Asymptotic behavior of the Rao–Nakra sandwich beam model with Kelvin–Voigt damping

Abstract

The Rao–Nakra sandwich beam is a coupled system consisting of two wave equations for the longitudinal displacements of the top and bottom layers and an Euler–Bernoulli beam equation for the transversal displacement. This paper concerns the system’s stability when the Kelvin–Voigt damping terms act on the first and third equations. Using the semigroup theory of linear operators, we prove the global well-posedness of the associated initial boundary value problem. And then, we prove the lack of exponential stability of the system. Because of this lack of exponential stability, we study the polynomial stability and prove that the system decays with rate $t^{-\frac{1}{4}}$ . We further prove that this decay rate is optimal.

2020 Mathematics Subject Classification: 35B35; 35B40; 93D20.

1. Introduction

The sandwich beam model is a three-layer beam consisting of stiff outer face plates and a more compliant inner core layer (see Figure 1). These layers are held together by some type of structural adhesive. These models are widely used in various fields of engineering, aerospace, automotive, and shipbuilding, among others. One of the main reasons for the importance of this type of material is that sandwich structures offer designers many advantages, perhaps the most important of which is the high strength-to-weight ratio. This fact is fundamental, for example, in aerospace design, where the weight reduction of a structure with maximum stiffness is always taken into account.

Figure 1.

(a) Purely elastic outer layers constraining a viscoelastic core layer [1] and (b) voltage or current-actuated piezoelectric outer layers [6, 7].

There are two main types of sandwich beams: the Mead–Markus beam and the Rao–Nakra beam. The derivation of various multilayer sandwich beam models can be found in Hansen [1], Mead and Markus [2], Yan and Dowell [3], and Rao and Nakra [4]. The Rao–Nakra model assumes continuous, piecewise linear displacements through the cross-sections, with the Euler–Bernoulli beam assumptions on the outer layers. Transverse (bending) and longitudinal motions are all retained in the modeling. The equations of motions for a three-layer clamped Rao–Nakra sandwich beam model developed in Rao and Nakra [4] and Liu et al. [5] are

$$\{\begin{array}{c}{\rho }_1{h}_1{u}_{tt}=E_1{h}_1{u}_{xx}+k(-u+v+\alpha w_x),\\ {\rho }_3{h}_3{v}_{tt}=E_3{h}_3{v}_{xx}-k(-u+v+\alpha w_x),\\ \rho h{w}_{tt}=-EI{w}_{xxxx}+k\alpha {(-u+v+\alpha w_x)}_x,\end{array}$$

(1)

where $u,v$ , and $w$ are the longitudinal displacements of the centerlines of the outer layers and uniform transverse displacement of the whole composite. The positive constants ${\rho }_i,h_i,E_i$ , and $I_i$ are physical parameters representing, respectively, density, thickness, Young’s modulus, and moments of inertia of the $i\mathrm{th}$ layer for $i=1,2,3;$ $k$ be the shear modulus of the middle layer. The new coefficients are $\rho h={\rho }_1{h}_1+{\rho }_2{h}_2+{\rho }_3{h}_3$ , $\mathrm{EI}=E_1{I}_1+E_3{I}_3$ , and $\alpha =h_2+\frac{1}{2}(h_1+h_3)$ .

In the literature, there are several results regarding the stabilization and controllability of the linear Rao–Nakra beam model. To begin, we can review some of the contributions related to the Rao–Nakra beam model with boundary control. Hansen and Imanuvilov [8, 9] studied a multilayer plate system with locally distributed control on the boundary and obtained exact controllability results using Carleman estimates. Additionally, Özer and Hansen [10, 11] achieved boundary feedback stabilization and the exact controllability of a multilayer Rao–Nakra sandwich beam, respectively.

Wang et al. [12] utilized the Riesz basis method to prove the exact controllability, observability, and exponential stability of a sandwich beam system with boundary control. Hansen and Rajaram [13] established the exact controllability for a Rao–Nakra sandwich beam with boundary controls using the multiplier approach. The Riesz basis property for a Rao–Nakra sandwich beam with different wave speeds was also analyzed in Hansen and Rajaram [13]. Furthermore, Wang [14] investigated the case of Rao–Nakra beam with boundary damping acting on a single end for two displacements and showed that the system’s semigroup is polynomially stable of order $1/2$ .

For the Rao–Nakra beam with internal damping term(s), such as viscous or Kelvin–Voigt type damping, it is known that if one of these dampings is added in each of the three equations of the system, it turns out to be exponentially stable, see Cabanillas et al. [15], Feng and Özer [16], Feng et al. [17], and Raposo [18].

Moreover, if two of the three equations are damped, only polynomial stability is possible [19, 20]. But, if internal viscous damping acts on only one of the three Rao–Nakra beam equations, then polynomial decay rate is sensitive to various boundary conditions and damping locations [21]. Guesmia [22] investigated a Rao–Nakra sandwich beam with a frictional damping or an infinite memory acting on the Euler–Bernoulli equation in an unbounded domain. The author stated that the solutions do not converge to zero when time goes to infinity and when the propagation speeds of the two wave equations are equal. The long-time dynamics of a fully damped non-autonomous Rao–Nakra beam with fractional Laplacian dissipation was studied by Aouadi [23]. Stability results of the Rao–Nakra sandwich beam with thermal damping can be found in Mukiawa et al. [24] and Raposo et al. [25].

This paper is motivated by Li et al. [20], where the following Rao–Nakra beam system with internal frictional damping or/and Kelvin–Voigt damping was considered:

$$\{\begin{array}{c}{\rho }_1{h}_1{u}_{\mathrm{tt}}-E_1{h}_1{u}_{\mathrm{xx}}-k(-u+v+\alpha w_x)-a_1{u}_{\mathrm{xxt}}+a_2{u}_t=0,\hfill \\ \multicolumn{1}{c}{{\rho }_3{h}_3{v}_{\mathrm{tt}}-E_3{h}_3{v}_{\mathrm{xx}}+k(-u+v+\alpha w_x)-b_1{v}_{\mathrm{xxt}}+b_2{v}_t=0,}\\ \multicolumn{1}{c}{\rho h{w}_{\mathrm{tt}}+\mathrm{EI}{w}_{\mathrm{xxxx}}-k\alpha {(-u+v+\alpha w_x)}_x+c_1{w}_{\mathrm{xxxxt}}+c_2{w}_t=0.}\end{array}$$

Li et al. [20] considered two main cases: $a_1=a_2=0$ and $c_1=c_2=0$ , that is, when two of the three equations are directly damped. In both cases, they obtained polynomial stability results of different orders and its optimality. More precisely, they obtained the following results: (1) when $c_1=c_2=0$ , the system is polynomially stable of order $\frac{1}{3}$ if only frictional dampings act on the two wave equations and of order $\frac{1}{2}$ if Kelvin–Voigt dampings act on both wave equations; (2) when $a_1=a_2=0$ , the system is polynomially stable of order $\frac{1}{2}$ if frictional dampings act on both the wave and beam equations and the propagation speeds of the two wave equations are equal. However, if the propagation speeds of the waves are different, the order of the polynomial decay is $\frac{1}{4}$ . The same result holds when $b_1=c_2=0,b_2,c_1>0$ . Whereas, the system is polynomially stable of order $\frac{1}{4}$ if $b_2=c_2=0,b_1,c_1>0$ or $b_2=c_1=0,b_1,c_2>0$ regardless of the relationship between the propagation speeds of the waves. However, they did not discuss the possibility of achieving exponential stability in the aforementioned cases, that is, the lack of exponential stability. The main goal of our work is to provide an answer to this question. More specifically, we study the stability of the following Rao–Nakra sandwich beam model with two Kelvin–Voigt dampers acting on the first and third equations

$${\rho }_1{h}_1{u}_{\mathrm{tt}}-E_1{h}_1{u}_{\mathrm{xx}}-k(-u+v+\alpha w_x)-{\delta }_1{u}_{\mathrm{xxt}}=0\mathrm{in}(0,L)\times {\mathbb{R}}^{+},$$

(2)

$${\rho }_3{h}_3{v}_{\mathrm{tt}}-E_3{h}_3{v}_{\mathrm{xx}}+k(-u+v+\alpha w_x)=0\mathrm{in}(0,L)\times {\mathbb{R}}^{+},$$

(3)

$$\rho h{w}_{\mathrm{tt}}+\mathrm{EI}{w}_{\mathrm{xxxx}}-k\alpha {(-u+v+\alpha w_x)}_x+{\delta }_3{w}_{\mathrm{xxxxt}}=0\mathrm{in}(0,L)\times {\mathbb{R}}^{+},$$

(4)

with initial data

$$\begin{array}{cc}(u(x,0),v(x,0),w(x,0))=(u_0\left(x\right),v_0\left(x\right),w_0\left(x\right))\hfill & \mathrm{in}(0,L),\hfill \\ \multicolumn{1}{c}{(u_t(x,0),v_t(x,0),w_t(x,0))=(u_1\left(x\right),v_1\left(x\right),w_1\left(x\right))}& \mathrm{in}(0,L),\hfill \end{array}$$

(5)

and mixed boundary conditions

$$\begin{array}{cc}u(0,t)=v(0,t)=w_x(0,t)=w_{\mathrm{xxx}}(0,t)=0\hfill & \mathrm{in}{\mathbb{R}}^{+},\hfill \\ \multicolumn{1}{c}{u(L,t)=v(L,t)=w_x(L,t)=w_{\mathrm{xxx}}(L,t)=0}& \mathrm{in}{\mathbb{R}}^{+}.\hfill \end{array}$$

(6)

The main contributions of our paper are the following: (1) The global well-posedness of system (2)–(6) is proved in detail by using the semigroup theory of linear operators approach in Section 2; (2) The lack of exponential stability is obtained through the Gerhart–Huang–Prüss theorem in Section 3; (3) Polynomial decay with rate $t^{-\frac{1}{4}}$ is obtained by direct proof and using Borichev and Tomilov’s theorem in Section 4. The optimality of the obtained rate is also proved. All this on a functional framework are different from the one used by Li et al. [20].

