Optimal stability for laminated beams with Kelvin–Voigt damping and Fourier’s law

Abstract

This article deals with the asymptotic behavior of a mathematical model for laminated beams with Kelvin–Voigt dissipation acting on the equations of transverse displacement and dimensionless slip. We prove that the evolution semigroup is exponentially stable if the damping is effective in the two equations of the model. Otherwise, we prove that the semigroup is polynomially stable and find the optimal decay rate when damping is effective only in the slip equation. Our stability approach is based on the Gearhart–Prüss–Huang Theorem, which characterizes exponential stability, while the polynomial decay rate is obtained using the Borichev and Tomilov Theorem.

Keywords

Laminated beam Kelvin–Voigt damping exponential stability polynomial stability

1. Introduction

In 1824 Fourier [11] studied the problem of heat conduction in solid bodies, which is also associated with linear transport equations. Since then, these equations have been used in describing such problems in heat conduction theory. This theory guarantees that the heat flux in a homogeneous body is proportional to the temperature gradient (Fourier’s law), i.e., $\begin{array}{r} (1.1) & q = - κ \nabla θ, \end{array}$ where $q$ , κ, $\nabla θ$ denote the thermal flow, thermal conductivity, and temperature gradient, respectively. On the other hand, keeping in mind the equilibrium equation given by $\begin{array}{r} (1.2) & ρ c \frac{\partial θ}{\partial t} = - div q, \end{array}$ (where ρ is the mass density, c is the specific heat capacity) and combining with equation (1.1), we get the heat equation given by $\begin{array}{r} (1.3) & \frac{\partial θ}{\partial t} - β Δ θ = 0, \end{array}$ where β is the thermal diffusivity coefficient. Due to the importance of studies associated with the theory of heat conduction, we found several works in the literature [15–17,19,31,32] that describe the thermoelastic behavior of hyperbolic systems coupled to Fourier’s law. For example in [15], Kim considered the problem $\begin{array}{c} (1.4) & u_{t t} + Δ^{2} u + α Δ θ = 0, \\ (1.5) & θ_{t} - β Δ θ + γ θ - α Δ u_{t} = 0, \end{array}$ where $α \neq 0$ , $β > 0$ , $γ ⩾ 0$ . The dependent variables u and θ denote vertical plate deflection and temperature, respectively. He used energy method combined with the multiplier techniques to prove the exponential decay of the system energy. On the other hand, the same problem was investigated by Liu and Zheng in [23] by adopting more general boundary conditions to obtain an exponential decay result.

In the context of Timoshenko beams, we highlight the pioneering work of Rivera and Racke [33] where the dynamics of heat conduction with wave propagation is described by $\begin{array}{c} (1.6) & ρ_{1} φ_{t t} - κ {(φ_{x} + ψ)}_{x} = 0, \\ (1.7) & ρ_{2} ψ_{t t} - b ψ_{x x} + κ (φ_{x} + ψ) + γ θ_{x} = 0, \\ (1.8) & ρ_{3} θ_{t} - κ θ_{x x} + γ ψ_{x t} = 0, \end{array}$ where φ, ψ and θ denote the transverse displacement, angle of rotation, temperature of the beam and moreover, $ρ_{1}$ , $ρ_{2}$ , b, κ, γ are positive constants. Using energy method, the authors proved exponential decay depending on the known speed equality relationship, i.e., $\begin{array}{r} (1.9) & \frac{κ}{ρ_{1}} = \frac{b}{ρ_{2}} . \end{array}$ Other results on the dynamics of heat conduction in Timoshenko systems with Fourier’s law can be found in [1,2]. However, we are interested in the dynamics of laminated beams (composite laminated in the form of beams) subject to heat conduction.

The term laminated composite is used to designate the combination of two or more materials distinct in their physical properties. It is a heterogeneous environment whose main objective is to obtain a material that, matching its components’ characteristics properly, presents a structural performance better than the components under specific conditions of use. In addition, in many of these materials, as part of the assembly process, the materials are subjected to the curing process with a hot press [25]. This motivates us to consider the problem of heat conduction in laminated beams (see Fig. 1).

Fig. 1.

Adhesive application with hot press (left) and plate in the hot press (right).

The laminated beam model was introduced by Hansen and Spies [13] and is described by the following system $\begin{matrix} (1.10) & \begin{array}{c} ρ w_{t t} + G {(ψ - w_{x})}_{x} = 0 in (0, l) \times R^{+}, \\ I_{ρ} (3 S_{t t} - ψ_{t t}) - D (3 S_{x x} - ψ_{x x}) - G (ψ - w_{x}) = 0 in (0, l) \times R^{+}, \\ 3 I_{ρ} S_{t t} - 3 D S_{x x} + 3 G (ψ - w_{x}) + 4 γ_{0} S + 4 β_{0} S_{t} = 0 in (0, l) \times R^{+}, \end{array} \end{matrix}$ where l represents the length of the beams.

This model consists of a two-layer beam, which is bonded together by an adhesive layer so that bonding them together gives rise to a restoring force that is assumed to be proportional to the amount of slip. The adhesive layer is assumed to have negligible mass and thickness, and thus does not contribute to the system’s kinetic energy. According to Hansen and Spies [13], the first and second equations in (1.10) are derived from the Timoshenko theory [34], while the third equation describes the dynamics of the interaction between the two layers and includes internal frictional damping, also known as structural damping.

In (1.10), the functions $w (x, t)$ and $ψ (x, t)$ represent the transverse displacement and the angle of rotation. In addition, $S (x, t)$ is proportional to the amount of slip along with the interface at time t and longitudinal spatial variable x, respectively. The positive parameters ρ, G, $I_{ρ}$ , D, $γ_{0}$ , and $β_{0}$ , represent the density, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness, and the adhesive structural damping parameter, respectively.

There is extensive literature about the system’s stability (1.10) under various damping mechanisms (frictional, Kelvin–Voigt, delay, history, etc.). We can also find some studies about the stability of the system (1.10) under thermal effects (see, for instance [3,4,8–10,20–22,26,29,30], etc.)

Recently, Liu and Zhao [22] studied the laminated beam system with and without structural dissipation, considering the thermal effect described by Fourier’s law. Specifically, the authors addressed the following system $\begin{matrix} (1.11) & \begin{array}{c} ρ w_{t t} + G {(ψ - w_{x})}_{x} = 0 in (0, l) \times R^{+}, \\ I_{ρ} {(3 S - ψ)}_{t t} - D {(3 S - ψ)}_{x x} - G (ψ - w_{x}) + δ θ_{x} = 0 in (0, l) \times R^{+}, \\ 3 I_{ρ} S_{t t} - 3 D S_{x x} + 3 G (ψ - w_{x}) + 4 γ_{0} S + 4 β S_{t} = 0 in (0, l) \times R^{+}, \\ ρ_{3} θ_{t} - α θ_{x x} + 3 δ {(3 S - ψ)}_{t x} = 0 in (0, l) \times R^{+}, \end{array} \end{matrix}$ where θ represents the temperature difference. The authors studied the stability of system (1.11) with structural damping (β $> 0$ ) and without structural damping ( $β = 0$ ), proving in both cases that the system is exponentially stable when the wave propagation speeds are equal (i.e., when $\frac{ρ}{G} = \frac{I_{ρ}}{D}$ ), for which they used the perturbed energy method. When the wave propagation speeds differ, they demonstrate the lack of exponential stability when $β ⩾ 0$ . Additionally, they obtained a polynomial stability result in the case β $> 0$ .

Motivated by the cited works and mainly by [22], we wonder what would happen if we introduce a Kelvin–Voigt dissipation on the transverse displacement in the model (1.11) without the presence of the structural damping (i.e., with $β = 0$ ). That is, $\begin{matrix} (1.12) & \begin{array}{c} ρ w_{t t} + G {(ψ - w_{x})}_{x} - μ_{1} w_{x x t} = 0 in (0, l) \times R^{+}, \\ I_{ρ} {(3 S - ψ)}_{t t} - D {(3 S - ψ)}_{x x} - G (ψ - w_{x}) + δ θ_{x} = 0 in (0, l) \times R^{+}, \\ 3 I_{ρ} S_{t t} - 3 D S_{x x} + 3 G (ψ - w_{x}) + 4 γ_{0} S = 0 in (0, l) \times R^{+}, \\ ρ_{3} θ_{t} - α θ_{x x} + 3 δ {(3 S - ψ)}_{t x} = 0 in (0, l) \times R^{+} . \end{array} \end{matrix}$ To be more explicit, according to the results proved in [22], the system $\begin{matrix} (1.13) & \begin{array}{c} ρ w_{t t} + G {(ψ - w_{x})}_{x} = 0 in (0, l) \times R^{+}, \\ I_{ρ} {(3 S - ψ)}_{t t} - D {(3 S - ψ)}_{x x} - G (ψ - w_{x}) + δ θ_{x} = 0 in (0, l) \times R^{+}, \\ 3 I_{ρ} S_{t t} - 3 D S_{x x} + 3 G (ψ - w_{x}) + 4 γ_{0} S = 0 in (0, l) \times R^{+}, \\ ρ_{3} θ_{t} - α θ_{x x} + 3 δ {(3 S - ψ)}_{t x} = 0 in (0, l) \times R^{+} . \end{array} \end{matrix}$ is exponentially stable. Usually, Kelvin–Voigt damping is known as strong damping, so if the system (1.13) is exponentially stable, the system (1.12) should also be so. However, surprisingly, the presence of Kelvin–Voigt damping in the first equation of (1.12) destroys the exponential stability of the system, even when the wave propagation speeds coincide.

More generally, we study the asymptotic behavior of the laminated beam system with Fourier’s law given by $\begin{array}{c} (1.14) & ρ_{1} w_{t t} + G {(3 S - ξ - w_{x})}_{x} - μ_{1} w_{x x t} = 0 in (0, l) \times R^{+}, \\ (1.15) & I_{ρ} ξ_{t t} - D ξ_{x x} - G (3 S - ξ - w_{x}) + δ θ_{x} = 0 in (0, l) \times R^{+}, \\ (1.16) & 3 I_{ρ} S_{t t} - 3 D S_{x x} + 3 G (3 S - ξ - w_{x}) + 4 γ_{0} S - {\tilde{μ}}_{2} S_{x x t} = 0 in (0, l) \times R^{+}, \\ (1.17) & ρ_{3} θ_{t} - α θ_{x x} + 3 δ ξ_{x t} = 0 in (0, l) \times R^{+}, \end{array}$ with $μ_{1} > 0$ and ${\tilde{μ}}_{2} ⩾ 0$ . In the last system, $ξ : = 3 S - ψ$ with ψ denoting the rotation angle of the filament. If we make the variable changes $s : = - 3 S$ , $ρ_{2} : = I_{ρ}$ , $k : = G$ , $b : = D$ , $3 γ : = 4 γ_{0}$ , ${\tilde{μ}}_{2} : = 3 μ_{2}$ , the system (1.14)–(1.17) takes the form $\begin{array}{c} (1.18) & ρ_{1} w_{t t} - k {(w_{x} + ξ + s)}_{x} - μ_{1} w_{x x t} = 0 in (0, l) \times R^{+}, \\ (1.19) & ρ_{2} ξ_{t t} - b ξ_{x x} + k (w_{x} + ξ + s) + δ θ_{x} = 0 in (0, l) \times R^{+}, \\ (1.20) & ρ_{2} s_{t t} - b s_{x x} + 3 k (w_{x} + ξ + s) + γ s - μ_{2} s_{x x t} = 0 in (0, l) \times R^{+}, \\ (1.21) & ρ_{3} θ_{t} - α θ_{x x} + 3 δ ξ_{x t} = 0 in (0, l) \times R^{+} . \end{array}$ For the above system, we will consider the initial conditions $\begin{matrix} (1.22) & \begin{array}{c} (w (x, 0), ξ (x, 0), s (x, 0), θ (x, 0)) = (w_{0} (x), ξ_{0} (x), s_{0} (x), θ_{0} (x)) in (0, l), \\ (w_{t} (x, 0), ξ_{t} (x, 0), s_{t} (x, 0)) = (w_{1} (x), ξ_{1} (x), s_{1} (x)) in (0, l), \end{array} \end{matrix}$ and the mixed boundary conditions $\begin{matrix} (1.23) & \begin{array}{c} w_{x} (0, t) = ξ (0, t) = s (0, t) = θ_{x} (0, t) = 0 in R^{+}, \\ w_{x} (l, t) = ξ (l, t) = s (l, t) = θ_{x} (l, t) = 0 in R^{+}, \end{array} \end{matrix}$ where $w_{0}$ , $w_{1}$ , $ξ_{0}$ , $ξ_{1}$ , $s_{0}$ , $s_{1}$ and $θ_{0}$ are functions given in appropriate spaces and the parameters $ρ_{1}$ , $ρ_{2}$ , $ρ_{3}$ , $μ_{1}$ , k, b, γ, α and δ are positive constants and $μ_{2} ⩾ 0$ . We notice that the dampings $w_{x x t}$ and $s_{x x t}$ correspond to Kelvin–Voigt type damping. According to Lazan [18], the materials with Kelvin–Voigt damping are characterized by the following constitutive equation $\begin{array}{r} (1.24) & σ = a ϵ + b \frac{\partial ϵ}{\partial t}, \end{array}$ where σ, ϵ represent the stresses and strains, respectively and the constants a, b are positive numbers. This paper deals with the stability of the system (1.18)–(1.23). We will use the semigroup theory of linear operators and the frequency domain method approach to address the following problems:

The well-posedness of the system (1.18)–(1.23), see Section 2;

The exponential stability of system (1.18)–(1.23) when $μ_{1} > 0$ and $μ_{2} > 0$ , independent of the wave propagation speeds, see Section 3;

The lack of exponential stability of system (1.18)–(1.23) when $μ_{1} > 0$ and $μ_{2} = 0$ , see Section 4.

