Stability Results for a Laminated Beam with Kelvin–Voigt Damping

Abstract

In this work, we consider a laminated beam subjected to Kelvin–Voigt damping. Under the semigroup theory approach, applying the Lumer–Phillips Theorem, we establish the well-posedness of the associated initial value problem. This paper aims to prove exponential and polynomial stability results when the system is fully and partially damped. First, using the method developed by Z. Liu and S. Zheng, we show that the semigroup associated with the fully damped system is analytic and, consequently, exponentially stable. On the other hand, we prove the lack of exponential stability when the system is partially damped, and then, using the Borichev and Tomilov Theorem, we prove its polynomial stability.

1 Introduction

The use of “laminated composites" materials has become more and more frequent due to their wide variety of applications in the aircraft, aerospace, automotive, sports equipment, offshore structures and building industries (see, e.g. [6, 8, 32]). Hansen [13] was the first to introduce a model for a beam consisting of two layers of laminated composite materials in which sliding can occur at the interface (Fig. 1).

Fig. 1

Adhesive application (left) and plate in the press (right)

In [13], the author assumed that a layer of “adhesive" of negligible thickness joins the board’s two adjacent surfaces so that the adhesive’s restoring force is proportional to the amount of slip. Applying the principle of virtual work, he obtained the following model of plates

$$\begin{aligned} \rho h \ddot{w} + \text{ div }\big (K(3s+\xi -\nabla w)\big )-f_{3}=0, \end{aligned}$$

(1)

$$\begin{aligned} I_{\rho }\ddot{\xi }+K(3s+\xi -\nabla w)-L\xi +M =0, \end{aligned}$$

(2)

$$\begin{aligned} I_{\rho }\ddot{s}+K(3s+\xi -\nabla w)+\frac{4}{3}\gamma h^2\,s-Ls-\mathcal {K}=0, \end{aligned}$$

(3)

where $\xi :=\psi -3s$ suggests that $\xi$ has an interpretation as an effective angle of rotation. In these equations, “$\dot{ \ \ \ }$” denotes differentiation with respect to time, L is defined as

$$\begin{aligned} L\phi :=(L_{1}\phi , L_{2}\phi )= \text{ div }\, \mathcal {M}[\phi ] \end{aligned}$$

where

$$\begin{aligned} \mathcal {M}[\phi ]:=M[\epsilon (\phi )]=D\Bigg (\begin{array}{cc} \epsilon _{11}+\mu \epsilon _{22} &{} (1-\mu )\epsilon _{12}\\ (1-\mu )\epsilon _{12} &{} \mu \epsilon _{11}+\epsilon _{22} \end{array}\Bigg ) \end{aligned}$$

(4)

is a $2\times 2$ symmetric matrix. In addition, it defines

$$\begin{aligned} \epsilon _{ij}(\phi ) =\frac{1}{2}\bigg (\frac{\partial \phi _{i}}{\partial x_{j}}+\frac{\partial \phi _{j}}{\partial x_{i}}\bigg ), \end{aligned}$$

(5)

thus obtaining the second-order operator which is given by

$$\begin{aligned} L_{i}\phi :=\;&\frac{\partial }{\partial x_{i}}\bigg (D\frac{\partial \phi _{i}}{\partial x_{i}}\bigg )+\frac{\partial }{\partial x_{i}}\bigg (\mu D\frac{\partial \phi _{j}}{\partial x_{j}}\bigg ) \nonumber \\ {}&+\frac{\partial }{\partial x_{j}}\bigg (\frac{1-\mu }{2}D\frac{\partial \phi _{i}}{\partial x_{j}}\bigg )+\frac{\partial }{\partial x_{j}}\bigg (\frac{1-\mu }{2}D\frac{\partial \phi _{j}}{\partial x_{i}}\bigg ), \end{aligned}$$

(6)

to $(i,j) \in \{(1,2), \, (2,1)\}$. According to Hansen [13], the laminated composite materials can be designed to produce either favorable damping characteristics or high strength-to-mass ratios, and often for a combination of each.

The search for damping mechanisms capable of stabilizing the most diverse mechanical structures has become a significant challenge. The most practical and straightforward strategy used to dampen oscillations and dissipate energy is based on the assumption that damping is viscous [41]. It is based on the physical concept that damping forces are opposed to speed. That is, the damping source is external.

However, dissipation mechanisms in real structures arise in the most complex forms, and therefore, viscous damping may fail to predict the actual effect on overall dynamics accurately. Unlike viscous damping, here is an assumption that damping is structural. That is, the damping source is internal.

The materials with structural damping are characterized by having stress $(\sigma )$ proportional to strain $(\epsilon )$ and strain rate $(\partial \epsilon /\partial t)$ (see e.g., [17]), i.e.,

$$\begin{aligned} \sigma= & {} a\epsilon +b\frac{\partial \epsilon }{\partial t}, \end{aligned}$$

(7)

with constant coefficients $a>0$ and $b>0$. For this reason, it is also known by strain rate damping or Kelvin-Voigt damping.

An interesting case, from a mathematical point of view, is the study of the type of stability (exponential or polynomial) of the system when structural damping acts only on the rotation angle $\xi$ or the sliding s (see, for example, Malacarne and Rivera [31], where such a study is conducted in the context of Timoshenko beams). In such a case, we consider

$$\begin{aligned} \widetilde{L}\phi :=(\widetilde{L}_{1}\phi , \widetilde{L}_{2}\phi )= \text{ div }\bigg (\mathcal {M}[\phi ]+b\frac{\partial \mathcal {M}[\phi ]}{\partial t}\bigg ), \end{aligned}$$

(8)

where $\widetilde{L}$ is the second-order operator given by

$$\begin{aligned} \widetilde{L}_{i}\phi :=L_{i}\phi +bL_{i}\dot{\phi }, \end{aligned}$$

(9)

which gives rise to the following model of plates

$$\begin{aligned}&\rho h \ddot{w} + \text{ div }\big (K(3s+\xi -\nabla w)\big )-f_{3}=0, \end{aligned}$$

(10)

$$\begin{aligned}&I_{\rho }\ddot{\xi }+K(3s+\xi -\nabla w)-L\xi -L\dot{\xi }+M=0, \end{aligned}$$

(11)

$$\begin{aligned}&I_{\rho }\ddot{s}+K(3s+\xi -\nabla w)+\frac{4}{3}\gamma h^2\,s-Ls-L\dot{s}-\mathcal {K}=0. \end{aligned}$$

(12)

Following the ideas of Hansen and Spies [14] and based on the plate model described by (10)–(12), we can consider the analogous model for beams, including the strain rate damping. The equations for this beam model in the absence of external forces are

$$\begin{aligned}&\rho \ddot{w}+G(\psi -w_{x})_{x}=0\text {,} \end{aligned}$$

(13)

$$\begin{aligned}&I_{\rho }\big (3\ddot{s}-\ddot{\psi }\big )-G\left( \psi -w_{x}\right) -D(3s_{x}-\psi _{x})_{x}-\mu _{1}(3\dot{s}_{x}-\dot{\psi }_{x})_{x} =0\text {,} \end{aligned}$$

(14)

$$\begin{aligned}&3I_{\rho }\ddot{s}+3G\left( \psi -w_{x}\right) +4\gamma s-3Ds_{xx}-\mu _{2}\dot{s}_{xx}=0\text {,} \end{aligned}$$

(15)

where the subscripted x denotes differentiation with respect to the longitudinal spatial variable, $\rho , G, I_\rho , D, \gamma$ and $\mu _{i}$ ($i=1,2$) are positive constants and represent density, shear stiffness, mass moment of inertia, flexural rigidity, adhesive stiffness and structural damping parameter, respectively. The function w(x, t) denotes the transversal displacement, $\psi (x,t)$ represents the rotational displacement, and s(x, t) is proportional to the amount of slip along the interface at instant t and point x on the beam. The first two equations are related to the well-known Timoshenko system (see [42]), and the third one describes the dynamic of the slip. Laminated structures with interfacial slip have become increasingly important in recent years due, among other reasons, to their ability to provide high strength, large energy absorption capacity, and the ability to customize them for specific applications.

So far, there are very few results on the stability of beams subjected to Kelvin-Voigt damping. For instance, in [18], the authors considered the three-layer laminated beam derived by Liu, Trogdon, and Yong [23]. This model consists of a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the beam’s top and bottom layers and the beam’s transverse motion. They consider the following Rao-Nakra model given by

$$\begin{aligned} \rho _1 h_1 u_{tt} - E_1h_1 u_{xx} - k(-u + v + \alpha w_x) - a_1 u_{txx} + a_2 u_t = 0, \end{aligned}$$

(16)

$$\begin{aligned} \rho _3 h_3 v_{tt} - E_3h_3 v_{xx} + k(-u + v + \alpha w_x) - b_1 v_{txx} + b_2 v_t = 0, \end{aligned}$$

(17)

$$\begin{aligned} \rho h w_{tt} + E I w_{xxxx} - k\alpha (-u + v + \alpha w_x)_x + c_1 w_{txxxx} + c_2 w_t = 0, \end{aligned}$$

(18)

where $a_i, b_i, c_i > 0, i=1,2$, $h_i, \rho _i, E_i, G_i, I > 0$ are the thickness, density, Young’s modulus, shear modulus, and moments of inertia of the each layer for $i=1,2,3,$ respectively, $k:= \frac{G_2}{h_2}$ and $\rho h = \rho _1 h_1 + \rho _2 h_2 + \rho _3 h_3$. In this model, u, v, are the longitudinal displacement and w is the transverse displacement of the beam. For the Rao-Nakra model (16)–(18), when the extensional motion of the bottom and top layers is neglected, the model of a two-layer laminated beam proposed by Hansen and Spies [14] is obtained. In [18], the authors showed that the system (16)–(18) is unstable if single damping is imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping is imposed on all three equations. Then they show the polynomial stability of the system where just two of the three equations are directly damped by viscous or Kelvin-Voigt damping. Later, in [27], the polynomial stability of the Rao-Nakra beam was proved when $a_{1}=b_{1}=c_{1}=0$ and two of the coefficients $a_{2}, b_{2}, c_{2}$, vanishes.

On the other hand, several works on two-layer laminated beams are in the literature. These results show the global well-posedness, stability, and long-time dynamics by adding several stabilization mechanisms other than Kelvin-Voigt damping. We recall some of them. Wang et al. [43] proved that the frictional damping $s_t$, created by the interfacial slip of the laminated beam system, is insufficient to stabilize the system exponentially to its equilibrium state. To achieve uniform decay, the authors implemented boundary feedback and used spectral analysis to obtain exponential decay rates under certain conditions on the system coefficients. Mustafa [33] completed the work done by Wang et al. by obtaining an exponential stability result without any conditions on the coefficients. Naturally, the question arises of studying the effect of other stabilizing mechanisms on the laminated beam model.

Later, Feng [9] established the exponential decay of energy, taking time-delay terms and boundary feedback into account. Raposo [37] considered the weak damping in the two first equations and established the exponential stability. The importance of this result is in the proof that the full damped laminated beam has the same behavior as the full damped Timoshenko system. Later, Apalara et al. [2] improved this result and proved that a single control in the form of frictional damping just in the second equation on the rotation angle is strong enough to stabilize the system exponentially. In [39], the authors studied a hybrid Timoshenko laminated beam model. In that paper, the authors assumed that the beam is fastened securely at its left end while remaining free and has a container attached to the right. Using the semigroup approach and a result of Borichev and Tomilov, they showed that the solution is polynomially stable.

