Two-grid $ H^1 $-Galerkin mixed finite elements combined with $ L1 $ scheme for nonlinear time fractional parabolic equations

Jun Pan; Yuelong Tang; Jun Pan; Yuelong Tang

doi:10.3934/era.2023365

Electronic Research Archive

2023, Volume 31, Issue 12: 7207-7223. doi: 10.3934/era.2023365

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Research article

Two-grid $H^1$ -Galerkin mixed finite elements combined with $L1$ scheme for nonlinear time fractional parabolic equations

Jun Pan ^{1
,
,},
Yuelong Tang ²

1.
School of Foundation Studies, Zhejiang Pharmaceutical University, Ningbo 315500, China
2.
College of Science, Hunan University of Science and Engineering, Yongzhou 425199, China

Academic Editor: Alain Miranville

Received: 08 September 2023 Revised: 27 October 2023 Accepted: 07 November 2023 Published: 10 November 2023

In this paper, we propose a two-grid algorithm for nonlinear time fractional parabolic equations by $H^1$ -Galerkin mixed finite element discreitzation. First, we use linear finite elements and Raviart-Thomas mixed finite elements for spatial discretization, and $L1$ scheme on graded mesh for temporal discretization to construct a fully discrete approximation scheme. Second, we derive the stability and error estimates of the discrete scheme. Third, we present a two-grid method to linearize the nonlinear system and discuss its stability and convergence. Finally, we confirm our theoretical results by some numerical examples.

Keywords:

two-grid algorithm,
$H^1$ -Galerkin mixed finite element,
nonlinear time fractional partial differential equations

Citation: Jun Pan, Yuelong Tang. Two-grid $H^1$ -Galerkin mixed finite elements combined with $L1$ scheme for nonlinear time fractional parabolic equations[J]. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365

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Abstract

1. Introduction

Since Hirsch and Smale^[1] proposed the necessity of structural stability, this topic has received sufficient attention from scholars. This type of research focuses on whether small disturbances in the coefficients, initial data, and geometric models in the equations will cause significant disturbances in the solutions. At the beginning, people were mainly keen on dealing with the continuous dependence and convergence of fluid in porous media defined in two-dimensional or three-dimensional bounded regions. Freitas et al. ^[2] studied the long-term behavior of porous-elastic systems and proved that solutions depend continuously on the initial data. Payne and Straughan^[3] established a prior bounds and maximum principles for the solutions and obtained the structural stability of Darcy fluid in porous media, where they assumed that the temperature satisfies Newton's cooling conditions at the boundary. Scott^[4] considered the situation where Darcy fluid undergoes exothermic reactions at the boundary and obtained the continuous dependence of the solutions on the boundary parameters. Li et al.^[5] studied the interface connection between Brinkman–Forchheimer fluid and Darcy fluid in a bounded region, and obtained the continuous dependence on the heat source and Forchheimer coefficient. For more papers, on can see ^[6,7,8,9,10].

With the continuous development of technology and progress in the field of engineering, the necessity of studying the structural stability of fluid equations on a semi-infinite cylinder is even more urgent. The semi-infinite cylinder refers to a cylinder whose generatrix is parallel to the coordinate axis and its base is located on the coordinate plane, i.e.,

$R = \Big\{(x_1, x_2, x_3)\Big|(x_1, x_2)\in D,\ x_3\geq0\Big\},$

where $D$ is a bounded domain on $x_1Ox_2$ .

Li et al. have already done some work on this topic. Li and Lin^[11] proved the continuous dependence on the Forchheimer coefficient of the Brinkman–Forchheimer equations in $R$ . Papers ^[12] and ^[13] obtained structural stability for Forchheimer fluid and temperature-dependent bidispersive flow in $R$ , respectively.

In this paper, we introduce a new cylinder with a disturbed base, which has been considered in ^[14]. Let $D(f)$ represent the disturbed base, i.e.,

$D(f) = \Big\{(x_1, x_2, x_3)\Big|x_3 = f(x_1, x_2)\geq0,\ (x_1, x_2)\in D\Big\},$

where the given function $f$ satisfies

$\begin{align*} |f(x_1, x_2)| < \epsilon,\ \epsilon > 0. \end{align*}$

$\epsilon$ is called the perturbation parameter. The cylinder with a disturbed base is defined as

$R(f) = \Big\{(x_1, x_2, x_3)\Big|(x_1, x_2)\in D,\ x_3\geq f(x_1, x_2)\geq0\Big\}.$

Different from ^[14], we study the heat conduction equation applicable to the study of layered composite materials in binary mixtures^[15]

$\begin{align} b_1u_t& = k_1\triangle u-\gamma(u-v), \ \text{in} \ R\times\{t > 0\}, \end{align}$

(1.1)

$\begin{align} b_2v_t& = k_2\triangle v+\gamma(u-v), \ \text{in} \ R\times\{t > 0\}, \end{align}$

(1.2)

$\begin{align} u& = v = 0, \ \text{on}\ \partial D\times\{x_3 > 0\}\times\{t > 0\}, \end{align}$

(1.3)

$\begin{align} u& = v = 0, \ \text{in}\ R\times\{t = 0\}, \end{align}$

(1.4)

where $k_1, k_2, b_1, b_2$ and $\gamma$ are positive constants. $u$ and $v$ are the temperature fields in each constituent. Papers ^[16,17,18] further discussed and generalized the application of Eqs (1.1) and (1.2).

In this paper, we shall also use the notations

$R(z) = \Big\{(x_1, x_2, x_3)\Big| (x_1, x_2)\in D, x_3\geq z\geq 0\Big\},$

$D(z) = \Big\{(x_1, x_2, x_3)\Big| (x_1, x_2)\in D, x_3 = z\geq 0\Big\}.$

The main work of this article investigates the continuous dependence of solutions to Eqs (1.1)–(1.4) on perturbation parameters and base data. Due to many practical constraints, it is very common for the base of the cylinder to experience minor disturbance. Therefore, studying the effects of these disturbances is essential. To this end, we assume that $u^*$ and $v^*$ are perturbed solutions of Eqs (1.1)–(1.4) on $R(f)$ , and then prove that the difference between the unperturbed solutions and the perturbed solutions satisfies a first-order differential inequality. By solving this inequality, we can obtain the continuous dependence of the solution.

