# Structural Stability for a Coupled System of Wave Plate Type

Authors: Jincheng Shi; Ze Wang; Yuanlong Chen
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

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 coefficientλ is established.


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## Body

## 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 [3–5] 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 [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. 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 [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].The problems of the present paper will defined in an unbounded region which is denoted byΩ0:(1)Ω0:=x1,x2x1>0,0<x2<h,where h>0 is a known constant. We also let(2)Lz=x1,x2x1=z≥0,0≤x2≤h.The time interval of our problems is denoted as0,T, where T>0.We note that Tang et al. [35] considered the spatial behavior of the following equations:(3)ρ1u,tt−Δu−μΔu,t+aΔv=0,(4)ρ2v,tt+γΔ2v+aΔu−λΔv,t=0.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 ofa, 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. The constants ρ1, ρ2, μ, a, γ, m, τ, and k are nonnegative.In the boundary ofΩ0 and the initial time, the solutions satisfy(5)ux1,0,t=vx1,0,t=v,2x1,0,t=0,x1>0,t>0,(6)ux1,h,t=vx1,h,t=v,2x1,h,t=0,x1>0,t>0,(7)u0,x2,t=g1x2,t,0≤x2≤h,t>0,(8)v0,x2,t=g2x2,t,0≤x2≤h,t>0,(9)v,10,x2,t=g3x2,t0≤x2≤h,t>0,(10)ux1,x2,0=vx1,x2,0=u,tx1,x2,0=ϕ,tx1,x2,0=0,0≤x2≤h,x1>0,where the symbol Δ is the harmonic operator and Δ2 is the biharmonic operator.The present paper will consider the classical solutions to the problem (3)–(10). g1,g2,g3 are known functions which satisfy the compatibility:(11)g10,t=g1h,t=g1,20,t=g1,2h,t=0,g20,t=g2h,t=g2,20,t=g2,2h,t=0,g30,t=g3h,t=g3,20,t=g3,2h,t=0,g1x2,0=g2x2,0=g3x2,0=0.We shall frequently make use of the following Poincare inequality:(12)∫0hu2dξ≤h2π2∫0hdudξ2dξ,where uξ is a smooth function which satisfies u0=0(see [38]).Throughout this paper, the Greek subscript is summed from 1 to 2. Comma is used to indicate partial differentiation, i.e.,ui,j=∂ui/∂xj,φ,αα=∑α=12∂2φ/∂xα2, and ui,t denotes ∂ui/∂t.The paper is structured as follows. In Section2, we define a weighted energy expression. Section 3 is devoted to seek the continuous dependence for the coefficient λ.