A thermoelastic laminated beam model with nonlinear weights and a time-varying delay was analyzed in [34]. On suitable premises about the time delay and the hypothesis of equal-speed wave propagation, the existence and uniqueness of the solution were obtained by combining semigroup theory with Kato’s variable norm technique [15]. In [34], the exponential stability was proved by the energy method in two cases, with and without the structural damping, through constructing an appropriate Lyapunov functional. Still in the context of thermoelastic laminated beams, Liu and Zhao [21] studied a thermoelastic laminated beam system with past history and thermal effect given by Fourier’s law. We can refer to [1, 10, 25, 26, 40] for more results on laminated beams under thermal laws. Laminated beams with memory were studied in [38], where it was proved that the effect of memory and frictional damping produces exponential stabilization. We refer the reader to [7, 11, 24, 28,29,30] for some stability results of laminated beams with structural memory.

To our knowledge, few works consider the effect of Kelvin-Voigt damping on a two-layer laminated beam (see, for example, [4, 5]). Motivated by the previous results, in this work, we study the well-posedness and the asymptotic behavior of a laminated beam with Kelvin-Voigt damping given by

$$\begin{aligned}&\rho w_{tt} + G( \psi - w_x)_x -\mu _{0} w_{xxt}=0, \end{aligned}$$

(19)

$$\begin{aligned}&I_\rho ( 3s_{tt} - \psi _{tt} ) - D(3s_{xx} - \psi _{xx}) - G(\psi - u_x) -\mu _{1}(3{s}_{xx}-{\psi }_{xx})_{t}= 0, \end{aligned}$$

(20)

$$\begin{aligned}&3I_{\rho }s_{tt}-3Ds_{xx}+ 3G(\psi - w_x)+4\gamma s -\mu _{2}s_{xxt}=0, \end{aligned}$$

(21)

where $(x,t) \in \left( 0,l\right) \times (0,\infty )$.

As in Lima and Fernández Sare [19], we will use the term mathematical system to refer to the system (19)–(21), where the dissipation mechanism $\mu _{0} w_{xxt}$ was introduced taking into account only mathematical aspects. On the other hand, we will use the term physical system to refer to the system (19)–(21) without the term $\mu _{0}w_{xxt}$, but considering the mechanisms of dissipation $\mu _{1}(3{s}_{xx}-{\psi }_{xx})_{t}$ and $\mu _{2}s_{xxt}$, introduced from the assumptions made in the derivation of the system (10)–(12), which take into account the ideas of Hansen and Spies [14].

This manuscript is organized as follows. Section 2, considers the fully damped laminated beam system with Kelvin-Voigt dampings. We prove the well-posedness of the corresponding initial value problem by applying the Lumer-Phillips Theorem. The analyticity of the associated semigroup is obtained using the method developed by Z. Liu and S. Zheng [22], which implies exponential stability. In Sect. 3, we consider the partially damped system, with Kelvin-Voigt dampings acting on the second and third equations of the system. In this case, we prove the lack of exponential stability, and then, using the Borichev and Tomilov’s Theorem, we prove the polynomial decay of the system.

2 Mathematical System – Full Damped System

In this section, we study the well-posedness and the asymptotic behavior of a fully Kelvin-Voigt damped laminated beam system. Based in [43] we consider $\xi :=3s-\psi$ and we write (19)–(21) as

$$\begin{aligned}&\rho w_{tt}\!+\!G(3s\!-\!\xi \!-\!w_{x})_{x}\!-\!\mu _{0} w_{xxt}=0\,\, \text {in} \,\, \left( 0,l\right) \times (0,\infty ), \end{aligned}$$

(22)

$$\begin{aligned}&I_{\rho }\xi _{tt}\!-\!D\xi _{xx}-G(3s\!-\!\xi \!-\!w_{x})\!-\!\mu _{1}\xi _{xxt}=0\,\, \text {in} \,\, \left( 0,l\right) \times (0,\infty ),\end{aligned}$$

(23)

$$\begin{aligned}&3I_{\rho }s_{tt}\!-\!3Ds_{xx}\!+\!3G(3s\!-\!\xi \!-\!w_{x})\!+\!4\gamma s\!-\!\mu _{2}s_{xxt}=0 \,\, \text {in} \,\, \left( 0,l\right) \times (0,\infty ), \end{aligned}$$

(24)

where $\xi =\xi (x,t)$ denotes the effective rotation angle.

We consider homogeneous Dirichlet–Neumann boundary conditions

$$\begin{aligned} w_{x}(0,t)=w(l,t)=\xi (0,t)=\xi _{x}(l,t)=s(0,t)=s_{x} (l,t)=0,\ \ t>0, \end{aligned}$$

(25)

and initial data

$$\begin{aligned} \big (w(x,0),\xi (x,0),s(x,0)\big )&=\big (w_{0}(x),\xi _{0}(x),s_{0}(x)\big ) \quad \text {in} \quad \left( 0,l\right) , \end{aligned}$$

(26)

$$\begin{aligned} \big (w_{t}(x,0),\xi _{t}(x,0),s_{t}(x,0)\big )&=\big (w_{1}(x),\xi _{1}(x),s_{1}(x)\big ) \quad \text {in} \quad \left( 0,l\right) . \end{aligned}$$

(27)

2.1 Setting of the Semigroup

We will denote by $\left\langle \cdot \ ,\ \cdot \right\rangle$ and $\left\| \ \cdot \ \right\|$ the usual inner product and norm in $L^{2}(0,l)$, respectively. Introducing the vector function $U=\big (w,u,\xi ,\zeta ,s,v\big )^{\top },$ system (22)–(27) can be written as an abstract Cauchy problem

$$\begin{aligned} \left\{ \begin{array}{l} \dfrac{d}{dt}U(t)=\mathcal {A}^{(k)}U(t),\ \ \ \ \ t>0,\\ \\ U(0)=U_{0}:=(w_{0},w_{1},\xi _{0},\xi _{1},s_{0},s_{1}), \end{array} \right. \end{aligned}$$

(28)

where the operator $\mathcal {A}^{(k)}:D\big (\mathcal {A}^{(k)}\big )\subset \mathcal {H}\rightarrow \mathcal {H}$ $(k= I, II)$ is defined on the phase space

$$\begin{aligned} \mathcal {H}:= H^{1}_{b}(0,l)\times L^{2}(0,l)\times H^{1}_{a}(0,l)\times L^{2}(0,l)\times H^{1}_{a}(0,l) \times L^{2}(0,l), \end{aligned}$$

where

$$\begin{aligned} H_{a}^{1}(0,l):=\Big \{f\in H^{1}(0,l); \ f(0)=0\Big \}, \, H_{b}^{1}(0,l):=\Big \{f\in H^{1}(0,l); \ f(l)=0\Big \}. \end{aligned}$$

For $U=\big (w,u,\xi ,\zeta ,s,v\big )^{\top }\in D\big (\mathcal {A}^{(k)}\big )$, we define

$$\begin{aligned} \mathcal {A}^{(I)}U=\left( \begin{array}{c} u\\ -\dfrac{G}{\rho }(3s-\xi -w_{x})_{x}+\dfrac{\mu _{0}}{\rho }u_{xx}\\ \zeta \\ \dfrac{D}{I_{\rho }}\xi _{xx}+\dfrac{G}{I_{\rho }}(3s-\xi -w_{x})+\dfrac{\mu _{1} }{I_{\rho }}\zeta _{xx}\\ v\\ \dfrac{D}{I_{\rho }}s_{xx}-\dfrac{G}{I_{\rho }}(3s-\xi -w_{x})-\dfrac{4\gamma }{3I_{\rho }}s+\dfrac{\mu _{2}}{3I_{\rho }}v_{xx} \end{array} \right) , \ \mu _{i}>0, \ i=0,1,2, \end{aligned}$$

(29)

where the domain of $\mathcal {A}^{(I)}$ is defined by

$$\begin{aligned} D(\mathcal {A}^{(I)})=\left\{ U \in \mathcal {H}\left| \begin{array}{l} -Gw+\mu _{0}u \in H^{2}(0,l), \ u\in H_{b}^{1}, \ w_{x}(0)=0, \\ D\xi +\mu _{1}\zeta , \ Ds+\mu _{2}v \in H^{2}(0,l), \\ \ \zeta , v\in H_{a}^{1}, \ \xi _{x}(l)=s_{x}(l)=0. \end{array} \right. \right\} . \end{aligned}$$

(30)

For $\mu _{0}=0$, we define

$$\begin{aligned} \mathcal {A}^{(II)}U=\left( \begin{array}{c} u\\ -\dfrac{G}{\rho }(3s-\xi -w_{x})_{x}\\ \zeta \\ \dfrac{D}{I_{\rho }}\xi _{xx}+\dfrac{G}{I_{\rho }}(3s-\xi -w_{x})+\dfrac{\mu _{1} }{I_{\rho }}\zeta _{xx}\\ v\\ \dfrac{D}{I_{\rho }}s_{xx}-\dfrac{G}{I_{\rho }}(3s-\xi -w_{x})-\dfrac{4\gamma }{3I_{\rho }}s+\dfrac{\mu _{2}}{3I_{\rho }}v_{xx} \end{array} \right) , \ \mu _{i}>0, \ i=1,2, \end{aligned}$$

(31)

where the domain of $\mathcal {A}^{(II)}$ is defined by

$$\begin{aligned} D(\mathcal {A}^{(II)})=\left\{ U \in \mathcal {H}\left| \begin{array}{l} w \in H^{2}(0,l), \ u\in H_{b}^{1}, \ w_{x}(0)=0, \\ D\xi +\mu _{1}\zeta , \ Ds+\mu _{2}v \in H^{2}(0,l), \\ \ \zeta , v\in H_{a}^{1}, \ \xi _{x}(l)=s_{x}(l)=0. \end{array} \right. \right\} . \end{aligned}$$

(32)

The phase space $\mathcal {H}$ is a Hilbert space with respect to the inner product

$$\begin{aligned} \langle U_{1}, \, {U}_{2}\rangle _{\mathcal {H}}:= & {} \rho \langle u_{1},\,{u}_{2}\rangle +I_{\rho }\langle \zeta _{1},\,{\zeta }_{2}\rangle +3I_{\rho }\langle v_{1},\,{v}_{2}\rangle \nonumber \\ {}{} & {} +D\langle \xi _{1,x},\,{\xi }_{2,x}\rangle +3D\langle s_{1,x},\,{s}_{2,x}\rangle \nonumber \\ {}{} & {} +4\gamma \langle s_{1},\,{s}_{2}\rangle +G\langle 3s_{1}-\xi _{1}-w_{1,x},\,{3s_{2}-\xi _{2}-w_{2,x}}\rangle \end{aligned}$$

(33)

and norm

$$\begin{aligned} \displaystyle \left\| U_{1}\right\| ^{2}_{\mathcal {H}}:= & {} \rho \left\| u_{1}\right\| ^{2}+I_{\rho }\left\| \zeta _{1}\right\| ^{2}+3I_{\rho }\left\| v_{1}\right\| ^{2} \nonumber \\ {}{} & {} + D\left\| \xi _{1,x}\right\| ^{2}+3D\left\| s_{1,x}\right\| ^{2} +4\gamma \left\| s_{1}\right\| ^{2} \nonumber \\ {}{} & {} + G\left\| 3s_{1}-\xi _{1}-w_{1,x}\right\| ^{2}, \end{aligned}$$

(34)

where $U_{1}=\big (w_{1} ,u_{1} ,\xi _{1} ,\zeta _{1} , s_{1} , v_{1} \big )^{\top }$, $U_{2}=\big (w_{2} ,u_{2} ,\xi _{2} ,\zeta _{2} , s_{2} , v_{2} \big )^{\top } \in \mathcal {H}$ .