2. A priori bounds

On the finite end $D$ , we assume that the solutions to (1.1)–(1.4) satisfy

$\begin{align} u(t, {\boldsymbol{x}}) = L_{11}(t, x_1, x_2),\ v(t, {\boldsymbol{x}})& = L_{12}(t, x_1, x_2), t > 0,\ x_3 = 0,\ (x_1, x_2)\in D(0), \end{align}$

(2.1)

$\begin{align} u^*(t, {\boldsymbol{x}}) = L_{21}(t, x_1, x_2),\ v^*(t, {\boldsymbol{x}})& = L_{22}(t, x_1, x_2), t > 0,\ x_3 = f(x_1, x_2),\ (x_1, x_2)\in D(0). \end{align}$

(2.2)

In (2.1) and (2.2), the known functions $L_{ij} (i, j = 1, 2)$ satisfy the compatibility conditions on $\partial D$ .

We let that $\mathcal{H}_1(t, \boldsymbol{x})$ and $\mathcal{H}_2(t, \boldsymbol{x})$ are specific functions who have the same boundary conditions as $u^*$ and $v^*$ , respectively. That is

$\begin{align} \mathcal{H}_1(t, {\boldsymbol{x}}) = L_{21}(t, x_1, x_2)\exp\{-\sigma(x_3-f)\},\ \mathcal{H}_2(t, {\boldsymbol{x}}) = L_{22}(t, x_1, x_2)\exp\{-\sigma(x_3-f)\}, \end{align}$

(2.3)

where $\sigma > 0$ .

We now derive some lemmas.

Lemma 2.1. If $L_{21}, L_{22}\in H^1([0, \infty)\times D(f))$ , then

$\begin{align*} \int_0^t\exp\{-\eta_1\tau\}&\Big[k_1||\nabla u^*(\tau)||_{L^2(R(f))}^2+k_2||\nabla v^*(\tau)||_{L^2(R(f))}^2\Big]d\tau\leq d_1(t), \end{align*}$

where

$\begin{align} d_1(t)& = \int_0^t\exp\{-\eta_1\tau\}\Big[k_1||\nabla \mathcal{H}_1||_{L^2(R(f))}^2+k_2||\nabla \mathcal{H}_2||_{L^2(R(f))}^2\Big]d\tau \\ &+\exp\{-\eta_1t\}\Big[b_1||\mathcal{H}_1(t)||_{L^2(R(f))}^2+b_2||\mathcal{H}_2(t)||_{L^2(R(f))}^2\Big] \\ &+\frac{1}{2}\int_0^t\exp\{-\eta_1\tau\}\Big[b_1\eta_1||\mathcal{H}_{1,\tau}(\tau)||_{L^2(R(f))}^2 +b_2\eta_1||\mathcal{H}_{2,\tau}(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &+\frac{1}{2}\gamma\int_0^t\exp\{-\eta_1\tau\}||(\mathcal{H}_1-\mathcal{H}_2)(\tau)||_{L^2(R(f))}^2d\tau. \end{align}$

(2.4)

Proof. Using (1.1)–(1.4), we begin with

$\begin{align} \int_0^t\int_{R(f)}\exp\{-\eta_1\tau\}\Big[b_1u^*_\tau-k_1\triangle u^*+\gamma(u^*-v^*)\Big]u^*dxd\tau& = 0, \\ \int_0^t\int_{R(f)}\exp\{-\eta_1\tau\}\Big[b_2v^*_\tau-k_2\triangle v^*-\gamma(u^*-v^*)\Big]v^*dxd\tau& = 0. \end{align}$

We compute

$\begin{align} \frac{1}{2}\exp\{-\eta_1t\}&\Big[b_1||u^*(t)||_{L^2(R(f))}^2+b_2||v^*(t)||_{L^2(R(f))}^2\Big] \\ &+\int_0^t\exp\{-\eta_1\tau\}\Big[b_1\eta_1||u^*(\tau)||_{L^2(R(f))}^2+b_2\eta_1||v^*(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &+\int_0^t\exp\{-\eta_1\tau\}\Big[k_1||\nabla u^*(\tau)||_{L^2(R(f))}^2+k_2||\nabla v^*(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &+\gamma\int_0^t\exp\{-\eta_1\tau\}||(u^*-v^*)(\tau)||_{L^2(R(f))}^2d\tau \\ & = -\int_0^t\int_{D(f)}\exp\{-\eta_1\tau\}\Big[k_1\frac{\partial u^*}{\partial x_3}u^*+k_2\frac{\partial v^*}{\partial x_3}v^*\Big]dAd\tau. \end{align}$

(2.5)

On the other hand, we use (2.3) to compute

$\begin{align} -\int_0^t\int_{D(f)}&\exp\{-\eta_1\tau\}\Big[k_1\frac{\partial u^*}{\partial x_3}u^*+k_2\frac{\partial v^*}{\partial x_3}v^*\Big]dAd\tau \\ & = -\int_0^t\int_{D(f)}\exp\{-\eta_1\tau\}\Big[k_1\frac{\partial u^*}{\partial x_3}\mathcal{H}_1+k_2\frac{\partial v^*}{\partial x_3}\mathcal{H}_2\Big]dAd\tau \\ & = \int_0^t\int_{R(f)}\exp\{-\eta_1\tau\}\Big[k_1\nabla\cdot\Big(\nabla u^*\mathcal{H}_1\Big) +k_2\nabla\cdot\Big(\nabla v^*\mathcal{H}_2\Big)dxd\tau \\ & = \int_0^t\int_{R(f)}\exp\{-\eta_1\tau\}\Big[k_1\nabla u^*\cdot\nabla\mathcal{H}_1+k_2\nabla v^*\cdot\nabla \mathcal{H}_2\Big]dxd\tau \\ &+\exp\{-\eta_1t\}\int_{R(f)}\Big[b_1u^*\mathcal{H}_1+b_2v^*\mathcal{H}_2\Big]dx \\ &+\eta_1\int_0^t\int_{R(f)}\exp\{-\eta_1\tau\}\Big[b_1u^*\mathcal{H}_{1,\tau}+b_2v^*\mathcal{H}_{2,\tau}\Big]dxd\tau \\ &+\gamma\int_0^t\int_{R(f)}\exp\{-\eta_1\tau\}(u^*-v^*)(\mathcal{H}_{1}-\mathcal{H}_{2})dxd\tau \\ &\doteq \mathcal{F}_1+\mathcal{F}_2+\mathcal{F}_3+\mathcal{F}_4. \end{align}$

