Convergence Results for the Double-Diffusion Perturbation Equations

Abstract

We study the structural stability for the double-diffusion perturbation equations. Using the a priori bounds, the convergence results on the reaction boundary coefficients $k_1$, $k_2$ and the Lewis coefficient $L_e$ could be obtained with the aid of some $Poincar\acute{e}$ inequalities. The results showed that the structural stability is valid for the the double-diffusion perturbation equations with reaction boundary conditions. Our results can be seen as a version of symmetry in inequality for studying the structural stability.

1. Introduction

Many papers in the literature have studied the continuous dependence or convergence of solutions of different equations in porous media on construction coefficients. We give these studies a new name. We call these stabilities structural stability. This kind of stability is different from the traditional stability. We do not care about the stability with the initial data, but about their structural stability with the model itself. For an introduction to the nature of this structural stability, please see book [1]. It is important to establish the result of the structural stability in the problem of the continuum mechanics. In [2], the authors studied a variety of equations and obtained many results on structural stability. We think it is very important to study structural stability. In the process of establishing the model, the error always exists. We want to know whether a small error will cause a sharp change in the solution

Straughan in paper [3] proposed a new type of double diffusion perturbation model in porous media. The Darcy approximation is used in the derivation of this type of equation. We usually call this type of equation Darcy equations. Details about such types of equations were introduced in [4,5].

There are many equations that describe fluids in porous media. In books [4,6,7], the authors studied many different types of equations. In [8,9,10], the saint-venant principle results were studied for the Brinkman, Darcy, and Forchheimer equations. The spatial decay results were obtained. In the literature, many results on the structural stability of equations in porous media have been obtained. Representative papers can be seen by [11,12,13,14,15,16]. It should be emphasized that some new results have also emerged recently, see [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48]. These results all belong to the category of the study of structural stability.

In this article, we continue to consider the structural stability of such types of equations. We consider the following double-diffusion perturbation equations with velocity, pressure, temperature, and concentration perturbations:

$$\left\{\begin{array}{c}\frac{\partial u_i}{\partial t}=C\phi l_i-R\theta l_i+{\pi }_{,i},\hfill \\ \frac{\partial u_i}{\partial x_i}=0,\hfill \\ \frac{\partial \theta }{\partial t}+u_i{\theta }_{,i}=u_3+\Delta \theta ,\hfill \\ {\epsilon }_1\frac{\partial \phi }{\partial t}+L_e{u}_i{\phi }_{,i}=u_3+\Delta \phi ,\hfill \end{array}\right.$$

(1)

where $u_i$, $\theta$, $\phi$, and $\pi$ are the velocity, temperature, concentration disturbance, and pressure, respectively. $\Delta$ is the Laplace operator. In Equation (1), R is the Rayleigh coefficient and C is the salinity Rayleigh coefficient, ${\epsilon }_1$ represents the porosity, and $L_e$ is the Lewis coefficient, $l=(0,0,1)=(l_1,l_2,l_3)$. The system of Equation (1) is established in the region $\mathsf{\Omega }\times [0,\tau ]$, where $\mathsf{\Omega }$ is bounded in the strictly convex region in $R^3$, and $\tau$ is a given constant and satisfies $0\le \tau <\infty$. The boundary conditions are:

$$u_i{n}_i=0,\frac{\partial \theta }{\partial n}=k_1\theta ,\frac{\partial \phi }{\partial n}=k_2\phi ,(x,t)\in \partial \mathsf{\Omega }\times [0,\tau ].$$

(2)

The initial conditions are:

$$u_i(x, 0)=u_{i0}\left(x\right), \phi (x, 0)={\phi }_0\left(x\right), \theta (x,0)={\theta }_0\left(x\right),x\in \mathsf{\Omega }.$$

(3)

There are significant differences between the double-diffusion perturbation equations and the Brinkman, Forchheimer, Darcy equations. The main difficulty is that we can not get the maximum value of the disturbance as the previous papers [11,12,13,14,15,16]. In the references, the maximum value of the disturbance is often used to obtain the required structural stability results. In this paper, we can not get the maximum estimates of disturbance. The structural stability results we need will not be obtained by using the previous methods. We must adopt a new method to overcome the difficulty of not getting the maximum value. We adapt the $L^4$ norm of the disturbance. In addition, since the velocity equation does not contain the Laplacian term, the estimates of the gradient of the velocity is not easy to obtain. There is no Laplacian term, so we can not get the gradient estimation of the velocity according to the conventional method. The estimation of gradient of the velocity is very important in this paper. How to get the gradient estimation of the velocity is the biggest innovation of this paper because the space and the inequalities used in this paper have the property of symmetry. Our results can be regarded as the application of the symmetry in the study of the structural stability. In this article, we will use other estimations to obtain the gradient estimation. The content of this paper is arranged as follows: First, some a prior estimates of the solutions are given, and then based on these a prior estimates, the differential inequality satisfied by the concentration difference of the solution is established, and the convergence results of the solutions are obtained by integrating the inequality. The following notational conventions are adopted in the text: A comma is used to indicate the partial derivative. For example, $i$ denotes the partial differentiation with respect to $x_i$. For example $u_{,i}=\frac{\partial u}{\partial x_i}$. The repeated Latin subscripts denotes summation. For example, $u_{i,i}={\displaystyle \sum _{i=1}^3}\frac{\partial u_i}{\partial x_i}$. The symbol $dx=d{x}_{1}d{x}_{2}d{x}_3$.

2. A Priori Bounds

In the course of producing the results of convergence on the coefficient of (1), we find it is easy if we can derive some a priori bounds for the solutions. We will give some Lemmas that are useful in proving our main results.

Lemma 1.