The polynomial stability and optimal polynomial decay rate of the system (1.18)–(1.23) when $μ_{1} > 0$ and $μ_{2} = 0$ , see Section 5.

2. Setting of the semigroup

To prove the well-posedness of system (1.18)–(1.21) using semigroup theory (see Liu and Zheng [24]), we define $u = w_{t}$ , $φ = ξ_{t}$ , $z = s_{t}$ and set $\begin{matrix} U = {(w, u, ξ, φ, s, z, θ)}^{T} and U_{0} = {(w_{0}, w_{1}, ξ_{0}, ξ_{1}, s_{0}, s_{1}, θ_{0})}^{T} . \end{matrix}$ Then, system (1.18)–(1.21) can be written as the following abstract Cauchy problem $\begin{matrix} (2.1) & \{\begin{array}{ll} U_{t} = A U, & t > 0, \\ U (0) = U_{0}, \end{array} \end{matrix}$ where the operator $A$ is defined by $\begin{matrix} (2.2) & A U = [\begin{array}{c} u \\ \frac{1}{ρ_{1}} [k {(w_{x} + ξ + s)}_{x} + μ_{1} u_{x x}] \\ φ \\ \frac{1}{ρ_{2}} [b ξ_{x x} - k (w_{x} + ξ + s) - δ θ_{x}] \\ z \\ \frac{1}{ρ_{2}} [b s_{x x} - 3 k (w_{x} + ξ + s) - γ s + μ_{2} z_{x x}] \\ \frac{1}{ρ_{3}} [α θ_{x x} - 3 δ φ_{x}] \end{array}], \end{matrix}$ para $U = {(w, u, ξ, φ, s, z, θ)}^{T}$ and $U^{*} = {(w^{*}, u^{*}, ξ^{*}, φ^{*}, s^{*}, z^{*}, θ^{*})}^{T}$ in $H$ .

To define the phase space $H$ , we consider the Lebesgue space $L^{2} (0, l)$ whose usual inner product and norm will be denoted, respectively, by $⟨ \cdot, \cdot ⟩$ and $‖ \cdot ‖$ . We will also use the following Hilbert spaces $\begin{matrix} L_{*}^{2} (0, l) = {w \in L^{2} (0, l); \int_{0}^{l} w (x) d x = 0}, H_{*}^{1} (0, l) = H^{1} (0, l) \cap L_{*}^{2} (0, l) \end{matrix}$ and $\begin{matrix} H_{*}^{2} (0, l) = {w \in H^{2} (0, l); w_{x} (0) = w_{x} (l) = 0} . \end{matrix}$ Then, we define the phase space as $\begin{matrix} H = H_{*}^{1} (0, l) \times L_{*}^{2} (0, l) \times {[H_{0}^{1} (0, l) \times L^{2} (0, l)]}^{2} \times L_{*}^{2} (0, l), \end{matrix}$ endowed with the inner product $\begin{aligned} {⟨ U, U^{*} ⟩}_{H} = & 3 ρ_{1} ⟨ u, u^{*} ⟩ + 3 ρ_{2} ⟨ φ, φ^{*} ⟩ + ρ_{2} ⟨ z, z^{*} ⟩ + 3 b ⟨ ξ_{x}, ξ_{x}^{*} ⟩ + b ⟨ s_{x}, s_{x}^{*} ⟩ \\ + γ ⟨ s, s^{*} ⟩ + 3 k ⟨ (w_{x} + ξ + s), (w_{x}^{*} + ξ^{*} + s^{*}) ⟩ + ρ_{3} ⟨ θ, θ^{*} ⟩ \end{aligned}$ and induced norm $\begin{aligned} ‖ U ‖_{H}^{2} = & 3 ρ_{1} ‖ u ‖^{2} + 3 ρ_{2} ‖ φ ‖^{2} + ρ_{2} ‖ z ‖^{2} + 3 b ‖ ξ_{x} ‖^{2} \\ + b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} + 3 k ‖ w_{x} + ξ + s ‖^{2} + ρ_{3} ‖ θ ‖^{2}, \end{aligned}$ for $U = {(w, u, ξ, φ, s, z, θ)}^{T}$ , $U^{*} = {(w^{*}, u^{*}, ξ^{*}, φ^{*}, s^{*}, z^{*}, θ^{*})}^{T} \in H$ .

The domain of operator $A : D (A) \subset H \to H$ is defined by $\begin{matrix} (2.3) & D (A) = \{{(w, u, ξ, φ, s, z, θ)}^{T} \in H |\begin{array}{c} u \in H_{*}^{1} (0, l), φ, z \in H_{0}^{1} (0, l), \\ (k w + μ_{1} u) \in H_{*}^{2} (0, l), θ \in H_{*}^{1} (0, l) \cap H_{*}^{2} (0, l) \\ (b s + μ_{2} z), ξ \in H^{2} (0, l) \end{array}\} . \end{matrix}$

Next, we will prove that the operator $A$ is dissipative and that system (1.18)–(1.22) is well-posed.

Proposition 2.1.
The operator $A$ defined in ( 2.2 )–( 2.3 ) is dissipative. Moreover, for $U = {(w, u, ξ, φ, s, z, θ)}^{T} \in D (A)$ we have $\begin{matrix} (2.4) & Re {⟨ A U, U ⟩}_{H} = - 3 μ_{1} ‖ u_{x} ‖^{2} - μ_{2} ‖ z_{x} ‖^{2} - α ‖ θ_{x} ‖^{2} . \end{matrix}$
Proof.
Using the definition of the inner product in $H$ and integration by parts, a straightforward calculation leads us to (2.4). □
Proposition 2.2.
If $ϱ (A)$ is the resolvent set of the operator $A$ , then $0 \in ϱ (A)$ .
Proof.
To begin with, let us prove that $A$ is a bijective operator. For this, let us show that for any $F = (f^{1}, f^{2}, f^{3}, f^{4}, f^{5}, f^{6}, f^{7}) \in H$ , there exists a unique $U = {(w, u, ξ, φ, s, z, θ)}^{T} \in D (A)$ such that $A U = F$ . This is equivalent to solving the system $\begin{array}{c} (2.5) & u = f^{1} in H_{}^{1} (0, l), \\ (2.6) & \frac{1}{ρ_{1}} [k {(w_{x} + ξ + s)}_{x} + μ_{1} u_{x x}] = f^{2} in L_{}^{2} (0, l), \\ (2.7) & φ = f^{3} in H_{0}^{1} (0, l), \\ (2.8) & \frac{1}{ρ_{2}} [b ξ_{x x} - k (w_{x} + ξ + s) - δ θ_{x}] = f^{4} in L^{2} (0, l), \\ (2.9) & z = f^{5} in H_{0}^{1} (0, l), \\ (2.10) & \frac{1}{ρ_{2}} [b s_{x x} - 3 k (w_{x} + ξ + s) - γ s + μ_{2} z_{x x}] = f^{6} in L^{2} (0, l), \\ (2.11) & \frac{1}{ρ_{3}} [α θ_{x x} - 3 δ φ_{x}] = f^{7} in L_{}^{2} (0, l) . \end{array}$ From (2.5), (2.7) and (2.9), we have $\begin{matrix} (2.12) & u \in H_{}^{1} (0, l), φ, z \in H_{0}^{1} (0, l) . \end{matrix}$ Therefore, the problem (2.5)–(2.11) reduces to solving the system $\begin{matrix} (2.13) & \{\begin{array}{l} - 3 k {(w_{x} + ξ + s)}_{x} = - 3 ρ_{1} f^{2} + 3 μ_{1} f_{x x}^{1}, \\ - 3 b ξ_{x x} + 3 k (w_{x} + ξ + s) + 3 δ θ_{x} = - 3 ρ_{2} f^{4}, \\ - b s_{x x} + 3 k (w_{x} + ξ + s) + γ s = - ρ_{2} f^{6} + μ_{2} f_{x x}^{5}, \\ - α θ_{x x} = - ρ_{3} f^{7} - 3 δ f_{x}^{3}, \end{array} \end{matrix}$ with mixed boundary conditions $\begin{matrix} (2.14) & \{\begin{array}{l} w_{x} (0) = ξ (0) = s (0) = θ_{x} (0) = 0, \\ w_{x} (l) = ξ (l) = s (l) = θ_{x} (l) = 0 . \end{array} \end{matrix}$ The system (2.13)–(2.14) admits the following weak formulation $\begin{matrix} (2.15) & \{\begin{array}{l} 3 k ⟨ (w_{x} + ξ + s), w_{x}^{} ⟩ = - 3 ρ_{1} ⟨ f^{2}, w^{} ⟩ - 3 μ_{1} ⟨ f_{x}^{1}, w_{x}^{} ⟩, \\ 3 b ⟨ ξ_{x}, ξ_{x}^{} ⟩ + 3 k ⟨ (w_{x} + ξ + s), ξ^{} ⟩ + 3 δ ⟨ θ_{x}, ξ^{} ⟩ = - 3 ρ_{2} ⟨ f^{4}, ξ^{} ⟩, \\ b ⟨ s_{x}, s_{x}^{} ⟩ + 3 k ⟨ (w_{x} + ξ + s), s^{} ⟩ + γ ⟨ s, s^{} ⟩ = - ρ_{2} ⟨ f^{6}, s^{} ⟩ - μ_{2} ⟨ f_{x}^{5}, s_{x}^{} ⟩, \\ α ϑ ⟨ θ_{x}, θ_{x}^{} ⟩ = - ρ_{3} ϑ ⟨ f^{7}, θ^{} ⟩ - 3 δ ϑ ⟨ f_{x}^{3}, θ^{} ⟩, \end{array} \end{matrix}$ for all $w^{}, θ^{} \in H_{}^{1} (0, l)$ , $ξ^{}, s^{} \in H_{0}^{1} (0, l)$ , where ϑ is a positive constant that will be defined later in a convenient form. These equations motivate us to define the forms $B_{ϑ} : H_{1} \times H_{1} \to C$ and $L_{ϑ} : H_{1} \to C$ by $\begin{aligned} B_{ϑ} ((w, ξ, s, θ), (w^{}, ξ^{}, s^{}, θ^{})) = & 3 k ⟨ (w_{x} + ξ + s), (w_{x}^{} + ξ^{} + s^{}) ⟩ + 3 b ⟨ ξ_{x}, ξ_{x}^{} ⟩ \\ + b ⟨ s_{x}, s_{x}^{} ⟩ + γ ⟨ s, s^{} ⟩ + 3 δ ⟨ θ_{x}, ξ^{} ⟩ + α ϑ ⟨ θ_{x}, θ_{x}^{} ⟩ \end{aligned}$ and $\begin{aligned} L_{ϑ} (w^{}, ξ^{}, s^{}, θ^{}) = & - 3 ρ_{1} ⟨ f^{2}, w^{} ⟩ - 3 μ_{1} ⟨ f_{x}^{1}, w_{x}^{} ⟩ - 3 ρ_{2} ⟨ f^{4}, ξ^{} ⟩ - ρ_{2} ⟨ f^{6}, s^{} ⟩ \\ - μ_{2} ⟨ f_{x}^{5}, s_{x}^{} ⟩ - ρ_{3} ϑ ⟨ f^{7}, θ^{} ⟩ - 3 δ ϑ ⟨ f_{x}^{3}, θ^{} ⟩, \end{aligned}$ where $H_{1}$ is the space defined by $\begin{matrix} H_{1} = H_{}^{1} (0, l) \times {[H_{0}^{1} (0, l)]}^{2} \times H_{}^{1} (0, l) . \end{matrix}$ Then, the system (2.15) can be written in the following variational form $\begin{matrix} (2.16) & B_{ϑ} ((w, ξ, s, θ), (w^{}, ξ^{}, s^{}, θ^{})) = L_{ϑ} (w^{}, ξ^{}, s^{}, θ^{}), \forall (w^{}, ξ^{}, s^{}, θ^{}) \in H_{1} . \end{matrix}$ Applying Hölder’s, Poincaré’s and Young’s inequalities, we obtain $\begin{matrix} 3 δ | Re ⟨ θ_{x}, ξ ⟩ | ⩽ 3 δ ‖ θ_{x} ‖ ‖ ξ ‖ ⩽ \frac{9 δ^{2} c_{p}}{4 b} ‖ θ_{x} ‖^{2} + b ‖ ξ_{x} ‖^{2}, \end{matrix}$ where $c_{p} > 0$ is the optimal constant of Poincaré’s inequality. To prove the continuity of seaquilinear form $B_{ϑ}$ and antilinear form $L_{ϑ}$ we proceed in the usual way. In order to obtain the coercivity of $B_{ϑ}$ , let us note that $\begin{aligned} Re B_{ϑ} ((w, ξ, s, θ), (w, ξ, s, θ)) = & 3 k ‖ w_{x} + ξ + s ‖^{2} + 3 b ‖ ξ_{x} ‖^{2} \\ + b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} + 3 δ Re ⟨ θ_{x}, ξ ⟩ + α ϑ ‖ θ_{x} ‖^{2} \\ ⩾ & 3 k ‖ w_{x} + ξ + s ‖^{2} + 2 b ‖ ξ_{x} ‖^{2} \\ + b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} + α (ϑ - \frac{9 δ^{2} c_{p}}{4 α b}) ‖ θ_{x} ‖^{2} . \end{aligned}$ Then, if we choose $ϑ > \frac{9 δ^{2} c_{p}}{4 α b}$ , we conclude that $B_{ϑ}$ is coercive.