In order to prove the existence and uniqueness of solutions of problem (22)–(27), we will use the Lumer-Phillips Theorem and Theorem 4.6 from Pazy [35]. We will begin by proving that operators $\mathcal {A}^{(I)}$ and $\mathcal {A}^{(II)}$ are dissipative.

Proposition 1

The operators $\mathcal {A}^{(k)}$, $(k=I, II)$, defined by (29 )–(30) and (31 )–(32), are dissipative and satisfy

$$\begin{aligned} {\text {Re}}\left\langle \mathcal {A}^{(k)}U,{U}\right\rangle _{\mathcal {H}}=\left\{ \begin{array}{ll} -\mu _{0} \left\| u_{x}\right\| ^{2}-\mu _{1}\left\| \zeta _{x}\right\| ^{2}-\mu _{2}\left\| v_{x}\right\| ^{2}, \quad &{}\text{ if } \quad k=I, \\ \\ -\mu _{1}\left\| \zeta _{x}\right\| ^{2}-\mu _{2}\left\| v_{x}\right\| ^{2}, \quad &{}\text{ if } \quad k=II. \end{array} \right. \end{aligned}$$

(35)

Proof

From the definition of the inner product in $\mathcal {H}$, the definitions (29) and (31) of the operators $\mathcal {A}^{(k)}$, $(k=I, II)$, defined in (29)–(30) and (31 )–(32), applying the integration by parts formula and by grouping the terms conveniently, we can show that

$$\begin{aligned} \left\langle \mathcal {A}^{(k)}U, \,{U}\right\rangle _{\mathcal {H}} \!=\!\left\{ \begin{array}{rll} &{}&{}\hspace{-0.3cm}2iD{\text {Im}} \left\langle \zeta _{x}, \,{\xi }_{x}\right\rangle \!-\!\mu _{0}\left\| u_{x}\right\| ^{2}\!+\!6iD{\text {Im}}\left\langle v_{x}, \,{s}_{x}\right\rangle \!+\!8i\gamma {\text {Im}}\left\langle v, \,{s}\right\rangle \\ {} &{}&{}\hspace{-0.3cm}+ \! 2iG{\text {Im}}\left\langle 3s\!-\!\xi \!-\!w_{x}, \, {3v\!-\!\zeta \!-\!u_{x}}\right\rangle \!-\!\mu _{1}\left\| \zeta _{x}\right\| ^{2}\!-\!\mu _{2}\left\| v_{x} \right\| ^{2}, \,\, k=I, \\ \\ {} &{}&{}\hspace{-0.3cm}2iD{\text {Im}} \left\langle \zeta _{x}, \,{\xi }_{x}\right\rangle \!+\!6iD{\text {Im}}\left\langle v_{x}, \,{s}_{x}\right\rangle \!+\!8i\gamma {\text {Im}}\left\langle v, \,{s}\right\rangle \\ &{}&{}\hspace{-0.3cm} +\!2iG{\text {Im}}\left\langle 3s\!-\!\xi \!-\!w_{x}, \, {3v\!-\!\zeta \!-\!u_{x}}\right\rangle \!-\!\mu _{1}\left\| \zeta _{x}\right\| ^{2}\!-\!\mu _{2}\left\| v_{x} \right\| ^{2}, \,\, k=II. \end{array} \right. \end{aligned}$$

Then, taking the real part we obtain (35), as desired. $\square$

Theorem 1

Let $U_{0}\in \mathcal {H}$, then there exists a unique weak solution of problem (22)–(27) satisfying

$$\begin{aligned} U\in C\big ([0,+\infty ); \mathcal {H}\big ). \end{aligned}$$

(36)

Moreover, if $U_{0}\in D(\mathcal {A}^{(k)})$, $(k=I, II)$ then

$$\begin{aligned} U\in C\big ([0,+\infty );D(\mathcal {A}^{(k)})\big )\cap C^{1}\big ([0,+\infty ); \mathcal {H}\big ). \end{aligned}$$

(37)

Proof

Proceeding as usual, it is not difficult to prove that $I-\mathcal {A}^{(k)}$ is surjective. This result, the dissipativity of the operator $\mathcal {A}^{(k)}$ and the reflexivity of the phase space $\mathcal {H}$ allow us to conclude that the domain of the operator $\mathcal {A}^{(k)}$ is dense in $\mathcal {H}$. Finally, the Lumer-Phillips Theorem implies that $\mathcal {A}^{(k)}$ is the infinitesimal generator of a linear $C_{0}$-semigroup of contractions $S(t)=e^{t\mathcal {A}^{(k)}}$ on $\mathcal {H}$. From semigroups theory, $U(t)=e^{t\mathcal {A}^{(k)}}U_{0}$ is the unique solution of problem (22)–(27) satisfying (36) and (37). $\square$

Now, we define the energy of the system (22)–(27) as

$$\begin{aligned} E(t) =\frac{1}{2}&\Big [ \rho \left\| w_{t}\right\| ^{2}+I_{\rho }\left\| \xi _{t}\right\| ^{2}+3I_{\rho }\left\| s_{t}\right\| ^{2} \nonumber \\ {}&+D\left\| \xi _{x}\right\| ^{2}+3D\left\| s_{x}\right\| ^{2} +4\gamma \left\| s\right\| ^{2} +G\left\| 3s-\xi -w_{x}\right\| ^{2}\Big ]. \end{aligned}$$

(38)

Proposition 2

The energy (38) of the system (22)–(27) is not increasing and satisfies

$$\begin{aligned} \frac{d}{dt}E(t)=-\mu _{0}\left\| w_{xt}\right\| ^{2}-\mu _{1}\left\| \xi _{xt}\right\| ^{2}-\mu _{2}\left\| s_{xt}\right\| ^{2}. \end{aligned}$$

(39)

Proof

Taking the inner product of the first equation of (22) by ${w}_{t}$ in $L^{2}(0,l),$ integrating by parts and using the boundary conditions (25) we have

$$\begin{aligned} \frac{1}{2}\rho \frac{d}{dt}\left\| w_{t}\right\| ^{2}-G\left\langle 3s-\xi -w_{x}, \,{w}_{xt}\right\rangle +\mu _{0}\left\| w_{xt}\right\| ^{2}=0. \end{aligned}$$

(40)

Next, multiplying equations of (23) and (24) by ${\xi }_{t}$ and ${s}_{t}$, respectively, and proceeding in the same way as in last step, we obtain

$$\begin{aligned} \frac{1}{2}I_{\rho }\frac{d}{dt}\left\| \xi _{t}\right\| ^{2}+\frac{D}{2}\frac{d}{dt}\left\| \xi _{x}\right\| ^{2}-G\left\langle 3s-\xi -w_{x}, \,{\xi }_{t}\right\rangle +\mu _{1}\left\| \xi _{xt}\right\| ^{2}=0, \end{aligned}$$

(41)

and

$$\begin{aligned} \frac{3}{2}I_{\rho }\frac{d}{dt}\left\| s_{t}\right\| ^{2}+\frac{3}{2}D\frac{d}{dt}\left\| s_{x}\right\| ^{2}+G\left\langle 3s-\xi -w_{x}, \,3{s}_{t}\right\rangle +2\gamma \frac{d}{dt}\left\| s\right\| ^{2}+\mu _{2}\left\| s_{xt}\right\| ^{2}=0. \end{aligned}$$

(42)

Finally, adding (40), (41) and (42) we get (39), as desired. $\square$

2.2 Analyticity

In this section we prove the analyticity of semigroup $S\left( t\right) =e^{t\mathcal {A}^{(I)}}$ generated by $\mathcal {A}^{(I)}$.

Proposition 3

Let $\rho ( \mathcal {A}^{(I)})$ be the resolvent set of the operator $\mathcal {A}^{(I)}$ defined in (29)–(30). Then

$$\begin{aligned} i\mathbb {R\subset }\rho ( \mathcal {A}^{(I)}) . \end{aligned}$$

(43)

Proof

If (43) is not true, then there exists $\omega \in \mathbb {R}$ with $\Vert (\mathcal {A}^{(I)})^{-1}\Vert \le \left| \omega \right| <\infty$, a sequence $(\lambda _{n})\subset \mathbb {R}$ such that $\lambda _{n}\rightarrow \omega$, $\left| \lambda _{n}\right| <\left| \omega \right|$ and a sequence $(U_{n})\subset D\big (\mathcal {A}^{(I)}\big )$, $U_{n}=(w_{n},u_{n},\xi _{n},\zeta _{n},s_{n},v_{n})$, with $\left\| U_{n}\right\| _{\mathcal {H}}=1$ such that

$$\begin{aligned} \left\| \left( i\lambda _{n}I-\mathcal {A}^{(I)}\right) U_{n}\right\| _{\mathcal {H}}\rightarrow 0 \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(44)

Denoting

$$\begin{aligned} \left( i\lambda _{n}I-\mathcal {A}^{(I)}\right) U_{n}=F_{n}, \quad n \in \mathbb {N}, \end{aligned}$$

with $F_{n}=\left( f_{n}^{1},\ .\ .\ .\ ,f_{n}^{6}\right) ^{\top }\in \mathcal {H}$, then $F_{n}\rightarrow 0$ in $\mathcal {H}$. That is,

$$\begin{aligned}{} & {} (f_{n}^{1},~f_{n}^{3},\ f_{n}^{5})\rightarrow (0,0,0) \,\, \text {in}\,H_{a}^{1}(0,l)\times H_{b}^{1}(0,l)\times H_{b}^{1}(0,l) \quad \text {as} \quad n\rightarrow \infty , \qquad \end{aligned}$$

(45)

$$\begin{aligned}{} & {} f_{n}^{2},\ f_{n}^{4},\ f_{n}^{6}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l) \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(46)

From (44) we get

$$\begin{aligned} \left\langle \left( i\lambda _{n}I-\mathcal {A}^{(I)}\right) U_{n}, {U}_{n} \right\rangle _{\mathcal {H}}=\left\langle F_{n},{U}_{n}\right\rangle _{\mathcal {H}}\rightarrow 0 \quad \text {as} \quad n\rightarrow \infty , \end{aligned}$$

that is

$$\begin{aligned} i\lambda _{n}-\left\langle \mathcal {A}^{(I)}U_{n},{U}_{n}\right\rangle _{\mathcal {H}} \rightarrow 0 \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(47)

Taking the real part, using the dissipativity identity (35), we have

$$\begin{aligned} \mu _{0}\left\| u_{n,x}\right\| ^{2}+\mu _{1}\left\| \zeta _{n,x}\right\| ^{2}+\mu _{2}\left\| v_{n,x}\right\| ^{2}\rightarrow 0, \end{aligned}$$

where $u_{n,x}$ denotes the derivative of $u_{n}$ with respect to x. Then

$$\begin{aligned} u_{n,x}\ ,\ \zeta _{n,x}\ ,\ v_{n,x}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l). \end{aligned}$$

(48)

So, thanks to Poincaré’s inequality, we have

$$\begin{aligned} u_{n}\ ,\ \zeta _{n}\ ,\ v_{n}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l). \end{aligned}$$

(49)

The convergence (44) leads to

$$\begin{aligned}&i\lambda _{n}w_{n}-u_{n} = f_{n}^{1}\,\, \text {in}\, H_{a}^{1} (0,l), \end{aligned}$$

(50)

$$\begin{aligned}&i\lambda _{n}u_{n}+\frac{G}{\rho }\left( 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}\right) _{x}+\frac{\mu _{0}}{\rho }u_{n,xx} = f_{n}^{2} \,\, \text {in}\, L^{2}(0,l), \end{aligned}$$

(51)

$$\begin{aligned}&i\lambda _{n}\xi _{n}-\zeta _{n} = f_{n}^{3} \,\, \text {in}\, H_{b}^{1}(0,l), \end{aligned}$$

(52)

$$\begin{aligned}&i\lambda _{n}\zeta _{n}-\frac{D}{I_{\rho }}\xi _{n,xx}-\frac{G}{I_{\rho }}\left( 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}\right) -\frac{\mu _{1}}{I_{\rho }}\zeta _{n,xx} = f_{n}^{4} \,\, \text {in}\, L^{2}(0,l), \end{aligned}$$

(53)

$$\begin{aligned}&i\lambda _{n}s_{n}-v_{n} = f_{n}^{5} \,\, \text {in}\, H_{b}^{1}(0,l), \end{aligned}$$

(54)

$$\begin{aligned}&i\lambda _{n}v_{n}-\frac{D}{I_{\rho }}s_{n,xx}+\frac{G}{I_{\rho }}\left( 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}\right) +\frac{4\gamma }{3I_{\rho }}s_{n}-\frac{\mu _{2} }{3I_{\rho }}v_{n,xx} = f_{n}^{6} \,\, \text {in}\, L^{2}(0,l), \end{aligned}$$

(55)

$F_{n}=\left( f_{n}^{1},\ .\ .\ .\ ,f_{n}^{6}\right) ^{\top }\rightarrow 0$ in $\mathcal {H}$.