(2.6)

An application of the Schwarz inequality leads to

$\begin{align} \mathcal{F}_1&\leq\frac{1}{2}\int_0^t\exp\{-\eta_1\tau\}\Big[k_1||\nabla u^*(\tau)||_{L^2(R(f))}^2+k_2||\nabla v^*(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &+ \frac{1}{2}\int_0^t\exp\{-\eta_1\tau\}\Big[k_1||\nabla \mathcal{H}_1||_{L^2(R(f))}^2+k_2||\nabla \mathcal{H}_2||_{L^2(R(f))}^2\Big]d\tau, \end{align}$

(2.7)

$\begin{align} \mathcal{F}_2&\leq\frac{1}{2}\exp\{-\eta_1t\}\Big[b_1||u^*(t)||_{L^2(R(f))}^2+b_2||v^*(t)||_{L^2(R(f))}^2\Big] \\ &+\frac{1}{2}\exp\{-\eta_1t\}\Big[b_1||\mathcal{H}_1(t)||_{L^2(R(f))}^2+b_2||\mathcal{H}_2(t)||_{L^2(R(f))}^2\Big], \end{align}$

(2.8)

$\begin{align} \mathcal{F}_3&\leq\int_0^t\exp\{-\eta_1\tau\}\Big[b_1\eta_1||u^*(\tau)||_{L^2(R(f))}^2+b_2\eta_1||v^*(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &+\frac{1}{4}\int_0^t\exp\{-\eta_1\tau\}\Big[b_1\eta_1||\mathcal{H}_{1,\tau}(\tau)||_{L^2(R(f))}^2 +b_2\eta_1||\mathcal{H}_{2,\tau}(\tau)||_{L^2(R(f))}^2\Big]d\tau, \end{align}$

(2.9)

$\begin{align} \mathcal{F}_4&\leq\gamma\int_0^t\exp\{-\eta_1\tau\}||(u^*-v^*)(\tau)||_{L^2(R(f))}^2d\tau \\ &+\frac{1}{4}\gamma\int_0^t\exp\{-\eta_1\tau\}||(\mathcal{H}_1-\mathcal{H}_2)(\tau)||_{L^2(R(f))}^2d\tau. \end{align}$

(2.10)

Inserting Eqs (2.7)–(2.10) into (2.6) and combining (2.5), it can be obtained

$\begin{align} \int_0^t\exp\{-\eta_1\tau\}&\Big[k_1||\nabla u^*(\tau)||_{L^2(R(f))}^2+k_2||\nabla v^*(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &\leq\int_0^t\exp\{-\eta_1\tau\}\Big[k_1||\nabla \mathcal{H}_1||_{L^2(R(f))}^2+k_2||\nabla \mathcal{H}_2||_{L^2(R(f))}^2\Big]d\tau \\ &+\exp\{-\eta_1t\}\Big[b_1||\mathcal{H}_1(t)||_{L^2(R(f))}^2+b_2||\mathcal{H}_2(t)||_{L^2(R(f))}^2\Big] \\ &+\frac{1}{2}\int_0^t\exp\{-\eta_1\tau\}\Big[b_1\eta_1||\mathcal{H}_{1,\tau}(\tau)||_{L^2(R(f))}^2 +b_2\eta_1||\mathcal{H}_{2,\tau}(\tau)||_{L^2(R(f))}^2\Big]d\tau \\ &+\frac{1}{2}\gamma\int_0^t\exp\{-\eta_1\tau\}||(\mathcal{H}_1-\mathcal{H}_2)(\tau)||_{L^2(R(f))}^2d\tau. \end{align}$

(2.11)

From (2.11), we can conclude that Lemma 2.1 holds.

We not only need a prior bounds for $v$ and $v^*$ , but also for $u$ and $u^*$ . Since $u$ and $u^*$ are undisturbed solutions of Eqs (1.1)–(1.4), in Lemma 2.1 we only need to set $f = 0$ and replace $L_{21}$ and $L_{22}$ with $L_{11}$ and $L_{12}$ , respectively, and then we can obtain the a prior bounds for $u$ and $u^*$ .

Lemma 2.2. If $L_{11}, L_{12}\in H^1([0, \infty)\times D)$ , then

$\begin{align} \int_0^t\exp\{-\eta_1\tau\}&\Big[k_1||\nabla u(\tau)||_{L^2(R)}+k_2||\nabla v(\tau)||_{L^2(R)}\Big]d\tau\leq d_2(t), \end{align}$

where

$\begin{align} d_2(t)& = \int_0^t\exp\{-\eta_1\tau\}\Big[k_1||\nabla \mathcal{H}_3||_{L^2(R)}^2+k_2||\nabla \mathcal{H}_4||_{L^2(R)}^2\Big]d\tau \\ &+\exp\{-\eta_1t\}\Big[b_1||\mathcal{H}_3(t)||_{L^2(R)}^2+b_2||\mathcal{H}_4(t)||_{L^2(R)}^2\Big] \\ &+\frac{1}{2}\int_0^t\exp\{-\eta_1\tau\}\Big[b_1\eta_1||\mathcal{H}_{3,\tau}(\tau)||_{L^2(R)}^2 +b_2\eta_1||\mathcal{H}_{4,\tau}(\tau)||_{L^2(R)}^2\Big]d\tau \\ &+\frac{1}{2}\gamma\int_0^t\exp\{-\eta_1\tau\}||(\mathcal{H}_3-\mathcal{H}_4)(\tau)||_{L^2(R)}^2d\tau \end{align}$

and

$\mathcal{H}_3(t, {\boldsymbol{x}}) = L_{11}(t, x_1, x_2)\exp\{-\sigma x_3\},\ \mathcal{H}_4(t, {\boldsymbol{x}}) = L_{12}(t, x_1, x_2)\exp\{-\sigma x_3\}.$

Remark 2.1. Lemmas 2.1 and 2.2 will provide a priori estimates for the proof of the lemmas in the next section.