## 2. Weighted Energy Expressions

We defineu,v is the solution of (3) and (4) with λ=λ1, u∗,v∗ is the solution of (3) and (4) with λ=λ2, we now define the differences π=u−u∗,ϕ=v−v∗; then, the difference π,ϕ satisfies(13)ρ1π,tt−Δπ−μΔπ,t+aΔϕ=0,(14)ρ2ϕ,tt+γΔ2ϕ+aΔπ−λ2Δϕ,t−λΔv,t∗=0,and the initial boundary conditions are(15)πx1,0,t=ϕx1,0,t=ϕ,2x1,0,t=0,x1>0,t>0,(16)πx1,h,t=ϕx1,h,t=ϕ,2x1,h,t=0,x1>0,t>0,(17)π0,x2,t=0,0≤x2≤h,t>0,(18)ϕ0,x2,t=0,0≤x2≤h,t>0,(19)ϕ,10,x2,t=0,0≤x2≤h,t>0,(20)πx1,x2,0=ϕx1,x2,0=π,tx1,x2,0=ϕ,tx1,x2,0=0,0≤x2≤h,x1>0.We add some conditions on the solutions:(21)π,π,η,π,α⟶0asz⟶∞,(22)ϕ,ϕ,η,ϕ,αη,ϕ,αβ⟶0asz⟶∞.From (13), we have(23)0=∫0t∫z∞∫Lξexp−wηπ,ηρ1π,ηη−π,αα−μπ,ααη+aϕ,ααdAdη=ρ1w2∫0t∫z∞∫Lξexp−wηπ,η2dAdη+ρ12∫z∞∫Lξexp−wtπ,tπ,tdA+∫0t∫Lzexp−wηπ,ηπ,1dx2dη+w2∫0t∫z∞∫Lξexp−wtπ,απ,αdAdη+12∫z∞∫Lξexp−wtπ,απ,αdA+μ∫0t∫z∞∫Lξexp−wηπ,αηπ,αηdAdη+μ∫0t∫Lzexp−wηπ,ηπ,1ηdx2dη+a∫0t∫z∞∫Lξexp−wηπ,ηϕ,ααdAdη.In this part, we will derive some useful energy expressions. From (14), we also have(24)0=∫0t∫z∞∫Lξexp−wηϕ,ηρ2ϕ,ηη+γϕ,ααββ+aπ,αα−λ2ϕ,ααη−λv,ααη∗dAdη=ρ2w2∫0t∫z∞∫Lξexp−wηϕ,η2dAdη+ρ22∫z∞∫Lξexp−wtϕ,tϕ,tdA+wγ2∫0t∫z∞∫Lξexp−wηϕ,αβϕ,αβdAdη+γ2∫z∞∫Lξexp−wtϕ,αβϕ,αβdA+γ∫0t∫Lzexp−wηϕ,αηϕ,α1dx2dη−γ∫0t∫Lzexp−wηϕ,ηϕ,1ββdx2dη+a∫0t∫z∞∫Lξexp−wηϕ,ηπ,ααdAdη+λ2∫0t∫z∞∫Lξexp−wηϕ,αηϕ,αηdAdη+λ2∫0t∫Lzexp−wηϕ,ηϕ,1ηdx2dη+λ∫0t∫z∞∫Lξexp−wηv,αη∗ϕ,αηdAdη+λ∫0t∫Lzexp−wηv,1η∗ϕ,ηdx2dη.We now tackle the item(25)a∫0t∫z∞∫Lξexp−wηϕ,ηπ,ααdAdη=−a∫0t∫z∞∫Lξexp−wηϕ,αηπ,αdAdη−a∫0t∫Lzexp−wηϕ,ηπ,1dx2dη=a∫0t∫z∞∫Lξexp−wηϕ,ααηπdAdη−a∫0t∫Lzexp−wηϕ,ηπ,1dx2dη+a∫0t∫Lzexp−wηϕ,1ηπdx2dη=−a∫0t∫z∞∫Lξexp−wηϕ,ααπ,ηdAdη+aw∫0t∫z∞∫Lξexp−wηϕ,ααπdAdη+a∫z∞∫Lξexp−wtϕ,ααπdA−a∫0t∫Lzexp−wηϕ,ηπ,1dx2dη+a∫0t∫Lzexp−wηϕ,1ηπdx2dη.We define a new function(26)ψz,t=ρ1w2∫0t∫z∞∫Lξexp−wηπ,η2dAdη+ρ12∫z∞∫Lξexp−wtπ,tπ,tdA+w2∫0t∫z∞∫Lξexp−wηπ,απ,αdAdη+12∫z∞∫Lξexp−wtπ,απ,αdA+μ∫0t∫z∞∫Lξexp−wηπ,αηπ,αηdAdη+ρ2w2∫0t∫z∞∫Lξexp−wηϕ,η2dAdη+ρ22∫z∞∫Lξexp−wtϕ,tϕ,tdA+wγ2∫0t∫z∞∫Lξexp−wηϕ,αβϕ,αβdAdη+γ2∫z∞∫Lξexp−wtϕ,αβϕ,αβdA+λ2∫0t∫z∞∫Lξexp−wηϕ,αηϕ,αηdAdη+aw∫0t∫z∞∫Lξexp−wηϕ,ααπdAdη+a∫z∞∫Lξexp−wtϕ,ααπdA.By combining (24)–(26), we can easily