The Lax-Milgram Theorem (ver Dautray and Lions [6]) guarantees the existence of a unique weak solution $(w, ξ, s, θ) \in H_{1}$ of the problem (2.13)–(2.14). From (2.6), (2.8), (2.10) and (2.11), we conclude that $\begin{matrix} ξ, (b s + μ_{2} z) \in H^{2} (0, l) \end{matrix}$ and $\begin{matrix} (2.17) & θ_{x x}, (k w_{x x} + μ_{1} u_{x x}) \in L_{}^{2} (0, l) . \end{matrix}$ From (2.17), we obtain $θ_{x} (0) = θ_{x} (l)$ and $(k w_{x} + μ_{1} u_{x}) (0) = (k w_{x} + μ_{1} u_{x}) (l)$ . On the other hand, replacing $(w^{}, ξ^{}, s^{}, θ^{}) = (w^{}, 0, 0, 0)$ in (2.16), we get $\begin{matrix} (2.18) & - ⟨ [k (w_{x} + ξ + s) + μ_{1} u_{x}], w_{x}^{} ⟩ = ρ_{1} ⟨ f^{2}, w^{} ⟩, \forall w^{} \in H_{}^{1} (0, l) . \end{matrix}$ And, taking $(w^{}, ξ^{}, s^{}, θ^{}) = (0, 0, 0, θ^{})$ in (2.16), we obtain $\begin{matrix} (2.19) & - ⟨ α θ_{x}, θ_{x}^{} ⟩ - 3 δ ⟨ φ_{x}, θ^{} ⟩ = ρ_{3} ⟨ f^{7}, θ^{} ⟩, \forall θ^{} \in H_{}^{1} (0, l) . \end{matrix}$ Applying integration by parts formula in (2.18), using (2.6) and since $s, ξ \in H_{0}^{1} (0, l)$ , it results $\begin{matrix} {[(k w_{x} + μ_{1} u_{x}) w^{}]}_{0}^{l} = 0, \forall w^{} \in H_{}^{1} (0, l) . \end{matrix}$ This implies $\begin{matrix} (k w_{x} + μ_{1} u_{x}) (0) = (k w_{x} + μ_{1} u_{x}) (l) = 0 \end{matrix}$ and then $\begin{matrix} (k w + μ_{1} u) \in H_{}^{2} (0, l) . \end{matrix}$ Proceeding analogously in (2.19) and applying (2.11), we get $\begin{matrix} θ \in H_{}^{2} (0, l) . \end{matrix}$ Therefore, there is a unique solution ${(w, u, ξ, φ, s, z, θ)}^{T} \in D (A)$ of system (2.5)–(2.11), and consequently, $A$ is bijective.

Let us prove the boundedness of $A^{- 1}$ . Given $F = {(f^{1}, f^{2}, f^{3}, f^{4}, f^{5}, f^{6}, f^{7})}^{T} \in H$ , let $U = {(w, u, ξ, φ, s, z, θ)}^{T} \in D (A)$ be the unique vectorial function that satisfies $U = A^{- 1} F$ . Taking $θ^{} = θ$ in the fourth equation of (2.15) and using Poincaré’s inequality, it follows that $\begin{matrix} (2.20) & ‖ θ_{x} ‖^{2} = \frac{1}{α} [- ρ_{3} ⟨ f^{7}, θ ⟩ - 3 δ ⟨ f_{x}^{3}, θ ⟩] ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} \end{matrix}$ and $\begin{matrix} (2.21) & ρ_{3} ‖ θ ‖^{2} ⩽ C ‖ F ‖_{H} ‖ U ‖_{H}, \end{matrix}$ where C denotes a generic positive constant, whose value may differ from one line to another.

If we take $w^{} = w$ , $ξ^{} = ξ$ , $s^{} = s$ and $θ^{} = 0$ in (2.16), integrate by parts and use Poincaré and Young’s inequalities with $ε > 0$ , we obtain. $\begin{aligned} 3 k ‖ w_{x} + ξ + s ‖^{2} + 3 b ‖ ξ_{x} ‖^{2} + b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} \\ = - 3 δ ⟨ θ_{x}, ξ ⟩ - 3 ρ_{1} ⟨ f^{2}, w ⟩ \\ - 3 μ_{1} ⟨ f_{x}^{1}, w_{x} ⟩ - 3 ρ_{2} ⟨ f^{4}, ξ ⟩ - ρ_{2} ⟨ f^{6}, s ⟩ - μ_{2} ⟨ f_{x}^{5}, s_{x}^{} ⟩ \\ ⩽ C ‖ θ ‖^{2} + \frac{ε}{2} ‖ ξ_{x} ‖^{2} + C ‖ f^{2} ‖ (‖ w_{x} + ξ + s ‖ + ‖ ξ ‖ + ‖ s ‖) \\ + C (‖ f_{x}^{1} + f^{3} + f^{5} ‖ + ‖ f^{3} ‖ + ‖ f^{5} ‖) (‖ w_{x} + ξ + s ‖ + ‖ ξ ‖ + ‖ s ‖) \\ + C ‖ f^{4} ‖ ‖ ξ_{x} ‖ + ρ_{2} ‖ f^{6} ‖ ‖ s ‖ + μ_{2} ‖ f_{x}^{5} ‖ ‖ s_{x} ‖ . \end{aligned}$ Taking $ε = 3 b$ and employing (2.21), we have $\begin{matrix} (2.22) & 3 k ‖ w_{x} + ξ + s ‖^{2} + 3 b ‖ ξ_{x} ‖^{2} + b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} . \end{matrix}$ Using (2.5), (2.7) and (2.9), we get $\begin{matrix} (2.23) & 3 ρ_{1} ‖ u ‖^{2} + 3 ρ_{2} ‖ φ ‖^{2} + ρ_{2} ‖ z ‖^{2} = 3 ρ_{1} {‖ f^{1} ‖}^{2} + 3 ρ_{2} {‖ f^{3} ‖}^{2} + ρ_{2} {‖ f^{5} ‖}^{2} ⩽ C ‖ F ‖_{H}^{2} . \end{matrix}$ Adding (2.21), (2.22) and (2.23), from Young’s inequality leads us to $\begin{aligned} ‖ U ‖_{H}^{2} = & 3 ρ_{1} ‖ u ‖^{2} + 3 ρ_{2} ‖ φ ‖^{2} + ρ_{2} ‖ z ‖^{2} + 3 b ‖ ξ_{x} ‖^{2} \\ + b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} + 3 k ‖ w_{x} + ξ + s ‖^{2} \\ ⩽ & C ‖ F ‖_{H} ‖ U ‖_{H} + C ‖ F ‖_{H}^{2} ⩽ \frac{1}{2} ‖ U ‖_{H}^{2} + C ‖ F ‖_{H}^{2} . \end{aligned}$ Then $\begin{matrix} {‖ A^{- 1} F ‖}_{H} = ‖ U ‖_{H} ⩽ C ‖ F ‖_{H} . \end{matrix}$ This means that $A^{- 1}$ is bounded. Therefore, $0 \in ϱ (A)$ . □

The well-posedness of (1.18)–(1.22) is ensured by the following theorem. Theorem 2.1.
For each $U_{0} \in H$ , there exists a unique weak solution U of problem ( 2.1 ) satisfying $\begin{matrix} (2.24) & U \in C ([0, + \infty); H) . \end{matrix}$ Moreover, if $U_{0} \in D (A)$ , then $\begin{matrix} (2.25) & U \in C ([0, + \infty); D (A)) \cap C^{1} ([0, + \infty); H) . \end{matrix}$ In this case, U is called the strong solution of ( 2.1 ).
Proof.
From Theorem 4.6, chapter 1, of Pazy [27], we have that $D (A)$ is dense in $H$ . From Propositions 2.1 and 2.2, $A$ is a dissipative operator and $0 \in ϱ (A)$ . Then, applying Theorem 1.2.4 of Liu and Zheng [24] we conclude that the operator $A$ is the infinitesimal generator of a $C_{0}$ -semigroup $e^{t A}$ of contractions in the space $H$ . Finally, $U (t) = e^{t A} U_{0}$ is the unique solution of problem (2.1). □

3. Exponential stability when $μ_{1} > 0$ and $μ_{2} > 0$

In this section, we will prove that the system (1.18)–(1.23) with $μ_{1} > 0$ and $μ_{2} > 0$ is exponentially stable. As we will see, our exponential stability result does not depend on the wave speeds. Our proof is based on Gearhart-Prüss-Huang’s theorem (see [12,14] and [28]). This theorem states the following

Theorem 3.1.
Let $S (t) = e^{t A}$ be the $C_{0}$ -semigroup of contractions on a Hilbert space $H$ . Then $S (t)$ is exponentially stable if and only if $\begin{matrix} (3.1) & i R \equiv {i λ; λ \in R} \subset ϱ (A) \end{matrix}$ and $\begin{matrix} (3.2) & \underset{| λ | \to \infty}{lim sup} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)} < \infty, \end{matrix}$ where $ϱ (A)$ is the resolvent set of the differential operator $A$ .