For the temperature θ and the concentration disturbance φ, we have the following estimates:

$${\int }_{\partial \mathsf{\Omega }}{\theta }^{2}dS\le \left(\frac{m_1}{m_0}+\frac{{\epsilon }_0{m}_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}{\theta }^{2}dS+\frac{1}{{\epsilon }_0}{\int }_{\mathsf{\Omega }}{\theta }_{,i}{\theta }_{,i}dS,$$

(4)

and

$${\int }_{\partial \mathsf{\Omega }}{\phi }^{2}dS\le \left(\frac{m_1}{m_0}+\frac{{\epsilon }_0{m}_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}{\phi }^{2}dS+\frac{1}{{\epsilon }_0}{\int }_{\mathsf{\Omega }}^{}{\phi }_{,i}{\phi }_{,i}dS,$$

(5)

where $m_0$, $m_1$, $m_2$ are positive constants, and ${\epsilon }_0$ is an arbitrary positive constant.

Proof. 

We defined a function ${\xi }_i$ on $\mathsf{\Omega }$. The function ${\xi }_i$ satisfies the following conditions:

$${\xi }_i{n}_i\ge m_0>0,\mathrm{x}\in \partial \mathsf{\Omega },$$

(6)

$$|{\xi }_{i,i}|\le m_1,\mathrm{x}\in \mathsf{\Omega },$$

(7)

$$|{\xi }_i|\le m_2,\mathrm{x}\in \mathsf{\Omega },$$

(8)

where $n_i$ is the unit outward normal vector.

From the divergence theorem, we have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill m_o{\oint }_{\partial \mathsf{\Omega }}{\theta }^{2}ds& \le {\oint }_{\partial \mathsf{\Omega }}{\xi }_i{n}_i{\theta }^{2}ds\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}^{}{\left({\xi }_i{\theta }^2\right)}_{i}dx\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}^{}{\xi }_{i,i}{\theta }^{2}dx+2{\int }_{\mathsf{\Omega }}^{}{\xi }_i\theta {\theta }_{,i}dx\hfill \\ \hfill & \le m_1{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx+2m_2{\int }_{\mathsf{\Omega }}^{}\theta \overline{x}\theta dx.\hfill \end{array}\end{array}$$

(9)

Using Schwarz’s inequality, we have:

$${\oint }_{\partial \mathsf{\Omega }}{\theta }^{2}ds\le \left(\frac{m_1}{m_0}+\frac{{\epsilon }_0{m_2}^2}{{m_0}^2}\right){\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx+\frac{1}{{\epsilon }_0}{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dx.$$

(10)

Following the same procedures, we can also get:

$${\int }_{\partial \mathsf{\Omega }}{\phi }^{2}dS\le \left(\frac{m_1}{m_0}+\frac{{\epsilon }_0{m}_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}{\phi }^{2}dS+\frac{1}{{\epsilon }_0}{\int }_{\mathsf{\Omega }}^{}{\phi }_{,i}{\phi }_{,i}dS.$$

(11)

□

Lemma 2.

For the velocity $u_i$, temperature θ, and the concentration disturbance φ, we have the following estimates:

$${\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\phi }^{2}dx\le n_1\left(t\right),$$

(12)

$${\int }_0^t{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dxd\eta \le n_2\left(t\right),$$

(13)

$${\int }_0^t{\int }_{\mathsf{\Omega }}^{}{\phi }_{,i}{\phi }_{,i}dxd\eta \le n_3\left(t\right),$$

(14)

where $n_1\left(t\right)$, $n_2\left(t\right)$, and $n_3\left(t\right)$ are non-negative monotonically increasing functions.

Proof. 

Multiplying both sides of the Equation (1) by $2u_i$, and integrating over $\mathsf{\Omega }$, we can get:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx& =2C{\int }_{\mathsf{\Omega }}^{}\phi l_i{u}_{i}dx-2R{\int }_{\mathsf{\Omega }}^{}\theta l_i{u}_{i}dx+2{\int }_{\mathsf{\Omega }}^{}{\pi }_{,i}{u}_{i}dx\hfill \\ \hfill & \le 2{\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx+C^2{\int }_{\mathsf{\Omega }}^{}{\phi }^{2}dx+R^2{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx.\hfill \end{array}\end{array}$$

(15)

Multiplying both sides of the Equation (1) by $2\theta$, and integrating over $\mathsf{\Omega }$, we can get:

$$\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx=2{\int }_{\mathsf{\Omega }}^{}{u}_3\theta dx+2{\int }_{\mathsf{\Omega }}^{}\theta \Delta \theta dx-2{\int }_{\mathsf{\Omega }}^{}{u}_i{\theta }_{,i}\theta dx.$$

(16)

Using (4) and taking ${\epsilon }_0=2k_1$, we can get:

$${\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS\le \left[\frac{m_1}{m_0}+\frac{2k_1{m}_2^2}{m_0^2}\right]{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx+\frac{1}{2k_1}{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dx.$$

(17)

For the second term on the right side of Equation (16), we have:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill 2{\int }_{\mathsf{\Omega }}^{}\theta \Delta \theta dx& =2{\int }_{\partial \mathsf{\Omega }}^{}\theta \frac{\partial \theta }{\partial n}dS-2{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dx\hfill \\ \hfill & =2k_1{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS-2{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dx\hfill \\ \hfill & \le \left(\frac{2k_1{m}_1}{m_0}+\frac{4k_1^2{m}_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx-{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dx.\hfill \end{array}\end{array}$$

(18)

Combining (16) and (18), and using the $H\ddot{o}lde{r}^{\prime }s$ inequality, we can get:

$$\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dx\le {\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx+(\frac{2k_1{m}_1}{m_0}+\frac{4k_1^2{m}_2^2}{m_0^2}+1){\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx.$$

(19)

Multiplying both sides of Equation (1) by 2$\phi$, and integrating over $\mathsf{\Omega }$, we can obtain:

$$\begin{array}{cc}\hfill & {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\phi }^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\phi }_{,i}{\phi }_{,i}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx+\left(\frac{2k_2{m}_1}{m_0}+\frac{4k_2^2{m}_2^2}{m_0^2}+1\right){\int }_{\mathsf{\Omega }}^{}{\phi }^{2}dx.\hfill \end{array}$$

(20)