get(27)ψz,t=−∫0t∫Lzexp−wηπ,ηπ,1dx2dη−μ∫0t∫Lzexp−wηπ,ηπ,1ηdx2dη−γ∫0t∫Lzexp−wηϕ,αηϕ,1αdx2dη+γ∫0t∫Lzexp−wηϕ,ηϕ,1ββdx2dη−λ2∫0t∫Lzexp−wηϕ,ηϕ,1ηdx2dη+a∫0t∫Lzexp−wηϕ,ηπ,1dx2dη−a∫0t∫Lzexp−wηϕ,1ηπdx2dη+λ∫0t∫z∞∫Lξexp−wηv,αη∗ϕ,αηdAdη+λ∫0t∫Lzexp−wηv,1η∗ϕ,ηdx2dη.Using Cauchy’s inequality, we obtain(28)aw∫0t∫z∞∫Lξexp−wηϕ,ααπdAdη≤wγ8∫0t∫z∞∫Lξexp−wηϕ,αβϕ,αβdAdη+2a2wh2γπ2∫0t∫z∞∫Lξexp−wηπ,απ,αdAdη,(29)a∫z∞∫Lξexp−wtϕ,ααπdA≤γ8∫z∞∫Lξexp−wtϕ,αβϕ,αβdA+2a2h2γπ2∫z∞∫Lξexp−wtπ,απ,αdA.We choose suitablea such that(30)a2h2π2γ≤18.By combining (26), (28), and (29), we have(31)ψz,t≥ρ1w4∫0t∫z∞∫Lξexp−wηπ,η2dAdη+ρ14∫z∞∫Lξexp−wtπ,tπ,tdA+w4∫0t∫z∞∫Lξexp−wηπ,απ,αdAdη+14∫z∞∫Lξexp−wtπ,απ,αdA+μ2∫0t∫z∞∫Lξexp−wηπ,αηπ,αηdAdη+ρ2w4∫0t∫z∞∫Lξexp−wηϕ,η2dAdη+ρ24∫z∞∫Lξexp−wtϕ,tϕ,tdA+wγ4∫0t∫z∞∫Lξexp−wηϕ,αβϕ,αβdAdη+γ4∫z∞∫Lξexp−wtϕ,αβϕ,αβdA+m22k∫0t∫z∞∫Lξexp−wηϕ,αηϕ,αηdAdη.From (27), we define(32)φz,t=∫z∞ψξ,tdξ=−∫0t∫z∞∫Lξexp−wηπ,ηπ,1dAdη−μ∫0t∫z∞∫Lξexp−wηπ,ηπ,1ηdAdη−γ∫0t∫z∞∫Lξexp−wηϕ,αηϕ,1αdAdη+γ∫0t∫z∞∫Lξexp−wηϕ,ηϕ,1ββdAdη−λ2∫0t∫z∞∫Lξexp−wηϕ,ηϕ,1ηdAdη+a∫0t∫z∞∫Lξexp−wηϕ,ηπ,1dAdη−a∫0t∫z∞∫Lξexp−wηϕ,1ηπdAdη+λ∫0t∫z∞∫Lξexp−wηξ−zv,αη∗ϕ,αηdAdη+λ∫0t∫z∞∫Lξexp−wηv,1η∗ϕ,ηdAdη.By combining (26) and (31), we also have(33)φz,t=ρ1w2∫0t∫z∞∫Lξξ−zexp−wηπ,η2dAdη+ρ12∫z∞∫Lξξ−zexp−wtπ,tπ,tdA+w2∫0t∫z∞∫Lξξ−zexp−wηπ,η2π,απ,αdAdη+12∫z∞∫Lξξ−zexp−wtπ,απ,αdA+μ∫0t∫z∞∫Lξξ−zexp−wηπ,η2π,αηπ,αηdAdη+ρ2w2∫0t∫z∞∫Lξξ−zexp−wηϕ,η2dAdη+ρ22∫z∞∫Lξξ−zexp−wtϕ,tϕ,tdA+wγ2∫0t∫z∞∫Lξξ−zexp−wηπ,η2ϕ,αβϕ,αβdAdη+γ2∫z∞∫Lξξ−zexp−wtϕ,αβϕ,αβdA+λ2∫0t∫z∞∫Lξξ−zexp−wηπ,η2ϕ,αηϕ,αηdAdη+aw∫0t∫z∞∫Lξξ−zexp−wηπ,η2ϕ,ααπdAdη+a∫z∞∫Lξξ−zexp−wtϕ,ααπdA.From (31), following Schwarz’s inequality, we can easily get(34)φz,t≥ρ1w4∫0t∫z∞∫Lξξ−zexp−wηπ,η2dAdη+ρ14∫z∞∫Lξξ−zexp−wtπ,tπ,tdA+w4∫0t∫z∞∫Lξξ−zexp−wηπ,απ,αdAdη+14∫z∞∫Lξξ−zexp−wtπ,απ,αdA+μ2∫0t∫z∞∫Lξξ−zexp−wηπ,αηπ,αηdAdη+ρ2w4∫0t∫z∞∫Lξξ−zexp−wηϕ,η2dAdη+ρ24∫z∞∫Lξξ−zexp−wtϕ,tϕ,tdA+wγ4∫0t∫z∞∫Lξξ−zexp−wηϕ,αβϕ,αβdAdη+γ4∫z∞∫Lξξ−zexp−wtϕ,αβϕ,αβdA+λ22∫0t∫z∞∫Lξξ−zexp−wηϕ,αηϕ,αηdAdη,and we also get(35)∂φz,t∂z=−ψz,t,(36)∂2φz,t∂z2≥ρ1w4∫0t∫Lzexp−wηπ,η2dAdη+ρ14∫Lzexp−wtπ,tπ,tdx2+w4∫0t∫Lzexp−wηπ,απ,αdx2dη+14∫Lzexp−wtπ,απ,αdx2+μ2∫0t∫Lzexp−wηπ,αηπ,αηdx2dη+ρ2w4∫0t∫Lzexp−wηϕ,η2dx2dη+ρ24∫Lzexp−wtϕ,tϕ,tdx2+wγ4∫0t∫Lzexp−wηϕ,αβϕ,αβdx2dη+γ4∫Lzexp−wtϕ,αβϕ,αβdx2+λ22∫0t∫Lzexp−wηϕ,αηϕ,αηdx2dη.