From (48) and (49) we have that

$$\begin{aligned} u_{n}\ ,\ \zeta _{n}\ ,\ v_{n}\rightarrow 0 \quad \text {in} \quad H_{a}^{1}(0,l), \ H_{b}^{1}(0,l), \ H_{b}^{1} (0,l). \end{aligned}$$

(56)

Then from (50), (52) and (54) we have

$$\begin{aligned} \left\| w_{n}\right\| _{H_{a}^{1}(0,l)}\le \frac{1}{\left| \lambda _{n}\right| }\left( \left\| u_{n}\right\| _{H_{a}^{1} (0,l)}+\Vert f_{n}^{1}\Vert _{H_{a}^{1}(0,l)}\right) , \end{aligned}$$

$$\begin{aligned}{} & {} \left\| \xi _{n}\right\| _{H_{b}^{1}(0,l)}\le \frac{1}{\left| \lambda _{n}\right| }\left( \left\| \zeta _{n}\right\| _{H_{b} ^{1}(0,l)}+\Vert f_{n}^{3}\Vert _{H_{b}^{1}(0,l)}\right) , \\{} & {} \left\| s_{n}\right\| _{H_{b}^{1}(0,l)}\le \frac{1}{\left| \lambda _{n}\right| }\left( \left\| v_{n}\right\| _{H_{b}^{1} (0,l)}+\Vert f_{n}^{5}\Vert _{H_{b}^{1}(0,l)}\right) . \end{aligned}$$

Then, making $n\rightarrow \infty$, using (56) and (45) we get

$$\begin{aligned}{} & {} w_{n}\rightarrow 0 \quad \text {in} \quad H_{a}^{1}(0,l), \end{aligned}$$

(57)

$$\begin{aligned}{} & {} \xi _{n}\rightarrow 0 \quad \text {in} \quad H_{b}^{1}(0,l), \end{aligned}$$

(58)

$$\begin{aligned}{} & {} s_{n}\rightarrow 0 \quad \text {in} \quad H_{b}^{1}(0,l). \end{aligned}$$

(59)

From (57), (58) and (59) we get

$$\begin{aligned} 3s_{n}-\xi _{n}-w_{n,x}\rightarrow 0\ \ \ \text {in }L^{2}(0,l). \end{aligned}$$

(60)

From convergences (56)– (60), we conclude that $\left\| U_{n}\right\| _{\mathcal {H}}\rightarrow 0$, which is a contradiction because $\left\| U_{n}\right\| _{\mathcal {H}}=1.$ Therefore (43) holds, and thus the proof of Proposition 3 ends. $\square$

The proof of the semigroup analyticity result is based on the following theorem.

Theorem 2

Let $S\left( t\right) =e^{t\mathcal {A}^{(I)}}$ be a $C_{0}$-semigroup of contractions in a Hilbert space $\mathcal {H}$. Suposse that

$$\begin{aligned} i\mathbb {R\subset }\rho ( \mathcal {A}^{(I)}) . \end{aligned}$$

(61)

Then S(t) is analytic if and only if

$$\begin{aligned} \underset{\vert \lambda \vert \rightarrow \infty }{\lim \sup }\left\| \lambda ( i\lambda I-\mathcal {A}^{(I)}) ^{-1}\right\| _{\mathcal {L(H)}} <\infty \end{aligned}$$

(62)

holds.

Proof

The proof can be found in [22], see Theorem 1.3.3. $\square$

The following theorem is one of the main results of this section.

Theorem 3

Let $\mathcal {A}^{(I)}$ be the linear operator defined by (29)–(30). The $C_{0}$-semigroup of contractions generated by $\mathcal {A}^{(I)}$ is analytic. In particular, it is exponentially stable.

Proof

Due to Proposition 3, it only remains to prove condition (62). We will use again a contradiction argument. In fact, if (62) is not true, there exists a real sequence $(\lambda _{n})_{n \in \mathbb {N}},$ with $\lambda _{n}\rightarrow \infty$ and a sequence of vectorial functions $U_{n}=\big (w_{n},u_{n},\xi _{n},\zeta _{n} ,s_{n},v_{n}\big )^{\top }\in D(\mathcal {A}^{(I)})$ with $\left\| U_{n}\right\| _{\mathcal {H}}=1,$ such that

$$\begin{aligned} \left\| \left( iI-\frac{1}{\lambda _{n}}\mathcal {A}^{(I)}\right) U_{n} \right\| _{\mathcal {H}}\rightarrow 0 \quad \text {as } \quad n\rightarrow \infty . \end{aligned}$$

(63)

Denoting

$$\begin{aligned} \left( iI-\frac{1}{\lambda _{n}}\mathcal {A}^{(I)}\right) U_{n}=F_{n}\ , \end{aligned}$$

(64)

with $F_{n}=\big ( f_{n}^{1},\ .\ .\ .\ ,f_{n}^{6}\big )^{\top } ,$ we rewrite equation (64) in terms of its components

$$\begin{aligned}&iw_{n}-\frac{1}{\lambda _{n}}u_{n}=f_{n}^{1}\,\, \text {in}\, H_{a}^{1}(0,l),\end{aligned}$$

(65)

$$\begin{aligned}&iu_{n}+\frac{G}{\rho \lambda _{n}}\left( 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}\right) _{x}-\frac{\mu _{0}}{\rho \lambda _{n}}u_{n,xx} =f_{n}^{2}\,\, \text {in}\, L^{2}(0,l), \end{aligned}$$

(66)

$$\begin{aligned}&i\xi _{n}-\frac{1}{\lambda _{n}}\zeta _{n} =f_{n}^{3}\,\, \text {in}\, H_{b}^{1}(0,l), \end{aligned}$$

(67)

$$\begin{aligned}&i\zeta _{n}-\frac{D}{I_{\rho }\lambda _{n}}\xi _{n,xx}-\frac{G}{I_{\rho } \lambda _{n}}\left( 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}\right) -\frac{\mu _{1}}{I_{\rho }\lambda _{n}}\zeta _{n,xx} =f_{n}^{4}\,\, \text {in}\, L^{2}(0,l), \end{aligned}$$

(68)

$$\begin{aligned}&is_{n}-\frac{1}{\lambda _{n}}v_{n} =f_{n}^{5}\,\, \text {in}\, H_{b} ^{1}(0,l), \end{aligned}$$

(69)

$$\begin{aligned}&iv_{n}-\frac{D}{I_{\rho }\lambda _{n}}s_{n,xx}+\frac{G}{I_{\rho }\lambda _{n} }\left( 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}\right) +\frac{4\gamma }{3I_{\rho }\lambda _{n} }s_{n}-\frac{\mu _{2}}{3I_{\rho }\lambda _{n}}v_{n,xx} =f_{n}^{6}\,\, \text {in}\, L^{2}(0,l), \end{aligned}$$

(70)

where

$$\begin{aligned} (f_{n}^{1},~f_{n}^{3},\ f_{n}^{5})\rightarrow (0,0,0) \,\, \text {in}\,H_{a}^{1}(0,l)\times H_{b}^{1}(0,l)\times H_{b}^{1}(0,l) \end{aligned}$$

(71)

and

$$\begin{aligned} f_{n}^{2},~f_{n}^{4},\ f_{n}^{6}\rightarrow 0 \,\, \text {in}\, L^{2}(0,l). \end{aligned}$$

(72)

Since $\left\| U_{n}\right\| _{\mathcal {H}}=1,$ then

$$\begin{aligned}&\left( u_{n}\right) ,\ \left( \zeta _{n}\right) \text { \ are bounded in }L^{2}\left( 0,l\right) \text {,} \end{aligned}$$

(73)

$$\begin{aligned}&\left( s_{n}\right) ,\ \left( v_{n}\right) \text { \ are bounded in }L^{2}\left( 0,l\right) \text {,} \end{aligned}$$

(74)

$$\begin{aligned}&\left( w_{n,x}\right) ,\ \left( \xi _{n,x}\right) ,\ \left( s_{n,x}\right) \text { \ are bounded in }L^{2}\left( 0,l\right) \text {,} \end{aligned}$$

(75)

$$\begin{aligned}&\left( 3s_{n}-\xi _{n}-w_{n,x}\right) \text { \ is bounded in }L^{2}\left( 0,l\right) \text {.} \end{aligned}$$

(76)

From (65), (67) and (69) we have

$$\begin{aligned}{} & {} \left\| w_{n}\right\| \le \left| \frac{1}{\lambda _{n}}\right| \left\| u_{n}\right\| +\left\| f_{n}^{1}\right\| , \\{} & {} \left\| \xi _{n}\right\| \le \left| \frac{1}{\lambda _{n}}\right| \left\| \zeta _{n}\right\| +\left\| f_{n}^{3}\right\| , \\{} & {} \left\| s_{n}\right\| \le \left| \frac{1}{\lambda _{n}}\right| \left\| v_{n}\right\| +\left\| f_{n}^{5}\right\| . \end{aligned}$$

Then, since $\lambda _{n}\rightarrow \infty$, using (71), (72), (73) and (74), we get

$$\begin{aligned} w_{n}\ ,\ \xi _{n}\ ,\ s_{n}\ \rightarrow 0 \quad \text {in} \quad L^{2}(0,l). \end{aligned}$$

(77)

Taking the inner product of (66) with ${w}_{n}$ in $L^{2}(0,l)$ and integrating by parts, we get

$$\begin{aligned} i\left\langle u_{n},\, {w}_{n}\right\rangle -\frac{G}{\rho \lambda _{n}}\left\langle 3s_{n}-\xi _{n}-w_{n,x}, \, {w}_{n,x}\right\rangle +\frac{\mu _{0}}{\rho \lambda _{n} }\left\langle u_{n,x},\, {w}_{n,x}\right\rangle =\left\langle f_{n}^{2} ,\, {w}_{n}\right\rangle . \end{aligned}$$

The convergences (72), (77), the boundedness (73) and (76) and the fact that $\lambda _{n}\rightarrow \infty$, leads to

$$\begin{aligned} \frac{1}{\lambda _{n}}\left\langle u_{n,x},{w}_{n,x}\right\rangle \rightarrow 0 \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

Then, replacing (65), we have

$$\begin{aligned} \left\langle \frac{1}{\lambda _{n}}u_{n,x},\,{w}_{n,x}\right\rangle \! = \!\left\langle iw_{n,x}-f_{n,x}^{1},\,{w}_{n,x}\right\rangle \!=\! i\left\| w_{n,x}\right\| ^{2}-\left\langle f_{n,x}^{1},\,{w}_{n,x}\right\rangle \rightarrow 0 \,\, \text {as} \,\,n\rightarrow \infty . \end{aligned}$$

