2.129)
from (2.58) and (2.60), we deduce that
$\begin{aligned} i λ \bar{φ_{x}} = - \bar{η_{x}} - λ^{- ℓ} \bar{(f_{5})_{x}} and i λ \bar{ψ_{x}} = - \bar{ξ_{x}} - λ^{- ℓ} \bar{(f_{7})_{x}} . \end{aligned}$
Inserting the above equations in (2.128) and (2.129), then using the fact that η and ξ are uniformly bounded in $L^{2} (L_{0}, L)$ in particular in $L^{2} (L_{0}, β_{1})$ and $‖ (f_{5})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , $‖ (f_{7})_{x} ‖_{L^{2} (L_{0}, L)} = o (1)$ , we get
$\begin{aligned} - \int_{L_{0}}^{β_{1}} h_{1} (| η |^{2} + a_{2} | φ_{x} |^{2})_{x} d x = o (λ^{- ℓ}), \end{aligned}$
(2.130)
and
$\begin{aligned} - \int_{L_{0}}^{β_{1}} h_{1} (| ξ |^{2} + | ψ_{x} |^{2})_{x} d x = o (λ^{- ℓ}) . \end{aligned}$
(2.131)
Adding (2.130) and (2.131), then using integration by parts, we get (2.127).

Step2. Taking $h_{1} (x) = (x - β_{1})$ in (2.127) and using (2.113), we obtain
$\begin{aligned} | η (L_{0}) |^{2} + a_{2} | φ_{x} (L_{0}) |^{2} + | ξ (L_{0}) |^{2} + | ψ_{x} (L_{0}) |^{2} \\ ⩽ \frac{K_{1}}{(β_{1} - L_{0})} | a_{2} - 1 | | λ | o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1) . \end{aligned}$
(2.132)
Using (2.58), (2.60) and the transmission conditions (1.3), we get (2.126).
Lemma 2.15.
Let $max (β_{2}, \frac{β_{3} - β_{2}}{5}) < δ < β_{4}$ . The solution $(u, v, y, z, φ, η, ψ, ξ) \in D (A)$ of (2.54)–(2.61) satisfies the following estimate
$\begin{aligned} \int_{0}^{L_{0}} (| v |^{2} + a_{1} | u_{x} |^{2} + | z |^{2} + | y_{x} |^{2}) d x \\ ⩽ L_{0} K_{3} | a_{2} - 1 | | λ | o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2})}) + o (1) . \end{aligned}$
(2.133)

Proof.
First, using the multipliers $2 x \bar{u_{x}}$ and $2 x \bar{y_{x}}$ for (2.55) and (2.57) respectively, integrating over $(0, L_{0})$ , taking the real part, then using the fact that $u_{x}$ and $y_{x}$ are uniformly bounded in $L^{2} (0, L_{0})$ and $‖ f_{2} ‖_{L^{2} (0, L_{0})} = o (1)$ , $‖ f_{4} ‖_{L^{2} (0, L_{0})} = o (1)$ , we obtain
$\begin{aligned} 2 ℜ {i λ \int_{0}^{L_{0}} x v \bar{u_{x}} d x} - a_{1} \int_{0}^{L_{0}} x (| u_{x} |^{2})_{x} d x + 2 ℜ {\int_{0}^{L_{0}} x c_{1} (x) y \bar{u_{x}} d x} = o (λ^{- ℓ}), \end{aligned}$
(2.134)
and
$\begin{aligned} 2 ℜ {i λ \int_{0}^{L_{0}} x z \bar{y_{x}} d x} - \int_{0}^{L_{0}} x (| y_{x} |^{2})_{x} d x + 2 ℜ {\int_{0}^{L_{0}} x c_{1} (x) u \bar{y_{x}} d x} = o (λ^{- ℓ}) . \end{aligned}$
(2.135)
Using Cauchy–Schwarz inequality, the fact that $λ u$ , $λ y$ , $u_{x}$ , $y_{x}$ are uniformly bounded in $L^{2} (0, L_{0})$ , we get
$\begin{aligned} 2 ℜ {\int_{0}^{L_{0}} x c_{1} (x) y \bar{u_{x}} d x} = O (λ^{- 1}) = o (1) and \\ 2 ℜ {\int_{0}^{L_{0}} x c_{1} (x) u \bar{y_{x}} d x} = O (λ^{- 1}) = o (1) . \end{aligned}$
(2.136)
Inserting (2.136) in (2.134) and (2.135), we obtain
$\begin{aligned} 2 ℜ {i λ \int_{0}^{L_{0}} x v \bar{u_{x}} d x} - a_{1} \int_{0}^{L_{0}} x (| u_{x} |^{2})_{x} d x = o (1), \end{aligned}$
(2.137)
and
$\begin{aligned} 2 ℜ {i λ \int_{0}^{L_{0}} x z \bar{y_{x}} d x} - \int_{0}^{L_{0}} x (| y_{x} |^{2})_{x} d x = o (1), \end{aligned}$
(2.138)
from (2.54) and (2.56), we deduce that
$\begin{aligned} i λ \bar{u_{x}} = - \bar{v_{x}} - λ^{- ℓ} \bar{(f_{1})_{x}} and i λ \bar{y_{x}} = - \bar{z_{x}} - λ^{- ℓ} \bar{(f_{3})_{x}} . \end{aligned}$
Inserting the above equations in (2.137) and (2.138), then using the fact that v and z are uniformly bounded in $L^{2} (0, L_{0})$ and $‖ (f_{1})_{x} ‖_{L^{2} (0, L_{0})} = o (1)$ , $‖ (f_{3})_{x} ‖_{L^{2} (0, L_{0})} = o (1)$ , we get
$\begin{aligned} - \int_{0}^{L_{0}} x (| v |^{2} + a_{1} | u_{x} |^{2})_{x} d x = o (1), \end{aligned}$
(2.139)
and
$\begin{aligned} - \int_{0}^{L_{0}} x (| z |^{2} + | y_{x} |^{2})_{x} d x = o (1) . \end{aligned}$
(2.140)
Adding (2.139) and (2.140), then using integration by parts, we obtain
$\begin{aligned} \int_{0}^{L_{0}} (| v |^{2} + a_{1} | u_{x} |^{2} + | z |^{2} + | y_{x} |^{2}) d x \\ = L_{0} (| v (L_{0}) |^{2} + a_{1} | u_{x} (L_{0}) |^{2} + | z (L_{0}) |^{2} + | y_{x} (L_{0}) |^{2}) + o (1) . \end{aligned}$
(2.141)
Using the above result and (2.126), we get (2.133).
Proof of Theorem 2.4. Proof of Theorem 2.4.

