Structural Stability for a Coupled System of Wave Plate Type

'e research on the structural stability of various types of partial differential equations has always been the focus of current research and received long-term attention. One can refer to the book of Ames and Straughan [1] and the monograph of Straughan [2]. For more papers, one can also see [3–5] and the papers listed therein. 'e so-called structural stability is to study the continuous dependence and convergence of the model on the coefficients. In the process of model building, simplification, and numerical calculation, some errors will inevitably appear. And these errors will not be completely avoided with the progress of measurement methods like errors. 'erefore, it is necessary to study the influence of these errors on the solution of the equation. Recently, there are many papers that study the structural stability for the fluid flow in porous medium (one could see [6–9]). In the field of differential equations, biharmonic equation represents a lot of physical models and has important applications in elastic mechanics and porous media. For the biharmonic equation on the two-dimensional pipe line, there are many articles studying the spatial properties of the solution and accumulating a lot of methods. For more specific results, readers can refer to Knowles [10, 11], Flavin [12], Flavin and Knops [13], Horgan [14], Payne and Scaefer [15], and Varlamov [16]. More references on structural stability can be found in literatures [17, 18]. Lin [19] and Knops and Lupoli [20] studied some time-dependent problems which involved the biharmonic operator. 'ey transformed the biharmonic operator into a fourth-order differential equation and studied the spatial attenuation estimation along the semi-infinite pipeline. Lin’s study [19] was improved by Song [21, 22] who studied the time-dependent Stokes flow. For more articles on the spatial behavior, see [23–27]. For the viscoelasticity equations, there are some recent contributions [28–31]. For a review of new contributions about practical application of plates of waves systems, one could see [32–34]. 'e problems of the present paper will defined in an unbounded region which is denoted by Ω0: Ω0 :� x1, x2 ( 􏼁 x1 􏼌􏼌􏼌 > 0, 0<x2 < h 􏽮 􏽯, (1)


Introduction
e research on the structural stability of various types of partial differential equations has always been the focus of current research and received long-term attention. One can refer to the book of Ames and Straughan [1] and the monograph of Straughan [2]. For more papers, one can also see [3][4][5] and the papers listed therein. e so-called structural stability is to study the continuous dependence and convergence of the model on the coefficients. In the process of model building, simplification, and numerical calculation, some errors will inevitably appear. And these errors will not be completely avoided with the progress of measurement methods like errors. erefore, it is necessary to study the influence of these errors on the solution of the equation. Recently, there are many papers that study the structural stability for the fluid flow in porous medium (one could see [6][7][8][9]).
In the field of differential equations, biharmonic equation represents a lot of physical models and has important applications in elastic mechanics and porous media. For the biharmonic equation on the two-dimensional pipe line, there are many articles studying the spatial properties of the solution and accumulating a lot of methods. For more specific results, readers can refer to Knowles [10,11], Flavin [12], Flavin and Knops [13], Horgan [14], Payne and Scaefer [15], and Varlamov [16]. More references on structural stability can be found in literatures [17,18]. Lin [19] and Knops and Lupoli [20] studied some time-dependent problems which involved the biharmonic operator. ey transformed the biharmonic operator into a fourth-order differential equation and studied the spatial attenuation estimation along the semi-infinite pipeline. Lin's study [19] was improved by Song [21,22] who studied the time-dependent Stokes flow. For more articles on the spatial behavior, see [23][24][25][26][27]. For the viscoelasticity equations, there are some recent contributions [28][29][30][31].
For a review of new contributions about practical application of plates of waves systems, one could see [32][33][34]. e problems of the present paper will defined in an unbounded region which is denoted by Ω 0 : where h > 0 is a known constant. We also let e time interval of our problems is denoted as [0, T], where T > 0.
We note that Tang et al. [35] considered the spatial behavior of the following equations: e model is mainly used to describe the evolution of the system composed of elastic film and elastic plate. e system is subject to elastic force, which attracts the film to the plate with coefficient of a, and is affected by the thermal effect (see [36,37]). In (3) and (4), u and v denote the vertical deflections of the membrane and of the plate, respectively. e constants ρ 1 , ρ 2 , μ, a, c, m, τ, and k are nonnegative.
In the boundary of Ω 0 and the initial time, the solutions satisfy where the symbol Δ is the harmonic operator and Δ 2 is the biharmonic operator. e present paper will consider the classical solutions to the problem (3)-(10). g 1 , g 2 , g 3 are known functions which satisfy the compatibility: We shall frequently make use of the following Poincare inequality: where u(ξ) is a smooth function which satisfies u(0) � 0(see [38]). roughout this paper, the Greek subscript is summed from 1 to 2. Comma is used to indicate partial differentiation, i.e., u i,j � zu i /zx j , φ ,αα � 2 α�1 (z 2 φ/zx 2 α ), and u i,t denotes zu i /zt. e paper is structured as follows. In Section 2, we define a weighted energy expression. Section 3 is devoted to seek the continuous dependence for the coefficient λ.

Weighted Energy Expressions
We define (u, v) is the solution of (3) and (4) with λ � λ 1 , (u * , v * ) is the solution of (3) and (4) with λ � λ 2 , we now define the differences π � u − u * , ϕ � v − v * ; then, the difference (π, ϕ) satisfies and the initial boundary conditions are We add some conditions on the solutions: From (13), we have (23) In this part, we will derive some useful energy expressions. From (14), we also have 2 Discrete Dynamics in Nature and Society We now tackle the item We define a new function By combining (24)-(26), we can easily get Discrete Dynamics in Nature and Society 3 Using Cauchy's inequality, we obtain We choose suitable a such that By combining (26), (28), and (29), we have From (27), we define By combining (26) and (31), we also have Discrete Dynamics in Nature and Society From (31), following Schwarz's inequality, we can easily get and we also get

Continuous Dependence for the Coefficient λ
In this part, we will use the following lemmas to prove our result.

Lemma 1.
We can get the following decay estimates: Proof. From (54) of [35], we have where c 1 and c 2 are positive constants.

Lemma 2.
e energy ψ(z, t) defined in (26) and (27) satisfies the following differential inequality: where k 3 and k 4 are positive constants.
Proof. From the definition of φ(z, t) in (31), using Schwarz's inequality and (34), we can get where k 1 is a positive constant. In order to give a bound for φ, we only need to give a bound for c t 0 Choosing ω > max (4/ρ 2 c), 4 and using Schwarz's inequality, (34) and (36), we obtain where k 2 is a positive constant, Using the Schwarz inequality, we have A combination of (42) (44), (45), and (37) leads to the result If we set k 3 � 2(k 1 + k 2 ) and k 4 � (2/λ 2 2 )(1 + (1/(k + χ))), we can get (47) can be rewritten as 6 Discrete Dynamics in Nature and Society where Λ(0, t) is a positive function and α 0 is a positive constant to be defined later.
Proof. Let where E(z, t) is defined as and α is an arbitrary constant to be chosen later.
(41) may be rewritten as if α satisfies the quadratic equation Solving (53), we have Choosing α � α 0 and integrating (53), we have the following two cases.
From (70), we can easily get that Summarizing all the above discussions, we can establish the following theorem.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.