We will prove that the semigroup associated with our problem satisfies conditions (3.1) and (3.2) of Theorem 3.1.
Proposition 3.1.
Let $ϱ (A)$ be the resolvent set of the operator $A$ defined in ( 2.2 )–( 2.3 ) with $μ_{1} > 0$ and $μ_{2} > 0$ . Then $\begin{matrix} (3.3) & i R \subset ϱ (A) . \end{matrix}$
Proof.
We will prove this inclusion using a contradiction argument. Since $A$ is the infinitesimal generator of a $C_{0}$ -semigroup of contractions, then $A$ is a closed operator. We already know that $0 \in ϱ (A)$ . Hence if (3.3) is not valid, then there exists $ω \in R$ with $‖ A^{- 1} ‖ ⩽ | ω |$ , a sequence $(λ_{n}) \subset R$ such that $λ_{n} \to ω$ , $| λ_{n} | < | ω |$ and a sequence $(U_{n}) \subset D (A)$ , denoted by $U_{n} = (u_{n}, w_{n}, ξ_{n}, v_{n}, s_{n}, z_{n}, θ_{n})$ , with $‖ U_{n} ‖_{H} = 1$ such that $\begin{matrix} {‖ (i λ_{n} I - A) U_{n} ‖}_{H} \to 0 . \end{matrix}$

Denoting $\begin{matrix} (3.4) & (i λ_{n} I - A) U_{n} = F_{n}, n \in N, \end{matrix}$ with $F_{n} = {(f_{n}^{1}, f_{n}^{2}, f_{n}^{3}, f_{n}^{4}, f_{n}^{5}, f_{n}^{6}, f_{n}^{7})}^{T} \in H$ , then $\begin{matrix} (3.5) & F_{n} \to 0 in H, \end{matrix}$ and it follows that $\begin{matrix} {⟨ (i λ_{n} I - A) U_{n}, U_{n} ⟩}_{H} = {⟨ F_{n}, U_{n} ⟩}_{H} \to 0 . \end{matrix}$ Taking the real part, using the dissipativity identity (2.4) of the operator $A$ and (3.5), we have $\begin{matrix} 3 μ_{1} ‖ u_{n, x} ‖^{2} + μ_{2} ‖ z_{n, x} ‖^{2} + α ‖ θ_{n, x} ‖^{2} \to 0, \end{matrix}$ then $\begin{matrix} (3.6) & u_{n, x}, z_{n, x}, θ_{n, x} \to 0 in L^{2} (0, l), \end{matrix}$ and thanks to Poincaré’s inequality, we have $\begin{matrix} (3.7) & u_{n}, z_{n}, θ_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Decomposing (3.4) in terms of its components, we obtain the system: $\begin{array}{c} (3.8) & i λ_{n} w_{n} - u_{n} = f_{n}^{1} in H_{}^{1} (0, l), \\ (3.9) & i λ_{n} u_{n} - \frac{k}{ρ_{1}} {(w_{n, x} + ξ_{n} + s_{n})}_{x} - \frac{μ_{1}}{ρ_{1}} u_{n, x x} = f_{n}^{2} in L_{}^{2} (0, l), \\ (3.10) & i λ_{n} ξ_{n} - φ_{n} = f_{n}^{3} in H_{0}^{1} (0, l), \\ (3.11) & i λ_{n} φ_{n} - \frac{b}{ρ_{2}} ξ_{n, x x} + \frac{k}{ρ_{2}} (w_{n, x} + ξ_{n} + s_{n}) + \frac{δ}{ρ_{2}} θ_{n, x} = f_{n}^{4} in L^{2} (0, l), \\ (3.12) & i λ_{n} s_{n} - z_{n} = f_{n}^{5} in H_{0}^{1} (0, l), \\ (3.13) & i λ_{n} z_{n} - \frac{b}{ρ_{2}} s_{n, x x} + \frac{3 k}{ρ_{2}} (w_{n, x} + ξ_{n} + s_{n}) + \frac{γ}{ρ_{2}} s_{n} - \frac{μ_{2}}{ρ_{2}} z_{n, x x} = f_{n}^{6} in L^{2} (0, l), \\ (3.14) & i λ_{n} θ_{n} - \frac{α}{ρ_{3}} θ_{n, x x} + \frac{3 δ}{ρ_{3}} φ_{n, x} = f_{n}^{7} in L_{}^{2} (0, l) . \end{array}$ Since $(λ_{n})$ is a bounded sequence, replacing (3.5), (3.6) and (3.7) in equations (3.8) and (3.12) yield $\begin{matrix} (3.15) & w_{n}, w_{n, x}, s_{n}, s_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$

Multiplying (3.13) with $ξ_{n}$ in $L^{2} (0, l)$ we yield $\begin{matrix} i λ_{n} ⟨ z_{n}, ξ_{n} ⟩ + \frac{b}{ρ_{2}} ⟨ s_{n, x}, ξ_{n, x} ⟩ + \frac{3 k}{ρ_{2}} ⟨ w_{n, x} + ξ_{n} + s_{n}, ξ_{n} ⟩ + \frac{γ}{ρ_{2}} ⟨ s_{n}, ξ_{n} ⟩ + \frac{μ_{2}}{ρ_{2}} ⟨ z_{n}, ξ_{n} ⟩ = ⟨ f_{n}^{6}, ξ_{n} ⟩ . \end{matrix}$ Applying convergences (3.5), (3.7) and (3.15) we obtain $\begin{matrix} ⟨ w_{n, x} + ξ_{n} + s_{n}, ξ_{n} ⟩ \to 0 . \end{matrix}$ Since $\begin{matrix} ⟨ w_{n, x} + ξ_{n} + s_{n}, ξ_{n} ⟩ = ⟨ w_{n, x}, ξ_{n} ⟩ + ‖ ξ_{n} ‖^{2} + ⟨ s_{n}, ξ_{n} ⟩ \to 0, \end{matrix}$ from (3.15) we conclude that $\begin{matrix} (3.16) & ξ_{n} \to 0 in L^{2} (0, l), \end{matrix}$ and from (3.5) and (3.10) follows that $\begin{matrix} (3.17) & φ_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Convergences (3.15) and (3.16) imply $\begin{matrix} (3.18) & w_{n, x} + ξ_{n} + s_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Finally, multiplying (3.11) by $ξ_{n}$ , and using all the previous convergences, we get $\begin{matrix} (3.19) & ξ_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$

From convergences (3.7), (3.15), (3.17), (3.18) and (3.19), we get $‖ U_{n} ‖_{H} \to 0$ , which is a contradiction because $‖ U_{n} ‖_{H} = 1$ . Therefore, condition (3.3) is proved. □
Proposition 3.2.
The operator* $A$ with $μ_{1} > 0$ and $μ_{2} > 0$ satisfies the following resolvent estimate $\begin{matrix} \underset{| λ | \to \infty}{lim sup} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)} < \infty . \end{matrix}$
Proof.
We again use a contradiction argument. If the above condition does not hold, then there is a real sequence $(λ_{n})$ , with $| λ_{n} | \to \infty$ and a sequence $(U_{n}) \subset D (A)$ with $‖ U_{n} ‖_{H} = 1$ such that $\begin{matrix} {‖ (i λ_{n} I - A) U_{n} ‖}_{H} \to 0 . \end{matrix}$ Setting $U_{n} = (u_{n}, w_{n}, ξ_{n}, v_{n}, s_{n}, z_{n}, θ_{n})$ and $\begin{matrix} (3.20) & (i λ_{n} I - A) U_{n} = F_{n}, \end{matrix}$ where $\begin{matrix} (3.21) & F_{n} = {(f_{n}^{1}, f_{n}^{2}, f_{n}^{3}, f_{n}^{4}, f_{n}^{5}, f_{n}^{6}, f_{n}^{7})}^{T} \to 0 in H . \end{matrix}$ Although the proof will be made by contradiction, as in Proposition 3.1, here the situation is more delicate because the sequence $(λ_{n})$ is not bounded.

Taking the inner product of (3.20) with $U_{n}$ in $H$ and using, once again, the estimation (2.4) of dissipativity, we obtain $\begin{matrix} 3 μ_{1} ‖ u_{n, x} ‖^{2} + μ_{2} ‖ z_{n, x} ‖^{2} + α ‖ θ_{n, x} ‖^{2} \to 0, \end{matrix}$ Then, by Poincaré inequality, we have $\begin{matrix} (3.22) & u_{n}, u_{n, x}, z_{n}, z_{n, x}, θ_{n}, θ_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$

If we rewrite the spectral equation (3.20) in terms of its components, we obtain the system (3.8)–(3.14) with (3.5) again. From (3.8), we get $\begin{matrix} w_{n} = - \frac{i}{λ_{n}} (u_{n} + f_{n}^{1}) . \end{matrix}$ Since $(u_{n} + f_{n}^{1})$ is a bounded sequence in $H^{1} (0, l)$ and $| λ_{n} | \to \infty$ , we have $\begin{matrix} (3.23) & w_{n}, w_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$ Analogously, from (3.10) and (3.12) we have $\begin{matrix} ξ_{n} = - \frac{i}{λ_{n}} (φ_{n} + f_{n}^{3}), s_{n} = - \frac{i}{λ_{n}} (z_{n} + f_{n}^{5}), s_{n, x} = - \frac{i}{λ_{n}} (z_{n, x} + f_{n, x}^{5}) . \end{matrix}$ Since $(φ_{n} + f_{n}^{3})$ and $(z_{n} + f_{n}^{5})$ are bounded sequences in $L^{2} (0, l)$ and $H^{1} (0, l)$ , respectively, we get $\begin{matrix} (3.24) & ξ_{n}, s_{n}, s_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$ It follows directly from (3.23) and (3.24) that $\begin{matrix} (3.25) & w_{n, x} + ξ_{n} + s_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Multiplying (3.11) by $ρ_{2} ξ_{n}$ in $L^{2} (0, l)$ , we obtain $\begin{matrix} i λ_{n} ρ_{2} ⟨ φ_{n}, ξ_{n} ⟩ + b ‖ ξ_{n, x} ‖^{2} + k ⟨ w_{n, x} + ξ_{n} + s_{n}, ξ_{n} ⟩ + δ ⟨ θ_{n, x}, ξ_{n} ⟩ = ρ_{2} ⟨ f_{n}^{4}, ξ_{n} ⟩ . \end{matrix}$ Then, convergences (3.21), (3.22) and (3.25) imply $\begin{matrix} (3.26) & i λ_{n} ρ_{2} ⟨ φ_{n}, ξ_{n} ⟩ + b ‖ ξ_{n, x} ‖^{2} \to 0 . \end{matrix}$ Since $(φ_{n})$ is bounded in $L^{2} (0, l)$ , equation (3.10) implies that sequence $(i λ_{n} ξ_{n})$ is also bounded in $L^{2} (0, l)$ . Then, to conclude that $ξ_{n, x} \to 0$ in $L^{2} (0, l)$ it will suffice to prove that $φ_{n} \to 0$ in $L^{2} (0, l)$ . In fact, multiplying (3.11) by $α θ_{n, x}$ in $L^{2} (0, l)$ , we obtain $\begin{matrix} i λ_{n} ρ_{2} ⟨ φ_{n}, α θ_{n, x} ⟩ + b ⟨ ξ_{n, x}, α θ_{n, x x} ⟩ + k ⟨ w_{n, x} + ξ_{n} + s_{n}, α θ_{n, x} ⟩ + δ α ‖ θ_{n, x} ‖^{2} = ⟨ f_{n}^{4}, α θ_{n, x} ⟩ . \end{matrix}$ From (3.21), (3.22) and boundedness of sequence $(w_{n, x} + ξ_{n} + s_{n})$ follows that $\begin{matrix} (3.27) & i λ_{n} ρ_{2} ⟨ φ_{n}, α θ_{n, x} ⟩ + b ρ_{3} ⟨ ξ_{n, x}, \frac{α}{ρ_{3}} θ_{n, x x} ⟩ \to 0 . \end{matrix}$ Now, if we multiply (3.11) by $φ_{n}$ in $L^{2} (0, l)$ give us $\begin{matrix} i λ_{n} ρ_{2} ‖ φ_{n} ‖^{2} + b ⟨ ξ_{n, x}, φ_{n, x} ⟩ + k ⟨ w_{n, x} + ξ_{n} + s_{n}, φ_{n} ⟩ + δ ⟨ θ_{n, x}, φ_{n} ⟩ = ρ_{2} ⟨ f_{n}^{4}, φ_{n} ⟩ . \end{matrix}$ Again, from (3.21), (3.22) and (3.25) follows that $\begin{matrix} (3.28) & 3 δ i λ_{n} ρ_{2} ‖ φ_{n} ‖^{2} + b ρ_{3} ⟨ ξ_{n, x}, \frac{3 δ}{ρ_{3}} φ_{n, x} ⟩ \to 0 . \end{matrix}$ Subtracting equation (3.28) from equation (3.27) it follows that $\begin{matrix} 3 δ i λ_{n} ρ_{2} ‖ φ_{n} ‖^{2} - i λ_{n} ρ_{2} ⟨ φ_{n}, α θ_{n, x} ⟩ + b ρ_{3} ⟨ ξ_{n, x}, - \frac{α}{ρ_{3}} θ_{n, x x} + \frac{3 δ}{ρ_{3}} φ_{n, x} ⟩ \to 0 . \end{matrix}$ Replacing (3.14) we have $\begin{matrix} 3 δ i λ_{n} ρ_{2} ‖ φ_{n} ‖^{2} - i λ_{n} ρ_{2} ⟨ φ_{n}, α θ_{n, x} ⟩ + b ρ_{3} ⟨ ξ_{n, x}, f_{n}^{7} - i λ_{n} θ_{n} ⟩ \to 0, \end{matrix}$ that is $\begin{matrix} 3 δ i λ_{n} ρ_{2} ‖ φ_{n} ‖^{2} - i λ_{n} ρ_{2} ⟨ φ_{n}, α θ_{n, x} ⟩ + b ρ_{3} ⟨ ξ_{n, x}, f_{n}^{7} ⟩ - b ρ_{3} ⟨ i λ_{n} ξ_{n}, θ_{n, x} ⟩ \to 0 . \end{matrix}$ From (3.21), (3.22) and since $(ξ_{n, x})$ , $(i λ_{n} ξ_{n})$ are bounded in $L^{2} (0, l)$ we get $\begin{matrix} 3 δ i λ_{n} ρ_{2} ‖ φ_{n} ‖^{2} - i λ_{n} ρ_{2} ⟨ φ_{n}, α θ_{n, x} ⟩ \to 0 . \end{matrix}$ Then $\begin{matrix} 3 δ ‖ φ_{n} ‖^{2} - ⟨ φ_{n}, α θ_{n, x} ⟩ \to 0 . \end{matrix}$ Convergence (3.22), again, implies that $\begin{matrix} (3.29) & φ_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Returning to the convergence (3.26), we see that this can be written as $\begin{matrix} - ρ_{2} ⟨ φ_{n}, i λ_{n} ξ_{n} ⟩ + b ‖ ξ_{n, x} ‖^{2} \to 0 . \end{matrix}$ Then, since $(i λ_{n} ξ_{n})$ is a bounded sequence in $L^{2} (0, l)$ , the convergence (3.29) implies that $\begin{matrix} (3.30) & ξ_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$