The proof of Theorem 2.4 is divided into three steps.

Step 1. By taking $a_{2} = 1$ and $ℓ = 0$ in Lemmas 2.6–2.9, we obtain

\begin{aligned} \int_{β_{2}}^{β_{4}} | η |^{2} d x = o (1), \int_{β_{2} + δ}^{β_{4} - δ} | φ_{x} |^{2} d x = o (1), \\ \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x = o (1) and \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | ξ |^{2} d x = o (1) . \end{aligned}

(2.142)

Consequently, we have

\begin{aligned} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (| η |^{2} + | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x = o (1) . \end{aligned}

(2.143)

Step 2. Using the fact that

a_{2} = 1

in Lemmas 2.13 and 2.15, we obtain

\begin{aligned} \int_{L_{0}}^{β_{2} + 3 δ} (| η |^{2} + | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x = o (1), \end{aligned}

(2.144)

\begin{aligned} \int_{β_{3} - 2 δ}^{L} (| η |^{2} + | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x = o (1), \end{aligned}

(2.145)

and

\begin{aligned} \int_{0}^{L_{0}} (| v |^{2} + a_{1} | u_{x} |^{2} + | z |^{2} + | y_{x} |^{2}) d x = o (1) . \end{aligned}

(2.146)

Step 3. According to Step 1 and Step 2, we obtain

‖ U ‖_{H} = o (1)

, which contradicts (2.52). This implies that

\begin{aligned} sup_{λ \in R} ‖ (i λ I - A)^{- 1} ‖_{L (H)} = O (1) . \end{aligned}

So by Theorem A.3, we deduce that system (1.1)–(1.4) is exponentially stable.

Proof of Theorem 2.5. Proof of Theorem 2.5.

The proof of Theorem 2.5 is divided into three steps.

Step 1. Taking $a_{2} \neq 1$ , then from Lemmas 2.8 and 2.9, we get

\begin{aligned} \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x = o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2}) + 1}) and \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | ξ |^{2} d x = o (λ^{- min (\frac{ℓ}{2}, \frac{ℓ}{4} + \frac{1}{2}) + 1}) . \end{aligned}

(2.147)

Taking

ℓ = 2

in the above estimations, we obtain

\begin{aligned} \int_{β_{2} + 2 δ}^{β_{3} - δ} | ψ_{x} |^{2} d x = o (1) and \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} | ξ |^{2} d x = o (1) . \end{aligned}

(2.148)

Taking

ℓ = 2

in Lemma 2.6 and 2.7, we get

\begin{aligned} \int_{β_{2}}^{β_{4}} | η |^{2} d x = o (λ^{- 2}) and \int_{β_{2} + δ}^{β_{4} - δ} | φ_{x} |^{2} d x = o (λ^{- 2}) . \end{aligned}

(2.149)