3. Auxiliary functions

Let $w$ and $s$ represent the difference between the perturbed solutions and the unperturbed solutions, i.e.,

$\begin{align} w = u-u^*,\ s = v-v^*, \end{align}$

(3.1)

then $w$ and $s$ satisfy

$\begin{align} b_1w_t = k_1\triangle w-\gamma(w-s),&\ \text{in}\ R(\epsilon)\times\{t > 0\}, \end{align}$

(3.2)

$\begin{align} b_2s_t = k_2\triangle s+\gamma(w-s),&\ \text{in} \ R(\epsilon)\times\{t > 0\}, \end{align}$

(3.3)

$\begin{align} w = s = 0, \ & \text{on}\ \partial D\times\{x_3 > \epsilon\}\times\{t > 0\}, \end{align}$

(3.4)

$\begin{align} w = s = 0, \ &\text{in}\ R(\epsilon)\times\{t = 0\}. \end{align}$

(3.5)

To obtain the continuous dependence of the solution on the perturbation parameter, we establish a new energy function

$\begin{align} V(t, x_3) = \int_0^t\Big[||w(\tau)||_{L^2(R(x_3))}^2+||s(\tau)||_{L^2(R(x_3))}^2\Big]d\tau,\ x_3\geq\epsilon. \end{align}$

(3.6)

Noting the definition of $R(x_3)$ , we can obtain the derivative of $V(t, x_3)$ as follows:

$\begin{align*} -\frac{\partial}{\partial x_3}V(t, x_3) = \int_0^t\Big[||w(\tau)||_{L^2(D(x_3))}^2+||s(\tau)||_{L^2(D(x_3))}^2\Big]d\tau. \end{align*}$

We introduce two auxiliary functions $\varphi$ and $\psi$ such that

$\begin{align} b_1\varphi_\tau+k_1\triangle\varphi = -w,\ b_2\psi_\tau+k_2\triangle\psi = -s,\ &\text{in}\ R(x_3), 0 < \tau < t, \end{align}$

(3.7)

$\begin{align} \varphi(\tau, x_1, x_2, x_3) = \psi(\tau, x_1, x_2, x_3) = 0, \ & \text{on}\ \partial D\times\{x_3\}, 0 < \tau < t, \end{align}$

(3.8)

$\begin{align} \varphi(\tau, x_1, x_2, x_3) = \psi(\tau, x_1, x_2, x_3) = 0,\ &(x_1, x_2)\in D, 0 < \tau < t, \end{align}$

(3.9)

$\begin{align} \varphi(t, {\boldsymbol{x}}) = \psi(t, {\boldsymbol{x}}) = 0, \ & \text{in}\ R(x_3), \end{align}$

(3.10)

$\begin{align} \varphi, \nabla\varphi, \psi, \nabla\psi\rightarrow0(\text{uniformly}\ \text{in} \ x_1, x_2, \tau) \ & \text{as}\ x_3\rightarrow \infty, \end{align}$

(3.11)

where $x_3 > \epsilon$ .

Next, we will derive some necessary properties of the auxiliary functions, which will play a crucial role in proving the continuous dependence of the solutions.

Lemma 3.1. If $\varphi, \psi\in H^1([0, t]\times R(x_3))$ , then

$\begin{align*} \int_0^t\Big[b_1||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2+b_2||\psi_\tau(\tau)||_{L^2(R(x_3))}^2\Big]d\tau\leq a_1V(t, x_3), \ x_3\geq\epsilon, \end{align*}$

where $a_1 = \max\{b_1^{-1}, b_2^{-1}\}$ .

Proof. We begin with

$\begin{align*} \int_0^t\int_{R(x_3)}\varphi_\tau\Big[b_1\varphi_\tau+k_1\triangle\varphi+w\Big]dxd\tau& = 0, \\ \int_0^t\int_{R(x_3)}\psi_\tau\Big[b_2\psi_\tau+k_2\triangle\psi+s\Big]dxd\tau& = 0. \end{align*}$

Using the divergence theorem $\oint_{\partial R(x_3)}Fds = \int_{R(x_3)}div F dx$ and (3.8)–(3.11), we have

$\begin{align} b_1\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau& = -\frac{1}{2}k_1||\nabla\varphi(0)||_{L^2(R(x_3))}^2 +\int_0^t\int_{R(x_3)}w\varphi_\tau dxd\tau \\ &\leq\Big[\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}, \end{align}$

(3.12)

and

$\begin{align} b_2\int_0^t||\psi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\leq\Big[\int_0^t||\psi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau \int_0^t||s(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}. \end{align}$

(3.13)

Using the Schwarz inequality, (3.12) and (3.13), Lemma 3.1 can be obtained.

Lemma 3.2. If $\varphi, \psi\in H^1(R(x_3))$ , then

$\begin{align*} \int_0^t\Big[k_1||\nabla\varphi(\tau)||_{L^2(R(x_3))}^2+k_2||\nabla\psi(\tau)||_{L^2(R(x_3))}^2\Big]d\tau \leq a_2V(t, x_3), \end{align*}$

where $a_2 = \frac{1}{\lambda}\max\{k_1^{-1}, k_2^{-1}\}$ .