Finally, thanks to convergences (3.22), (3.24), (3.25), (3.29) and (3.30), we conclude that $‖ U_{n} ‖_{H} \to 0$ , which is absurd, since $‖ U_{n} ‖_{H} = 1$ . This contradiction completes the proof. □

Now, we establish our exponential stability result. Theorem 3.2.
The semigroup $S (t) = e^{t A}$ generated by $A$ when $μ_{1} > 0$ and $μ_{2} > 0$ is exponentially stable.
Proof.
From Proposition 3.1 and Proposition 3.2, it follows that the conditions of Theorem 3.1 are satisfied, and then our semigroup $S (t) = e^{t A}$ generated by $A$ is exponentially stable. □

4. Lack of exponential stability when $μ_{1} > 0$ and $μ_{2} = 0$

The exponential stability result obtained in the preceding section leads us to ask whether it is possible to obtain exponential decay when single damping acts on the system. In this section, we will prove that this is not possible. We will prove that the system (2.1) with single damping acting on the transverse displacement equation is not exponentially stable. This lack of exponential stability is independent of the wave propagation speeds. To prove this result, we will use Theorem 3.1, showing that condition (3.2) is not satisfied.

Theorem 4.1.
If $μ_{1} > 0$ and $μ_{2} = 0$ , the semigroup $S (t) = e^{t A}$ generated by $A$ is not exponentially stable.
Proof.
Once again, we will prove by reductio ad absurdum. Our strategy consists in proving that condition (3.2) of Theorem 3.1 is not satisfied. Thus, we will prove that there exists a sequence $(λ_{n}) \subset R$ such that $\begin{matrix} (4.1) & | λ_{n} | \to \infty \end{matrix}$ and $\begin{matrix} lim_{n \to \infty} {‖ {(i λ_{n} I - A)}^{- 1} ‖}_{L (H)} = \infty . \end{matrix}$ This is equivalent to finding a sequence $(F_{n})$ in $H$ such that $\begin{matrix} (4.2) & ‖ F_{n} ‖_{H} ⩽ 1, n \in N \end{matrix}$ and $\begin{matrix} (4.3) & lim_{n \to \infty} {‖ {(i λ_{n} I - A)}^{- 1} F_{n} ‖}_{H} = \infty . \end{matrix}$ If we denote $U_{n} = {(i λ_{n} I - A)}^{- 1} F_{n}$ , $n \in N$ , the last limit is written as $\begin{matrix} (4.4) & lim_{n \to \infty} ‖ U_{n} ‖_{H} = \infty \end{matrix}$ and $\begin{matrix} (4.5) & i λ_{n} U_{n} - A U_{n} = F_{n}, n \in N . \end{matrix}$ Putting $U_{n} = {(w_{n}, u_{n}, ξ_{n}, φ_{n}, s_{n}, z_{n}, θ_{n})}^{T}$ and $F_{n} = {(f_{n}^{1}, f_{n}^{2}, f_{n}^{3}, f_{n}^{4}, f_{n}^{5}, f_{n}^{6}, f_{n}^{7})}^{T}$ the resolvent equation (4.5) gives the system $\begin{array}{c} (4.6) & i λ_{n} w_{n} - u_{n} = f_{n}^{1} in H_{}^{1} (0, l), \\ (4.7) & i λ_{n} u_{n} - \frac{1}{ρ_{1}} [k {(w_{n, x} + ξ_{n} + s_{n})}_{x} + μ_{1} u_{n, x x}] = f_{n}^{2} in L_{}^{2} (0, l), \\ (4.8) & i λ_{n} ξ_{n} - φ_{n} = f_{n}^{3} in H_{0}^{1} (0, l), \\ (4.9) & i λ_{n} φ_{n} - \frac{1}{ρ_{2}} [b ξ_{n, x x} - k (w_{n, x} + ξ_{n} + s_{n}) - δ θ_{n, x}] = f_{n}^{4} in L^{2} (0, l), \\ (4.10) & i λ_{n} s_{n} - z_{n} = f_{n}^{5} in H_{0}^{1} (0, l), \\ (4.11) & i λ_{n} z_{n} - \frac{1}{ρ_{2}} [b s_{n, x x} - 3 k (w_{n, x} + ξ_{n} + s_{n}) - γ s_{n}] = f_{n}^{6} in L^{2} (0, l), \\ (4.12) & i λ_{n} θ_{n} - \frac{1}{ρ_{3}} [α θ_{n, x x} - 3 δ φ_{n, x}] = f_{n}^{7} in L_{*}^{2} (0, l) . \end{array}$ Let us define the sequences $\begin{matrix} (4.13) & α_{n} = \frac{n π}{l}, n \in N \end{matrix}$ and $F_{n} = {(f_{n}^{1}, f_{n}^{2}, f_{n}^{3}, f_{n}^{4}, f_{n}^{5}, f_{n}^{6}, f_{n}^{7})}^{T}$ by $\begin{matrix} (4.14) & f_{n}^{1} = f_{n}^{2} = f_{n}^{3} = f_{n}^{4} = f_{n}^{5} = f_{n}^{7} = 0 and f_{n}^{6} = c sin α_{n} x, \end{matrix}$ where c is a constant satisfying $\begin{matrix} (4.15) & 0 < | c | ⩽ \sqrt{\frac{2}{ρ_{2} l}} . \end{matrix}$ For (4.14) and (4.15) we have that $(F_{n}) \subset H$ and $\begin{matrix} (4.16) & ‖ F_{n} ‖_{H}^{2} = ρ_{2} {‖ f_{n}^{6} ‖}^{2} = ρ_{2} ‖ c sin α_{n} x ‖^{2} = \frac{1}{2} ρ_{2} | c |^{2} l ⩽ 1, \end{matrix}$ so (4.2) is satisfied. On the other hand, the system (4.6)–(4.12) is transformed into the system $\begin{array}{c} (4.17) & u_{n} = i λ_{n} w_{n}, φ_{n} = i λ_{n} ξ_{n}, z_{n} = i λ_{n} s_{n}, \\ (4.18) & - λ_{n}^{2} ρ_{1} w_{n} - k {(w_{n, x} + ξ_{n} + s_{n})}_{x} - i λ_{n} μ_{1} w_{n, x x} = 0, \\ (4.19) & - λ_{n}^{2} ρ_{2} ξ_{n} - b ξ_{n, x x} + k (w_{n, x} + ξ_{n} + s_{n}) + δ θ_{n, x} = 0, \\ (4.20) & - λ_{n}^{2} ρ_{2} s_{n} - b s_{n, x x} + 3 k (w_{n, x} + ξ_{n} + s_{n}) + γ s_{n} = ρ_{2} c sin α_{n} x, \\ (4.21) & i λ_{n} ρ_{3} θ_{n} - α θ_{n, x x} + 3 i δ λ_{n} ξ_{n, x} = 0 . \end{array}$ Then, we choose the sequences $(w_{n})$ , $(ξ_{n})$ , $(s_{n})$ and $(θ_{n})$ of the following form $\begin{array}{c} (4.22) & w_{n} (x) = A_{n} cos α_{n} x, ξ_{n} (x) = B_{n} sin α_{n} x, \\ (4.23) & s_{n} (x) = C_{n} sin α_{n} x, θ_{n} (x) = D_{n} cos α_{n} x, \end{array}$ where $(A_{n})$ , $(B_{n})$ , $(C_{n})$ and $(D_{n})$ are sequences of real numbers to be fixed later. Choice (4.22)–(4.23) together with equations (4.17) completes the definition of the sequence $(U_{n}) \subset D (A)$ . Then $w_{n}$ , $ξ_{n}$ , $s_{n}$ and $θ_{n}$ must satisfy equations (4.18)–(4.21). By replacing (4.22)–(4.23) in (4.18)–(4.21) we obtain the system: $\begin{array}{c} (4.24) & (k α_{n}^{2} - λ_{n}^{2} ρ_{1} + i μ_{1} λ_{n} α_{n}^{2}) A_{n} - k α_{n} B_{n} - k α_{n} C_{n} = 0, \\ (4.25) & - α_{n} k A_{n} + (α_{n}^{2} b - λ_{n}^{2} ρ_{2} + k) B_{n} + k C_{n} - δ α_{n} D_{n} = 0, \\ (4.26) & - 3 α_{n} k A_{n} + 3 k B_{n} + (b α_{n}^{2} - λ_{n}^{2} ρ_{2} + 3 k + γ) C_{n} = ρ_{2} c, \\ (4.27) & 3 i δ λ_{n} α_{n} B_{n} + (α α_{n}^{2} + i ρ_{3} λ_{n}) D_{n} = 0 . \end{array}$ If we choose $\begin{matrix} (4.28) & λ_{n} = \sqrt{\frac{b}{ρ_{2}} α_{n}^{2} + \frac{3 k + γ}{ρ_{2}}}, n \in N, \end{matrix}$ we see that (4.1) is verified and then it remains to identify the coefficients $A_{n}$ , $B_{n}$ , $C_{n}$ and $D_{n}$ that solve the system (4.24)–(4.27). Replacing (4.28) in (4.24)–(4.27), we get $\begin{array}{c} (4.29) & α_{n}^{2} P_{n} A_{n} - k B_{n} - k C_{n} = 0, \\ (4.30) & - k α_{n} A_{n} - Q B_{n} + k C_{n} - δ α_{n} D_{n} = 0, \\ (4.31) & - 3 k α_{n} A_{n} + 3 k B_{n} = ρ_{2} c, \\ (4.32) & 3 R_{n} B_{n} + S_{n} D_{n} = 0, \end{array}$ where $\begin{array}{c} P_{n} = (k - \frac{b ρ_{1}}{ρ_{2}}) \frac{1}{α_{n}} - \frac{(3 k + γ) ρ_{1}}{ρ_{2} α_{n}^{3}} + i μ_{1} \sqrt{\frac{b}{ρ_{2}} + \frac{3 k + γ}{ρ_{2} α_{n}^{2}}}, \\ Q = 2 k + γ, R_{n} = i δ \sqrt{\frac{b}{ρ_{2}} + \frac{3 k + γ}{ρ_{2} α_{n}^{2}}} and S_{n} = α + i ρ_{3} \sqrt{\frac{b}{ρ_{2}} + \frac{3 k + γ}{ρ_{2} α_{n}^{2}}} \frac{1}{α_{n}} . \end{array}$ By solving the system (4.29)–(4.32), we obtain $\begin{array}{c} (4.33) & A_{n} = \frac{- ρ_{2} c (Q S_{n} + k S_{n} - 3 δ α_{n} R_{n})}{3 k α_{n} (Q S_{n} + 2 k S_{n} - 3 δ α_{n} R_{n} - α_{n} P_{n} S_{n})}, \\ (4.34) & B_{n} = \frac{ρ_{2} c S_{n} (k - α_{n} P_{n})}{3 k (Q S_{n} + 2 k S_{n} - 3 δ α_{n} R_{n} - α_{n} P_{n} S_{n})}, \\ (4.35) & C_{n} = \frac{- ρ_{2} c (k^{2} S_{n} - 3 δ α_{n}^{2} P_{n} R_{n} + Q α_{n} P_{n} S_{n})}{3 k^{2} (Q S_{n} + 2 k S_{n} - 3 δ α_{n} R_{n} - α_{n} P_{n} S_{n})}, \\ (4.36) & D_{n} = \frac{- ρ_{2} c R_{n} (k - α_{n} P_{n})}{k (Q S_{n} + 2 k S_{n} - 3 δ α_{n} R_{n} - α_{n} P_{n} S_{n})} . \end{array}$ Since $c \neq 0$ , the formula for $C_{n}$ and the definition of $α_{n}$ in (4.13), allow us to obtain $\begin{matrix} (4.37) & lim_{n \to \infty} | C_{n} | = \infty, \end{matrix}$ and consequently $\begin{matrix} (4.38) & lim_{n \to \infty} | λ_{n} C_{n} | = \infty . \end{matrix}$ Then, we have $\begin{matrix} (4.39) & ‖ U_{n} ‖_{H}^{2} ⩾ ρ_{2} ‖ z_{n} ‖^{2} = ρ_{2} ‖ i C_{n} λ_{n} sin α_{n} x ‖^{2} = ρ_{2} | λ_{n} C_{n} |^{2} ‖ sin α_{n} x ‖^{2} = \frac{l}{2} ρ_{2} | λ_{n} C_{n} |^{2} . \end{matrix}$ Then (4.38) and (4.39) imply the limit (4.4). Therefore, the $C_{0}$ -semigroup $S (t)$ associated with problem (1.18)–(1.23) is not exponentially stable. This completes the proof. □