2. Well-posedness of the problem

As usual, let us begin by defining the energy associated with the system (2)–(6). Taking the inner product in $L^2(0,L)$ of equation (2) with $u_t$ , of equation (3) with $v_t$ , and of equation (4) with $w_t$ , integrating by parts and applying the boundary conditions (6), we obtain

$$\frac{1}{2}{\rho }_1{h}_1\frac{d}{\mathrm{dt}}\Vert u_t{\Vert }^2+\frac{1}{2}{E}_1{h}_1\frac{d}{\mathrm{dt}}\Vert u_x{\Vert }^2+k〈-u+v+\alpha w_x,-u_t〉=-{\delta }_1\Vert u_{\mathrm{xt}}{\Vert }^2,$$

(7)

$$\frac{1}{2}{\rho }_3{h}_3\frac{d}{\mathrm{dt}}\Vert v_t{\Vert }^2+\frac{1}{2}{E}_3{h}_3\frac{d}{\mathrm{dt}}\Vert v_x{\Vert }^2+k〈-u+v+\alpha w_x,v_t〉=0$$

(8)

and

$$\frac{1}{2}\rho h\frac{d}{\mathrm{dt}}\Vert w_t{\Vert }^2+\frac{1}{2}\mathrm{EI}\frac{d}{\mathrm{dt}}\Vert w_{\mathrm{xx}}{\Vert }^2+k〈-u+v+\alpha w_x,\alpha w_{\mathrm{tx}}〉=-{\delta }_3\Vert w_{\mathrm{xxt}}{\Vert }^2.$$

(9)

Then, we define the associated energy with the system (2)–(6) as

$$\begin{array}{c}E\left(t\right)=\frac{1}{2}[{\rho }_1{h}_1{\Vert u_t\Vert }^2+E_1{h}_1{\Vert u_x\Vert }^2+{\rho }_3{h}_3{\Vert v_t\Vert }^2+E_3{h}_3{\Vert v_x\Vert }^2\hfill \\ +\rho h{\Vert w_t\Vert }^2+\mathrm{EI}{\Vert w_{\mathrm{xx}}\Vert }^2+k{\Vert -u+v+\alpha w_x\Vert }^2].\hfill \end{array}$$

Adding equations (7), (8), and (9), we conclude that

$$\frac{d}{\mathrm{dt}}E\left(t\right)=-{\delta }_1\Vert u_{\mathrm{xt}}{\Vert }^2-{\delta }_3\Vert w_{\mathrm{xxt}}{\Vert }^2\le 0.$$

(10)

The identity (10) indicates that the energy $E\left(t\right)$ of the system (2)–(6) is non-increasing.

The remainder of this section is devoted to proving the well-posedness of system (2)–(6) using semigroup theory. We define $\phi =u_t$ , $\psi =v_t$ , $z=w_t$ , and introduce the vector functions

$$U=(u,\phi ,v,\psi ,w,z{)}^T\mathrm{and}{U}_0={(u_0,u_1,v_0,v_1,w_0,w_1)}^T.$$

With this notation, the system (2)–(6) can be written as the abstract Cauchy problem

$$\{\begin{array}{cc}{U}_t=\mathcal{A}U,\hfill & t>0,\hfill \\ \multicolumn{1}{c}{U\left(0\right)=U_0,}& \hfill \end{array}$$

(11)

where $\mathcal{A}$ is the differential operator defined by

$$\mathcal{A}U=\left[\begin{array}{c}\phi \hfill \\ \multicolumn{1}{c}{\frac{1}{{\rho }_1{h}_1}[E_1{h}_1{u}_{\mathrm{xx}}+k(-u+v+\alpha w_x)+{\delta }_1{\phi }_{\mathrm{xx}}]}\\ \multicolumn{1}{c}{\psi }\\ \multicolumn{1}{c}{\frac{1}{{\rho }_3{h}_3}[E_3{h}_3{v}_{\mathrm{xx}}-k(-u+v+\alpha w_x)]}\\ \multicolumn{1}{c}{z}\\ \multicolumn{1}{c}{\frac{1}{\rho h}[-\mathrm{EI}{w}_{\mathrm{xxxx}}+k\alpha {(-u+v+\alpha w_x)}_x-{\delta }_3{z}_{\mathrm{xxxx}}]}\end{array}\right].$$

(12)

Our goal in this section is to show that $\mathcal{A}$ generates a $C_0$ -semigroup of contractions on a Hilbert space $\mathcal{H}$ . To define the phase space $\mathcal{H}$ , we will consider the Lebesgue space $L^2(0,L)$ with its usual inner product 〈·,·〉 and induced norm ‖·‖, and the following subspaces

$$\begin{array}{c}{L}_{*}^2(0,L)=\{w\in L^2(0,L);\underset{0}{\overset{L}{\int }}w\left(x\right)\mathrm{dx}=0\},\hfill \\ H_{*}^1(0,L)=H^1(0,L)\cap L_{*}^2(0,L),\hfill \\ H_{*}^2(0,L)=\{w\in H^2(0,L);w_x\left(0\right)=w_x\left(L\right)=0\}\hfill \end{array}$$

and

$$H_{*}^4(0,L)=\{w\in H^4(0,L);w_{\mathrm{xxx}}\left(0\right)=w_{\mathrm{xxx}}\left(L\right)=0\}.$$

Then, we define the phase space as

$$\mathcal{H}={[H_0^1(0,L)\times L^2(0,L)]}^2\times H_{*}^2(0,L)\cap H_{*}^1(0,L)\times L_{*}^2(0,L)$$

endowed with the inner product

$$\begin{array}{c}〈U,U^{\#}{〉}_{\mathcal{H}}={\rho }_1{h}_1〈\phi ,{\phi }^{\#}〉+{\rho }_3{h}_3〈\psi ,{\psi }^{\#}〉+\rho h〈z,z^{\#}〉+E_1{h}_1〈u_x,u_x^{\#}〉\hfill \\ +E_3{h}_3〈v_x,v_x^{\#}〉+\mathrm{EI}〈w_{\mathrm{xx}},w_{\mathrm{xx}}^{\#}〉+k〈(-u+v+\alpha w_x),(-u^{\#}+v^{\#}+\alpha w_x^{\#})〉\hfill \end{array}$$

and induced norm

$$\begin{array}{c}\Vert U{\Vert }_{\mathcal{H}}=({\rho }_1{h}_1{\Vert \phi \Vert }^2+{\rho }_3{h}_3{\Vert \psi \Vert }^2+\rho h{\Vert z\Vert }^2+E_1{h}_1{\Vert u_x\Vert }^2\hfill \\ +{E_3{h}_3{\Vert v_x\Vert }^2+\mathrm{EI}{\Vert w_{\mathrm{xx}}\Vert }^2+k{\Vert -u+v+\alpha w_x\Vert }^2)}^{1/2},\hfill \end{array}$$

for $U=(u,\phi ,v,\psi ,w,z{)}^T$ , $U^{\#}=(u^{\#},{\phi }^{\#},v^{\#},{\psi }^{\#},w^{\#},z^{\#}{)}^T\in \mathcal{H}$ . Thus, $\mathcal{H}$ is a Hilbert space.

Since $\mathcal{A}:D(\mathcal{A})\subset \mathcal{H}\to \mathcal{H}$ , according to the conditions of system (2)–(6), the domain of operator $\mathcal{A}$ is defined as follows:

$$D(\mathcal{A})=\{{(u,\phi ,v,\psi ,w,z)}^T\in \mathcal{H}|\begin{array}{c}\phi ,\psi \in H_0^1(0,L),w\in H_{*}^4(0,L)\hfill \\ \multicolumn{1}{c}{z\in H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L),}\\ \multicolumn{1}{c}{(E_1{h}_{1}u+{\delta }_1\phi ),v\in H^2(0,L)}\end{array}\}.$$

(13)

We shall now prove some properties of operator $\mathcal{A}$ .