Proof. We begin with

$\begin{align*} \int_0^t\int_{R(x_3)}\varphi\Big[b_1\varphi_\tau+k_1\triangle\varphi+w\Big]dxd\tau& = 0, \\ \int_0^t\int_{R(x_3)}\varphi\Big[b_2\psi_\tau+k_2\triangle\psi+s\Big]dxd\tau& = 0. \end{align*}$

Using the divergence theorem and Lemma 2.2, we have

$\begin{align} k_1\int_0^t||\nabla\varphi(\tau)||_{L^2(R(x_3))}^2d\tau& = -\frac{1}{2}b_1||\varphi(0)||_{L^2(R(x_3))}^2 +\int_0^t\int_{R(x_3)}w\varphi dxd\tau \\ &\leq\Big[\int_0^t||\varphi(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2} \\ &\leq\frac{1}{\sqrt{\lambda}}\Big[\int_0^t||\nabla_2\varphi(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2} \end{align}$

(3.14)

and

$\begin{align} k_2\int_0^t||\nabla\psi(\tau)||^2_{L^2(R(x_3))}d\tau \leq\frac{1}{\sqrt{\lambda}}\Big[\int_0^t||\nabla_2\psi(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||s(\tau)||^2_{L^2(R(x_3))}d\tau\Big]^\frac{1}{2}. \end{align}$

(3.15)

Using the following inequality

$\begin{align} \sqrt{ab}+\sqrt{cd}\leq\sqrt{(a+c)(b+d)},\ \text{for}\ \ a,b,c,d > 0, \end{align}$

(3.16)

the Young inequality and Lemma 3.1, we can have from (3.14) and (3.15)

$\begin{align} \int_0^t\Big[k_1||\nabla\varphi(\tau)||_{L^2(R(x_3))}^2&+k_2||\nabla\psi(\tau)||_{L^2(R(x_3))}^2\Big]d\tau \\ &\leq\frac{1}{\sqrt{\lambda}}\Big\{\int_0^t\Big[k_1||\nabla_2\varphi(\tau)||_{L^2(R(x_3))}^2+k_2||\nabla_2\psi(\tau)||_{L^2(R(x_3))}^2\Big]d\tau \\ &\cdot\int_0^t\Big[k_1^{-1}||w(\tau)||_{L^2(R(x_3))}^2+k_2^{-1}||s(\tau)||^2_{L^2(R(x_3))}\Big]d\tau\Big\}^\frac{1}{2}. \end{align}$

(3.17)

From (3.17) we can obtain Lemma 3.2.

Lemma 3.3. If $\varphi, \psi\in H^1(R(x_3))$ , then

$\begin{align} k_1\int_0^t&||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau +k_2\int_0^t||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau \leq a_3V(t, x_3), \end{align}$

where $a_3$ is a positive constant.

Proof. Letting $\delta$ be a positive constant. We compute

$\begin{align} \int_0^t\int_{R(x_3)}\Big[\frac{\partial\varphi}{\partial x_3}-\delta\varphi_\tau\Big] \Big[b_1\varphi_\tau+k_1\triangle\varphi+w\Big]dxd\tau& = 0, \end{align}$

(3.18)

$\begin{align} \int_0^t\int_{R(x_3)}\Big[\frac{\partial\psi}{\partial x_3}-\delta\psi_\tau\Big] \Big[b_2\psi_\tau+k_2\triangle\psi+s\Big]dxd\tau& = 0. \end{align}$

(3.19)

Using the divergence theorem and (3.8)–(3.10) in (3.18) and (3.19), we obtain

$\begin{align} \frac{1}{2}k_1\delta||\nabla\varphi(0)||_{L^2(R(x_3))}^2d\tau&+b_1\delta\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau +\frac{1}{2}k_1\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau \\ & = \int_0^t\int_{R(x_3)}\frac{\partial\varphi}{\partial x_3}\varphi_\tau dxd\tau +\int_0^t\int_{R(x_3)}\Big[\frac{\partial\varphi}{\partial x_3}-\delta\varphi_\tau\Big] wdxd\tau. \end{align}$

(3.20)

Using the Schwarz inequality, we obtain

$\begin{align} \int_0^t\int_{R(x_3)}\frac{\partial\varphi}{\partial x_3}\varphi_\tau dxd\tau &\leq\Big[\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}, \end{align}$

(3.21)

$\begin{align} \int_0^t\int_{R(x_3)}\frac{\partial\varphi}{\partial x_3}wdxd\tau& \leq\Big[\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}, \end{align}$

(3.22)

$\begin{align} -\delta\int_0^t\int_{R(x_3)}\varphi_\tau wdxd\tau& \leq\delta\Big[\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}. \end{align}$

(3.23)

Inserting (3.21)–(3.23) into (3.20) and dropping the first two terms in the left of (3.20), we have

$\begin{align} \frac{1}{2}k_1\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau &\leq\Big[\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2} \\ &+\Big[\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2} \\ &+\delta\Big[\int_0^t||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||w(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}. \end{align}$

(3.24)

Similar, we can also have from (3.19)

$\begin{align} \frac{1}{2}k_2\int_0^t||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau &\leq\Big[\int_0^t||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||\psi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2} \\ &+\Big[\int_0^t||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||s(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2} \\ &+\delta\Big[\int_0^t||\psi_\tau(\tau)||_{L^2(R(x_3))}^2d\tau\int_0^t||s(\tau)||_{L^2(R(x_3))}^2d\tau\Big]^\frac{1}{2}. \end{align}$

(3.25)

Using (3.16) and Lemmas 3.1 and 3.2, we obtain

$\begin{align} k_1\int_0^t||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau &+k_2\int_0^t||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(D(x_3))}^2d\tau \\ &\leq 2a_1a_2\Big\{\int_0^t\Big[b_1||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2+b_2||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2\Big]d\tau \\ &\cdot\int_0^t\Big[k_1||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2+k_2||\psi_\tau(\tau)||_{L^2(R(x_3))}^2\Big]d\tau\Big\}^\frac{1}{2} \\ &+2a_2\Big\{\int_0^t\Big[b_1||\frac{\partial\varphi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2+b_2||\frac{\partial\psi}{\partial x_3}(\tau)||_{L^2(R(x_3))}^2\Big]d\tau \\ &\cdot\int_0^t\Big[||w(\tau)||_{L^2(R(x_3))}^2+||s(\tau)||_{L^2(R(x_3))}^2\Big]d\tau\Big\}^\frac{1}{2} \\ &+2a_1\delta\Big\{\int_0^t\Big[k_1||\varphi_\tau(\tau)||_{L^2(R(x_3))}^2+k_2||\psi_\tau(\tau)||_{L^2(R(x_3))}^2\Big]d\tau \\ &\cdot\int_0^t\Big[||w(\tau)||_{L^2(R(x_3))}^2+||s(\tau)||_{L^2(R(x_3))}^2\Big]d\tau\Big\}^\frac{1}{2} \\ &\leq a_3V(t, x_3), \end{align}$

(3.26)

where $a_3 = 2a_1^2a_2^2+2a_2^2+2a_1^2$ .