5. Polynomial stability when $μ_{1} > 0$ and $μ_{2} = 0$

In the absence of exponential decay of the system (1.18)–(1.23) when $μ_{1} > 0$ and $μ_{2} = 0$ , we wonder if it decays in some other way. In this section, we will prove that the aforementioned system decays polynomially when $μ_{1} > 0$ and $μ_{2} = 0$ with rate $t^{- \frac{1}{2}}$ , and that this decay rate is the best rate that can be obtained. In our proof we use two results: the first one due to Borichev and Tomilov [5] will help us to obtain the polynomial decay, while the second one, due to Fatori and Muñoz [7], will allow us to prove the optimality of our decay rate. In the sequel, we state these two results.

Theorem 5.1.

Let $S (t) = e^{t A}$ be a bounded $C_{0}$ -semigroup of contractions on Hilbert space $H$ with infinitesimal generator $A$ such that $i R \subset ϱ (A)$ . Then, for fixed $α > 0$ , the following conditions are equivalent:

${sup}_{| λ | > δ} \frac{1}{| λ |^{α}} ‖ {(i λ I - A)}^{- 1} ‖_{L (H)} ⩽ C$ for any $δ > 0$ .

There exists a constant $C > 0$ such that $‖ S (t) U_{0} ‖_{H} ⩽ C t^{- \frac{1}{α}} ‖ U_{0} ‖_{H}$ , for any $U_{0} \in D (A)$ .

Theorem 5.2.

Let $S (t)$ be a $C_{0}$ -semigroup of contractions of linear operators on Hilbert space $H$ with infinitesimal generator A and such that $i R \subset ϱ (A)$ . If $\begin{matrix} {‖ S (t) U_{0} ‖}_{H} ⩽ \frac{C}{t^{γ}} ‖ U_{0} ‖_{D (A)}, \end{matrix}$ then, for any $ϵ > 0$ , there exists a constant $C_{ϵ} > 0$ such that $\begin{matrix} \frac{1}{| λ |^{ϵ + \frac{1}{γ}}} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)} ⩽ C_{ϵ} . \end{matrix}$

As we can see, to apply either of these two results, we need to prove that the resolvent set $ϱ (A)$ of the operator $A$ associated with problem (1.18)–(1.23) with $μ_{1} > 0$ and $μ_{2} = 0$ contains the imaginary axis.

Proposition 5.1.

Let $ϱ (A)$ be the resolvent set of the operator $A$ associated with the system ( 1.18 )–( 1.23 ) with $μ_{1} > 0$ and $μ_{2} = 0$ . Then $\begin{matrix} (5.1) & i R \subset ϱ (A) . \end{matrix}$

Proof.

Let us suppose, by contradiction, that $i R ⊄ ϱ (A)$ . Since $A$ is an infinitesimal generator of a $C_{0}$ -semigroup of contractions $S (t)$ in $H$ , then $A$ is closed, see Pazy [27, Corollary 2.5]. Moreover, since $0 \in ϱ (A)$ , there exists a sequence $(λ_{n}) \subset R$ and a sequence $(U_{n}) \subset D (A)$ with $‖ U_{n} ‖_{H} = 1$ such that $\begin{matrix} (5.2) & λ_{n} \to λ with | λ_{n} | < | λ | \end{matrix}$ and $\begin{matrix} {‖ (i λ_{n} I - A) U_{n} ‖}_{H} \to 0 . \end{matrix}$ Putting $\begin{matrix} (5.3) & (i λ_{n} I - A) U_{n} = F_{n}, \end{matrix}$ where $F_{n} = {(f_{n}^{1}, f_{n}^{2}, f_{n}^{3}, f_{n}^{4}, f_{n}^{5}, f_{n}^{6}, f_{n}^{7})}^{T} \in H$ and $U_{n} = {(w_{n}, u_{n}, ξ_{n}, φ_{n}, s_{n}, z_{n}, θ_{n})}^{T} \in D (A)$ , we have $\begin{matrix} (5.4) & F_{n} \to 0 en H . \end{matrix}$ Taking the inner product of (5.3) with $U_{n}$ in $H$ , considering the real part and using (2.4), we get $\begin{matrix} 3 μ_{1} ‖ u_{n, x} ‖^{2} + α ‖ θ_{n, x} ‖^{2} = - Re {⟨ A U_{n}, U_{n} ⟩}_{H} = Re {⟨ F_{n}, U_{n} ⟩}_{H} . \end{matrix}$ Then, the convergence (5.4) and Poincaré’s inequality imply that $\begin{aligned} (5.5) & u_{n, x} \to 0 in L^{2} (0, l) and u_{n} \to 0 in L^{2} (0, l), \\ (5.6) & θ_{n, x} \to 0 in L^{2} (0, l) and θ_{n} \to 0 in L^{2} (0, l) . \end{aligned}$

The resolvent equation (5.3) and the convergence (5.4), give origin to the system $\begin{array}{c} (5.7) & i λ_{n} w_{n} - u_{n} = f_{n}^{1} \to 0 in H_{*}^{1} (0, l), \\ (5.8) & i λ_{n} u_{n} - \frac{1}{ρ_{1}} [k {(w_{n, x} + ξ_{n} + s_{n})}_{x} + μ_{1} u_{n, x x}] = f_{n}^{2} \to 0 in L_{*}^{2} (0, l), \\ (5.9) & i λ_{n} ξ_{n} - φ_{n} = f_{n}^{3} \to 0 in H_{0}^{1} (0, l), \\ (5.10) & i λ_{n} φ_{n} - \frac{1}{ρ_{2}} [b ξ_{n, x x} - k (w_{n, x} + ξ_{n} + s_{n}) - δ θ_{n, x}] = f_{n}^{4} \to 0 in L^{2} (0, l), \\ (5.11) & i λ_{n} s_{n} - z_{n} = f_{n}^{5} \to 0 in H_{0}^{1} (0, l), \\ (5.12) & i λ_{n} z_{n} - \frac{1}{ρ_{2}} [b s_{n, x x} - 3 k (w_{n, x} + ξ_{n} + s_{n}) - γ s_{n}] = f_{n}^{6} \to 0 in L^{2} (0, l), \\ (5.13) & i λ_{n} θ_{n} - \frac{1}{ρ_{3}} [α θ_{n, x x} - 3 δ φ_{n, x}] = f_{n}^{7} \to 0 in L_{*}^{2} (0, l) . \end{array}$ Since $‖ U_{n} ‖_{H} = 1$ , the sequences $(w_{n, x} + ξ_{n} + s_{n})$ , $(u_{n})$ , $(φ_{n})$ , $(z_{n})$ , $(w_{n, x})$ , $(ξ_{n, x})$ , $(s_{n, x})$ and $(θ_{n})$ are bounded in $L^{2} (0, l)$ . From (5.9), (5.5), (5.7) and Poincaré’s inequality, it follows that $\begin{matrix} (5.14) & w_{n, x} \to 0 in L^{2} (0, l) and w_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Consider the function $v_{n} : (0, l) \to R$ defined by $\begin{matrix} (5.15) & v_{n} (x) = \int_{0}^{x} φ_{n} (r) d r . \end{matrix}$ We note that $(v_{n})$ is a bounded sequence in $L^{2} (0, l)$ and $v_{n, x} = φ_{n}$ . Moreover, since $φ_{n} \in H_{0}^{1} (0, L)$ , then $v_{n} \in H^{2} (0, l)$ and $v_{n, x} (0) = v_{n, x} (l) = 0$ . That is, $v_{n} \in H_{*}^{2} (0, l)$ , $n \in N$ . Then, multiplying equation (5.13) by $(α θ_{n} - 3 δ v_{n})$ in $L^{2} (0, l)$ we get $\begin{matrix} i λ_{n} ⟨ θ_{n}, α θ_{n} - 3 δ v_{n} ⟩ + \frac{1}{ρ_{3}} ‖ α θ_{n, x} - 3 δ φ_{n} ‖^{2} = ⟨ f_{n}^{7}, α θ_{n} - 3 δ v_{n} ⟩ . \end{matrix}$ Since $(α θ_{n} - 3 δ v_{n})$ is a bounded sequence in $L^{2} (0, l)$ , convergences (5.4) and (5.6) imply that $\begin{matrix} α θ_{n, x} - 3 δ φ_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ From (5.6), $θ_{n, x} \to 0$ in $L^{2} (0, l)$ , then we conclude that $\begin{matrix} (5.16) & φ_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Replacing (5.4) and (5.16) in (5.9) it follows that $\begin{matrix} (5.17) & ξ_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Multiplying (5.10) by $ξ_{n}$ in $L^{2} (0, l)$ and applying convergences (5.4), (5.16) and (5.17) we get $\begin{matrix} (5.18) & ξ_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$ Now, taking the inner product of (5.10) with $s_{n}$ in $L^{2} (0, l)$ and integrating by parts gives $\begin{matrix} i ρ_{2} λ_{n} ⟨ φ_{n}, s_{n} ⟩ + b ⟨ ξ_{n, x}, s_{n, x} ⟩ + k ⟨ Φ_{n}, s_{n} ⟩ + δ ⟨ θ_{n, x}, s_{n} ⟩ = ⟨ f_{n}^{4}, s_{n} ⟩, \end{matrix}$ where $Φ_{n} = w_{n, x} + ξ_{n} + s_{n}$ . By the above convergences, the first, second, fourth and fifth terms of this expression go to zero when $n \to \infty$ . Then $\begin{matrix} ⟨ Φ_{n}, s_{n} ⟩ = ⟨ w_{n, x} + ξ_{n}, s_{n} ⟩ + ‖ s_{n} ‖^{2} \to 0 . \end{matrix}$ Then, convergences (5.14) and (5.17) yield $\begin{matrix} (5.19) & s_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ From (5.4), (5.11), (5.14) and (5.19) we conclude that $\begin{matrix} (5.20) & z_{n} \to 0 in L^{2} (0, l), \end{matrix}$ and $\begin{matrix} (5.21) & Φ_{n} = w_{n, x} + ξ_{n} + s_{n} \to 0 in L^{2} (0, l) . \end{matrix}$ Finally, taking the inner product of (5.12) with $s_{n}$ in $L^{2} (0, l)$ using the above convergences, we obtain $\begin{matrix} (5.22) & s_{n, x} \to 0 in L^{2} (0, l) . \end{matrix}$ Convergences (5.5), (5.6), (5.16), (5.18), (5.19), (5.20), (5.21) and (5.22) imply that $‖ U_{n} ‖_{H} \to 0$ , which is a contradicction with $‖ U_{n} ‖_{H} = 1$ . Therefore, $i R \subset ϱ (A)$ . □