We define a new function:

$$F_1\left(t\right)={\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\theta }^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\phi }^{2}dx.$$

Combining (15), (19), and (20), we obtain:

$$F_1\left(t\right)\le m_4+m_3{\int }_0^t{F}_1\left(\eta \right)d\eta ,$$

(21)

where

$$m_3=max\left\{3+\frac{1}{{\epsilon }_1},R^2+\frac{2k_1{m}_1}{m_0}+\frac{4k_1^2{m}_2^2}{m_0^2}+1,C^2+\frac{2k_2{m}_1}{m_0{\epsilon }_1}+\frac{4k_2^2{m}_2^2}{m_0^2{\epsilon }_1}+\frac{1}{{\epsilon }_1}\right\},$$

and

$$m_4={\int }_{\mathsf{\Omega }}^{}{u}_{i0}{u}_{i0}dx+{\int }_{\mathsf{\Omega }}^{}{\theta }_0{\theta }_{0}dx+{\int }_{\mathsf{\Omega }}^{}{\phi }_0{\phi }_{0}dx.$$

Using Gronwall’s inequality, we can get:

$$F_1\left(t\right)\le m_3{m}_4{e}^{m_{3}t}{\int }_0^t{e}^{-m_3\eta }d\eta =n_1\left(t\right).$$

(22)

Inserting (22) into (19), we can get:

$${\int }_0^t{\int }_{\mathsf{\Omega }}^{}{\theta }_{,i}{\theta }_{,i}dxd\eta \le \left(\frac{2k_1{m}_1}{m_0}+\frac{4k_1^2{m}_2^2}{m_0^2}+1\right){\int }_0^t{n}_1\left(\eta \right)d\eta =n_2\left(t\right).$$

Inserting (22) into (20), we can get:

$${\int }_0^t{\int }_{\mathsf{\Omega }}^{}{\phi }_{,i}{\phi }_{,i}dxd\eta \le \left(\frac{2k_2{m}_1}{m_0}+\frac{4k_2^2{m}_2^2}{m_0^2}+1\right){\int }_0^t{n}_1\left(\eta \right)d\eta =n_3\left(t\right).$$

□

Lemma 3.

For velocity $u_i$, we have the following estimates:

$${\left[{\int }_{\mathsf{\Omega }}^{}{\left(u_i{u}_i\right)}^{2}dx\right]}^{\frac{1}{2}}\le m_1\left(t\right),$$

(23)

where $m_1\left(t\right)$ is a positive function to be defined later.

Proof. 

We have the following identity:

$${\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{i,j}dx={\int }_{\mathsf{\Omega }}^{}{u}_{i,j}(u_{i,j}-u_{j,i})dx+{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{j,i}dx.$$

(24)

Since $\partial \mathsf{\Omega }$ is bounded, we know from the result of [44]:

$$|{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{j,i}dx|\le k_0{\int }_{\partial \mathsf{\Omega }}^{}{u}_i{u}_{i}ds,$$

where $k_0$ is the Gaussian curvature depending on $\partial \mathsf{\Omega }$.

Taking $\theta$=$u_i$, ${\epsilon }_0=2k_0$ in (4), we have:

$$\begin{array}{cc}\hfill |{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{j,i}dx|& \le k_0\left(\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{u}_i{u}_{i}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{i,j}dx\hfill \\ & \le k_0\left(\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right)n_1\left(t\right)+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{i,j}dx.\hfill \end{array}$$

(25)

Combining (24) and (25), we get:

$${\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{j,i}dx\le 2{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}(u_{i,j}-u_{j,i})dx+2k_0\left(\frac{3}{m}+\frac{2k_0{d}^2}{2m^2}\right)n_1\left(t\right).$$

(26)

Using Equation (1), we obtain:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}(u_{i,j}-u_{j,i})dx=2{\int }_{\mathsf{\Omega }}^{}(u_{i,j}-u_{j,i})u_{i,jt}dx\hfill \\ \hfill & \le 4{\int }_{\mathsf{\Omega }}^{}{u}_{i,j}(u_{i,j}-u_{j,i})dx+C^2{\int }_{\mathsf{\Omega }}^{}{\phi }_{,j}{\phi }_{,j}dx+R^2{\int }_{\mathsf{\Omega }}^{}{\theta }_{,j}{\theta }_{,j}dx.\hfill \end{array}$$

(27)

We define $E\left(t\right)={\int }_{\mathsf{\Omega }}^{}{u}_{i,j}(u_{i,j}-u_{j,i})dx$. From (26), we obtain:

$$E\left(t\right)\le 4e^{4t}{\int }_0^{t}m\left(y\right)e^{-4y}dy+R^2{n}_2\left(t\right)+C^2{n}_3\left(t\right)=m_2\left(t\right),$$

(28)

with $m\left(t\right)={\int }_{\mathsf{\Omega }}^{}{u}_{i,j}(x,0)\left[(u_{i,j}(x,0)-u_{j,i}(x,0)\right]dx$.

Inserting (28) into (26), we have:

$${\int }_{\mathsf{\Omega }}^{}{u}_{i,j}{u}_{i,j}dx\le 2m_2\left(t\right)+2k_0\left(\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right)n_1\left(t\right).$$

(29)

Using the result of (B.17) in [26], we have:

$$\begin{array}{cc}\hfill & {\left({\int }_{\mathsf{\Omega }}^{}{\left|u\right|}^{4}dx\right)}^{\frac{1}{2}}\le M\left(\frac{5}{4}{\int }_{\mathsf{\Omega }}^{}{\left|u\right|}^{2}dx+\frac{3}{4}{\int }_{\mathsf{\Omega }}^{}{|\nabla u|}^{2}dx\right)\hfill \\ \hfill & \le M\left(\frac{5}{4}{n}_1\left(t\right)+\frac{6}{4}{m}_2\left(t\right)\right)+\frac{6k_0}{4}\left(\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right)n_1\left(t\right)=m_1\left(t\right),\hfill \end{array}$$

(30)

where M is a positive constant. □

Lemma 4.