In particular, we have

\begin{aligned} \int_{β_{2} + 3 δ}^{β_{3} - 2 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x = o (1) . \end{aligned}

(2.150)

Step 2. Using the fact that

a_{2} \neq 1

and

ℓ = 2

, then from Lemmas 2.13 and 2.15, we obtain

\begin{aligned} \int_{L_{0}}^{β_{2} + 3 δ} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x = o (1), \end{aligned}

(2.151)

\begin{aligned} \int_{β_{3} - 2 δ}^{L} (| η |^{2} + a_{2} | φ_{x} |^{2} + | ξ |^{2} + | ψ_{x} |^{2}) d x = o (1), \end{aligned}

(2.152)

and

\begin{aligned} \int_{0}^{L_{0}} (| v |^{2} + a_{1} | u_{x} |^{2} + | z |^{2} + | y_{x} |^{2}) d x = o (1) . \end{aligned}

(2.153)

Step 3. According to Step 1 and Step 2, we obtain

‖ U ‖_{H} = o (1)

, which contradicts (2.52). This implies that

\begin{aligned} sup_{λ \in R} ‖ (i λ I - A)^{- 1} ‖_{L (H)} = O (λ^{2}) . \end{aligned}

So by Theorem A.4, we deduce that system (1.1)–(1.4) is polynomially stable.

3. .Conclusion and future works

We have studied the stabilization of a locally transmission problems of two wave systems. We proved the strong stability of the system by using Arendt and Batty criteria. We established the exponential stability of the solution if and only if the waves of the second coupled equations have the same speed propagation (i.e., $a_{2} = 1$ ). In the case $a_{2} \neq 1$ , we proved that the energy of our problem decays polynomially with the rate $t^{- 1}$ . Finally, we present some open problems:

(1)

Prove that the energy decay rate $t^{- 1}$ is optimal.

(2)

Study system (1.1)–(1.4) in the multidimensional case.

(3)

Can we get stability results if $d_{1} (x) \neq 0$ in $(0, L_{0})$ .

Footnotes

To make this paper more self-contained, we provide a brief appendix that recalls some key concepts and stability results used in this work.

To show the strong stability of a C₀-semigroup of contraction $(e^{t A})_{t ⩾ 0}$ we rely on the following result due to Arendt–Batty [15].

Concerning the characterization of exponential stability of a C₀-semigroup of contraction $(e^{t A})_{t ⩾ 0}$ we rely on the following result due to Huang [21] and Prüss [31].

Also, concerning the characterization of polynomial stability of a C₀-semigroup of contraction $(e^{t A})_{t ⩾ 0}$ we rely on the following result due to Borichev and Tomilov [16].

References

Akil

, Stability of piezoelectric beam with magnetic effect under (Coleman or Pipkin) – Gurtin thermal law, Zeitschrift für angewandte Mathematik und Physik 73(6) (2022), 236. doi:10.1007/s00033-022-01867-w.

Akil

Badawi

Nicaise

, Stability results of locally coupled wave equations with local Kelvin–Voigt damping: Cases when the supports of damping and coupling coefficients are disjoint, Computational and Applied Mathematics 41 (2022), 240.

Akil

Badawi

Nicaise

Régnier

, Stabilization of coupled wave equations with viscous damping on cylindrical and non-regular domains: Cases without the geometric control condition, Mediterr. J. Math. 19 (2022), 271. doi:10.1007/s00009-022-02164-6.

Akil

Badawi

Nicaise

Wehbe

, Stability results of coupled wave models with locally memory in a past history framework via nonsmooth coefficients on the interface, Math. Methods Appl. Sci. 44(8) (2021), 6950–6981. doi:10.1002/mma.7235.

Akil

Badawi

Wehbe

, Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay, Commun Pure Appl Anal 20(9) (2021), 2991–3028. doi:10.3934/cpaa.2021092.

Akil

Ghader

Wehbe

, The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization, SeMA 78 (2021), 287–333. doi:10.1007/s40324-020-00233-y.

Akil

Hajjej

, Exponential stability and exact controllability of a system of coupled wave equations by second order terms (via Laplacian) with only one non-smooth local damping, Mathematical Methods in the Applied Sciences 47 (2024), 1883–1902.

Akil

Issa

Wehbe

, A N-dimensional elastic/viscoelastic transmission problem with Kelvin–Voigt damping and non smooth coefficient at the interface, SeMA 80 (2023), 425–462. doi:10.1007/s40324-022-00297-y.

Akil

Wehbe

, Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions, Math. Control Relat. Fields 8 (2018), 1–20. doi:10.3934/mcrf.2018001.

10.

Akil

Wehbe

, Indirect stability of a multidimensional coupled wave equations with one locally boundary fractional damping, 2021.

11.