From Proposition 5.1, for every $F \in H$ there exists a unique $U \in D (A)$ such that $\begin{matrix} (5.23) & (i λ I - A) U = F . \end{matrix}$ Denoting $F = {(f^{1}, f^{2}, f^{3}, f^{4}, f^{5}, f^{6}, f^{7})}^{T}$ and $U = {(w, u, ξ, φ, s, z, θ)}^{T}$ , the resolvent equation (5.23) gives us the following system of equations $\begin{array}{c} (5.24) & i λ w - u = f^{1} in H_{*}^{1} (0, l), \\ (5.25) & i λ u - \frac{1}{ρ_{1}} [k {(w_{x} + ξ + s)}_{x} + μ_{1} u_{x x}] = f^{2} in L_{*}^{2} (0, l), \\ (5.26) & i λ ξ - φ = f^{3} in H_{0}^{1} (0, l), \\ (5.27) & i λ φ - \frac{1}{ρ_{2}} [b ξ_{x x} - k (w_{x} + ξ + s) - δ θ_{x}] = f^{4} in L^{2} (0, l), \\ (5.28) & i λ s - z = f^{5} in H_{0}^{1} (0, l), \\ (5.29) & i λ z - \frac{1}{ρ_{2}} [b s_{x x} - 3 k (w_{x} + ξ + s) - γ s] = f^{6} in L^{2} (0, l), \\ (5.30) & i λ θ - \frac{1}{ρ_{3}} [α θ_{x x} - 3 δ φ_{x}] = f^{7} in L_{*}^{2} (0, l) . \end{array}$ Our polynomial stability result is based on six lemmas, which we will present below. Hereafter, we will use the letter C to denote a generic positive constant whose value may not be the same from one line to another. For short, we will denote $Φ = w_{x} + ξ + s$ .

Lemma 5.3.

There exists a constant $C > 0$ independent of λ such that $\begin{array}{c} (5.31) & ‖ θ_{x} ‖^{2}, ‖ θ ‖^{2} ⩽ C ‖ F ‖_{H} ‖ U ‖_{H}, \\ (5.32) & ‖ u_{x} ‖^{2}, ‖ u ‖^{2} ⩽ C ‖ F ‖_{H} ‖ U ‖_{H}, \\ (5.33) & ‖ w_{x} ‖^{2}, ‖ w ‖^{2} ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} + C ‖ F ‖_{H}^{2}, \end{array}$ with $| λ | ⩾ 1$ .

Proof.

Taking the inner product of (5.23) with U in $H$ , considering the real part and using (2.4), we get $\begin{matrix} 3 μ_{1} ‖ u_{x} ‖^{2} + α ‖ θ_{x} ‖^{2} = - Re {⟨ A U, U ⟩}_{H} ⩽ ‖ F ‖_{H} ‖ U ‖_{H} . \end{matrix}$ From this estimate and Poincaré’s inequality, it follows (5.31) and (5.32). Equation (5.24) and estimate (5.32) yield $\begin{matrix} ‖ w_{x} ‖^{2} ⩽ \frac{C}{λ^{2}} (‖ u_{x} ‖^{2} + {‖ f_{x}^{1} ‖}^{2}) ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} + C ‖ F ‖_{H}^{2} . \end{matrix}$ One more application of Poincaré’s inequality leads us to (5.33). □

Lemma 5.4.

There exists a constant $C > 0$ independent of λ such that $\begin{matrix} (5.34) & 3 ρ_{2} ‖ φ ‖^{2} ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H}, \end{matrix}$ with $| λ | ⩾ 1$ .

Proof.

Taking the inner product of (5.30) with v in $L^{2} (0, l)$ , and integrating by parts, we get $\begin{matrix} 3 δ ‖ φ ‖^{2} = i ρ_{3} λ ⟨ θ, v ⟩ + α ⟨ θ_{x}, φ ⟩ - ρ_{3} ⟨ f^{7}, v ⟩, \end{matrix}$ where $v (x) = \int_{0}^{x} φ (r) d r$ . Similar to what we observed in the proof of Proposition 5.1, $v \in H_{*}^{2} (0, l)$ and moreover $‖ v ‖ ⩽ L ‖ φ ‖$ . Applying Cauchy-Schwarz, and Young’s inequalities we get $\begin{matrix} 3 δ ‖ φ ‖^{2} ⩽ C | λ |^{2} ‖ θ ‖^{2} + \frac{ε}{2} ‖ φ ‖^{2} + C ‖ θ_{x} ‖^{2} + \frac{ε}{2} ‖ φ ‖^{2} + ρ_{3} ‖ f^{7} ‖ ‖ v ‖ . \end{matrix}$ Then, inequality (5.31) yields $\begin{matrix} 3 δ ‖ φ ‖^{2} ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H} + ε ‖ φ ‖^{2} . \end{matrix}$ Taking $ε = \frac{3 δ}{2}$ we obtain the desired inequality. □

Lemma 5.5.

There exists a constant $C > 0$ independent of λ such that $\begin{matrix} (5.35) & 3 k ‖ Φ ‖^{2} ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H}, \end{matrix}$ with $| λ | ⩾ 1$ .

Proof.

We begin by defining the function $\begin{matrix} ψ (x) = \int_{0}^{x} Φ (r) d r . \end{matrix}$ Since Φ is an element in $L^{2} (0, l)$ , then ψ also belongs to $L^{2} (0, l)$ and moreover $\begin{matrix} ψ_{x} = Φ and ‖ ψ ‖ ⩽ C ‖ Φ ‖ . \end{matrix}$ Then, multiplying both sides of equation (5.25) by ψ, we get $\begin{matrix} k ‖ Φ ‖^{2} = ρ_{1} ⟨ f^{2}, ψ ⟩ - ρ_{1} i λ ⟨ u, ψ ⟩ - μ_{1} ⟨ u_{x}, Φ ⟩ . \end{matrix}$ Applying Cauchy-Schwarz and Young’s inequalities we obtain $\begin{aligned} 3 k ‖ Φ ‖^{2} & ⩽ C ‖ f^{2} ‖ ‖ Φ ‖ + C | λ | ‖ u ‖ ‖ Φ ‖ + C ‖ u_{x} ‖ ‖ Φ ‖ \\ ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} + C | λ |^{2} ‖ u ‖^{2} + C ‖ u_{x} ‖^{2} + ε ‖ Φ ‖^{2} . \end{aligned}$ Replacing (5.32) we obtain $\begin{matrix} 3 k ‖ Φ ‖^{2} ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H} + ε ‖ Φ ‖^{2} . \end{matrix}$ Taking $ε = \frac{3 k}{2}$ we obtain (5.35). □

Lemma 5.6.

There exists a constant $C > 0$ independent of λ such that $\begin{matrix} (5.36) & 3 b ‖ ξ_{x} ‖^{2} ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H}, \end{matrix}$ with $| λ | ⩾ 1$ .

Proof.

Taking the inner product of (5.27) with ξ in $L^{2} (0, l)$ and integrating by parts we get $\begin{matrix} ρ_{2} i λ ⟨ φ, ξ ⟩ + b ‖ ξ_{x} ‖^{2} + k ⟨ Φ, ξ ⟩ + δ ⟨ θ_{x}, ξ ⟩ = ρ_{2} ⟨ f^{4}, ξ ⟩ . \end{matrix}$ Then, replacing (5.26), and applying Cauchy-Schwarz and Young’s inequalities we get $\begin{aligned} b ‖ ξ_{x} ‖^{2} & = ρ_{2} ⟨ f^{4}, ξ ⟩ + ρ_{2} ⟨ φ, φ + f^{3} ⟩ - k ⟨ Φ, ξ ⟩ - δ ⟨ θ_{x}, ξ ⟩ \\ ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} + C ‖ φ ‖^{2} + C ‖ φ ‖ ‖ f_{x}^{3} ‖ + C ‖ Φ ‖ ‖ ξ ‖ + C ‖ θ_{x} ‖ ‖ ξ ‖ \\ ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} + C ‖ φ ‖^{2} + C ‖ Φ ‖^{2} + C ‖ θ_{x} ‖^{2} + ε ‖ ξ_{x} ‖^{2} . \end{aligned}$ Applying (5.31), (5.34), (5.35) and taking $ε = \frac{b}{2}$ we have $\begin{matrix} 3 b ‖ ξ_{x} ‖^{2} ⩽ C ‖ F ‖_{H} ‖ U ‖_{H} + C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H} . \end{matrix}$ Since $| λ | ⩾ 1$ , inequality (5.36) follows. □

Lemma 5.7.

There exists a constant $C > 0$ independent of λ such that $\begin{matrix} (5.37) & ρ_{2} ‖ z ‖^{2} ⩽ C λ^{2} ‖ F ‖_{H} ‖ U ‖_{H} + C λ^{2} ‖ F ‖_{H}^{2}, \end{matrix}$ with $| λ | ⩾ 1$ .

Proof.