In the next section, we will use Lemma 3.3 to derive the continuous dependence of the solutions.

4. Main results

In this section, we first derive a bound for $V(t, \epsilon)$ . To do this, we define

$\begin{align} u(t, \mathit{\boldsymbol{x}}) = L_{11}(t, x_1, x_2),\ v(t, \mathit{\boldsymbol{x}})& = L_{12}(t, x_1, x_2), \ -\epsilon\leq x_3\leq 0, (x_1, x_2)\in D, t\in[0, +\infty), \end{align}$

(4.1)

$\begin{align} u^*(t, \mathit{\boldsymbol{x}}) = L_{21}(t, x_1, x_2),\ v^*(t, \mathit{\boldsymbol{x}})& = L_{22}(t, x_1, x_2), \ -\epsilon\leq x_3\leq f(x_1, x_2), (x_1, x_2)\in D, t\in[0, +\infty). \end{align}$

(4.2)

When $-\epsilon\leq x_3\leq\epsilon$ , we let

$\begin{align} w(t, \mathit{\boldsymbol{x}}) = u(t, \mathit{\boldsymbol{x}})-u^*(t, \mathit{\boldsymbol{x}}),\ s(t, \mathit{\boldsymbol{x}}) = v(t, \mathit{\boldsymbol{x}})-v^*(t, \mathit{\boldsymbol{x}}), (x_1, x_2)\in D, t\in[0, +\infty). \end{align}$

(4.3)

In view of (3.1) and (4.3), using the triangle inequality, it can be obtained that

$\begin{align} k_1\int_0^t\int_{R(-\epsilon)}\Big(\frac{\partial w}{\partial x_3}\Big)^2dxd\tau& +k_2\int_0^t\int_{R(-\epsilon)}\Big(\frac{\partial s}{\partial x_3}\Big)^2dxd\tau \\ &\leq \int_0^t\int_{R(-\epsilon)}\Big[k_1\Big(\frac{\partial u}{\partial x_3}\Big)^2 +k_2\Big(\frac{\partial v}{\partial x_3}\Big)^2\Big]dxd\tau \\ & +\int_0^t\int_{R(-\epsilon)}\Big[k_1\Big(\frac{\partial u^*}{\partial x_3}\Big)^2 +k_2\Big(\frac{\partial v^*}{\partial x_3}\Big)^2\Big]dxd\tau. \end{align}$

(4.4)

Using Lemmas 2.1 and 2.2, (4.1) and (4.2), from (4.4), we obtain

$\begin{align} k_1\int_0^t\int_{R(-\epsilon)}\Big(\frac{\partial w}{\partial x_3}\Big)^2dxd\tau& +k_2\int_0^t\int_{R(-\epsilon)}\Big(\frac{\partial s}{\partial x_3}\Big)^2dxd\tau \\ &\leq \int_0^t\int_{R}\Big[k_1\Big(\frac{\partial u}{\partial x_3}\Big)^2 +k_2\Big(\frac{\partial v}{\partial x_3}\Big)^2\Big]dxd\tau \\ & +\int_0^t\int_{R(f)}\Big[k_1\Big(\frac{\partial u^*}{\partial x_3}\Big)^2 +k_2\Big(\frac{\partial v^*}{\partial x_3}\Big)^2\Big]dxd\tau. \\ &\leq e^{\eta_1t}[d_1(t)+d_2(t)]\doteq d_3(t). \end{align}$

(4.5)

Now, we write the main theorem as:

Theorem 4.1. If $L_{11}, L_{12}\in H^1([0, \infty)\times R), L_{21}, L_{22}\in H^1([0, \infty)\times R(f))$ and $t < \frac{\pi}{4a_1\gamma}$ , then

$\begin{align} V(t, x_3)&\leq\exp\{-d_4(x_3-\epsilon)\}\Big\{ \frac{32}{d_4\pi}\max\{\frac{1}{k_1}, \frac{1}{k_2}\}d_3(t)\epsilon \\ &+d_5\int_0^t\Big[||(L_{11}-L_{21})(\tau)||^2_{L^2(D)}+||(L_{12}-L_{22})(\tau)||^2_{L^2(D)}\Big]d\tau\Big\}, x_3\geq\epsilon \end{align}$

holds, where $d_4 = a_3^{-1}\max\{k_1, k_2\}^{-1}$ and $d_5 = \frac{d_4\pi}{2}+\frac{2}{d_4}$ .

Proof. Let $x_3\geq\epsilon$ be a fixed point on the coordinate axis $x_ 3$ . Using (3.7)–(3.11) and the divergence theorem, we can have

$\begin{align} V(x_3, t)& = -\int_0^t\int_{R(x_3)}w\Big[b_1\varphi_\tau+k_1\triangle\varphi\Big]dxd\tau -\int_0^t\int_{R(x_3)}s\Big[b_2\psi_\tau+k_2\triangle\psi\Big]dxd\tau \\ & = -\int_0^t\int_{R(x_3)}\Big[b_1\varphi_\tau w+b_2\psi_\tau s\Big]dxd\tau +\int_0^t\int_{R(x_3)}\Big[k_1\nabla w\cdot\nabla \varphi+k_2\nabla s\cdot\nabla \psi\Big]dxd\tau \\ &+\int_0^t\int_{D(x_3)}\Big[k_1w\frac{\partial \varphi}{\partial x_3}+k_2s\frac{\partial \psi}{\partial x_3}\Big]dAd\tau \\ & = -\int_0^t\int_{R(x_3)}\Big[b_1\varphi_\tau w+b_2\psi_\tau s\Big]dxd\tau -\int_0^t\int_{R(x_3)}\Big[k_1\triangle w \varphi+k_2\triangle s\psi\Big]dxd\tau \\ &+\int_0^t\int_{D(x_3)}\Big[k_1w\frac{\partial \varphi}{\partial x_3}+k_2s\frac{\partial \psi}{\partial x_3}\Big]dAd\tau \\ & = -\int_0^t\int_{R(x_3)}\Big[b_1\varphi_\tau w+b_2\psi_\tau s\Big]dxd\tau-\int_0^t\int_{R(x_3)}\Big[b_1\varphi w_\tau+b_2\psi s_\tau\Big]dxd\tau \\ &+\int_0^t\int_{D(x_3)}\Big[k_1w\frac{\partial \varphi}{\partial x_3}+k_2s\frac{\partial \psi}{\partial x_3}\Big]dAd\tau -\gamma\int_0^t\int_{R(x_3)}\Big(\varphi-\psi\Big)\Big(w-s\Big)dxd\tau. \end{align}$

