This paper focuses on the spatial properties of a coupled system of wave plate type in a two-dimensional pipe. Using the technique of differential inequality and the method of energy estimation, the effect of the coefficient is established.

1. Introduction

The 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 [35] and the papers listed therein. The 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. Therefore, 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 [69]).

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. They 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 [2327]. For the viscoelasticity equations, there are some recent contributions [2831].

For a review of new contributions about practical application of plates of waves systems, one could see [3234].

The problems of the present paper will defined in an unbounded region which is denoted by :where is a known constant. We also let

The time interval of our problems is denoted as , where .

We note that Tang et al. [35] considered the spatial behavior of the following equations:

The model is mainly used to describe the evolution of the system composed of elastic film and elastic plate. The system is subject to elastic force, which attracts the film to the plate with coefficient of , and is affected by the thermal effect (see [36, 37]). In (3) and (4), and denote the vertical deflections of the membrane and of the plate, respectively. The constants , , , , , , , and are nonnegative.

In the boundary of and the initial time, the solutions satisfywhere the symbol is the harmonic operator and is the biharmonic operator.

The present paper will consider the classical solutions to the problem (3)–(10). are known functions which satisfy the compatibility:

We shall frequently make use of the following Poincare inequality:where is a smooth function which satisfies (see [38]).

Throughout this paper, the Greek subscript is summed from 1 to 2. Comma is used to indicate partial differentiation, i.e., , and denotes .

The 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 .

2. Weighted Energy Expressions

We define is the solution of (3) and (4) with , is the solution of (3) and (4) with , we now define the differences ; then, the difference satisfiesand the initial boundary conditions are

We add some conditions on the solutions:

From (13), we have

In this part, we will derive some useful energy expressions. From (14), we also have

We now tackle the item

We define a new function

By combining (24)–(26), we can easily get

Using Cauchy’s inequality, we obtain

We choose suitable such that

By combining (26), (28), and (29), we have

From (27), we define

By combining (26) and (31), we also have

From (31), following Schwarz’s inequality, we can easily getand we also get

3. Continuous Dependence for the Coefficient

In this part, we will use the following lemmas to prove our result.

Lemma 1. We can get the following decay estimates:with , and .

Proof. From (54) of [35], we havewhere and are positive constants.
We rewrite (38) aswith and .
Integrating (39) from 0 to givesFrom the definition of in [35] (51), we can get the desired result (37).

Lemma 2. The energy defined in (26) and (27) satisfies the following differential inequality:where and are positive constants.

Proof. From the definition of in (31), using Schwarz’s inequality and (34), we can getwhere is a positive constant.
In order to give a bound for , we only need to give a bound for . We haveChoosing and using Schwarz’s inequality, (34) and (36), we obtainwhere is a positive constant,
Using the Schwarz inequality, we haveA combination of (42) (44), (45), and (37) leads to the resultIf we set and , we can get(47) can be rewritten asInequality (48) is the result of Lemma 2.

Lemma 3. For the energy expression defined in (26), we have the following estimates:Case 1: if , we getCase 2: if , we getwhere is a positive function and is a positive constant to be defined later.

Proof. Letwhere is defined asand is an arbitrary constant to be chosen later.
(41) may be rewritten asif satisfies the quadratic equationSolving (53), we haveChoosing and integrating (53), we have the following two cases.Case 1: if , we getFrom (51), we may deduce thatCase 2: if , we getFrom (51), we can easily get thatInequalities (57) and (59) show the desired results (49) and (50).

Lemma 4. In order to make inequalities (49) and (50) explicit, we can give a bound for :where is a positive function.

Proof. We knowWe only need to bound and .
From (27), we haveFrom (62), we obtainSimilarly, from (26), we haveFrom (48), we haveCombining (63)–(65), we havewith .
Using (51), we getwith .
Inequality (67) is the desired result (60).
Inserting (60) into (49) and (50), we obtain the following.Case 1: if , we getFrom (68), we may deduce thatCase 2: if , we getFrom (70), we can easily get thatSummarizing all the above discussions, we can establish the following theorem.

Theorem 1. Let and be the classical solutions of equations (13)–(22) for different values of and , respectively, and be the difference of and ; the estimates (68) and (71) are satisfied. Inequalities (68) and (71) exhibit not only exponential decay in but also show that the amplitude terms in (68) and (71) become small as .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this study. All authors read and approved the final manuscript.


This study was supported by the National Natural Science Foundation of China (grant no. 61907010), the Foundation for Natural Science in Higher Education of Guangdong, China (grant no. 2018KZDXM048), the General Project of Science Research of Guangzhou (grant no. 201707010126), and the Science Foundation of Huashang College Guangdong University of Finance and Economics (grant no. 2019HSDS28).