Taking the inner product of (5.29) with $(k Φ + μ_{1} u_{x})$ in $L^{2} (0, l)$ , integrating by parts and replacing (5.25), we get $\begin{aligned} i λ ρ_{2} k ⟨ z, Φ ⟩ + i λ ρ_{2} μ_{1} ⟨ z, u_{x} ⟩ + b ρ_{1} ⟨ s_{x}, i λ u ⟩ - b ρ_{1} ⟨ s_{x}, f^{2} ⟩ \\ (5.38) & + 3 k^{2} ‖ Φ ‖^{2} + 3 k μ_{1} ⟨ Φ, u_{x} ⟩ + γ ⟨ s, k Φ + μ_{1} u_{x} ⟩ = ρ_{2} ⟨ f^{6}, k Φ + μ_{1} u_{x} ⟩ . \end{aligned}$ Let us estimate the first, third and seventh terms of (5.38). Integrating by parts and replacing (5.28), we obtain $\begin{matrix} (5.39) & b ρ_{1} ⟨ s_{x}, i λ u ⟩ = b ρ_{1} ⟨ i λ s, u_{x} ⟩ = b ρ_{1} ⟨ z, u_{x} ⟩ - b ρ_{1} ⟨ f_{x}^{5}, u ⟩ . \end{matrix}$ Replacing (5.26) and (5.28) in the first term of (5.38), give us $\begin{aligned} i λ ρ_{2} k ⟨ z, Φ ⟩ & = - ρ_{2} k ⟨ z, i λ w_{x} ⟩ - ρ_{2} k ⟨ z, i λ ξ ⟩ - ρ_{2} k ⟨ z, i λ s ⟩ \\ (5.40) & = - ρ_{2} k ⟨ z, i λ w_{x} ⟩ - ρ_{2} k ⟨ z, φ ⟩ - ρ_{2} k ⟨ z, f^{3} ⟩ - ρ_{2} k ‖ z ‖^{2} - ρ_{2} k ⟨ z, f^{5} ⟩, \end{aligned}$ and replacing (5.28) in the seventh term of (5.38) yields $\begin{aligned} γ ⟨ s, k Φ + μ_{1} u_{x} ⟩ & = \frac{γ}{i λ} ⟨ z + f^{5}, k Φ + μ_{1} u_{x} ⟩ \\ (5.41) & = \frac{γ}{i λ} ⟨ z, k Φ + μ_{1} u_{x} ⟩ + \frac{γ}{i λ} ⟨ f^{5}, k Φ + μ_{1} u_{x} ⟩ . \end{aligned}$ Substituting (5.39), (5.40) and (5.41) in (5.38), we obtain $\begin{aligned} ρ_{2} k ‖ z ‖^{2} = & - ρ_{2} ⟨ f^{6}, k Φ + μ_{1} u_{x} ⟩ - ρ_{2} k ⟨ z, i λ w_{x} ⟩ - ρ_{2} k ⟨ z, φ ⟩ - ρ_{2} k ⟨ z, f^{3} ⟩ - ρ_{2} k ⟨ z, f^{5} ⟩ \\ - ρ_{2} μ_{1} ⟨ z, i λ u_{x} ⟩ + b ρ_{1} ⟨ z, u_{x} ⟩ - b ρ_{1} ⟨ f_{x}^{5}, u ⟩ - b ρ_{1} ⟨ s_{x}, f^{2} ⟩ \\ + 3 k^{2} ‖ Φ ‖^{2} + 3 k μ_{1} ⟨ Φ, u_{x} ⟩ + \frac{γ}{i λ} ⟨ z, k Φ + μ_{1} u_{x} ⟩ + \frac{γ}{i λ} ⟨ f^{5}, k Φ + μ_{1} u_{x} ⟩, \end{aligned}$ and applying Cauchy-Schwarz inequality we get $\begin{aligned} ρ_{2} ‖ z ‖^{2} ⩽ & C ‖ f^{6} ‖ ‖ Φ ‖ + C ‖ f^{6} ‖ ‖ u_{x} ‖ + C | λ | ‖ z ‖ ‖ w_{x} ‖ + C ‖ z ‖ ‖ φ ‖ \\ + C ‖ z ‖ ‖ f^{3} ‖ + C ‖ z ‖ ‖ f^{5} ‖ + C | λ | ‖ z ‖ ‖ u_{x} ‖ + C ‖ z ‖ ‖ u_{x} ‖ \\ + C ‖ f_{x}^{5} ‖ ‖ u ‖ + C ‖ s_{x} ‖ ‖ f^{2} ‖ + 3 k^{2} ‖ Φ ‖^{2} + 3 k μ_{1} ‖ Φ ‖ ‖ u_{x} ‖ \\ + \frac{C}{| λ |} ‖ z ‖ ‖ Φ ‖ + \frac{C}{| λ |} ‖ z ‖ ‖ u_{x} ‖ + \frac{C}{| λ |} ‖ f^{5} ‖ ‖ Φ ‖ + \frac{C}{| λ |} ‖ f_{x}^{5} ‖ ‖ u ‖ . \end{aligned}$ From (5.32), (5.33), (5.34), (5.35) and Young’s inequality we have $\begin{aligned} ρ_{2} ‖ z ‖^{2} ⩽ & C ‖ F ‖_{H} ‖ U ‖_{H} + C ‖ F ‖_{H}^{2} + C | λ |^{2} ‖ w_{x} ‖^{2} + C ‖ φ ‖^{2} + C | λ |^{2} ‖ u_{x} ‖^{2} \\ + C ‖ Φ ‖^{2} + \frac{ε}{2} ‖ z ‖^{2} \\ ⩽ & C λ^{2} ‖ F ‖_{H} ‖ U ‖_{H} + C | λ |^{2} ‖ F ‖_{H}^{2} + \frac{ε}{2} ‖ z ‖^{2} . \end{aligned}$ Choosing $ε = ρ_{2}$ , we obtain (5.37). □

Lemma 5.8.

There exists a constant $C > 0$ independent of λ such that $\begin{matrix} (5.42) & b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H} + C | λ |^{2} ‖ F ‖_{H}^{2}, \end{matrix}$ with $| λ | ⩾ 1$ .

Proof.

Multiplying (5.29) by s in $L^{2} (0, l)$ we get $\begin{matrix} b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} = ρ_{2} ⟨ f^{6}, s ⟩ + ρ_{2} ‖ z ‖^{2} + ρ_{2} ⟨ z, f^{5} ⟩ - 3 k ⟨ Φ, s ⟩ . \end{matrix}$ By Cauchy-Schwarz, Young’s inequalities and estimates (5.35) and (5.37) we have $\begin{aligned} b ‖ s_{x} ‖^{2} + γ ‖ s ‖^{2} & ⩽ C ‖ f^{6} ‖ ‖ s ‖ + ρ_{2} ‖ z ‖^{2} + C ‖ z ‖ ‖ f^{5} ‖ + C ‖ Φ ‖^{2} + \frac{ε}{2} ‖ s ‖^{2} \\ ⩽ C | λ |^{2} ‖ F ‖_{H} ‖ U ‖_{H} + C | λ |^{2} ‖ F ‖_{H}^{2} + \frac{ε}{2} ‖ s ‖^{2} . \end{aligned}$ The choice $ε = γ$ leads us to (5.42), and the proof is complete. □

Theorem 5.9.

If $U_{0} \in D (A)$ , the $C_{0}$ -semigroup $S (t) = e^{t A}$ associated to problem ( 1.18 )–( 1.23 ) with $μ_{1} > 0$ and $μ_{2} = 0$ is polynomially stable with decay rate $t^{- \frac{1}{2}}$ . That is, there exists a constant $C > 0$ such that $\begin{matrix} (5.43) & {‖ S (t) U_{0} ‖}_{H} ⩽ \frac{C}{t^{\frac{1}{2}}} ‖ U_{0} ‖_{D (A)}, \forall t > 0 . \end{matrix}$ Furthermore, this decay rate is optimal.

Proof.

From the definition of the norm in $H$ , applying the Lemmas 5.3–5.8, we get $\begin{matrix} ‖ U ‖_{H}^{2} ⩽ C λ^{2} ‖ F ‖_{H} ‖ U ‖_{H} + C λ^{2} ‖ F ‖_{H}^{2} . \end{matrix}$ Applying Young’s inequality yields $\begin{matrix} ‖ U ‖_{H}^{2} ⩽ C λ^{4} ‖ F ‖_{H}^{2} + \frac{1}{2} ‖ U ‖_{H}^{2} + C λ^{2} ‖ F ‖_{H}^{2}, \end{matrix}$ and consequently $\begin{matrix} (5.44) & ‖ U ‖_{H}^{2} ⩽ C λ^{4} ‖ F ‖_{H}^{2} . \end{matrix}$ The relationship (5.44) is equivalent to write $\begin{matrix} (5.45) & {‖ {(i λ I - A)}^{- 1} F ‖}_{H} ⩽ C λ^{2} ‖ F ‖_{H}, F \in H, | λ | ⩾ 1, \end{matrix}$ where the constant $C > 0$ is independent of λ. In addition, the estimate (5.45) can be expressed as $\begin{matrix} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)} = O (| λ |^{2}), | λ | \to \infty . \end{matrix}$ Applying Theorem 5.1, we obtain $\begin{matrix} (5.46) & {‖ S (t) A^{- 1} ‖}_{L (H)} = O (t^{- \frac{1}{2}}), t \to \infty, \end{matrix}$ and this means that there exists a constant $C > 0$ such that $\begin{matrix} (5.47) & {‖ S (t) A^{- 1} F ‖}_{H} ⩽ \frac{C}{t^{\frac{1}{2}}} ‖ F ‖_{H}, F \in H, t > 0 . \end{matrix}$ From (5.1) we know that $0 \in ϱ (A)$ , then $\begin{matrix} (5.48) & A^{- 1} F = U_{0} and ‖ U_{0} ‖_{D (A)} = ‖ A U_{0} ‖_{H} + ‖ U_{0} ‖_{H} . \end{matrix}$ From (5.47) and (5.48) we conclude that $\begin{matrix} {‖ S (t) U_{0} ‖}_{H} ⩽ \frac{C}{t^{\frac{1}{2}}} ‖ U_{0} ‖_{D (A)}, t > 0, \end{matrix}$ and this means that the $C_{0}$ -semigroup $S (t) = e^{t A}$ decays polynomially with a rate of $t^{- \frac{1}{2}}$ .

To prove the optimality of the decay rate, we will once again argue by contradiction. Suppose that the decay rate $t^{- \frac{1}{2}}$ can be improved, say to $t^{- \frac{1}{(2 - ε)}}$ , for some $0 < ε < 2$ . For $ϵ = \frac{ε}{2}$ , Theorem 5.2, implies $\begin{matrix} (5.49) & \frac{1}{| λ |^{2 - \frac{ε}{2}}} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)} ⩽ C_{ε} . \end{matrix}$ On the other hand, from (4.35), we obtain $\begin{matrix} (5.50) & C_{n} = N_{n} λ_{n}, \end{matrix}$ where $\begin{matrix} N_{n} = \frac{- ρ_{2} c (\frac{k^{2} S_{n}}{λ_{n}^{2}} - 3 δ (\frac{ρ_{2}}{b} - \frac{(3 k + γ)}{b λ_{n}^{2}}) P_{n} R_{n} + Q \sqrt{\frac{ρ_{2}}{b} - \frac{(3 k + γ)}{b λ_{n}^{2}}} \frac{1}{λ_{n}} P_{n} S_{n})}{3 k^{2} (\frac{(Q + 2 k)}{λ_{n}} S_{n} - (3 δ R_{n} + P_{n} S_{n}) \sqrt{\frac{ρ_{2}}{b} - \frac{(3 k + γ)}{b λ_{n}^{2}}})} \end{matrix}$ and $\begin{matrix} lim_{n \to \infty} N_{n} = - \frac{i b c μ_{1} δ^{2}}{k^{2} (3 δ^{2} + α μ_{1})} \neq 0 . \end{matrix}$ From (4.16), (4.39) and (5.50), we get $\begin{matrix} ‖ U_{n} ‖_{H}^{2} ⩾ \frac{L}{2} ρ_{2} | λ_{n} C_{n} |^{2} = λ_{n}^{4} \frac{| N_{n} |^{2}}{| c |^{2}} ‖ F_{n} ‖_{H}^{2} . \end{matrix}$ This inequality implies $\begin{matrix} (5.51) & \frac{1}{| λ_{n} |^{2 - \frac{ε}{2}}} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)} ⩾ \frac{1}{| λ_{n} |^{2 - \frac{ε}{2}}} λ_{n}^{2} \frac{| N_{n} |}{| c |} = | λ_{n} |^{\frac{ε}{2}} \frac{| N_{n} |}{| c |} . \end{matrix}$ Since $\begin{matrix} lim_{n \to \infty} (| λ_{n} |^{\frac{ε}{2}} \frac{| N_{n} |}{| c |}) = \infty, \end{matrix}$ we have $\begin{matrix} (5.52) & lim_{n \to \infty} (\frac{1}{| λ_{n} |^{2 - \frac{ε}{2}}} {‖ {(i λ I - A)}^{- 1} ‖}_{L (H)}) = \infty . \end{matrix}$ The limit (5.52) is a contradiction with (5.49). Therefore, the decay rate cannot be improved. This completes the proof of the Theorem. □

Footnotes

Acknowledgements

The authors would like to thank the anonymous referees for their careful comments and suggestions that helped to improve this paper.

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