Proposition 1. The operator $\mathcal{A}$ defined in (12)–(13) is dissipative. More precisely, for each $U=(u,\phi ,v,\psi ,w,z{)}^T\in D(\mathcal{A})$ we have

$$\mathrm{Re}〈\mathcal{A}U,U{〉}_{\mathcal{H}}=-{\delta }_1\Vert {\phi }_x{\Vert }^2-{\delta }_3\Vert z_{\mathrm{xx}}{\Vert }^2\le 0.$$

(14)

Proof. The application of the definition of the inner product in $\mathcal{H}$ , the integration by parts formula, and the boundary conditions lead us to the following identity:

$$\begin{array}{c}〈\mathcal{A}U,U{〉}_{\mathcal{H}}=2E_1{h}_{1}i\mathrm{Im}〈{\phi }_x,u_x〉+2E_3{h}_{3}i\mathrm{Im}〈{\psi }_x,v_x〉+2\mathrm{EIi}\mathrm{Im}〈z_{\mathrm{xx}},w_{\mathrm{xx}}〉\hfill \\ +2\mathrm{ki}\mathrm{Im}〈(-\phi +\psi +\alpha z_x),(-u+v+\alpha w_x)〉-{\delta }_1\Vert {\phi }_x{\Vert }^2-{\delta }_3\Vert z_{\mathrm{xx}}{\Vert }^2.\hfill \end{array}$$

Taking the real part, we obtain equation (14). Therefore, operator $\mathcal{A}$ is dissipative. □

Proposition 2. If $\varrho \left(\mathcal{A}\right)$ is the resolvent set of operator $\mathcal{A}$ , then $0\in \varrho \left(\mathcal{A}\right)$ .

Proof. Let us prove that $\mathcal{A}$ is injective, with inverse ${\mathcal{A}}^{-1}$ bounded and $\overline{R\left(\mathcal{A}\right)}=\mathcal{H}$ , where $R\left(\mathcal{A}\right)$ is the image of $\mathcal{A}$ . To prove that $\mathcal{A}$ is injective and $\overline{R\left(\mathcal{A}\right)}=\mathcal{H}$ , it is sufficient to verify that $\mathcal{A}$ is bijective, for this, let us show that for any $F=(f^1,f^2,f^3,f^4,f^5,f^6)\in \mathcal{H}$ , there exists a unique $U=(u,\phi ,v,\psi ,w,z{)}^T\in D\left(\mathcal{A}\right)$ such that

$$\mathcal{A}U=F.$$

(15)

Equality (15) is equivalent to the system of equations

$$\phi =f^1\in H_0^1,$$

(16)

$$\frac{1}{{\rho }_1{h}_1}[E_1{h}_1{u}_{\mathrm{xx}}+k(-u+v+\alpha w_x)+{\delta }_1{\phi }_{\mathrm{xx}}]=f^2\in L^2,$$

(17)

$$\psi =f^3\in H_0^1,$$

(18)

$$\frac{1}{{\rho }_3{h}_3}[E_3{h}_3{v}_{\mathrm{xx}}-k(-u+v+\alpha w_x)]=f^4\in L^2,$$

(19)

$$z=f^5\in H_{*}^2\cap H_{*}^1,$$

(20)

$$\frac{1}{\rho h}[-\mathrm{EI}{w}_{\mathrm{xxxx}}+k\alpha {(-u+v+\alpha w_x)}_x-{\delta }_3{z}_{\mathrm{xxxx}}]=f^6\in L_{*}^2.$$

(21)

From equations (16), (18), and (20), we deduce that

$$\phi ,\psi \in H_0^1(0,L),z\in H_{*}^2(0,L)\cap H_{*}^1(0,L).$$

(22)

The problem (16)–(21) is reduced to solving a system of three equations

$$\{\begin{array}{c}-E_1{h}_1{u}_{\mathrm{xx}}-k(-u+v+\alpha w_x)=-{\rho }_1{h}_1{f}^2+{\delta }_1{f}_{\mathrm{xx}}^1,\hfill \\ \multicolumn{1}{c}{-E_3{h}_3{v}_{\mathrm{xx}}+k(-u+v+\alpha w_x)=-{\rho }_3{h}_3{f}^4,}\\ \multicolumn{1}{c}{\mathrm{EI}{w}_{\mathrm{xxxx}}-k\alpha {(-u+v+\alpha w_x)}_x=-\rho h{f}^6-{\delta }_3{f}_{\mathrm{xxxx}}^5,}\end{array}$$

(23)

with boundary conditions

$$\{\begin{array}{c}u\left(0\right)=v\left(0\right)=w_x\left(0\right)=w_{\mathrm{xxx}}\left(0\right)=0,\hfill \\ \multicolumn{1}{c}{u\left(L\right)=v\left(L\right)=w_x\left(L\right)=w_{\mathrm{xxx}}\left(L\right)=0.}\end{array}$$

(24)

Then, the variational formulation of the system (23)–(24) takes the form

$$\{\begin{array}{c}{E}_1{h}_1〈u_x,u_x^{*}〉+k〈(-u+v+\alpha w_x),-u^{*}〉=-{\rho }_1{h}_1〈f^2,u^{*}〉-{\delta }_1〈f_x^1,u_x^{*}〉,\hfill \\ \multicolumn{1}{c}{E_3{h}_3〈v_x,v_x^{*}〉+k〈(-u+v+\alpha w_x),v^{*}〉=-{\rho }_3{h}_3〈f^4,v^{*}〉,}\\ \multicolumn{1}{c}{\mathrm{EI}〈w_{\mathrm{xx}},w_{\mathrm{xx}}^{*}〉+k〈(-u+v+\alpha w_x),\alpha w_x^{*}〉=-\rho h〈f^6,w^{*}〉-{\delta }_3〈f_{\mathrm{xx}}^5,w_{\mathrm{xx}}^{*}〉,}\end{array}$$

(25)

for any $w^{*}\in H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L)$ , $u^{*},v^{*}\in H_0^1(0,L)$ .

The system (25) motivates the definition of the sesquilinear form $\mathcal{B}:H_1\times H_1\to \mathbb{C}$ as

$$\begin{array}{c}\mathcal{B}((u,v,w),(u^{*},v^{*},w^{*}))=E_1{h}_1〈u_x,u_x^{*}〉+E_3{h}_3〈v_x,v_x^{*}〉+\mathrm{EI}〈w_{\mathrm{xx}},w_{\mathrm{xx}}^{*}〉\hfill \\ +k〈(-u+v+\alpha w_x),(-u^{*}+v^{*}+\alpha w_x^{*})〉\hfill \end{array}$$

and the antilinear form $\mathcal{L}:H_1\to \mathbb{C}$ as

$$\begin{array}{c}\mathcal{L}(u^{*},v^{*},w^{*})=-{\rho }_1{h}_1〈f^2,u^{*}〉-{\delta }_1〈f_x^1,u_x^{*}〉-{\rho }_3{h}_3〈f^4,v^{*}〉\hfill \\ -\rho h〈f^6,w^{*}〉-{\delta }_3〈f_{\mathrm{xx}}^5,w_{\mathrm{xx}}^{*}〉,\hfill \end{array}$$

where

$$H_1={[H_0^1(0,L)]}^2\times (H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L)).$$

In this way, we can write equation (25) in its variational form

$$\mathcal{B}((u,v,w),(u^{*},v^{*},w^{*}))=\mathcal{L}(u^{*},v^{*},w^{*}),\mathrm{for}\mathrm{all}(u^{*},v^{*},w^{*})\in H_1.$$

(26)

It can be easily proved that $\mathcal{B}$ is continuous and coercive, while $\mathcal{L}$ is continuous. Then, applying the Lax–Milgram Theorem (see Dautray and Lions [26]), problem (23)–(24) have a unique weak solution:

$$(u,v,w)\in H_1.$$

From equations (17), (19), and (21), we obtain

$$\begin{array}{c}(E_1{h}_{1}u+{\delta }_1\phi ),v\in H^2(0,L),\hfill \\ \mathrm{EIw}+{\delta }_{3}z\in H^4(0,L)\mathrm{and}{[\mathrm{EI}{w}_{\mathrm{xxx}}+{\delta }_3{z}_{\mathrm{xxx}}]}_0^L=0.\hfill \end{array}$$

(27)

Since $w\in H_{*}^4(0,L)$ , and thanks to equation (27), we get that $z\in H^4(0,L)$ and $z_{\mathrm{xxx}}\left(0\right)=z_{\mathrm{xxx}}\left(L\right).$ On the other hand, making $(u^{*},v^{*})=(0,0)$ in equation (26) and using equation (20), we have

$$\mathrm{EI}〈w_{\mathrm{xx}},w_{\mathrm{xx}}^{*}〉+k〈(-u+v+\alpha w_x),\alpha w_x^{*}〉+{\delta }_3〈z_{\mathrm{xx}},w_{\mathrm{xx}}^{*}〉=-\rho h〈f^6,w^{*}〉,$$

(28)

for any $w^{*}\in H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L)$ .