For the temperature θ, concentration disturbance φ, we have the following estimates:

$${\left({\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx\right)}^{\frac{1}{2}}\le n_4\left(t\right),$$

(31)

$${\left({\int }_{\mathsf{\Omega }}^{}{\phi }^{4}dx\right)}^{\frac{1}{2}}\le n_5\left(t\right),$$

(32)

with $n_4\left(t\right)$ and $n_5\left(t\right)$ are all monotonically increasing functions greater than zero.

Proof. 

Multiplying both sides of the (1)₃ by ${\theta }^3$ and integrating over $\mathsf{\Omega }$, we have:

$$\begin{array}{cc}\hfill \frac{1}{4}\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx+{\int }_{\mathsf{\Omega }}^{}{u}_i{\theta }_{,i}{\theta }^{3}dx& ={\int }_{\mathsf{\Omega }}{u}_3{\theta }^{3}dx+{\int }_{\mathsf{\Omega }}^{}{\theta }^3{\theta }_{,ii}dx\hfill \\ \hfill & ={\int }_{\mathsf{\Omega }}{u}_3{\theta }^{3}dx-3{\int }_{\mathsf{\Omega }}^{}{\theta }^2{\theta }_{,i}{\theta }_{,i}dx+k_1{\int }_{\partial \mathsf{\Omega }}{\theta }^{4}dS\hfill \\ \hfill & \le \frac{1}{4}{\int }_{\mathsf{\Omega }}{u}_3^{4}dx+\frac{3}{4}{\int }_{\mathsf{\Omega }}{\theta }^{4}dx-3{\int }_{\mathsf{\Omega }}^{}{\theta }^2{\theta }_{,i}{\theta }_{,i}dx+k_1{\int }_{\partial \mathsf{\Omega }}{\theta }^{4}dx.\hfill \end{array}$$

(33)

Replacing $\theta$ by ${\theta }^2$ and choosing ${\epsilon }_0=2k_1$ in (4), we get:

$$k_1{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{4}dx\le \frac{k_1{m}_1}{m_0}{\int }_n^{}{\theta }^{4}dx+\frac{2k_1^2{m}_2^2}{m_0^2}{\int }_n^{}{\theta }^{4}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\left({\theta }^2\right)}_{,i}{\left({\theta }^2\right)}_{,i}dx.$$

(34)

Inserting (34) into (33), we obtain:

$$\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx\le {\int }_{\mathsf{\Omega }}^{}{u}_3^{4}dx+3{\int }_n^{}{\theta }^{4}dx+\frac{4m_1{k}_1}{m_0}{\int }_n^{}{\theta }^{4}dx+\frac{8k_1^2{m}_2^2}{m_0^2}{\int }_n^{}{\theta }^{4}dx.$$

(35)

Inserting (23) into (35), we obtain:

$$\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx\le m_1^2\left(t\right)+\left(3+\frac{4m_1{k}_1}{m_0}+\frac{8k_1^2{m}_2^2}{m_0^2}\right){\int }_n^{}{\theta }^{4}dx.$$

(36)

An integration of (36) leads to

$${\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx\le e^{\left(3+\frac{4m_1{k}_1}{m_0}+\frac{8k_1^2{m}_2^2}{m^2}\right)t}\left[{\int }_{\mathsf{\Omega }}^{}{\theta }_0^{4}dx+{\int }_0^t{\left(m_1\left(\eta \right)\right)}^{2}d\eta \right].$$

(37)

We obtain:

$${\left({\int }_n^{}{\theta }^{4}dx\right)}^{\frac{1}{2}}\le n_4\left(t\right).$$

(38)

Following the same procedures, we can also get:

$${\left({\int }_{\mathsf{\Omega }}^{}{\psi }^{4}dx\right)}^{\frac{1}{2}}\le n_5\left(t\right),$$

(39)

with

$$n_4\left(t\right)=e^{\left(3+\frac{4m_1{k}_1}{m_0}+\frac{8k_1^2{m}_2^2}{m_0^2}\right)t}\left[{\int }_{\mathsf{\Omega }}{\theta }_0^{2}dx+{\int }_0^t{\left(m_1\left(\eta \right)\right)}^{2}d\eta \right],$$

and

$$n_5\left(t\right)=e^{\left(3+\frac{4m{k}_2}{m_0}+\frac{8k_2^2{m}_2^2}{m_0^2}\right)t}\left[{\int }_{\mathsf{\Omega }}{\psi }^{4}dx+{\int }_0^t{\left(m_1\left(\eta \right)\right)}^{2}d\eta \right].$$

□

3. Convergence Result for the Reaction Boundary Coefficients ${\mathit{k}}_{\mathbf{1}}$ and ${\mathit{k}}_{\mathbf{2}}$

Let $(u_i,\theta ,\phi ,\pi )$ be the solution of (1)–(3) with $k_1=\widehat{k_1}$, $k_2=\widehat{k_2}$$(u_i^{*},{\theta }^{*},{\phi }^{*},{\pi }^{*})$ be the solution of (1)–(3) with $k_1=0$, $k_2=0$. We define ${\omega }_i=u_i-u_i^{*}$, $\widehat{\theta }=\theta -{\theta }^{*}$, $\hat{\phi }=\phi -{\phi }^{*}$, $\hat{\pi }=\pi -{\pi }^{*}$, then $({\omega }_i,\hat{\theta },\hat{\phi },\hat{\pi })$ satisfies the following equations:

$$\begin{array}{c}\hfill \frac{\partial {\omega }_i}{\partial t}=C\widehat{\phi }{l}_i-R\widehat{\theta }{l}_i+{\widehat{\pi }}_{,i},\\ \hfill \frac{\partial {\omega }_i}{\partial x_i}=0,\\ \hfill \frac{\partial \widehat{\theta }}{\partial t}+{\omega }_i{\theta }_{,i}+u_i^{*}{\widehat{\theta }}_{,i}={\omega }_3+\Delta \widehat{\theta },\\ \hfill {\epsilon }_1\frac{\partial \widehat{\phi }}{\partial t}+L_e({\omega }_i{\phi }_{,i}+u_i^{*}{\widehat{\phi }}_{,i})={\omega }_3+\Delta \widehat{\phi }.\end{array}$$

(40)

The boundary conditions are:

$${\omega }_i{n}_i=0,\frac{\partial \widehat{\theta }}{\partial n}=\widehat{k_1}\theta ,\frac{\partial \widehat{\phi }}{\partial n}=\widehat{k_2}\phi ,(x,t)\in \partial \mathsf{\Omega }\times [0,\tau ].$$

(41)

The initial conditions are:

$${\omega }_i(x,0)=0,\widehat{\phi }(x,0)=0,\widehat{\theta }(x,0)=0,x\in \mathsf{\Omega }.$$

(42)

In deducing our main result, we will use the following Lemma.