## 3. Continuous Dependence for the Coefficientλ

In this part, we will use the following lemmas to prove our result.Lemma 1.
We can get the following decay estimates:(37)∫0t∫z∞∫Lξexp−wηξ−zv,αη∗v,αη∗dAdη+∫0t∫z∞∫Lξexp−wηv,αη∗v,αη∗dAdη≤1λ21+1k˜+χc10,te−k˜z,with k˜=−γ1+γ12+4γ2/2, χ=γ1 and c10,t=−∂F0,t/∂z+k˜+χF0,t.Proof.
From (54) of [35], we have(38)∂2F∂z2−γ1∂F∂z−γ2F≥0,where γ1 and γ2 are positive constants.
We rewrite (38) as(39)ek˜zF′−k˜+χF′≥0,with k˜=−γ1+γ12+4γ2/2 and χ=γ1.
Integrating (39) from 0 to z gives(40)−∂F∂z+k˜+χF≤−∂F0,t∂z+k˜+χF0,te−k˜z=c10,te−k˜z.
From the definition ofFz,t in [35] (51), we can get the desired result (37).Lemma 2.
The energyψz,t defined in (26) and (27) satisfies the following differential inequality:(41)∫z∞ψξ,tdξ+∂ψ∂z≤k3ψz,t+λ2k4c10,te−k˜z,where k3 and k4 are positive constants.Proof.
From the definition ofφz,t in (31), using Schwarz’s inequality and (34), we can get(42)−∫0t∫z∞∫Lξexp−wηπ,ηπ,1dAdη+−μ∫0t∫z∞∫Lξexp−wηπ,ηπ,1ηdAdη+−γ∫0t∫z∞∫Lξexp−wηϕ,αηϕ,1αdAdη+−λ2∫0t∫z∞∫Lξexp−wηϕ,ηϕ,1ηdAdη+a∫0t∫z∞∫Lξexp−wηϕ,ηπ,1dAdη+−a∫0t∫z∞∫Lξexp−wηϕ,1ηπdAdη≤k1−∂φ∂z,where k1 is a positive constant.
In order to give a bound forφ, we only need to give a bound for γ∫0t∫z∞∫Lξexp−wηϕ,ηϕ,1ββdAdη. We have(43)γ∫0t∫z∞∫Lξexp−wηϕ,ηϕ,1ββdAdη=−γ∫0t∫z∞∫Lξexp−wηϕ,βηϕ,1βdAdη−γ∫0t∫Lzexp−wηϕ,ηϕ,11dx2dη≤γ2∫0t∫z∞∫Lξexp−wηϕ,βηdAdη+γ2∫0t∫z∞∫Lξexp−wηϕ,1βϕ,1βdAdη+γ2∫0t∫Lzexp−wηϕ,ηϕ,ηdx2dη+γ2∫0t∫Lzexp−wηϕ,11ϕ,11dx2dη.
Choosingω>max4/ρ2γ,4 and using Schwarz’s inequality, (34) and (36), we obtain(44)γ∫0t∫z∞∫Lξexp−wηϕ,ηϕ,1ββdAdη≤−k2∂φ∂z+12∂2φ∂z2,where k2 is a positive constant,
Using the Schwarz inequality, we have(45)λ∫0t∫z∞∫Lξexp−wηξ−zv,αη∗ϕ,αηdAdη+λ∫0t∫z∞∫Lξexp−wηv,1η∗ϕ,ηdAdη≤λ24∫0t∫z∞∫Lξexp−wηξ−zϕ,αηϕ,αηdAdη+λ2λ2∫0t∫z∞∫Lξexp−wηξ−zv,αη∗v,αη∗dAdη+λ24∫0t∫z∞∫Lξexp−wηϕ,ηϕ,ηdAdη+λ2λ2∫0t∫z∞∫Lξexp−wηv,1η∗v,1η∗dAdη.
A combination of (42) (44), (45), and (37) leads to the result(46)φz,t≤φz,t2+−k1−k2∂φ∂z+12∂2φ∂z2+λ2λ221+1k˜+χc10,te−k˜z.
If we setk3=2k1+k2 and k4=2/λ221+1/k˜+χ, we can get(47)φz,t+k3∂φ∂z≤∂2φ∂z2+λ2k4c10,te−k˜z.
(47) can be rewritten as(48)∫z∞ψξ,tdξ+∂ψ∂z≤k3ψz,t+λ2k4c10,te−k˜z.
Inequality (48) is the result of Lemma 2.Lemma 3.
For the energy expressionψz,t defined in (26), we have the following estimates:Case 1: ifα0−k3−k˜≠0, we get(49)ψz,t≤Λ0,t−1α0−k3−k˜k4λ2c10,te−α0−k3z+1α0−k3−k˜k4λ2c10,te−k˜z.Case 2: ifα0−k3−k˜=0, we get(50)ψz,t≤Λ0,te−α0−k3z+k4zλ2c10,te−α0−k3z,whereΛ0,t is a positive function and α0 is a positive constant to be defined later.Proof.