Applying the integration by parts formula in equation (28), using equation (21), and taking into account that $u,v\in H_0^1(0,L)$ and $w\in H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L)$ , we obtain

$${\left[z_{\mathrm{xxx}}{w}^{*}\right]}_0^L=0,\mathrm{for}\mathrm{all}{w}^{*}\in H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L).$$

This implies that

$$z_{\mathrm{xxx}}\left(0\right)=z_{\mathrm{xxx}}\left(L\right)=0,$$

and consequently, $z\in H_{*}^4(0,L)$ . Moreover, from equation (22) we get

$$z\in H_{*}^4(0,L)\cap H_{*}^2(0,L)\cap H_{*}^1(0,L).$$

Therefore, there exists a unique solution ${(u,\phi ,v,\psi ,w,z)}^T\in D\left(\mathcal{A}\right)$ of system (16)–(21), and consequently, $\mathcal{A}$ is bijective.

It remains to prove that the inverse ${\mathcal{A}}^{-1}$ is bounded. In fact, we know that for any $F={(f^1,f^2,f^3,f^4,f^5,f^6)}^T\in \mathcal{H}$ , there exists a unique $U={(u,\phi ,v,\psi ,w,z)}^T\in D\left(\mathcal{A}\right)$ such that

$$\mathcal{A}U=F,$$

equivalently,

$$U={\mathcal{A}}^{-1}F.$$

(29)

We will prove that there exists a constant $C>0$ such that

$$\Vert {\mathcal{A}}^{-1}F{\Vert }_{\mathcal{H}}=\Vert U{\Vert }_{\mathcal{H}}\le C\Vert F{\Vert }_{\mathcal{H}}.$$

Taking $(u^{*},v^{*},w^{*})=(u,v,w)$ in equation (26) and using equation (24), we obtain

$$\begin{array}{c}{E}_1{h}_1\Vert u_x{\Vert }^2+E_3{h}_3\Vert v_x{\Vert }^2+\mathrm{EI}\Vert w_{\mathrm{xx}}{\Vert }^2+k\Vert -u+v+\alpha w_x{\Vert }^2\hfill \\ =-{\rho }_1{h}_1〈f^2,u〉-{\delta }_1〈f_x^1,u_x〉-{\rho }_3{h}_3〈f^4,v〉-\rho h〈f^6,w〉-{\delta }_3〈f_{\mathrm{xx}}^5,w_{\mathrm{xx}}〉.\hfill \end{array}$$

Applying the Cauchy–Schwarz inequality, we get

$$\begin{array}{c}{E}_1{h}_1\Vert u_x{\Vert }^2+E_3{h}_3\Vert v_x{\Vert }^2+\mathrm{EI}\Vert w_{\mathrm{xx}}{\Vert }^2+k\Vert -u+v+\alpha w_x{\Vert }^2\hfill \\ \le C_1[\Vert f^2\Vert \Vert u_x\Vert +\Vert f_x^1\Vert \Vert u_x\Vert +\Vert f^4\Vert \Vert v_x\Vert +\Vert f^6\Vert \Vert \alpha w_x\Vert +\Vert f_{\mathrm{xx}}^5\Vert \Vert w_{\mathrm{xx}}\Vert ].\hfill \end{array}$$

Then,

$$E_1{h}_1\Vert u_x{\Vert }^2+E_3{h}_3\Vert v_x{\Vert }^2+\mathrm{EI}\Vert w_{\mathrm{xx}}{\Vert }^2+k\Vert -u+v+\alpha w_x{\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}.$$

(30)

Furthermore, using equations (16), (18), and (20) and the Poincaré’s inequality, we obtain

$$\begin{array}{c}{\rho }_1{h}_1\Vert \phi {\Vert }^2+{\rho }_3{h}_3\Vert \psi {\Vert }^2+\rho h\Vert z{\Vert }^2\hfill \\ \le C_1[{\Vert f_x^1\Vert }^2+{\Vert f_x^3\Vert }^2+{\Vert \alpha f_x^5\Vert }^2]\hfill \\ \le C_2[{\Vert f_x^1\Vert }^2+{\Vert f_x^3\Vert }^2+{\Vert -f^1+f^3+\alpha f_x^5\Vert }^2]\hfill \\ \le C\Vert F{\Vert }_{\mathcal{H}}^2.\hfill \end{array}$$

(31)

Adding equations (30) and (31), using equation (29) and Young’s inequality, we get

$$\Vert U{\Vert }_{\mathcal{H}}^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2\le \frac{1}{2}\Vert U{\Vert }_{\mathcal{H}}^2+C\Vert F{\Vert }_{\mathcal{H}}^2.$$

Consequently,

$$\Vert {\mathcal{A}}^{-1}F{\Vert }_{\mathcal{H}}=\Vert U{\Vert }_{\mathcal{H}}\le \sqrt{2C}\Vert F{\Vert }_{\mathcal{H}}.$$

This means that ${\mathcal{A}}^{-1}$ is bounded. Therefore, $0\in \varrho \left(\mathcal{A}\right)$ . □

Therefore, the well-posedness of problem (2)–(6) is guaranteed by the following theorem.

Theorem 1. For each $U_0\in \mathcal{H}$ , there exists a unique weak solution $U$ of the problem (11) that satisfies

$$U\in C([0,\infty [;\mathcal{H}).$$

(32)

Moreover, if $U_0\in D\left(\mathcal{A}\right)$ , then

$$U\in C([0,\infty [;D\left(\mathcal{A}\right))\cap C^1([0,\infty [;\mathcal{H}).$$

(33)

In this case, $U$ is called a strong solution.

Proof. Thanks to Propositions 1 and 2, by applying Theorem 1.2.4 from Liu and Zheng’s [27] book, we obtain that $\mathcal{A}$ is the infinitesimal generator of a $C_0$ -semigroup of contractions $S(t)=e^{t\mathcal{A}}$ in $\mathcal{H}$ . By semigroup theory, $U(t)=e^{t\mathcal{A}}{U}_0$ is the unique solution of equation (11) satisfying (32) and (33). This proves the theorem. □

3. Lack of exponential stability

We prove in this section that the system (11) is not exponentially stable, independently of the propagation speeds of the waves. For this purpose, we will use the following Gerhart– Huang–Prüss theorem [28–30].

Theorem 2. Let $\varrho \left(A\right)$ be the resolvent set of the operator $A$ and $S\left(t\right)=e^{\mathrm{tA}}$ be the $C_0$ -semigroup of contractions generated by $A$ in a Hilbert space $H$ . Then S(t) is exponentially stable if and only if

$$i\mathbb{R}\subset \varrho (A)$$

(34)

and

$$\underset{|\beta |\to \infty }{lim\sup }\parallel (i\beta I-A{)}^{-1}\parallel <\infty .$$

(35)

Our main result is as follows.

Theorem 3. Let $S\left(t\right)=e^{t\mathcal{A}}$ be a $C_0$ -semigroup of contractions over the Hilbert space $\mathcal{H}$ , associated with problem (2)–(6). Then $S\left(t\right)$ is not exponentially stable.

Proof. The idea is to prove that condition (35) of Theorem 2 is not satisfied. This would imply the lack of exponential stability of the system. Proving that the condition (35) is not valid is equivalent to finding a sequence $\left({\lambda }_n\right)\subset \mathbb{R}$ such that

$$\left|{\lambda }_n\right|\to \infty$$

(36)

and

$$\underset{n\to \infty }{lim}\parallel (i{\lambda }_{n}I-\mathcal{A}{)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}=\infty .$$

This last limit is equivalent to finding a sequence $\left(F_n\right)\subset \mathcal{H}$ such that

$$\Vert F_n{\Vert }_{\mathcal{H}}\le 1,n\in \mathbb{N}$$

(37)

and

$$\underset{n\to \infty }{lim}\parallel (i{\lambda }_{n}I-\mathcal{A}{)}^{-1}{F}_n{\parallel }_{\mathcal{H}}=\infty .$$

(38)

If we denote

$$U_n=(i{\lambda }_{n}I-\mathcal{A}{)}^{-1}{F}_n,n\in \mathbb{N}.$$

(39)

Then, condition (38) takes the form

$$\underset{n\to \infty }{lim}\parallel U_n{\parallel }_{\mathcal{H}}=\infty$$

(40)

The remainder of this section is devoted to proving the existence of a sequence $\left(U_n\right)\subset D\left(\mathcal{A}\right)$ satisfying the relation (40). Let us denote

$$i{\lambda }_n{U}_n-\mathcal{A}{U}_n=F_n,n\in \mathbb{N},$$

(41)

with

$$U_n=(u_n,{\phi }_n,v_n,{\psi }_n,w_n,z_n{)}^T\mathrm{and}{F}_n={(f_n^1,f_n^2,f_n^3,f_n^4,f_n^5,f_n^6)}^T.$$