By (71) and (75) we conclude that

$$\begin{aligned} {w}_{n,x}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l). \end{aligned}$$

(78)

Repeating the same argument, multiplying (68) with ${\xi }_{n}$ in $L^{2}(0,l)$

$$\begin{aligned} i\left\langle \zeta _{n},\,{\xi }_{n}\right\rangle +\frac{D}{I_{\rho }\lambda _{n} }\left\| \xi _{n,x}\right\| ^{2}-\frac{G}{I_{\rho }\lambda _{n} }\left\langle 3s_{n}-\xi _{n}-w_{n,x},\,{\xi }_{n}\right\rangle +\frac{\mu _{1} }{I_{\rho }\lambda _{n}}\left\langle \zeta _{n,x},\,{\xi }_{n,x}\right\rangle =\left\langle f_{n}^{4},\,{\xi }_{n}\right\rangle \end{aligned}$$

and applying the above convergences and the bounded nature of the sequences involved, we get

$$\begin{aligned} \frac{1}{\lambda _{n}}\left\langle \zeta _{n,x},\,{\xi }_{n,x}\right\rangle \rightarrow 0 \quad \text {as} \quad n\rightarrow \infty , \end{aligned}$$

then, replacing equation (67) we obtain

$$\begin{aligned} {\xi }_{n,x}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l) \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(79)

Now, taking the inner product of (70) with $s_{n}$ in $L^{2}(0,l)$, applying the convergences (71), (72), (77) and the boundedness (73)–(76), we obtain

$$\begin{aligned} \frac{1}{\lambda _{n}}\left\langle v_{n,x},\,{s}_{n,x}\right\rangle \rightarrow 0 \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

This, combined with the equation (69), yields

$$\begin{aligned} {s}_{n,x}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l). \end{aligned}$$

(80)

On the other hand, taking the inner product of (64) with $U_{n}$ in $\mathcal {H}$ it follows that

$$\begin{aligned} i\left\| U_{n}\right\| ^{2}_{\mathcal {H}}-\frac{1}{\lambda _{n}}\left\langle \mathcal {A}U_{n},\,{U}_{n}\right\rangle _{\mathcal {H}}=\left\langle F_{n},\,{U}_{n}\right\rangle _{\mathcal {H}}\rightarrow 0. \end{aligned}$$

Taking the real part and using the dissipativity estimative (35), we obtain

$$\begin{aligned} \frac{1}{\lambda _{n}}\left( \mu _{0}\left\| u_{n,x}\right\| ^{2}+\mu _{1}\left\| \zeta _{n,x}\right\| ^{2}+\mu _{2}\left\| v_{n,x} \right\| ^{2}\right) ={\text {Re}}\left\langle F_{n},\,{U}_{n} \right\rangle _{\mathcal {H}}\rightarrow 0. \end{aligned}$$

This means

$$\begin{aligned} \frac{1}{\lambda _{n}}\left\| u_{n,x}\right\| ^{2}\ ,\ \frac{1}{\lambda _{n}}\left\| \zeta _{n,x}\right\| ^{2}\ ,\ \frac{1}{\lambda _{n} }\left\| v_{n,x}\right\| ^{2}\ \rightarrow 0. \end{aligned}$$

(81)

In particular

$$\begin{aligned} \frac{1}{\lambda _{n}}\left\| u_{n,x}\right\| ^{2}=\left\langle \frac{1}{\lambda _{n}}u_{n,x},\,{u}_{n,x}\right\rangle =i\left\langle w_{n,x} ,\,{u}_{n,x}\right\rangle -\left\langle f_{n,x}^{1},\,{u}_{n}\right\rangle \rightarrow 0, \end{aligned}$$

which, by (71), (72) implies

$$\begin{aligned} \left\langle w_{n,x},\,{u}_{n,x}\right\rangle \rightarrow 0 \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(82)

Similarly, replacing (67) and (69) in the last two convergences in (81) allow us to prove that

$$\begin{aligned} \left\langle \xi _{n,x},\,{\zeta }_{n,x}\right\rangle \rightarrow 0 \quad \text {and} \quad \left\langle s_{n,x},\,{v}_{n,x}\right\rangle \rightarrow 0 \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(83)

Taking the inner product of (68) with ${\zeta }_{n}$ in $L^{2}(0,l)$

$$\begin{aligned} i\left\| \zeta _{n}\right\| ^{2}+\frac{D}{I_{\rho }\lambda _{n} }\!\left\langle \xi _{n,x}, {\zeta }_{n,x}\right\rangle -\frac{G}{I_{\rho }\lambda _{n}}\!\left\langle 3s_{n}\!-\!\xi _{n}\!-\!w_{n,x}, {\zeta }_{n}\right\rangle +\frac{\mu _{1}}{I_{\rho }}\left( \frac{1}{\lambda _{n}}\!\left\| \zeta _{n,x}\right\| ^{2}\right) \! =\! \left\langle f_{n}^{4}, {\zeta }_{n}\right\rangle . \end{aligned}$$

Thanks to the convergences and boundedness (71)–(76), (81) and (83), as $n\rightarrow \infty$, we obtain

$$\begin{aligned} \zeta _{n}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l). \end{aligned}$$

(84)

Similarly, multiplying (66) with ${u}_{n}$, (70) by ${v}_{n}$ in $L^{2}(0,l)$, and using (71), (72), (73)–(76), (81) and (83), we obtain

$$\begin{aligned}{} & {} {u}_{n}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l) \quad \text {as} \quad n\rightarrow \infty , \end{aligned}$$

(85)

$$\begin{aligned}{} & {} {v}_{n}\rightarrow 0 \quad \text {in} \quad L^{2}(0,l) \quad \text {as} \quad n\rightarrow \infty . \end{aligned}$$

(86)

Finally, thanks to convergences (77)–(80) and (84)–(86), we conclude that $\left\| U_{n}\right\| _{\mathcal {H}}\rightarrow 0$, which is a contradiction, since $\left\| U_{n}\right\| _{\mathcal {H}}=1,$ and the proof of the theorem is completed. $\square$

3 Physical System – Partially Damped System

In this section, we consider a partially damped system with Kelvin-Voigt damping

$$\begin{aligned}&\rho w_{tt}+G(3s-\xi -w_{x})_{x}\! =\!0\,\, \text{ in } \, (0,l)\times (0,\infty ), \end{aligned}$$

(87)

$$\begin{aligned}&I_{\rho } \xi _{tt}-D\xi _{xx}-G(3s\!-\!\xi \!-\!w_{x})-\mu _{1}\xi _{xxt} \!=\!0 \,\, \text{ in } \, (0,l)\times (0,\infty ), \end{aligned}$$

(88)

$$\begin{aligned}&3I_{\rho } s_{tt}-3Ds_{xx}+3G(3s\!-\!\xi \!-\!w_{x})+4\gamma s-\mu _{2} s_{xxt}\!=\! 0 \,\, \text{ in } \, (0,l)\times (0,\infty ). \end{aligned}$$

(89)

The boundary conditions are given by

$$\begin{aligned} w_{x}(0,t)=w(l,t)=\xi (0,t)=\xi _{x}(l,t)=s(0,t)=s_{x}(l,t)=0, \quad t> 0, \end{aligned}$$

(90)

and the initial conditions are

$$\begin{aligned} \begin{aligned}&w(x,0)=w_0(x), \ w_t(x,0)=w_1(x), \,\,\, 0< x< l, \\&\xi (x,0)=\xi _0(x), \ \xi _t(x,0)=\xi _1(x), \,\,\, 0< x< l, \\&s(x,0)=s_0(x), \ s_t(x,0)=s_1(x), \,\,\, 0< x < l. \end{aligned} \end{aligned}$$

(91)

3.1 Lack of Exponential Decay

In this subsection, we study the qualitative behavior of the system solution (87)–(91) and we obtain sufficient conditions to guarantee the result of lack of exponential decay.

To do this, let us consider the following well-known result from semigroup theory (see e.g., [12, 36]).

Theorem 4

Let $S(t)=e^{\mathcal {A}^{(II)}t}$ be a $C_0$–semigroup of contractions on Hilbert space $\mathcal {H}$. Then S(t) is exponentially stable if and only if

$$\begin{aligned} i\mathrm{I\!R}:= \big \{i\lambda ; \ \lambda \in \mathrm{I\!R}\big \} \subset \rho \big (\mathcal {A}^{(II)}\big ) \end{aligned}$$

and

$$\begin{aligned} \limsup _{ \left| \lambda \right| \rightarrow \infty } \Vert (i\lambda I-\mathcal {A}^{(II)})^{-1}\Vert _{\mathcal {L}(\mathcal {H})}<\infty , \quad \lambda \in \mathrm{I\!R}. \end{aligned}$$

The first result of this subsection is given by the following theorem.

Theorem 5

The semigroup $S(t)=e^{\mathcal {A}^{(II)}t}$ associated with the system (87)–(91) is not exponentially stable.

Proof

We will use the Theorem 4. To prove the lack of exponential stability we will prove that there exists a sequence $(\lambda _n)_{n\in \mathrm{I\!N}} \subset \mathrm{I\!R}$ with $\lambda _n \rightarrow \infty$ and $(U_n)_{n\in \mathrm{I\!N}}\subset D(\mathcal {A}^{(II)})$ for $(F_n)_{n\in \mathrm{I\!N}}\subset \mathcal {H}$, with $\Vert F_n\Vert _{\mathcal {H}}<\infty$ such that

$$\begin{aligned} \Big (i \lambda _nI-\mathcal {A}^{(II)}\Big )U_{n}=F_{n}, \end{aligned}$$

(92)

where $(F_n)$ is bounded in $\mathcal {H}$, but $\Vert U_{n}\Vert _{\mathcal {H}}$ tends to infinity. Rewriting the resolvent equation (92) in term of its components with $U_{n}=\big (w_{n},u_{n},\xi _{n},\zeta _{n}, s_{n}, v_{n}\big )^{\top }$ and $F_n =\big (0, -\rho ^{-1}\cos (\beta _nx), 0, 0, 0, 0\big )^{\top }$, where $\beta _n:=\frac{(2n+1)\pi }{2l}$, we have

$$\begin{aligned}&i\lambda _{n} w_{n}-u_{n} = 0, \end{aligned}$$

(93)

$$\begin{aligned}&i\lambda _{n}\rho u_{n}+G(3s_{n}\!-\!\xi _{n}\!-\!w_{n,x})_{x} = -\cos (\beta _nx), \end{aligned}$$

(94)

$$\begin{aligned}&i\lambda _{n}\xi _{n}-\zeta _{n} = 0, \end{aligned}$$

(95)

$$\begin{aligned}&i\lambda _{n} I_{\rho } \zeta _{n}-D\xi _{n,xx}-G(3s_{n}\!-\!\xi _{n}\!-\!w_{n,x})-\mu _{1} \zeta _{n,xx} = 0, \end{aligned}$$

(96)

$$\begin{aligned}&i\lambda _{n} s_{n}-v_{n} = 0, \end{aligned}$$

(97)

$$\begin{aligned}&i\lambda _{n}3I_{\rho } v_{n}-3Ds_{n,xx}+3G(3s_{n}\!-\!\xi _{n}\!-Vw_{n,x})+4\gamma s_{n}-\mu _{2} v_{n,xx} = 0. \end{aligned}$$

(98)