Let(51)Λz,t=Ez,t+α˜∫z∞ek3ξ−zEξ,tdξ,where Ez,t is defined as(52)Ez,t=e−k3zψz,t,and α˜ is an arbitrary constant to be chosen later.
(41) may be rewritten as(53)∂Λz,t∂z+α˜Λz,t≤k4λ2c10,te−k3ze−k˜z,if α˜ satisfies the quadratic equation(54)α˜2−k3α˜−1=0.
Solving (53), we have(55)α˜=α0=k3+k32+42.
Choosingα˜=α0 and integrating (53), we have the following two cases.Case 1: ifα0−k3−k˜≠0, we get(56)Λz,t≤Λ0,t−1α0−k3−k˜k4λ2c10,te−α0z+1α0−k3−k˜k4λ2c10,te−k3+k˜z.From (51), we may deduce that(57)ψz,t≤Λ0,t−1α0−k3−k˜k4λ2c10,te−α0−k3z+1α0−k3−k˜k4λ2c10,te−k˜z.Case 2: ifα0−k3−k˜=0, we get(58)Λz,t≤Λ0,te−α0z+k4zλ2c10,te−α0z.From (51), we can easily get that(59)ψz,t≤Λ0,te−α0−k3z+k4zλ2c10,te−α0−k3z.Inequalities (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 Λ0,t:(60)Λ0,t≤λ2c30,t,where c30,t is a positive function.Proof.
We know(61)Λ0,t=ψ0,t+α0∫0∞ψξ,tdξ.
We only need to boundψ0,t and ∫0∞ψξ,tdξ.
From (27), we have(62)ψ0,t=λ∫0t∫0∞∫Lξexp−wηv,αη∗ϕ,αηdAdη≤λλ2ψ0,tF0,t.
From (62), we obtain(63)ψ0,t≤λ2λ22F0,t.
Similarly, from (26), we have(64)−∂ψ0,t∂z≤λ2λ22∂F0,t∂z.
From (48), we have(65)∫0∞ψz,tdz≤−∂ψ0,t∂z+k3ψ0,t+k4λ2c10,t.
Combining (63)–(65), we have(66)∫0∞ψz,tdz≤λ2c20,t,with c20,t=F0,t/λ22+1/λ22∂F0,t/∂z+k4c10,t.
Using (51), we get(67)Λ0,t≤λ2c30,t,with c30,t=F0,t/λ22+α0c20,t.
Inequality (67) is the desired result (60).
Inserting (60) into (49) and (50), we obtain the following.Case 1: ifα0−k3−k˜≠0, we get(68)Λz,t≤λ2c30,t−1α0−k3−k˜k4λ2c10,te−α0z+1α0−k3−k˜k4λ2c10,te−k3+k˜z.From (68), we may deduce that(69)ψz,t≤λ2c40,te−α0−k3z+1α0−k3−k˜k4λ2c10,te−k˜z.Case 2: ifα0−k3−k˜=0, we get(70)Λz,t≤λ2c20,te−α0z+k4zλ2c10,te−α0z.From (70), we can easily get that(71)ψz,t≤λ2c20,te−α0−k3z+k4zλ2c10,te−α0−k3z.Summarizing all the above discussions, we can establish the following theorem.Theorem 1.
Letu,v and u∗,v∗ be the classical solutions of equations (13)–(22) for different values of λ=λ1 and λ=λ2, respectively, and π,ϕ be the difference of u,v and u∗,v∗; the estimates (68) and (71) are satisfied. Inequalities (68) and (71) exhibit not only exponential decay in z but also show that the amplitude terms in (68) and (71) become small as λ⟶0.



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