Lemma 5.

For the difference of the velocity ${\omega }_i$, we can get the following estimates:

$${\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx\le 2{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx+2k_0\left[\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right]{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx,$$

(43)

with$k_0$as a positive constant.

Proof. 

We know the fact:

$${\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx={\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{j,i}dx.$$

(44)

Since the boundary of $\mathsf{\Omega }$ is bounded, we have:

$$|{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{j,i}dx|\le k_0{\int }_{\partial \mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dS,$$

(45)

with $k_0$ as a positive constant depending on the Gaussian curvature of $\partial \mathsf{\Omega }$ (see [44]).

Using the result (4) with ${\epsilon }_0=2k_0$, we can obtain:

$${\int }_{\partial \mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dS\le \left[\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right]{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+\frac{1}{2k_0}{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx.$$

(46)

Inserting (45) and (46) into (44), we can get:

$${\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx\le 2{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx+2k_0\left[\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right]{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx.$$

In this part, we will get the following Theorem. □

Theorem 1.

Let $(u_i,\theta ,\phi ,\pi )$ be the classical solution of the initial value problem (1)–(3) with $k_1=\widehat{k_1}$, $k_2=\widehat{k_2}$, and $(u_i^{*},{\theta }^{*},{\phi }^{*},{\pi }^{*})$ be the classical solution of the initial boundary value problem (1)–(3) with $k_1=0$, $k_2=0$. $({\omega }_i,\widehat{\theta },\widehat{\phi },\widehat{\pi })$ is the difference of these two solutions. When $\widehat{k_1}$ and $\widehat{k_2}$ tend to zero, the solution $(u_i,\theta ,\phi ,\pi )$ converges to the solution $(u_i^{*},{\theta }^{*},{\phi }^{*},{\pi }^{*})$. The difference of the solution $({\omega }_i,\widehat{\theta },\widehat{\phi },\widehat{\pi })$ satisfies:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\epsilon }_1{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx\hfill \\ \hfill & \le {\widehat{k}}_1^2{m}_8{e}^{m_{8}t}{n}_6\left(t\right)+{\widehat{k}}_2^2{m}_8{e}^{m_{8}t}{n}_7\left(t\right),\hfill \end{array}$$

(47)

where $m_8$ is a positive constant and $n_6\left(t\right)$ and $n_7\left(t\right)$ are positive functions.

Proof. 

Multiplying both sides of Equation (40) by 2${\omega }_i$, and integrating over $\mathsf{\Omega }$, we can get:

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx& =2C{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }{l}_i{\omega }_{i}dx-2R{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }{l}_i{\omega }_{i}dx+2{\int }_{\mathsf{\Omega }}^{}{\widehat{\pi }}_{,i}{\omega }_{i}dx\hfill \\ \hfill & \le 2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+C^2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+R^2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx.\hfill \end{array}$$

(48)

From Equation (40), we know:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx\hfill \\ \hfill & =2{\int }_{\mathsf{\Omega }}^{}({\omega }_{i,j}-{\omega }_{j,i}){\omega }_{i,jt}dx\hfill \\ \hfill & =2C{\int }_{\mathsf{\Omega }}^{}({\omega }_{i,j}-{\omega }_{j,i}){\widehat{\phi }}_{,j}{l}_{i}dx+2{\int }_{\mathsf{\Omega }}^{}({\omega }_{i,j}-{\omega }_{j,i}){\widehat{\pi }}_{,ij}dx-2R{\int }_{\mathsf{\Omega }}^{}({\omega }_{i,j}-{\omega }_{j,i}){\widehat{\theta }}_{,j}{l}_{i}dx.\hfill \end{array}$$

(49)

Using the divergence theorem and $H\ddot{o}lde{r}^{\prime }s$ inequality, we can get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx\hfill \\ \hfill & \le (2C^2+2R^2){\int }_{\mathsf{\Omega }}^{}({\omega }_{i,j}-{\omega }_{j,i})({\omega }_{i,j}-{\omega }_{j,i})dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,j}{\widehat{\phi }}_{,j}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,j}{\widehat{\theta }}_{,j}dx\hfill \\ \hfill & =(4C^2+4R^2){\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,j}{\widehat{\phi }}_{,j}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,j}{\widehat{\theta }}_{,j}dx.\hfill \end{array}$$

(50)

Multiplying both sides of Equation (40) by $2\widehat{\theta }$, and integrating over $\mathsf{\Omega }$, we can get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx=2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\theta }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }\Delta \widehat{\theta }dx-2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\theta }_{,i}\widehat{\theta }dx-2{\int }_{\mathsf{\Omega }}^{}{u}_i^{*}{\widehat{\theta }}_{,i}\widehat{\theta }dx\hfill \\ \hfill & =2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\theta }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }\Delta \widehat{\theta }dx+2{\int }_{\mathsf{\Omega }}^{}{\omega }_i\theta {\widehat{\theta }}_{,i}dx.\hfill \end{array}$$

(51)

The first term on the right side of Equation (51) can be bounded by:

$$2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\theta }dx\le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx.$$

(52)

Using the result (4), and taking ${\epsilon }_0=1$, we can get:

$${\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dS\le (\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}){\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx.$$

(53)

We now take the second term on the right side of Equation (51):

$$\begin{array}{cc}\hfill 2{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }\Delta \widehat{\theta }dx& =2{\int }_{\partial \mathsf{\Omega }}^{}\widehat{\theta }\frac{\partial \widehat{\theta }}{\partial n}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & =2\widehat{k_1}{\int }_{\partial \mathsf{\Omega }}^{}\theta \widehat{\theta }dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+{\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+\left(\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx-{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx.\hfill \end{array}$$

(54)

Combining (51), (52), and (54), we can get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_i{\theta }^{2}dx\hfill \\ \hfill & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2{\left({\int }_{\mathsf{\Omega }}^{}{\left({\omega }_i{\omega }_i\right)}^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx\right)}^{\frac{1}{2}},\hfill \end{array}$$

(55)

where $m_5=\left(\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}\right)+1$.