From (93), (95) and (97), we get $u_{n}=i\lambda _nw_{n}$, $\zeta _{n}=i\lambda _n\xi _{n}$ and $v_{n}=i\lambda _ns_{n}$. So we can write

$$\begin{aligned}&-\lambda ^{2}_{n}\rho w_{n}+G(3s_{n}\!-\!\xi _{n}\!-\!w_{n,x})_{x}=-\cos (\beta _nx), \end{aligned}$$

(99)

$$\begin{aligned}&-\lambda ^{2}_{n} I_{\rho } \xi _{n}-D\xi _{n,xx}-G(3s_{n}\!-\!\xi _{n}\!-\!w_{n,x})-i\lambda _{n}\mu _{1} \xi _{n,xx}=0, \end{aligned}$$

(100)

$$\begin{aligned}&-\lambda ^{2}_{n}3I_{\rho } s_{n}-3Ds_{n,xx}+3G(3s_{n}\!-\!\xi _{n}\!-\!w_{n,x})+4\gamma s_{n}-i\lambda _{n}\mu _{2} s_{n,xx}=0. \end{aligned}$$

(101)

Due to the boundary conditions (90), the functions given by

$$\begin{aligned} w_{n}(x)={\textbf {A}}_n\cos (\beta _nx), \quad \xi _{n}(x)={\textbf {B}}_n\sin (\beta _nx), \quad s_{n}(x)={\textbf {C}}_n\sin (\beta _nx), \end{aligned}$$

(102)

solve the system (99)–(101) if only if ${\textbf {A}}_n, {\textbf {B}}_n$ and ${\textbf {C}}_n$ satisfy

$$\begin{aligned}&\Big (\lambda _n^2\rho -\beta _{n}^{2}G\Big ){\textbf {A}}_{n}+\beta _{n}G {\textbf {B}}_{n}-3\beta _{n}G {\textbf {C}}_{n}=1, \end{aligned}$$

(103)

$$\begin{aligned}&\beta _{n}G {\textbf {A}}_{n}+\Big (\lambda _n^2I_{\rho }-\beta _{n}^{2}D-G-i\lambda _{n}\mu _{1} \beta _{n}^{2}\Big ){\textbf {B}}_{n}+3G{\textbf {C}}_{n}=0, \end{aligned}$$

(104)

$$\begin{aligned}&3\beta _{n}G {\textbf {A}}_{n}-3G{\textbf {B}}_{n}-\Big (\lambda _n^23I_{\rho }-3\beta _{n}^{2}D-9G-4\gamma -i\lambda _{n}\mu _{2} \beta _{n}^{2}\Big ){\textbf {C}}_{n}=0. \end{aligned}$$

(105)

We choose the sequence of real numbers defined by

$$\begin{aligned} \lambda _{n}:=\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}, \quad \forall \, n\in \mathrm{I\!N}, \end{aligned}$$

which gives $\lambda _n^2\rho -\beta _{n}^{2}G=G$. Therefore,

$$\begin{aligned} G {\textbf {A}}_{n}+\beta _{n}G {\textbf {B}}_{n}-3\beta _{n}G {\textbf {C}}_{n}= & {} 1, \end{aligned}$$

(106)

$$\begin{aligned} \beta _{n}G {\textbf {A}}_{n}+R_{n}{} {\textbf {B}}_{n}+3G{\textbf {C}}_{n}= & {} 0, \end{aligned}$$

(107)

$$\begin{aligned} 3\beta _{n}G {\textbf {A}}_{n}-3G{\textbf {B}}_{n}-S_{n}{} {\textbf {C}}_{n}= & {} 0, \end{aligned}$$

(108)

where we denote,

$$\begin{aligned}{} & {} R_{n}:=I_{\rho }\bigg [\bigg (\frac{G}{\rho }-\frac{G}{I_{\rho }}\bigg ) +\bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }}\bigg )\beta _{n}^{2}-\frac{\mu _{1}}{I_{\rho }} i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}\beta _{n}^{2}\bigg ], \end{aligned}$$

(109)

$$\begin{aligned}{} & {} S_{n}:=3I_{\rho }\bigg [\bigg (\frac{G}{\rho }-\frac{3G}{I_{\rho }} \bigg )+\bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }}\bigg ) \beta _{n}^{2}-\frac{4\gamma }{3I_{\rho }}-\frac{\mu _{2}}{3I_{\rho }} i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}\beta _{n}^{2}\bigg ].\qquad \quad \end{aligned}$$

(110)

Solving the equations (107)–(108) we have

$$\begin{aligned} {\textbf {A}}_{n}=Q_{n}^{1}{} {\textbf {C}}_{n} \qquad \text{ and } \qquad {\textbf {B}}_{n}=Q_{n}^{2}{} {\textbf {C}}_{n}, \end{aligned}$$

(111)

where,

$$\begin{aligned} Q_{n}^{1}:= & {} \frac{R_{n}S_{n}-9G^{2}}{3I_{\rho }G\beta _{n}\bigg [\bigg (\frac{G}{\rho }-\frac{G}{I_{\rho }}\bigg ) +\bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }}\bigg )\beta _{n}^{2}-\frac{\mu _{1}}{I_{\rho }} i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}\beta _{n}^{2}\bigg ]+3G^{2}\beta _{n}} \nonumber \\\sim & {} {-}\frac{\mu _{1}\mu _{2}\frac{G}{\rho } \big (1+\beta _{n}^{2}\big )\beta _{n}+(3\mu _{1}+\mu _{2})I_{\rho } \bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }}\bigg )i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )} \beta _{n}}{3I_{\rho }G\bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }} \bigg )-3i\mu _{1}G\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}}, \end{aligned}$$

(112)

$$\begin{aligned} Q_{n}^{2}:= & {} -\frac{\beta _{n}(S_{n} +9G)G}{3I_{\rho }G\beta _{n}\bigg [\bigg (\frac{G}{\rho }-\frac{G}{I_{\rho }}\bigg )+\bigg (\frac{G}{\rho } -\frac{D}{I_{\rho }}\bigg )\beta _{n}^{2}-\frac{\mu _{1}}{I_{\rho }} i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}\beta _{n}^{2}\bigg ]+3G^{2}\beta _{n}} \nonumber \\\sim & {} -\frac{3I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big )-\mu _{2} i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}}{3I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big ) -3\mu _{1}i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}}. \end{aligned}$$

(113)

Combining the equations (106) and (111), we obtain

$$\begin{aligned} \scriptstyle \frac{\mu _{1}\mu _{2}\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )\beta _{n} +\Big ((3\mu _{1}+\mu _{2})I_{\rho }\Big (\frac{G}{\rho } -\frac{D}{I_{\rho }}\Big )-\big (9\mu _{1}+\mu _{2}\big )G\Big )i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}\beta _{n}+12I_{\rho }\Big (\frac{G}{\rho } -\frac{D}{I_{\rho }}\Big )G\beta _{n}}{3I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big ) -3\mu _{1}i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}} {\textbf {C}}_{n}\sim -1 \end{aligned}$$

and

$$\begin{aligned} \scriptstyle {\textbf {C}}_{n}\sim -\frac{3I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big )-3\mu _{1}i \sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}}{\mu _{1}\mu _{2}\frac{G}{\rho } \big (1+\beta _{n}^{2}\big )\beta _{n}+\Big ((3\mu _{1}+\mu _{2})I_{\rho }\Big (\frac{G}{\rho } -\frac{D}{I_{\rho }}\Big )-\big (9\mu _{1}+\mu _{2}\big )G\Big )i\sqrt{\frac{G}{\rho } \big (1+\beta _{n}^{2}\big )}\beta _{n}+12I_{\rho }\Big (\frac{G}{\rho } -\frac{D}{I_{\rho }}\Big )G\beta _{n}}. \end{aligned}$$

From (111), we obtain

$$\begin{aligned} \scriptstyle {\textbf {A}}_{n}\sim \frac{\mu _{1}\mu _{2}\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )+(3\mu _{1}+\mu _{2})I_{\rho } \Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big )i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}}{\mu _{1}\mu _{2} \frac{G}{\rho }\big (1+\beta _{n}^{2}\big )+\Big ((3\mu _{1}+\mu _{2})I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }} \Big )-\big (9\mu _{1}+\mu _{2}\big )G \Big )i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}+12I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big )G} \end{aligned}$$

and

$$\begin{aligned} \scriptstyle {\textbf {B}}_{n}\sim \frac{3I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big )-\mu _{2} i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )}}{\mu _{1}\mu _{2} \frac{G}{\rho }\big (1+\beta _{n}^{2}\big )\beta _{n}+\Big ((3\mu _{1} +\mu _{2})I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big ) -\big (9\mu _{1}+\mu _{2}\big )G\Big )i\sqrt{\frac{G}{\rho }\big (1+\beta _{n}^{2}\big )} \beta _{n}+12I_{\rho }\Big (\frac{G}{\rho }-\frac{D}{I_{\rho }}\Big )G\beta _{n}}, \end{aligned}$$

Finally, since $\mu _{1}>0$ and $\mu _{2}>0$ we have ${\textbf {A}}_{n}\sim 1$. Consequently

$$\begin{aligned} \Vert U_{n}\Vert _{\mathcal {H}}^{2}\ge \rho \Vert u_{n}\Vert ^{2}=\lambda _{n}^{2} \rho \Vert w_{n}\Vert ^{2}= \lambda _{n}^{2}\rho \left| A_{n}\right| ^{2} \int _{0}^{L} \left| \cos ( \beta _n x) \right| ^{2}dx\sim \mathcal {O}(n^2). \qquad \end{aligned}$$

(114)

Then, as $n\rightarrow \infty$, we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert U_{n}\Vert _{\mathcal {H}}^{2}\ge \rho \lim _{n\rightarrow \infty }\Vert u_{n}\Vert ^{2}=\infty . \end{aligned}$$

(115)

Applying the Theorem 4, we conclude that the semigroup S(t) associated with the system (87)–(91) does not have exponential decay. $\square$

Remark 6

From the equations (111)–(113) we can infer other results of lack of exponential decay of the system (87)–(91) depending on where the damping is acting. The table below summarizes this information.

We can see that in the two cases: (i) $\mu _{1}=0$ and $\mu _{2}>0$ or (ii) $\mu _{1}>0$ and $\mu _{2}=0$, where the damping acts on a single equation, we obtain lack of exponential stability since

$$\begin{aligned} \chi _{0}:=\frac{G}{\rho }-\frac{D}{I_{\rho }} \ne 0, \,\, \chi _{1}:=I_{\rho }\bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }}\bigg )-G \ne 0, \,\, \chi _{2}:=I_{\rho }\bigg (\frac{G}{\rho }-\frac{D}{I_{\rho }}\bigg )-3G \ne 0. \end{aligned}$$

For the case (iii) $\mu _{1}>0$ and $\mu _{2}>0$, where damping acts in both equations, the lack of exponential decay does not depend on $\chi _{i} \ne 0$, $i=0, 1, 2$ (Table 1).

Table 1 Lack of exponential decay

3.2 Polynomial Stability

This section will show that the semigroup associated with the system (87)–(91) decays polynomially to zero with rate $t^{-1/4}$. For this purpose, we will use a polynomial stability result due to Borichev and Tomilov [3]:

Theorem 7

Let $S(t) = e^{\mathcal {A}^{(II)}t}$ be a bounded $C_0$-semigroup on Hilbert space $\mathcal {H}$ such that $i\mathrm{I\!R}\subset \rho (\mathcal {A}^{(II)})$. Then, to fixed $\alpha >0$, the following conditions are equivalent:

• (i) $\displaystyle \sup _{\left| \lambda \right| >\delta }\left| \lambda \right| ^{-\alpha }\Vert (i\lambda I-\mathcal {A}^{(II)})^{-1}F\Vert _{\mathcal {L}(\mathcal {H})}\le C$ for any $\delta >1;$

• (ii) There exists a constant $C>0$ such that $\displaystyle \Vert S(t)U_{0}\Vert _{\mathcal {H}}\le \frac{C}{t^{1/\alpha }}\Vert U_{0}\Vert _{D(\mathcal {A}^{(II)})}$.