Replacing definition (12), we may decompose the resolvent equation (41) as

$$i{\lambda }_n{u}_n-{\phi }_n=f_n^1\mathrm{in}{H}_0^1,$$

(42)

$$i{\lambda }_n{\phi }_n-\frac{1}{{\rho }_1{h}_1}[E_1{h}_1{u}_{n,\mathrm{xx}}+k(-u_n+v_n+\alpha w_{n,x})+{\delta }_1{\phi }_{n,\mathrm{xx}}]=f_n^2\mathrm{in}{L}^2,$$

(43)

$$i{\lambda }_n{v}_n-{\psi }_n=f_n^3\mathrm{in}{H}_0^1,$$

(44)

$$i{\lambda }_n{\psi }_n-\frac{1}{{\rho }_3{h}_3}[E_3{h}_3{v}_{n,\mathrm{xx}}-k(-u_n+v_n+\alpha w_{n,x})]=f_n^4\mathrm{in}{L}^2,$$

(45)

$$i{\lambda }_n{w}_n-z_n=f_n^5\mathrm{in}{H}_{*}^2\cap H_{*}^1,$$

(46)

$$i{\lambda }_n{z}_n-\frac{1}{\rho h}[-\mathrm{EI}{w}_{n,\mathrm{xxxx}}+k\alpha {(-u_n+v_n+\alpha w_{n,x})}_x-{\delta }_3{z}_{n,\mathrm{xxxx}}]=f_n^6\mathrm{in}{L}_{*}^2.$$

(47)

Consider

$$f_n^1=f_n^2=f_n^3=f_n^5=f_n^6=0\mathrm{and}{f}_n^4=c_2\sin {\alpha }_{n}x,$$

(48)

where

$${\alpha }_n=\frac{n\pi }{L},n\in \mathbb{N},$$

(49)

and $c_2$ is a constant satisfying

$$0<\left|c_2\right|\le \sqrt{\frac{2}{3L{\rho }_2}}.$$

(50)

In this way, $\left(F_n\right)\subset \mathcal{H}$ and satisfies condition (37). On the other hand, by replacing the functions in (48) in the system (42)–(47), we have

$${\phi }_n=i{\lambda }_n{u}_n,{\psi }_n=i{\lambda }_n{v}_n,z_n=i{\lambda }_n{w}_n,$$

(51)

$$-{\lambda }_n^2{\rho }_1{h}_1{u}_n-E_1{h}_1{u}_{n,\mathrm{xx}}-k(-u_n+v_n+\alpha w_{n,x})-i{\delta }_1{\lambda }_n{u}_{n,\mathrm{xx}}=0,$$

(52)

$$-{\lambda }_n^2{\rho }_3{h}_3{v}_n-E_3{h}_3{v}_{n,\mathrm{xx}}+k(-u_n+v_n+\alpha w_{n,x})={\rho }_3{h}_3{c}_2\sin {\alpha }_{n}x,$$

(53)

$$-{\lambda }_n^2\rho h{w}_n+\mathrm{EI}{w}_{n,\mathrm{xxxx}}-k\alpha {(-u_n+v_n+\alpha w_{n,x})}_x+i{\delta }_3{\lambda }_n{w}_{n,\mathrm{xxxx}}=0.$$

(54)

Now, we choose

$$u_n\left(x\right)=A_n\sin {\alpha }_{n}x,v_n\left(x\right)=B_n\sin {\alpha }_{n}x,\mathrm{and}{w}_n\left(x\right)=C_n\cos {\alpha }_{n}x,$$

(55)

where $\left(A_n\right)$ , $\left(B_n\right),$ and $\left(C_n\right)$ are sequences to be fixed later. Replacing these functions in equation (51), we have completely defined the sequence $\left(U_n\right)\subset D\left(\mathcal{A}\right)$ and

$${\phi }_n\left(x\right)=i{\lambda }_n{A}_n\sin {\alpha }_{n}x,{\psi }_n\left(x\right)=i{\lambda }_n{B}_n\sin {\alpha }_{n}x,{}_n\left(x\right)=i{\lambda }_n{C}_n\cos {\alpha }_{n}x.$$

Replacing these functions in equations (52)–(54), we obtain the system of algebraic equations:

$$(E_1{h}_1{\alpha }_n^2-{\lambda }_n^2{\rho }_1{h}_1+k+i{\delta }_1{\lambda }_n{\alpha }_n^2)A_n-k{B}_n+\alpha k{\alpha }_n{C}_n=0,$$

(56)

$$-k{A}_n+(E_3{h}_3{\alpha }_n^2-{\lambda }_n^2{\rho }_3{h}_3+k)B_n-\alpha k{\alpha }_n{C}_n={\rho }_3{h}_3{c}_2,$$

(57)

$$\alpha k{\alpha }_n{A}_n-\alpha k{\alpha }_n{B}_n+(\mathrm{EI}{\alpha }_n^4-{\lambda }_n^2\rho h+{\alpha }^{2}k{\alpha }_n^2+i{\delta }_3{\lambda }_n{\alpha }_n^4)C_n=0.$$

(58)

Then, we choose the sequence

$${\lambda }_n=\sqrt{\frac{E_3}{{\rho }_3}{\alpha }_n^2+\frac{k}{{\rho }_3{h}_3}},$$

(59)

which clearly verifies condition (36). Now, we can determine $A_n$ , $B_n$ , and $C_n$ by solving the system (56)–(58). Replacing the formula (59) in equations (56)–(58), we find

$${\alpha }_n^3{P}_n{A}_n-k{B}_n+\alpha k{\alpha }_n{C}_n=0,$$

(60)

$$-k{A}_n-\alpha k{\alpha }_n{C}_n={\rho }_3{h}_3{c}_2,$$

(61)

$$\alpha k{A}_n-\alpha k{B}_n+{\alpha }_n^4{Q}_n{C}_n=0,$$

(62)

where

$$P_n=(E_1{h}_1-\frac{E_3{\rho }_1{h}_1}{{\rho }_3})\frac{1}{{\alpha }_n}-\frac{k{\rho }_1{h}_1}{{\rho }_3{h}_3{\alpha }_n^3}+\frac{k}{{\alpha }_n^3}+i{\delta }_1\sqrt{\frac{E_3}{{\rho }_3}+\frac{k}{{\rho }_3{h}_3{\alpha }_n^2}}$$

and

$$Q_n=\frac{\mathrm{EI}}{{\alpha }_n}-\frac{E_3\rho h}{{\rho }_3{\alpha }_n^3}-\frac{k\rho h}{{\rho }_3{h}_3{\alpha }_n^5}+\frac{{\alpha }^{2}k}{{\alpha }_n^3}+i{\delta }_3\sqrt{\frac{E_3}{{\rho }_3}+\frac{k}{{\rho }_3{h}_3{\alpha }_n^2}}.$$

Solving the system (60)–(62), we obtain

$$A_n=-\frac{{\rho }_3{h}_3{c}_2(k{\alpha }^2-{\alpha }_n^3{Q}_n)}{k(2k{\alpha }^2-{\alpha }_n^3{Q}_n-{\alpha }^2{\alpha }_n^3{P}_n)},$$

(63)

$$B_n=-\frac{{\rho }_3{h}_3{c}_2(k^2{\alpha }^2-{\alpha }_n^6{P}_n{Q}_n)}{k^2(2k{\alpha }^2-{\alpha }_n^3{Q}_n-{\alpha }^2{\alpha }_n^3{P}_n)},$$

(64)

and

$$C_n=-\frac{\alpha {\rho }_3{h}_3{c}_2(k-{\alpha }_n^3{P}_n)}{k{\alpha }_n(2k{\alpha }^2-{\alpha }_n^3{Q}_n-{\alpha }^2{\alpha }_n^3{P}_n)}.$$

(65)

Since $c_2\ne 0$ , from formula (64) and the definition (49) of ${\alpha }_n$ , we obtain

$$\underset{n\to \infty }{lim}\left|B_n\right|=\infty ,$$

(66)

and consequently,

$$\underset{n\to \infty }{lim}\left|{\lambda }_n{B}_n\right|=\infty .$$

(67)

Finally, we have

$$\begin{array}{c}\Vert U_n{\Vert }_{\mathcal{H}}^2\ge {\rho }_3{h}_3\Vert \psi {\Vert }^2={\rho }_3{h}_3{\left|{\lambda }_n\right|}^2\Vert v_n{\Vert }^2\hfill \\ ={\rho }_3{h}_3{\left|{\lambda }_n{B}_n\right|}^2\Vert \sin {\alpha }_{n}x{\Vert }^2=\frac{1}{2}{\rho }_3{h}_{3}L{\left|{\lambda }_n{B}_n\right|}^2\to \infty ,\hfill \end{array}$$

(68)

which proves condition (40). Therefore, the $C_0$ -semigroup $S\left(t\right)$ associated with problem (2)–(6) is not exponentially stable. □

4. Polynomial stability

In this section, we will show that the system (2)–(6) decays polynomially with a rate $t^{-\frac{1}{4}}$ . Furthermore, we will prove that this decay rate is optimal. The proof of our result is based on the following theorem due to Borichev and Tomilov [31].