Using the result of (B.17) in [26] and (31), we can get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2{\left({\int }_{\mathsf{\Omega }}^{}{\left({\omega }_i{\omega }_i\right)}^{2}dx\right)}^{\frac{1}{2}}{n}_4\left(t\right)\hfill \\ & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2M\left\{\frac{5}{4}{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+\frac{3}{4}{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx\right\}{n}_4\left(t\right)\hfill \\ & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+m_6{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+3M{n}_4\left(\tau \right){\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx,\hfill \end{array}$$

(56)

where $m_6=M{n}_4\left(\tau \right)\left[3k_0(\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2})+\frac{5}{2}\right]+1$.

Multiplying both sides of Equation (40) by $2\widehat{\phi }$, and integrating over $\mathsf{\Omega }$ we can get:

$$\begin{array}{cc}\hfill {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx& =2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\phi }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }\Delta \widehat{\phi }dx-2L_e{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\phi }_{,i}\widehat{\phi }dx-2L_e{\int }_{\mathsf{\Omega }}^{}{u}_i^{*}\widehat{\psi }\widehat{\psi ,i}dx\hfill \\ \hfill & =2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\phi }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }\Delta \widehat{\phi }dx+2L_e{\int }_{\mathsf{\Omega }}^{}{\omega }_i\phi {\widehat{\phi }}_{,i}dx.\hfill \end{array}$$

(57)

The first term on the right side of Equation (57) can be bounded by:

$$2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\phi }dx\le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx.$$

(58)

Using (4), and taking ${\epsilon }_0=1$, we can get:

$${\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dS\le \left(\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx.$$

(59)

We now take the second term on the right side of Equation (57). We have:

$$\begin{array}{cc}\hfill 2{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }\Delta \widehat{\phi }dx& =2{\int }_{\partial \mathsf{\Omega }}^{}\widehat{\phi }\frac{\partial \widehat{\phi }}{\partial n}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & =2{\widehat{k}}_2{\int }_{\partial \mathsf{\Omega }}^{}\phi \widehat{\phi }dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+\left(\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx-{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx.\hfill \end{array}$$

(60)

Combining (57)–(60), we can obtain:

$$\begin{array}{cc}\hfill & {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+2L_e^2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_i{\phi }^{2}dx\hfill \\ \hfill & \le \widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+2L_e^2{\left({\int }_{\mathsf{\Omega }}^{}{\left({\omega }_i{\omega }_i\right)}^{2}dx\right)}^{\frac{1}{2}}{({\int }_{\mathsf{\Omega }}^{}{\phi }^{4}dx)}^{\frac{1}{2}}.\hfill \end{array}$$

(61)

We can also get:

$$\begin{array}{cc}\hfill & {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ & \le \widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_5{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+2L_e^{2}M\left\{\frac{5}{4}{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+\frac{3}{4}{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx\right\}{n}_5\left(t\right)\hfill \\ & \le \widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+m_7{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_7{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+3M{L}_e^2{n}_5\left(\tau \right){\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx,\hfill \end{array}$$

(62)

where $m_5=M{L}_e^2{n}_5\left(\tau \right)\left[3k_0\left(\frac{m_1}{m_0}+\frac{2k_0{m}_2^2}{m_0^2}\right)+\frac{5}{2}\right]+1$.

Combining (48), (50), (56), (61), and (62), we can get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}[{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\epsilon }_1{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx]\hfill \\ & \le (m_6+m_7+2){\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_i+(R^2+m_5){\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+(C^2+m_5){\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx\hfill \\ & +\widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+\widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+(3M{n}_4\left(\tau \right)+3M{L}_e^2{n}_5\left(\tau \right)+4C^2+4R^2){\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx.\hfill \end{array}$$

(63)

Let

$$F_2\left(t\right)={\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\epsilon }_1{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx,$$

$$m_8=max\{m_6+m_7+2,R^2+m_5,\frac{c^2+m_5}{{\epsilon }_1},3M{n}_4\left(\tau \right)+3M{L}_e^2{n}_5\left(\tau \right)+4C^2+4R^2.$$

From (63), it can be seen that:

$$\frac{d}{dt}{F}_2\left(t\right)\le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\theta }^{2}dS+\widehat{k_2^2}{\int }_{\partial \mathsf{\Omega }}^{}{\phi }^{2}dS+m_8{F}_2\left(t\right).$$

(64)

Integrating (64), and using (4), (5), (13), and (14), we can get:

$$F_2\left(t\right)\le {\widehat{k}}_1^2{m}_8{e}^{m_{8}t}{n}_6\left(t\right)+{\widehat{k}}_2^2{m}_8{e}^{m_{8}t}{n}_7\left(t\right),$$

(65)

with $n_6\left(t\right)={\int }_0^t\left(\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}\right)n_1\left(\eta \right)d\eta +n_2\left(t\right)$ and $n_7\left(t\right)={\int }_0^t\left(\frac{m_1}{m_0}+\frac{m_2^2}{m_0^2}\right)n_1\left(\eta \right)d\eta +n_3\left(t\right)$.