To start, let us consider here the resolvent equation written as

$$\begin{aligned} \Big (i\lambda -\mathcal {A}^{(II)}\Big )U=F, \end{aligned}$$

(116)

where $U = \big (w ,u ,\xi ,\zeta , s , v \big )^{\top }$ and $F = (f_{1}, f_{2}, f_{3}, f_{4}, f_{5}, f_{6})^{\top }$. In terms of its coordinates, we have

$$\begin{aligned} i\lambda w-u= & {} f_{1}, \end{aligned}$$

(117)

$$\begin{aligned} i\lambda \rho u+G(3s-\xi -w_{x})_{x}= & {} \rho f_{2}, \end{aligned}$$

(118)

$$\begin{aligned} i\lambda \xi -\zeta= & {} f_{3}, \end{aligned}$$

(119)

$$\begin{aligned} i\lambda I_{\rho } \zeta -D\xi _{xx} -G(3s -\xi -w_{x} )-\mu _{1} \zeta _{xx}= & {} I_{\rho }f_{4}, \end{aligned}$$

(120)

$$\begin{aligned} i\lambda s -v= & {} f_{5}, \end{aligned}$$

(121)

$$\begin{aligned} i\lambda 3I_{\rho } v -3Ds_{xx} +3G(3s -\xi -w_{x} )+4\gamma s -\mu _{2} v_{xx}= & {} 3I_{\rho }f_{6}. \end{aligned}$$

(122)

From (35)$_{2}$, it is easy to see that we have

$$\begin{aligned} \mu _{1}\Vert \zeta _{x}\Vert ^2 +\mu _{2}\Vert v_{x}\Vert ^2\le C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}. \end{aligned}$$

(123)

To prove the polynomial stability of the system, we will use the following technical lemmas.

Lemma 1

Let $\big (w ,u ,\xi ,\zeta , s , v \big )^{\top }$ be a solution of system (117)–(122). Then, there exists a positive constant C, independent of $\lambda$, such that

$$\begin{aligned} G\Vert 3s -\xi -w_{x}\Vert ^{2}\le&\, \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2}\\&+ \lambda ^{4}C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+\mu _{1}C\Vert F\Vert _{\mathcal {H}}^{2}. \end{aligned}$$

Proof

Multiplying Eq. (120) by ${3s-\xi -w_{x}}$ and performing an integration by parts, we have

$$\begin{aligned}&\displaystyle i\lambda I_{\rho } \Big<\zeta , \,{3s-\xi -w_{x}}\Big>+\underbrace{D\Big<\xi _{x}, (3s-\xi -w_{x})_{x} \Big>}_{J_{1}:=}-G\Vert 3s-\xi -w_{x}\Vert ^{2}&\nonumber \\&\displaystyle +\underbrace{\mu _{1} \Big<\zeta _{x}, (3s-\xi -w_{x})_{x}}\Big>_{J_{2}:=}=I_{\rho }\Big <f_{4}, \,{3s-\xi -w_{x}}\Big >.&\end{aligned}$$

(124)

From Eq. (118) and (119) we have

$$\begin{aligned} J_{1}:= & {} D\Big<\xi _{x}, (3s-\xi -w_{x})_{x} \Big>=i\lambda \frac{\rho D}{G}\Big<\xi _{x}, \,{u}\,\Big>+ \frac{\rho D}{G}\Big<\xi _{x}, \,{f}_{2}\Big> \nonumber \\ J_{1}= & {} \frac{\rho D}{G}\Big<\zeta _{x}, \,{u}\,\Big>+\frac{\rho D}{G}\Big<f_{3,x}, \,{u}\,\Big>+ \frac{\rho D}{G}\Big <\xi _{x}, \,{f}_{2}\Big >. \end{aligned}$$

(125)

On the other hand, using Eq. (118) and (119) in $J_{2}$ we have

$$\begin{aligned} J_{2}:= & {} \mu _{1} \Big<\zeta _{x}, (3s-\xi -w_{x})_{x} \Big>=-i\lambda \frac{\rho \mu _{1}}{G}\Big<\zeta , \,{u}_{x}\Big>+\frac{\rho \mu _{1}}{G}\Big<{f}_{2}, \,\zeta _{x}\Big> \nonumber \\ J_{2}= & {} -i\lambda \frac{\rho \mu _{1}}{G}\Big<\zeta , \, {u}_{x}\Big>+i\lambda \frac{\rho \mu _{1}}{G}\Big<{f}_{2}, \,\xi _{x}\Big>-\frac{\rho \mu _{1}}{G}\Big <{f}_{2}, \,f_{3,x}\Big >. \end{aligned}$$

(126)

From Eq. (117) we obtain

$$\begin{aligned} J_{2}= & {} -\lambda ^{2}\frac{\rho \mu _{1}}{G}\Big<\zeta , \,{w}_{x}\,\Big>-i\lambda \frac{\rho \mu _{1}}{G}\Big<\zeta _{x}, \,{f}_{1}\,\Big>+i\lambda \frac{\rho \mu _{1}}{G}\Big<{f}_{2}, \,\xi _{x}\Big>-\frac{\rho \mu _{1}}{G}\Big<{f}_{2}, \,f_{3,x}\Big> \nonumber \\= & {} \lambda ^{2}\frac{\rho \mu _{1}}{G}\Big<\zeta , \,{3s-\xi -w_{x}}\,\Big>-\lambda ^{2}\frac{3\rho \mu _{1}}{G}\Big<\zeta , \,{s}\,\Big>+\lambda ^{2}\frac{\rho \mu _{1}}{G}\Big<\zeta , \,{\xi }\Big>+i\lambda \frac{\rho \mu _{1}}{G}\Big<\zeta _{x}, \,{f}_{1}\,\Big> \nonumber \\ {}{} & {} +i\lambda \frac{\rho \mu _{1}}{G}\Big<{f}_{2}, \,\xi _{x}\Big>-\frac{\rho \mu _{1}}{G}\Big <{f}_{2}, \,f_{3,x}\Big >. \end{aligned}$$

(127)

Consequently, from (124), (125) and (127), we obtain

$$\begin{aligned} G\Vert 3s-\xi -w_{x}\Vert ^{2}=&\,i\lambda I_{\rho } \Big<\zeta , \,{3s-\xi -w_{x}}\Big>+\frac{\rho D}{G}\Big<\zeta _{x}, \,{u}\,\Big>+\frac{\rho D}{G}\Big<f_{3,x}, \,{u}\,\Big> \nonumber \\&+ \frac{\rho D}{G}\Big<\xi _{x}, \,{f}_{2}\Big> +\lambda ^{2}\frac{\rho \mu _{1}}{G}\Big<\zeta , \,{3s-\xi -w_{x}}\,\Big> \nonumber \\ {}&-I_{\rho }\Big<f_{4}, \,{3s-\xi -w_{x}}\Big> -\lambda ^{2}\frac{3\rho \mu _{1}}{G}\Big<\zeta , \,{s}\Big>+\lambda ^{2}\frac{\rho \mu _{1}}{G}\Big<\zeta , \,{\xi }\,\Big>&\nonumber \\ {}&-i\lambda \frac{\rho \mu _{1}}{G}\Big<\zeta _{x}, \,{f}_{1}\,\Big>+i\lambda \frac{\rho \mu _{1}}{G}\Big<{f}_{2}, \,\xi _{x}\Big>-\frac{\rho \mu _{1}}{G}\Big <{f}_{2}, \,f_{3,x}\Big >.&\end{aligned}$$

(128)

By using the Young’s inequality and estimate (123), we obtain

$$\begin{aligned} G\Vert 3s-\xi -w_{x}\Vert ^{2}\le&\, \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2} \nonumber \\ {}&+\lambda ^{4}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+\lambda ^{2}C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}&\nonumber \\ {}&+ \left| \lambda \right| C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+ C\Vert F\Vert _{\mathcal {H}}^{2}.&\end{aligned}$$

(129)

This concludes the proof. $\square$

Lemma 2

Let $\big (w ,u ,\xi ,\zeta , s , v \big )^{\top }$ be a solution of system (117)–(122). Then, there exists a positive constant C, independent of $\lambda$, such that

$$\begin{aligned} D\Vert \xi _{x}\Vert ^{2}\le&\, \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2}\\&+ \lambda ^{4}C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert F\Vert _{\mathcal {H}}^{2}. \end{aligned}$$

Proof

Multiplying Eq. (120) by ${\xi }$, we have

$$\begin{aligned} D\Vert \xi _{x}\Vert ^{2}=-i\lambda I_{\rho } \Big<\zeta , \,{\xi }\Big>+G\Big<3s-\xi -w_{x}, \,{\xi }\Big>+\mu _{1} \Big<\zeta _{x}, \,{\xi }_{x}\Big>+I_{\rho }\Big <f_{4}, \,{\xi }\Big >. \end{aligned}$$

(130)

Then, by using the Cauchy-Schwarz, Young, Poincaré inequalities and estimate (123), we get

$$\begin{aligned}&\displaystyle D\Vert \xi _{x}\Vert ^{2}\le C\Vert 3s-\xi -w_{x}\Vert ^{2}+\lambda ^{2}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} +C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}&\end{aligned}$$

(131)

Finally, using the Lemma 1 we have

$$\begin{aligned} D\Vert \xi _{x}\Vert ^{2}\le&\, \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2} \\ {}&\lambda ^{4}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} +\lambda ^{2}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} +C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert F\Vert _{\mathcal {H}}^{2}.&\end{aligned}$$

This concludes the proof. $\square$

Lemma 3

Let $\big (w ,u ,\xi ,\zeta , s , v \big )^{\top }$ be a solution of system (117)–(122). Then, there exists a positive constant C, independent of $\lambda$, such that

$$\begin{aligned} \rho \Vert u\Vert ^{2} \le&\, \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2} \\ {}&+\lambda ^{4}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert F\Vert _{\mathcal {H}}^{2}, \end{aligned}$$

for $\varepsilon _{i}>0$, $(i=1,2,3)$ and $\left| \lambda \right| > 1$ large enough.