Theorem 4. Let $S\left(t\right)=e^{t\mathcal{A}}$ be a bounded $C_0$ -semigroup of contractions on a Hilbert space $\mathcal{H}$ with infinitesimal generator $\mathcal{A}$ such that $i\mathbb{R}\subset \varrho (\mathcal{A}).$ Then, for any fixed $\alpha >0,$ the following conditions are equivalent:

• $\parallel (i\lambda I-\mathcal{A}{)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}=\mathcal{O}\left({\left|\lambda \right|}^{\alpha }\right),\lambda \in \mathbb{R},\left|\lambda \right|\to \infty$ .

• $\parallel S(t){\mathcal{A}}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}=\mathcal{O}\left(t^{-\frac{1}{\alpha }}\right)$ , $t\to \infty$ .

For the proof of the optimality of the polynomial decay rate, we will use the following theorem due to Fatori and Muñoz [32, Theorem 5.4].

Theorem 5. Let $S(t)$ be a $C_0$ -semigroup of contractions of linear operators on Hilbert space $\mathcal{H}$ with infinitesimal generator $\mathcal{A}$ and such that $i\mathbb{R}\subset \varrho (\mathcal{A})$ . If

$$\Vert S(t)U_0{\Vert }_{\mathcal{H}}\le \frac{C}{t^{\gamma }}\Vert U_0{\Vert }_{D(\mathcal{A})},$$

then, for any $ϵ>0$ , there exists a constant $C_{ϵ}>0$ such that

$$\frac{1}{{\left|\lambda \right|}^{ϵ+\frac{1}{\gamma }}}\parallel (i\lambda I-\mathcal{A}{)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}\le C_{ϵ}.$$

We must verify that the system (2)–(6) satisfies the first hypothesis of Theorem 4, that is, $i\mathbb{R}\subset \varrho \left(\mathcal{A}\right)$ . This result is proved in the following proposition.

Proposition 3. If $\varrho \left(A\right)$ is the resolvent set of the operator $\mathcal{A}$ , then

$$i\mathbb{R}\subset \varrho \left(\mathcal{A}\right).$$

(69)

Proof. Suppose that $i\mathbb{R}⊈\varrho \left(\mathcal{A}\right)$ . Since $\mathcal{A}$ is the infinitesimal generator of a $C_0$ -semigroup of contractions $S\left(t\right)$ on $\mathcal{H}$ , then $\mathcal{A}$ is closed (see Pazy [33, Corollary 2.5]). Moreover, since $0\in \varrho \left(\mathcal{A}\right)$ , applying the same argument stated in the proof of Theorem 2.2.1 of Liu and Zheng’s [27] book, there exists a real sequence $\left({\lambda }_n\right)$ and a sequence of vector functions $\left(U_n\right)\subset D\left(\mathcal{A}\right)$ with $\Vert U_n{\Vert }_{\mathcal{H}}=1$ such that

$${\lambda }_n\to \lambda \mathrm{with}\left|{\lambda }_n\right|<\left|\lambda \right|$$

(70)

and

$$\Vert (i{\lambda }_{n}I-\mathcal{A})U_n{\Vert }_{\mathcal{H}}\to 0.$$

As before, let

$$(i{\lambda }_{n}I-\mathcal{A})U_n=F_n,$$

(71)

where $F_n={(f_n^1,f_n^2,f_n^3,f_n^4,f_n^5,f_n^6)}^T\in \mathcal{H}$ and $U_n=(u_n,{\phi }_n,v_n,{\psi }_n,w_n,z_n{)}^T\in D(\mathcal{A})$ . Then,

$$F_n\to 0\mathrm{in}\mathcal{H}.$$

(72)

Taking the inner product of equation (71) with $U_n$ in $\mathcal{H}$ , it follows that

$$i{\lambda }_n\Vert U_n{\Vert }_{\mathcal{H}}^2-{〈\mathcal{A}{U}_n,U_n〉}_{\mathcal{H}}={〈F_n,U_n〉}_{\mathcal{H}}\to 0.$$

Taking the real part and using the dissipativity identity (14), we have

$${\delta }_1\Vert {\phi }_{n,x}{\Vert }^2+{\delta }_3\Vert z_{n,\mathrm{xx}}{\Vert }^2\to 0.$$

Thanks to Poincaré’s inequality, the above convergence implies

$${\phi }_{n,x}\to 0\mathrm{in}{L}^2(0,L),{\phi }_n\to 0\mathrm{in}{L}^2(0,L),$$

(73)

and

$$z_{n,\mathrm{xx}}\to 0\mathrm{in}{L}^2(0,L),z_{n,x}\to 0\mathrm{in}{L}^2(0,L),z_n\to 0\mathrm{in}{L}^2(0,L).$$

(74)

By decomposing equation (71) into its components and applying the convergence (72), we obtain the following system:

$$i{\lambda }_n{u}_n-{\phi }_n=f_n^1\to 0\mathrm{in}{H}_0^1,$$

(75)

$$i{\lambda }_n{\phi }_n-\frac{1}{{\rho }_1{h}_1}[E_1{h}_1{u}_{n,\mathrm{xx}}+k(-u_n+v_n+\alpha w_{n,x})+{\delta }_1{\phi }_{n,\mathrm{xx}}]=f_n^2\to 0\mathrm{in}{L}^2,$$

(76)

$$i{\lambda }_n{v}_n-{\psi }_n=f_n^3\to 0\mathrm{in}{H}_0^1,$$

(77)

$$i{\lambda }_n{\psi }_n-\frac{1}{{\rho }_3{h}_3}[E_3{h}_3{v}_{n,\mathrm{xx}}-k(-u_n+v_n+\alpha w_{n,x})]=f_n^4\to 0\mathrm{in}{L}^2,$$

(78)

$$i{\lambda }_n{w}_n-z_n=f_n^5\to 0\mathrm{in}{H}_{*}^2\cap H_{*}^1,$$

(79)

$$i{\lambda }_n{z}_n-\frac{1}{\rho h}[-\mathrm{EI}{w}_{n,\mathrm{xxxx}}+k\alpha {(-u_n+v_n+\alpha w_{n,x})}_x-{\delta }_3{z}_{n,\mathrm{xxxx}}]=f_n^6\to 0\mathrm{in}{L}_{*}^2.$$

(80)

Since $\Vert U_n{\Vert }_{\mathcal{H}}=1$ , it follows that the sequences $\left({\phi }_n\right)$ , $\left({\psi }_n\right)$ , $\left(z_n\right)$ , $\left(u_{n,x}\right)$ , $\left(v_{n,x}\right)$ , $\left(w_{n,\mathrm{xx}}\right)$ , and $(-u_n+v_n+\alpha w_{n,x})$ are bounded on $L^2(0,L)$ . Then, applying (73) in (75) we get

$$u_{n,x}\to 0\mathrm{in}{L}^2(0,L),u_n\to 0\mathrm{in}{L}^2(0,L).$$

(81)

Analogously, applying convergences (74) in equation (79) yields

$$w_{n,\mathrm{xx}}\to 0\mathrm{in}{L}^2(0,L),w_{n,x}\to 0\mathrm{in}{L}^2(0,L)\mathrm{and}{w}_n\to 0\mathrm{in}{L}^2(0,L).$$

(82)

Taking the inner product of equation (76) with $v_n$ in $L^2(0,L)$ yields

$$\begin{array}{c}i{\lambda }_n〈{\phi }_n,v_n〉+\frac{1}{{\rho }_1{h}_1}[E_1{h}_1〈u_{n,x},v_{n,x}〉-k〈(-u_n+\alpha w_{n,x}),v_n〉\hfill \\ -k{\Vert v_n\Vert }^2+{\delta }_1〈{\phi }_{n,x},v_{n,x}〉]=〈f_n^2,v_n〉\hfill \end{array}$$

and using convergences (73), (76), (81), and (82), we conclude that

$$v_n\to 0\mathrm{in}{L}^2(0,L).$$

(83)

And consequently, replacing the convergence (83) in equation (77) we get

$${\psi }_n\to 0\mathrm{in}{L}^2(0,L).$$

(84)

Multiplying equation (78) by $v_n$ in $L^2(0,L)$

$$i{\lambda }_n〈{\psi }_n,v_n〉+\frac{1}{{\rho }_3{h}_3}[E_3{h}_3{\Vert v_{n,x}\Vert }^2+k〈-u_n+v_n+\alpha w_{n,x},v_n〉]=〈f_n^4,v_n〉$$

and applying convergence (83), we obtain

$$v_{n,x}\to 0\mathrm{in}{L}^2(0,L).$$

(85)