(4.6)

In light of (1.4) and (3.10), it is clear that

$\begin{align} \int_0^t\int_{R(x_3)}\Big[b_1\varphi_\tau w+b_1\varphi w_\tau\Big]dxd\tau = 0,\ \int_0^t\int_{R(x_3)}\Big[b_2\psi_\tau s+b_2\psi s_\tau\Big]dxd\tau = 0. \end{align}$

(4.7)

A combination of the Hölder inequality, (3.16) and Lemma 3.3 leads to

$\begin{align} \int_0^t\int_{D(x_3)}\Big[k_1w\frac{\partial \varphi}{\partial x_3}&+k_2s\frac{\partial \psi}{\partial x_3}\Big]dAd\tau \\ &\leq k_1\Big[\int_0^t||\frac{\partial \varphi}{\partial x_3}(\tau)||^2_{L^2(D(x_3))}d\tau\int_0^t||w(\tau)||^2_{L^2(D(x_3))}d\tau\Big]^\frac{1}{2} \\ &+k_2\Big[\int_0^t||\frac{\partial \psi}{\partial x_3}(\tau)||^2_{L^2(D(x_3))}d\tau\int_0^t||s(\tau)||^2_{L^2(D(x_3))}d\tau\Big]^\frac{1}{2} \\ &\leq \max\{\sqrt{k_1}, \sqrt{k_2}\}\Big[\int_0^t\Big(k_1||\frac{\partial \varphi}{\partial x_3}(\tau)||^2_{L^2(D(x_3))}+k_2||\frac{\partial \psi}{\partial x_3}(\tau)||^2_{L^2(D(x_3))}\Big) d\tau\Big]^\frac{1}{2} \\ &\cdot\Big[\int_0^t\Big(||w(\tau)||^2_{L^2(D(x_3))}+||s(\tau)||^2_{L^2(D(x_3))}\Big)d\tau\Big]^\frac{1}{2} \\ &\leq\sqrt{a_3}\max\{\sqrt{k_1}, \sqrt{k_2}\}\sqrt{V(t, x_3)}\Big[-\frac{\partial}{\partial x_3}V(t, x_3)\Big]^\frac{1}{2}. \end{align}$

(4.8)

For the fourth term in the right of (4.6), we compute

$\begin{align} -\gamma\int_0^t\int_{R(x_3)}&\Big(\varphi-\psi\Big)\Big(w-s\Big)dxd\tau \\ & \leq\gamma\Big[\int_0^t\Big(||\varphi(\tau)||^2_{L^2(R(x_3))}+||\psi(\tau)||^2_{L^2(R(x_3))}\Big)d\tau \\ &\cdot\int_0^t \Big(||w(\tau)||^2_{L^2(R(x_3))}+||s(\tau)||^2_{L^2(R(x_3))}\Big)d\tau\Big]^\frac{1}{2}. \end{align}$

(4.9)

Using the inequality (see p182 in ^[19])

$\begin{align} \int_0^1\phi^{2}dx\leq \frac{4}{\pi^2}\int_0^1(\phi')^{2}dx,\ \text{for}\ \phi(0) = 0, \end{align}$

(4.10)

we have from (4.9)

$\begin{align} -\gamma&\int_0^t\int_{R(x_3)}\Big(\varphi-\psi\Big)\Big(w-s\Big)dxd\tau \\ & \leq\gamma\frac{2t}{\pi}\Big[\int_0^t\Big(||\varphi_\tau(\tau)||^2_{L^2(R(x_3))}+||\psi_\tau(\tau)||^2_{L^2(R(x_3))}\Big)d\tau V(t, x_3)\Big]^\frac{1}{2} \\ &\leq\gamma\frac{2t}{\pi}a_1V(t, x_3), \end{align}$

(4.11)

where we have also used Lemma 3.1. Combining (4.6), (4.7), (4.8) and (4.11) and choosing $t < \frac{\pi}{4a_1\gamma}$ , we can have

$\begin{align} V(t, x_3)&\leq-\frac{1}{d_4}\frac{\partial}{\partial x_3}V(t, x_3), x_3 > \epsilon. \end{align}$

(4.12)

Integrating (4.12) from $\epsilon$ to $x_3$ , we have

$\begin{align} V(t, x_3)\leq V(t, \epsilon)\exp\{-d_4(x_3-\epsilon)\}, x_3\geq\epsilon. \end{align}$

(4.13)

Equation (4.13) only indicates that the solutions to (1.1)–(1.4) decay exponentially as $x_3\rightarrow \infty$ . This decay result is not rigorous because we do not yet know whether $V(t, \epsilon)$ depends on the perturbation parameter $\epsilon$ . Therefore, we derive the explicit bound of $V(t, \epsilon)$ in terms of $\epsilon$ and $L_{ij}(ij = 1, 2)$ .