Inequality (65) shows that when ${\widehat{k}}_1$, ${\widehat{k}}_2$ simultaneously tend to zero, the energy $F_2\left(t\right)$ also tends to zero as the indicated norm. □

4. Convergence Result for the Lewis Coefficient ${\mathbf{L}}_{\mathbf{e}}$

Let $(u_i,\theta ,\phi ,\pi )$ be the solution of (1)–(3) when $Le=\widehat{Le}$, and let $(u_i^{*},{\theta }^{*},{\phi }^{*},{\pi }^{*})$ be the solution of (1)–(3) when $Le=0$. We assume ${\omega }_i=u_i-u_i^{*},\widehat{\theta }=\theta -{\theta }^{*},\widehat{\phi }=\phi -{\phi }^{*},\widehat{\pi }=\pi -{\pi }^{*}$, then $({\omega }_i,\widehat{\theta },\widehat{\phi },\widehat{\pi })$ satisfies the following equations:

$$\begin{array}{c}\hfill \frac{\partial {\omega }_i}{\partial t}=C\widehat{\phi }{l}_i-R\widehat{\theta }{l}_i+{\widehat{\pi }}_{,i},\\ \hfill \frac{\partial {\omega }_i}{\partial x_i}=0,\\ \hfill \frac{\partial \widehat{\theta }}{\partial t}+{\omega }_i{\theta }_{,i}+u_i^{*}{\widehat{\theta }}_{,i}={\omega }_3+\Delta \widehat{\theta },\\ \hfill {\epsilon }_1\frac{\partial \widehat{\phi }}{\partial t}+\widehat{L_e}{u}_i{\phi }_{,i}={\omega }_3+\Delta \widehat{\phi }.\end{array}$$

(66)

The boundary conditions are:

$${\omega }_i{n}_i=0,\frac{\partial \widehat{\theta }}{\partial n}=k_1\widehat{\theta },\frac{\partial \widehat{\phi }}{\partial n}=k_2\widehat{\phi },(x,t)\in \partial \mathsf{\Omega }\times [0,\tau ].$$

(67)

The initial conditions are:

$${\omega }_i(x,0)=0,\widehat{\phi }(x,0)=0,\widehat{\theta }(x,0)=0,x\in \mathsf{\Omega }.$$

(68)

Theorem 2.

Let $(u_i,\theta ,\phi ,\pi )$ be the solution of (1)–(3) when $Le=\widehat{Le}$, $(u_i^{*},{\theta }^{*},{\phi }^{*},{\pi }^{*})$ be the solution of (1)–(3) when $Le=0$. We assume ${\omega }_i=u_i-u_i^{*},\widehat{\theta }=\theta -{\theta }^{*},\widehat{\phi }=\phi -{\phi }^{*},\widehat{\pi }=\pi -{\pi }^{*}$, then $({\omega }_i,\widehat{\theta },\widehat{\phi },\widehat{\pi })$ satisfies the following estimates:

$$\begin{array}{cc}\hfill & {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\epsilon }_1{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx\hfill \\ \hfill & \le 2{\widehat{L_e}}^2{m}_{11}{e}^{m_{11}t}{\int }_0^t{m}_1\left(\eta \right)n_5\left(\eta \right)e^{-m_{11}\eta }d\eta ,\hfill \end{array}$$

(69)

where$m_{11}$is a constant greater than zero.

Proof. 

Multiplying both sides of Equation (66) by 2${\omega }_i$, and integrating over $\mathsf{\Omega }\times [0,t]$, we can obtain:

$$\begin{array}{cc}\hfill \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx& =2C{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }{l}_i{\omega }_{i}dx-2R{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }{l}_i{\omega }_{i}dx+2{\int }_{\mathsf{\Omega }}^{}{\widehat{\pi }}_{,i}{\omega }_{i}dx\hfill \\ \hfill & \le 2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+C^2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+R^2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx.\hfill \end{array}$$

(70)

□

Multiplying both sides of Equation (66) by $2\widehat{\theta }$, and integrating over $\mathsf{\Omega }$, we can get,

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx=2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\theta }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }\Delta \widehat{\theta }dx-2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\theta }_{,i}\widehat{\theta }dx-2{\int }_{\mathsf{\Omega }}^{}{u}_i^{*}{\widehat{\theta }}_{,i}\widehat{\theta }dx\hfill \\ \hfill & =2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\theta }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }\Delta \widehat{\theta }dx+2{\int }_{\mathsf{\Omega }}^{}{\omega }_i\theta {\widehat{\theta }}_{,i}dx.\hfill \end{array}$$

(71)

The first term on the right side of Equation (71) can be obtained from H$\ddot{o}$lder’s inequality:

$$2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\theta }dx\le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx.$$

(72)

Using a method similar to (4), and taking ${\epsilon }_0=2k_1$, we can get:

$${\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dS\le \left(\frac{m_1}{m_0}+2k_1\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx.$$

(73)

We now deal with the second term on the right side of Equation (71). We have:

$$\begin{array}{cc}\hfill 2{\int }_{\mathsf{\Omega }}^{}\widehat{\theta }\Delta \widehat{\theta }dx& =2{\int }_{\partial \mathsf{\Omega }}^{}\widehat{\theta }\frac{\partial \widehat{\theta }}{\partial n}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & =2k_1{\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le \widehat{k_1^2}{\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le 2k_1\left(\frac{m_1}{m_0}+2k_1\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx-{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx.\hfill \end{array}$$

(74)

Combining (71)–(74), we can get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_9{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_i{\theta }^{2}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_9{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2{\left({\int }_{\mathsf{\Omega }}^{}{\left({\omega }_i{\omega }_i\right)}^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\mathsf{\Omega }}^{}{\theta }^{4}dx\right)}^{\frac{1}{2}},\hfill \end{array}$$

(75)

where $m_9=2k_1(\frac{m_1}{m_0}+2k_1\frac{m_2^2}{m_0^2})+1$.