Convergences (81), (82), and (83) imply that

$$-u_n+v_n+\alpha w_{n,x}\to 0\mathrm{in}{L}^2(0,L).$$

(86)

Finally, convergences (73), (74), (81), (82), (84), (85), and (86) imply that

$$\Vert U{\Vert }_{\mathcal{H}}\to 0,$$

which is a contradiction with $\Vert U{\Vert }_{\mathcal{H}}=1$ . Therefore, condition (69) is verified. □

According to Proposition 3, for every $F\in \mathcal{H}$ , there exists a unique $U\in D\left(\mathcal{A}\right)$ such that

$$(i\lambda I-\mathcal{A})U=F,$$

(87)

where $F={(f^1,f^2,f^3,f^4,f^5,f^6)}^T$ and $U=(u,\phi ,v,\psi ,w,z{)}^T$ . By decomposing the resolvent equation (87), we obtain the system

$$i\lambda u-\phi =f^1\mathrm{in}{H}_0^1,$$

(88)

$$i\lambda \phi -\frac{1}{{\rho }_1{h}_1}[E_1{h}_1{u}_{\mathrm{xx}}+k(-u+v+\alpha w_x)+{\delta }_1{\phi }_{\mathrm{xx}}]=f^2\mathrm{in}{L}^2,$$

(89)

$$i\lambda v-\psi =f^3\mathrm{in}{H}_0^1,$$

(90)

$$i\lambda \psi -\frac{1}{{\rho }_3{h}_3}[E_3{h}_3{v}_{\mathrm{xx}}-k(-u+v+\alpha w_x)]=f^4\mathrm{in}{L}^2,$$

(91)

$$i\lambda w-z=f^5\mathrm{in}{H}_{*}^2\cap H_{*}^1,$$

(92)

$$i\lambda z-\frac{1}{\rho h}[-\mathrm{EI}{w}_{\mathrm{xxxx}}+k\alpha {(-u+v+\alpha w_x)}_x-{\delta }_3{z}_{\mathrm{xxxx}}]=f^6\mathrm{in}{L}_{*}^2.$$

(93)

To prove the main result of this section, we need some technical lemmas. In what follows, as before, and in order not to overload the writing, we will denote by $C$ a positive generic constant whose value is not necessarily the same in each line.

Lemma 6. There exists a constant $C>0$ independent of $\lambda$ such that

$$\Vert {\phi }_x{\Vert }^2,\Vert \phi {\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}},$$

(94)

$$\Vert z_{\mathrm{xx}}{\Vert }^2,\Vert z_x{\Vert }^2,\Vert z{\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}},$$

(95)

$$\Vert u_x{\Vert }^2,\Vert u{\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2$$

(96)

and

$$\Vert w_{\mathrm{xx}}{\Vert }^2,\Vert w_x{\Vert }^2,\Vert w{\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2,$$

(97)

with $\left|\lambda \right|\ge 1$ .

Proof. Taking the inner product of equation (87) with $U$ in $\mathcal{H}$ and using (14), we get

$${\delta }_1\Vert {\phi }_x{\Vert }^2+{\delta }_3\Vert z_{\mathrm{xx}}{\Vert }^2=-\mathrm{Re}〈\mathcal{A}U,U{〉}_{\mathcal{H}}\le \Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}.$$

Applying Poincaré’s inequality, we obtain inequalities (94) and (95) directly. From equation (88) and relation (94), we obtain

$$\Vert u_x{\Vert }^2\le \frac{C}{{\left|\lambda \right|}^2}({\Vert {\phi }_x\Vert }^2+{\Vert f_x^1\Vert }^2)\le C\Vert F\Vert \Vert U\Vert +C\Vert F{\Vert }^2.$$

Analogously, from equation (92) and estimate (95), we get

$$\Vert w_{\mathrm{xx}}{\Vert }^2\le \frac{C}{{\left|\lambda \right|}^2}({\Vert z_{\mathrm{xx}}\Vert }^2+{\Vert f_{\mathrm{xx}}^5\Vert }^2)\le C\Vert F\Vert \Vert U\Vert +C\Vert F{\Vert }^2.$$

Finally, we obtain the estimates (96) and (97) after applying the Poincaré inequality. □

Lemma 7. There exists a constant $C>0$ independent of $\lambda$ such that

$$E_3{h}_3\Vert v_x{\Vert }^2+k\Vert v{\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2+2{\rho }_3{h}_3\Vert \psi {\Vert }^2$$

(98)

and

$$k\Vert -u+v+\alpha w_x{\Vert }^2\le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2+4{\rho }_3{h}_3\Vert \psi {\Vert }^2,$$

(99)

with $\left|\lambda \right|\ge 1$ .

Proof. Taking the inner product of equation (91) with $v$ in $L^2(0,L)$ , integrating by parts and replacing equation (90), we find

$$E_3{h}_3\Vert v_x{\Vert }^2+k\Vert v{\Vert }^2={\rho }_3{h}_3〈f^4,v〉+{\rho }_3{h}_3\Vert \psi {\Vert }^2+{\rho }_3{h}_3〈\psi ,f^3〉+k〈u,v〉-k\alpha 〈w_x,v〉.$$

Using the estimates (96) and (97) and Young’s inequality with $\epsilon >0$ , we obtain

$$\begin{array}{c}{E}_3{h}_3\Vert v_x{\Vert }^2+k\Vert v{\Vert }^2\le C\Vert F\Vert \Vert U\Vert +{\rho }_3{h}_3\Vert \psi {\Vert }^2+C\Vert u{\Vert }^2+C\Vert w_x{\Vert }^2+\frac{\epsilon }{2}\Vert v{\Vert }^2\hfill \\ \le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2+{\rho }_3{h}_3\Vert \psi {\Vert }^2+\frac{\epsilon }{2}\Vert v{\Vert }^2.\hfill \end{array}$$

Taking $\epsilon =k$ , we obtain relation (98). On the other hand, using the estimates (96), (97), and (98), we have

$$\begin{array}{c}k\Vert -u+v+\alpha w_x{\Vert }^2\le 4k\Vert u{\Vert }^2+4k\Vert w_x{\Vert }^2+2k\Vert v{\Vert }^2\hfill \\ \le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2+4{\rho }_3{h}_3\Vert \psi {\Vert }^2,\hfill \end{array}$$

which proves estimate (99). □

Lemma 8. There exists a constant $C>0$ independent of $\lambda$ such that

$${\rho }_3{h}_3\Vert \psi {\Vert }^2\le {\left|\lambda \right|}^4\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\left|\lambda \right|}^4\Vert F{\Vert }_{\mathcal{H}}^2,$$

(100)

$$E_3{h}_3\Vert v_x{\Vert }^2\le {\left|\lambda \right|}^4\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\left|\lambda \right|}^4\Vert F{\Vert }_{\mathcal{H}}^2$$

(101)

and

$$k\Vert -u+v+\alpha w_x{\Vert }^2\le {\left|\lambda \right|}^4\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\left|\lambda \right|}^4\Vert F{\Vert }_{\mathcal{H}}^2,$$

(102)

with $\left|\lambda \right|\ge 1$ .

Proof. Multiplying equation (89) by $v$ in $L^2(0,L)$ and integrating by parts yields the following:

$$k\Vert v{\Vert }^2=-{\rho }_1{h}_1〈f^2,v〉+i\lambda {\rho }_1{h}_1〈\phi ,v〉+E_1{h}_1〈\lambda u_x,\frac{v_x}{\lambda }〉+k〈u,v〉-k\alpha 〈w_x,v〉+{\delta }_1〈\lambda {\phi }_x,\frac{v_x}{\lambda }〉.$$

Using the estimates (94), (96), and (97), and the Young’s inequality with $\epsilon ,{\epsilon }_1>0$ , we obtain

$$\begin{array}{c}k\Vert v{\Vert }^2\le C\Vert f^2\Vert \Vert v\Vert +C\left|\lambda \right|\Vert \phi \Vert \Vert v\Vert +\Vert \lambda u_x\Vert \Vert \frac{v_x}{\lambda }\Vert \hfill \\ +C\Vert u\Vert \Vert v\Vert +C\Vert w_x\Vert \Vert v\Vert +C\Vert \lambda {\phi }_x\Vert \Vert \frac{v_x}{\lambda }\Vert \hfill \\ \le C\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2+C{\lambda }^2\Vert \phi {\Vert }^2+C{\lambda }^2\Vert u_x{\Vert }^2\hfill \\ +C{\lambda }^2\Vert {\phi }_x{\Vert }^2+\frac{{\epsilon }_1}{2}\Vert v{\Vert }^2+\frac{\epsilon }{4}\Vert \frac{v_x}{\lambda }{\Vert }^2\hfill \\ \le C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}^2+\frac{{\epsilon }_1}{2}\Vert v{\Vert }^2+\frac{\epsilon }{4{\lambda }^2}\Vert v_x{\Vert }^2.\hfill \end{array}$$