Proof

Multiplying Eq. (118) by $-i\lambda ^{-1}{u}$ we have

$$\begin{aligned} \rho \Vert u\Vert ^{2}+\frac{iG}{\lambda }\Big<3s-\xi -w_{x}, \,{u}_{x}\Big>=-\frac{i\rho }{\lambda }\Big <f_{2}, \,{u}\Big >. \end{aligned}$$

(132)

From (117) we have ${u}_{x}=-(i\lambda {w}_{x}+{f}_{1,x} )$, and consequently,

$$\begin{aligned} \rho \Vert u\Vert ^{2}= & {} G\Big<3s-\xi -w_{x}, \,{w}_{x}\Big>-\frac{iG}{\lambda }\Big<{f}_{1,x}, \,3s-\xi -w_{x}\Big>-\frac{i\rho }{\lambda }\Big<f_{2}, \,{u}\Big> \nonumber \\= & {} -G\Vert 3s-\xi -w_{x}\Vert ^{2}+G\Big<3s-\xi -w_{x}, \,3s-\xi \Big> \nonumber \\{} & {} -\frac{iG}{\lambda }\Big<{f}_{1,x}, \,3s-\xi -w_{x}\Big>-\frac{i\rho }{\lambda }\Big <f_{2}, \,{u}\Big >. \end{aligned}$$

By using the Cauchy-Schwarz and Young inequalities and applying Lemma 1, we have

$$\begin{aligned} \rho \Vert u\Vert ^{2} \le C_{\varepsilon _{2,3}}\Vert 3s-\xi -w_{x}\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2} +D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2}+\frac{C}{\left| \lambda \right| } \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} \end{aligned}$$

and

$$\begin{aligned}&\displaystyle \rho \Vert u\Vert ^{2} \le \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2} +\lambda ^{4}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}&\nonumber \\&\displaystyle +\frac{C}{\left| \lambda \right| } \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} +\mu _{1}C \Vert F\Vert _{\mathcal {H}}^{2}.&\end{aligned}$$

This completes the proof of Lemma. $\square$

Lemma 4

Let $\big (w ,u ,\xi ,\zeta , s , v \big )^{\top }$ be a solution of system (117)–(122). Then, there exists a positive constant C, independent of $\lambda$, such that

$$\begin{aligned} \displaystyle 3D\Vert s_{x}\Vert ^{2}+4\gamma \Vert s\Vert ^{2}{} & {} \le \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2} \nonumber \\ {}{} & {} \quad +\lambda ^{4}C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert F\Vert _{\mathcal {H}}^{2} \end{aligned}$$

(133)

Proof

Multiplying Eq. (122) by s, we have

$$\begin{aligned}&\displaystyle 3D\Vert s_{x}\Vert ^{2}+4\gamma \Vert s\Vert ^{2}=-i\lambda 3I_{\rho }\Big<v, \,{s}\Big>-3G\Big<3s -\xi -w_{x}, \,{s}\Big>&\nonumber \\&\displaystyle -\mu _{2}\Big<v_{x}, \,{s}_{x}\Big> +3I_{\rho }\Big <f_{6}, \,{s}\Big >.&\end{aligned}$$

(134)

By using the Cauchy-Schwarz, Young and Poincaré inequalities and estimate (123) we get

$$\begin{aligned} 3D\Vert s_{x}\Vert ^{2}+4\gamma \Vert s\Vert ^{2}\le \lambda C\Vert v\Vert ^{2}+C\Vert 3s -\xi -w_{x}\Vert ^{2}+C\Vert v_{x}\Vert ^{2} +C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}\nonumber \\ \end{aligned}$$

(135)

and

$$\begin{aligned} 3D\Vert s_{x}\Vert ^{2}+4\gamma \Vert s\Vert ^{2}\le C\Vert 3s -\xi -w_{x}\Vert ^{2}+ \left| \lambda \right| C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H} }+ C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}.\nonumber \\ \end{aligned}$$

(136)

Finally, using the Lemma 1, we obtain

$$\begin{aligned}{} & {} \displaystyle 3D\Vert s_{x}\Vert ^{2}+4\gamma \Vert s\Vert ^{2}\le \rho \varepsilon _{1}C\Vert u\Vert ^{2}+3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2} +D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2}+\lambda ^{4}C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} \nonumber \\{} & {} \quad \displaystyle + \left| \lambda \right| C \Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}} +C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+ C\Vert F\Vert _{\mathcal {H}}^{2}. \end{aligned}$$

(137)

This completes the proof of Lemma. $\square$

Before using the above lemmas to prove polynomial decay, we need to ensure that $i\mathrm{I\!R}\subset \rho (\mathcal {A}^{(II)})$. In this case we will reason on the above system (117)–(122) with $F = 0$.

Proposition 4

$D(\mathcal {A})\Subset \mathcal {H}$, i.e., the embedding $D(\mathcal {A})\subset \mathcal {H}$ is compact.

Proof

Let $U_{n}=\big (w_{n} ,u_{n} ,\xi _{n} ,\zeta _{n} , s_{n} , v_{n} \big )^{\top }$ be a sequence bounded in $D(\mathcal {A})$. In particular, we have

$$\begin{aligned}{} & {} w_{n} \ \ \text{ is } \text{ bounded } \text{ in } \ \ H^{2}(0,l)\Subset H^{1}_{b}(0,l), \quad \zeta _{n}, \ v_{n} \ \ \text{ is } \text{ bounded } \text{ in } \ \ H^{1}_{a}(0,l)\Subset L^{2}(0,l),\\{} & {} u_{n} \ \, \text{ is } \text{ bounded } \text{ in } \ \ H^{1}_{b}(0,l)\Subset L^{2}(0,l). \end{aligned}$$

Consequently, there are $w \in H^{1}_{b}(0,l)$ and $\zeta , v, u\in L^{2}(0,l)$ such that, up to a subsequence,

$$\begin{aligned} w_{n}\rightarrow w \ \ \text{ in } \ \ H^{1}_{b}(0,l), \\ \zeta _{n}\ \rightarrow \ \zeta \ \ \text{ in } \ \ L^{2}(0,l),\\ \quad v_{n}\ \rightarrow \ v \ \ \text{ in } \ \ L^{2}(0,l),\\ \quad u_{n}\ \rightarrow \ u \ \ \text{ in } \ \ L^{2}(0,l). \end{aligned}$$

It remains to prove the convergence $\xi _{n}\ \rightarrow \ \xi$ in $H^{1}_{a}(0,l)$ for some $\xi$ in $H^{1}_{a}(0,l)$. Indeed, knowing that

$$\begin{aligned} D\xi _{n}+\mu _{1}\zeta _{n}, \ Ds_{n}+\mu _{2}v_{n} \ \ \text{ is } \text{ bounded } \text{ in } \ \ H^{2}(0,l)\Subset H^{1}_{a}(0,l), \end{aligned}$$

we get the convergence, up to a subsequence

$$\begin{aligned} D\xi _{n}+\mu _{1}\zeta _{n} \ \ \rightarrow \ \ \sigma _{1} \ \ \text{ in } \ \ H^{1}_{a}(0,l) \quad \text{ and } \quad Ds_{n}+\mu _{2}v_{n} \ \ \rightarrow \ \ \sigma _{2} \ \ \text{ in } \ \ H^{1}_{a}(0,l),\nonumber \\ \end{aligned}$$

(138)

From (138), we obtain $\xi _{n} \ \rightarrow \ \xi :=D^{-1}\big (\sigma _{1} -\mu _{1}\zeta \big ) \ \ \text{ in } \ \ H^{1}_{a}(0,l)$ and $s_{n} \ \rightarrow \ \xi :=D^{-1}\big (\sigma _{2} -\mu _{2}v\big ) \ \ \text{ in } \ \ H^{1}_{a}(0,l)$. $\square$

Remark 8

Since $D(\mathcal {A})\Subset \mathcal {H}$, the inverse $\mathcal {A}^{-1}$ is compact. It follows immediately from Lemma 5 (below) that the spectrum of $\mathcal {A}$ consists entirely of isolated eigenvalues.

Lemma 5

(Kato [16], Theorem 6.29) Let $\mathcal {A} : D(\mathcal {A}) \subset X \, \rightarrow \, X$ a closed linear operator acting on a complex Banach space X. If $\mathcal {A}$ is invertible and the inverse operator $\mathcal {A}^{-1}$ is compact, then the spectrum of $\mathcal {A}$ consists entirely of isolated eigenvalues.

Lemma 6

Under the above notations we have that $i\mathrm{I\!R}\subset \rho (\mathcal {A}^{(II)})$.

Proof

Let us suppose that $\mathcal {A}^{(II)}$ has an imaginary eigenvalue. Then we have that

$$\begin{aligned} (i\lambda -\mathcal {A}^{(II)})U=0, \quad \lambda \in \mathrm{I\!R}. \end{aligned}$$

(139)

From estimate (123) we get $\zeta _{x}= 0$ and $v_{x}=0$ in $L^{2}(0,l)$. Using Poincaré’s inequality we have $\zeta = 0$ and $v =0$ in $L^{2}(0,l)$, which implies that $\xi _{x}= 0$ and $s_{x}=0$ also in $L^{2}(0,l)$ (see Eqs. (119) and (121)). On the other hand, from Lemma 3 with $\varepsilon _{1}=1/2C$ we have

$$\begin{aligned} \frac{\rho }{2}\Vert u\Vert ^{2} \le 3D\varepsilon _{2}C\Vert s_{x}\Vert ^{2}+D\varepsilon _{3}C\Vert \xi _{x}\Vert ^{2}, \quad \forall \, \varepsilon _{i}>0, \quad i=2,3, \end{aligned}$$

which implies that $u=0$ in $L^{2}(0,l)$ and from (117) we obtain $w=0$ too in $L^{2}(0,l)$. Finally from (117)–(122) we have

$$\begin{aligned} \quad 3s-\xi -w_{x}=0 \quad \text{ in } \quad L^{2}(0,l). \end{aligned}$$

(140)

This implies that $U=0$. But this is a contradiction, therefore there are no imaginary eigenvalues. $\square$

We are now in a position to establish our polynomial decay result.

Theorem 9

(Polynomial decay) The semigroup S(t) associated to system (117)–(122) satisfies

$$\begin{aligned} \Vert S(t)U_{0}\Vert _{\mathcal {H}}\le \frac{C}{t^{1/4}}\Vert U_{0}\Vert _{D(\mathcal {A}^{(II)})}, \quad \forall \, t>0, \ \ U_{0}\in D(\mathcal {A}^{(II)}). \end{aligned}$$

(141)

Proof

To show the polynomial stability, we use Theorem 7. It follows from Lemmas 1, 2, 3 and 4 that

$$\begin{aligned}&\displaystyle \rho (1-\varepsilon _{1}C)\Vert u\Vert ^{2}+ D(1-\varepsilon _{3}C)\Vert \xi _{x}\Vert ^{2}+3D(1-\varepsilon _{2}C)\Vert s_{x}\Vert ^{2}+4\gamma \Vert s\Vert ^{2}&\nonumber \\&\displaystyle +G\Vert 3s-\xi -w_{x}\Vert ^{2}\le \lambda ^{4}C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert F\Vert _{\mathcal {H}}^{2}.&\end{aligned}$$

(142)

Using Poincaré’s inequality in estimate (123) we have

$$\begin{aligned} \Vert \zeta \Vert ^{2}+\Vert v\Vert ^{2}\le C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}. \end{aligned}$$

(143)

Adding (142), (143) and choosing $\varepsilon _{1}=\varepsilon _{2}=\varepsilon _{3}=\frac{1}{2C}$ we have

$$\begin{aligned} \Vert U\Vert _{\mathcal {H}}^{2}\le \lambda ^{4}C\Vert U\Vert _{\mathcal {H}}\Vert F\Vert _{\mathcal {H}}+C\Vert F\Vert _{\mathcal {H}}^{2}\le \eta C\Vert U\Vert _{\mathcal {H}}^{2}+\lambda ^{16}C\Vert F\Vert _{\mathcal {H}}^{2}+C\Vert F\Vert _{\mathcal {H}}^{2}.\nonumber \\ \end{aligned}$$

(144)

Consequently, for $\left| \lambda \right| > 1$ we have

$$\begin{aligned} (1-\eta C)\Vert U\Vert _{\mathcal {H}}^{2}\le \lambda ^{16}C\Vert F\Vert _{\mathcal {H}}^{2}. \end{aligned}$$

(145)

We choose $\eta$ small enough such that $1-\eta C>0$. Then we get after using (116)

$$\begin{aligned} \frac{1}{\lambda ^{4}}\Vert (i\lambda I-\mathcal {A}^{(II)})^{-1}F\Vert _{\mathcal {H}}\le C\Vert F\Vert _{\mathcal {H}}. \end{aligned}$$

(146)

Therefore, from Borichev and Tomilov Theorem, we prove that the solution decays polynomially (slow) as $t^{-1/4}$ as time goes to infinity. $\square$