Using the result of (B.17) in [26] again, we can also get:

$$\begin{array}{cc}\hfill & \frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}_{,i}{\widehat{\theta }}_{,i}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_9{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2{\left({\int }_{\mathsf{\Omega }}^{}{\left({\omega }_i{\omega }_i\right)}^{2}dx\right)}^{\frac{1}{2}}{n}_4\left(t\right)\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_9{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+2M\left\{\frac{5}{4}{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+\frac{3}{4}{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}{\omega }_{i,j}dx\right\}{n}_4\left(t\right)\hfill \\ \hfill & \le m_6{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_9{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+3M{n}_4\left(\tau \right){\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx.\hfill \end{array}$$

(76)

Multiplying both sides of Equation (66)₃ by $2\widehat{\phi }$, and integrating over $\mathsf{\Omega }\times [0,t]$, we can get,

$$\begin{array}{cc}\hfill {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx& =2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\phi }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }\Delta \widehat{\phi }dx-2\widehat{L_e}{\int }_{\mathsf{\Omega }}^{}{u}_i{\phi }_{,i}\widehat{\phi }dx\hfill \\ \hfill & =2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\phi }dx+2{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }\Delta \widehat{\phi }dx+2\widehat{L_e}{\int }_{\mathsf{\Omega }}^{}{u}_i\phi {\widehat{\phi }}_{,i}dx.\hfill \end{array}$$

(77)

The first term on the right side of Equation (77) can be bounded by:

$$2{\int }_{\mathsf{\Omega }}^{}{\omega }_3\widehat{\phi }dx\le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx.$$

(78)

Using the result (4), and taking ${\epsilon }_0=2k_2$, we can get:

$${\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dS\le \left(\frac{m_1}{m_0}+2k_2\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx.$$

(79)

The second term on the right side of (77) can be bounded as follows:

$$\begin{array}{cc}\hfill 2{\int }_{\mathsf{\Omega }}^{}\widehat{\phi }\Delta \widehat{\phi }dx& =2{\int }_{\partial \mathsf{\Omega }}^{}\widehat{\phi }\frac{\partial \widehat{\phi }}{\partial n}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & =2k_2{\int }_{\partial \mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dS-2{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & \le 2k_2\left(\frac{m_1}{m_0}+2k_2\frac{m_2^2}{m_0^2}\right){\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx-{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx.\hfill \end{array}$$

(80)

Combining (77), (78), and (80), we can obtain:

$$\begin{array}{cc}\hfill & {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_{10}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+2{\widehat{L_e}}^2{\int }_{\mathsf{\Omega }}^{}{u}_i{u}_i{\phi }^{2}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_{10}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+2{\widehat{L_e}}^2{\left({\int }_{\mathsf{\Omega }}^{}{\left(u_i{u}_i\right)}^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\mathsf{\Omega }}^{}{\phi }^{4}dx\right)}^{\frac{1}{2}},\hfill \end{array}$$

(81)

where $m_{10}=2k_2\left(\frac{m_1}{m_0}+2k_2\frac{m_2^2}{m_0^2}\right)+1$.

Using the results (23) and (32), we can obtain:

$$\begin{array}{cc}\hfill & {\epsilon }_1\frac{d}{dt}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+\frac{1}{2}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}_{,i}{\widehat{\phi }}_{,i}dx\hfill \\ \hfill & \le {\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+m_{10}{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+2{\widehat{L_e}}^2{m}_1\left(t\right)n_5\left(t\right).\hfill \end{array}$$

(82)

A combination of (70), (76), (82), and (50) gives:

$$\begin{array}{cc}\hfill & \frac{d}{dt}[{\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\epsilon }_1{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx]\hfill \\ \hfill & \le (m_6+3){\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_i+(R^2+m_9){\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+(C^2+m_{10}){\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx\hfill \\ \hfill & +(3M{n}_4\left(\tau \right)+4C^2+4R^2){\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx+2{\widehat{L_e}}^2{m}_1\left(t\right)n_5\left(t\right).\hfill \end{array}$$

(83)

Let,

$$F_3\left(t\right)={\int }_{\mathsf{\Omega }}^{}{\omega }_i{\omega }_{i}dx+{\int }_{\mathsf{\Omega }}^{}{\widehat{\theta }}^{2}dx+{\epsilon }_1{\int }_{\mathsf{\Omega }}^{}{\widehat{\phi }}^{2}dx+{\int }_{\mathsf{\Omega }}^{}{\omega }_{i,j}({\omega }_{i,j}-{\omega }_{j,i})dx,$$

$$m_{11}=max\left\{m_6+3,R^2+m_9,\frac{C^2+m_{10}}{{\epsilon }_1},3M{n}_4\left(\tau \right)+4C^2+4R^2\right\}.$$

From (83), it can be seen that:

$$\frac{d}{dt}{F}_3\left(t\right)\le 2{\widehat{L_e}}^2{m}_1\left(t\right)n_5\left(t\right)+m_{11}{F}_2\left(t\right).$$

(84)

by an integration of (84) leads to:

$$F_3\left(t\right)\le 2{\widehat{L_e}}^2{m}_{11}{e}^{m_{11}t}{\int }_0^t{m}_1\left(\eta \right)n_5\left(\eta \right)e^{-m_{11}\eta }d\eta .$$

(85)

Inequality (85) shows that when $\widehat{L_e}$ tends to zero, the energy $F_3\left(t\right)$ also tends to zero.

5. Conclusions

In this paper, we studied the convergence results for the double-diffusion perturbation equations in a bounded domain. The convergence result of solutions were gained for the reaction boundary coefficients $k_1$, $k_2$ and the Lewis coefficient $L_e$. Using the method in this paper, similar results for other coefficients could also be gained. Our method is useful for studying the structural stability of bounded regions. However, for unbounded regions, because the regions become more complex, and the inequalities that can be used in bounded regions cannot be used in unbounded regions, essential difficulties will arise. Methods of dealing with stress terms will be the biggest difficulty in unbounded areas. It is an open problem now that we could solve by constructing special functions in relevant future research. In this paper, we only give a theoretical proof and a numerical simulation will be given in another paper.