Choosing ${\epsilon }_1=k$ , we get

$$k\Vert v{\Vert }^2\le C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}^2+\frac{\epsilon }{2{\lambda }^2}\Vert v_x{\Vert }^2$$

and applying estimate (98) and replacing equation (90), we have

$$\begin{array}{c}k\Vert v{\Vert }^2\le C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}^2+\frac{\epsilon }{2E_3{h}_3{\lambda }^2}[C{\Vert F\Vert }_{\mathcal{H}}{\Vert U\Vert }_{\mathcal{H}}+C\Vert F{\Vert }_{\mathcal{H}}^2]\hfill \\ +\frac{{\rho }_3\epsilon }{E_3{\lambda }^2}\Vert i\lambda v-f^3{\Vert }^2\hfill \\ \le C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}^2+\frac{2{\rho }_3\epsilon }{E_3}\Vert v{\Vert }^2.\hfill \end{array}$$

Here, we choose $\epsilon =\frac{k{E}_3}{4{\rho }_3}$ and obtain

$$k\Vert v{\Vert }^2\le C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^2\Vert F{\Vert }_{\mathcal{H}}^2.$$

(103)

Equation (90) and estimate (103) yield

$$\begin{array}{c}{\rho }_3{h}_3\Vert \psi {\Vert }^2\le 2{\rho }_3{h}_3{\lambda }^2\Vert v{\Vert }^2+2{\rho }_3{h}_3\Vert f^3{\Vert }^2\hfill \\ \le C{\lambda }^4\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^4\Vert F{\Vert }_{\mathcal{H}}^2.\hfill \end{array}$$

This proves (100). Finally, to prove inequalities (101) and (102), just replace estimate (100) in (98) and (99). □

Theorem 9. Let $U_0\in D\left(\mathcal{A}\right)$ and let $S\left(t\right)=e^{t\mathcal{A}}$ be the $C_0$ -semigroup of contractions associated with problem (2)–(6). Then, S(t) is polynomially stable with decay rate $t^{-\frac{1}{4}}$ , that is, there exists a constant $C>0$ such that

$$\parallel S\left(t\right)U_0{\parallel }_{\mathcal{H}}\le \frac{C}{t^{\frac{1}{4}}}\parallel U_0{\parallel }_{D(\mathcal{A})},\forall t>0.$$

(104)

Moreover, this decay rate is optimal.

Proof. From the definition of norm in $\mathcal{H}$ , applying Lemmas 6 and 8, we get

$$\Vert U{\Vert }^2\le {\lambda }^4\Vert F{\Vert }_{\mathcal{H}}\Vert U{\Vert }_{\mathcal{H}}+C{\lambda }^4\Vert F{\Vert }_{\mathcal{H}}^2.$$

Applying Young’s inequality, we find that

$$\Vert U{\Vert }^2\le C{\lambda }^8\Vert F{\Vert }^2+\frac{1}{2}\Vert U{\Vert }^2+C{\lambda }^4\Vert F{\Vert }^2$$

and then

$$\Vert U{\Vert }^2\le C{\lambda }^8\Vert F{\Vert }^2.$$

(105)

Equivalently,

$$\Vert {(i\lambda I-\mathcal{A})}^{-1}F{\Vert }_{\mathcal{H}}\le C{\lambda }^4\Vert F{\Vert }_{\mathcal{H}},F\in \mathcal{H},\left|\lambda \right|\ge 1,$$

(106)

where the constant $C>0$ is independent of $\lambda$ . The relation (106) is written in the form

$$\Vert {(i\lambda I-\mathcal{A})}^{-1}{\Vert }_{\mathcal{L}\left(\mathcal{H}\right)}=\mathcal{O}\left({\left|\lambda \right|}^4\right),\left|\lambda \right|\to \infty .$$

Applying Theorem 4, we obtain

$$\parallel S(t){\mathcal{A}}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}=\mathcal{O}\left(t^{-\frac{1}{4}}\right),t\to \infty ,$$

(107)

and this means that there exists a constant $C>0$ such that

$$\Vert S(t){\mathcal{A}}^{-1}F{\Vert }_{\mathcal{H}}\le \frac{C}{t^{\frac{1}{4}}}\Vert F{\Vert }_{\mathcal{H}},F\in \mathcal{H},t>0.$$

(108)

From Proposition 3, we know that $0\in \varrho (\mathcal{A})$ , then there exists $U_0\in D\left(\mathcal{A}\right)$ such that

$${\mathcal{A}}^{-1}F=U_0\mathrm{y}\Vert U_0{\Vert }_{D\left(\mathcal{A}\right)}=\parallel U_0{\parallel }_{\mathcal{H}}+\parallel \mathcal{A}{U}_0{\parallel }_{\mathcal{H}}.$$

(109)

From equations (108) and (109), we get

$$\Vert S(t)U_0{\Vert }_{\mathcal{H}}\le \frac{C}{t^{\frac{1}{4}}}\Vert U_0{\Vert }_{D\left(\mathcal{A}\right)},t>0,$$

and this means that the $C_0$ -semigroup $S\left(t\right)=e^{t\mathcal{A}}$ decays polynomially with a rate of $t^{-\frac{1}{4}}$ .

To prove that the decay rate is optimal, we will use Theorem 5 and argue by contradiction. Suppose that the rate $t^{-\frac{1}{4}}$ can be improved. For example, let $t^{-\frac{1}{(4-\epsilon )}}$ be the new decay rate, for some $0<\epsilon <4$ . Taking $ϵ=\frac{3\epsilon }{4}$ in Theorem 5, we have

$$\frac{1}{{\left|\lambda \right|}^{4-\frac{\epsilon }{4}}}\parallel (i\lambda I-\mathcal{A}{)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{H}\right)}\le C_{\epsilon }.$$

(110)

On the other hand, from the formula (64) we obtain

$$B_n=N_n{\lambda }_n^3,$$

(111)

where

$$N_n=-\frac{{\rho }_3{h}_3{c}_2(\frac{k^2{\alpha }^2}{{\lambda }_n^6}-{(\frac{{\rho }_3}{E_3}-\frac{k}{E_3{h}_3{\lambda }_n^2})}^3{P}_n{Q}_n)}{k^2(\frac{2k{\alpha }^2}{{\lambda }_n^3}-(\frac{{\rho }_3}{E_3}-\frac{k}{E_3{h}_3{\lambda }_n^2})\sqrt{\frac{{\rho }_3}{E_3}-\frac{k}{E_3{h}_3{\lambda }_n^2}}{Q}_n-{\alpha }^2(\frac{{\rho }_3}{E_3}-\frac{k}{E_3{h}_3{\lambda }_n^2})\sqrt{\frac{{\rho }_3}{E_3}-\frac{k}{E_3{h}_3{\lambda }_n^2}}{P}_n)}$$

and

$$\underset{n\to \infty }{lim}{N}_n=\frac{{\rho }_3^2{h}_3{c}_2{\delta }_1{\delta }_3}{i{k}^2{E}_3({\delta }_3+{\alpha }^2{\delta }_1)}\ne 0.$$

By (37), (68), and (111), it follows that

$$\Vert U_n{\Vert }_{\mathcal{H}}^2\ge \frac{1}{2}{\rho }_3{h}_{3}L{\left|{\lambda }_n{B}_n\right|}^2={\lambda }_n^8{K}^2{\left|N_n\right|}^2\Vert F_n{\Vert }_{\mathcal{H}}^2,$$

where $K=\sqrt{\frac{{\rho }_3{h}_3}{3{\rho }_2{\left|c_2\right|}^2}}$ . This implies

$$\frac{1}{{\left|{\lambda }_n\right|}^{4-\frac{\epsilon }{4}}}\Vert {(i\lambda I-\mathcal{A})}^{-1}{\Vert }_{\mathcal{H}}\ge \frac{1}{{\left|{\lambda }_n\right|}^{4-\frac{\epsilon }{4}}}{\lambda }_n^{4}K\left|N_n\right|={\left|{\lambda }_n\right|}^{\frac{\epsilon }{4}}K\left|N_n\right|.$$

(112)

Since

$$\underset{n\to \infty }{lim}\left({\left|{\lambda }_n\right|}^{\frac{\epsilon }{4}}K\left|N_n\right|\right)=\infty$$

we have

$$\underset{n\to \infty }{lim}\left(\frac{1}{{\left|{\lambda }_n\right|}^{4-\frac{\epsilon }{4}}}{\Vert {(i\lambda I-\mathcal{A})}^{-1}\Vert }_{\mathcal{H}}\right)=\infty .$$

(113)

The limit (113) is a contradiction with estimate (110). Therefore, the decay rate cannot be improved. This completes the proof. □