Alabau-Boussouira

, Indirect boundary stabilization of weakly coupled hyperbolic systems, Mathematics SIAM J. Control. Optim. 41 (2002), 511–541.

12.

Alabau-Boussouira

, A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control. Optim. 42 (2003), 871–906.

13.

Alabau-Boussouira

Léautaud

, Indirect controllability of locally coupled wave-type systems and applications, Mathematics (2013).

14.

Ali Wehbe

N.N.

Nasser

, Stability of n-d transmission problem in viscoelasticity with localized Kelvin–Voigt damping under different types of geometric conditions, Math. Control Relat. Fields 11(4) (2021), 885–904. doi:10.3934/mcrf.2020050.

15.

Arendt

Batty

C.J.K.

, Tauberian theorems and stability of one-parameter semigroups, Trans. Am. Math. Soc. 306(2) (1988), 837–852. doi:10.1090/S0002-9947-1988-0933321-3.

16.

Borichev

Tomilov

, Optimal polynomial decay of functions and operator semigroups, Math. Annal. 347(2) (2009), 455–478. doi:10.1007/s00208-009-0439-0.

17.

Boutaayamou

Fragnelli

Mugnai

, Boundary controllability for a degenerate wave equation in non divergence form with drift, submitted, 33 pages, arXiv:2109.1253.

18.

Gerbi

Kassem

Mortada

et al., Exact controllability and stabilization of locally coupled wave equations: Theoretical results, Zeitschrift für Analysis und ihre Anwendungen 40 (2021), 67–96. doi:10.4171/zaa/1673.

19.

Hassine

Souayeh

, Stability for coupled waves with locally disturbed Kelvin–Voigt damping, 2019, arXiv:1909.09838.

20.

Hayek

Nicaise

Salloum

Wehbe

, A transmission problem of a system of weakly coupled wave equations with Kelvin–Voigt dampings and non-smooth coefficient at the interface, SeMA J. 77(3) (2020), 305–338. doi:10.1007/s40324-020-00218-x.

21.

Huang

F.L.

, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ. 1(1) (1985), 43–56.

22.

Lebeau

, Équation des ondes amorties, in: Algebraic and Geometric Methods in Mathematical Physics (Kaciveli, 1993), Mathematical Physics Studies, Vol. 19, Kluwer Academic Publishers, Dordrecht, 1996.

23.

Lebeau

, Équations des ondes amorties, Séminaire Équations aux dérivées partielles (Polytechnique), 1993–1994, talk:15.

24.

Liu

, Locally distributed control and damping for the conservative systems, SIAM J. Control Optim. 35(5) (1997), 1574–1590. doi:10.1137/S0363012995284928.

25.

Liu

Rao

, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys. 56(4) (2005), 630–644. doi:10.1007/s00033-004-3073-4.

26.

Liu

Zheng

, Semigroups Associated with Dissipative Systems, Research Notes in Mathematics, Vol. 398, Champman and Hall/CRC.

27.

Najdi

, Study of the exponential and polynomial stability of some systems of coupled equations with indirect bounded or unbounded control, PhD thesis, 2016.

28.

Nasser

Noun

Wehbe

, Stabilization of the wave equations with localized Kelvin-Voigt type damping under optimal geometric conditions, C. R. Math. 357(3) (2019), 272–277. doi:10.1016/j.crma.2019.01.005.

29.

Nicaise

Pignotti

, Stability of the wave equation with localized Kelvin–Voigt damping and boundary delay feedback, Discret. Contin. Dyn. Syst. Ser 9(3) (2016), 791–813. doi:10.3934/dcdss.2016029.

30.

Pazy

, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 1983.

31.

Prüss

, On the spectrum of C0-semigroups, Trans. Am. Math. Soc. 284(2) (1984), 847–857. doi:10.2307/1999112.

32.

Tebou

, A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping, Comptes Rendus Math. 350 (2012), 603–608. doi:10.1016/j.crma.2012.06.005.

33.

Wehbe

Issa

Akil

, Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients, Acta Appl. Math. 171(1) (2021), 1–46. doi:10.1007/s10440-021-00384-8.

34.

Wehbe

Youssef

, Indirect locally internal observability of weakly coupled wave equations, Differential Equations and Applications – DEA 3(3) (2011), 449–462. doi:10.7153/dea-03-28.

Stabilization of a locally transmission problems of two strongly-weakly coupled wave systems

Abstract

Keywords

1. .Introduction

(1) If U 0 ∈ D ( A ) , then problem (2.4) admits a unique strong solution U satisfying: U ∈ C 1 ( R + , H ) ∩ C 0 ( R + , D ( A ) ) . (2) If U 0 ∈ H , then problem (2.4) admits a unique weak solution U satisfying: U ∈ C 0 ( R + , H ) . 2.1. Strong stability

Footnotes

References