After letting $x_3 = \epsilon$ in (4.12), we have

$\begin{align} V(\epsilon, t)&\leq \frac{1}{d_4}\int_0^t\Big[||w(\tau)||^2_{L^2(D(\epsilon))}+||s(\tau)||^2_{L^2(D(\epsilon))}\Big]dAd\tau \\ & = \frac{2}{d_4}\int_0^t\int_{-\epsilon}^\epsilon\int_{D(x_3)}\Big[w\frac{\partial w}{\partial x_3}+s\frac{\partial s}{\partial x_3}\Big]dxd\tau \\ &+\frac{2}{d_4}\int_0^t\Big[||(L_{11}-L_{21})(\tau)||^2_{L^2(D)}+||(L_{12}-L_{22})(\tau)||^2_{L^2(D)}\Big]d\tau \\ &\leq\frac{2}{d_4}\Big[\int_0^t||w(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau\int_0^t||\frac{\partial w}{\partial x_3}(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau\Big]^\frac{1}{2} \\ &+\frac{2}{d_4}\Big[\int_0^t||s(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau\int_0^t||\frac{\partial s}{\partial x_3}(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau\Big]^\frac{1}{2} \\ &+\frac{2}{d_4}\int_0^t\Big[||(L_{11}-L_{21})(\tau)||^2_{L^2(D)}+||(L_{12}-L_{22})(\tau)||^2_{L^2(D)}\Big]d\tau. \end{align}$

(4.14)

Using (4.10) again, we have

$\begin{align} \int_0^t||w(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau& \leq\frac{16\epsilon^2}{\pi^2}\int_0^t||\frac{\partial w}{\partial x_3}(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau \\ &+2\epsilon\int_0^t||(L_{11}-L_{21})(\tau)||^2_{L^2(D)}d\tau, \end{align}$

(4.15)

$\begin{align} \int_0^t||s(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau& \leq\frac{16\epsilon^2}{\pi^2}\int_0^t||\frac{\partial s}{\partial x_3}(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2d\tau \\ &+2\epsilon\int_0^t||(L_{12}-L_{22})(\tau)||^2_{L^2(D)}d\tau. \end{align}$

(4.16)

Inserting (4.15) into (4.16) and combining the Schwarz inequality, we obtain

$\begin{align} V(\epsilon, t) &\leq\frac{32}{d_4\pi}\epsilon\int_0^t\Big[||\frac{\partial w}{\partial x_3}(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2 +||\frac{\partial s}{\partial x_3}(\tau)||_{L^2(D(x_3)\times[-\epsilon, \epsilon])}^2\Big]d\tau \\ &+\Big[\frac{d_4\pi}{2}+\frac{2}{d_4}\Big]\int_0^t\Big[||(L_{11}-L_{21})(\tau)||^2_{L^2(D)}+||(L_{12}-L_{22})(\tau)||^2_{L^2(D)}\Big]d\tau \\ &\leq\frac{32}{d_4\pi}\max\{\frac{1}{k_1}, \frac{1}{k_2}\}\epsilon\int_0^t\Big[k_1||\frac{\partial w}{\partial x_3}(\tau)||_{L^2(R(-\epsilon))}^2 +k_2||\frac{\partial s}{\partial x_3}(\tau)||_{L^2(R(-\epsilon))}^2\Big]d\tau \\ &+\Big[\frac{d_4\pi}{2}+\frac{2}{d_4}\Big]\int_0^t\Big[||(L_{11}-L_{21})(\tau)||^2_{L^2(D)}+||(L_{12}-L_{22})(\tau)||^2_{L^2(D)}\Big]d\tau. \end{align}$

(4.17)

In view of (4.5) and (4.13), from (4.17) we have Theorem 4.1.

Remark 4.1. Theorem 4.1 indicates that $V(t, x_3)$ continuously depends on $\epsilon$ and the base data. That is, when $\epsilon$ approaches 0, then $u(t, x_3)$ and $v(t, x_3)$ approach 0. If $\epsilon = 0$ , Theorem 4.1 is the Saint-Venant's principle type decay result.

Remark 4.2. In any cross-section of $R$ , the continuous dependence result can still be obtained. We compute

$\begin{align} \int_0^t\Big[||w(\tau)||^2_{L^2(D(x_3))}&+||s(\tau)||^2_{L^2(D(x_3))}\Big]d\tau \\ & = -2\int_0^t\int_{R(x_3)}\Big[w\frac{\partial w}{\partial x_3}+s\frac{\partial s}{\partial x_3}\Big]dxd\tau \\ &\leq2 \sqrt{V(x_3)}\Big[\int_0^t\Big[k_1||\frac{\partial w}{\partial x_3}(\tau)||_{L^2(R(-\epsilon))}^2 +k_2||\frac{\partial s}{\partial x_3}(\tau)||_{L^2(R(-\epsilon))}^2\Big]d\tau\Big]^\frac{1}{2}. \end{align}$

(4.18)

Using (4.18) and Theorem 4.1, we can obtain the continuous dependence result.

5. Conclusions

This article adopts the methods of the a prior estimates and energy estimate to obtain the continuous dependence of the solution on the base. This method can be further extended to other linear partial differential equation systems, such as pseudo-parabolic equation

$u_t = \Delta u+\delta \Delta u_t,$

where $\delta$ is a positive constant. However, for nonlinear equations (e.g., the Darcy equations), due to the inability to control nonlinear terms and derive a prior bounds for nonlinear terms, Lemma 3.3 will be difficult to obtain. This is a difficult problem we need to solve next.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Research Team Project of Guangzhou Huashang College (2021HSKT01).

Conflict of interest

The author declares there is no conflict of interest.

References

[1]	I. Podlubny, Fractional differential equations, in Mathematics in Science and Engineering, Academic Press, San Diego, 1999.
[2]	Z. Sun, G. Gao, The Finite Difference Methods for Fractional Differential Equations, Science Press, Beijing, 2015.
[3]	C. Li, F. Zeng, Numerical Methods for Fractional Calculas, Chapman and Hall/CRC Press, Boca Raton, 2015. https://doi.org/10.1201/b18503
[4]	F. Liu, P. Zhuang, Q. Liu, Numerical Methods for Fractional Partial Differential Equations and Their Applications, Science Press, Beijing, 2015.
[5]	Y. Lin, C. Xu, Finite difference/spectral approximation for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), 1533–1552. https://doi.org/10.1016/j.jcp.2007.02.001 doi: 10.1016/j.jcp.2007.02.001
[6]	M. Stynes, E. O'riordan, J. Gracia, Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation, SIAM J. Numer. Anal., 55 (2017), 1057–1079. https://doi.org/10.1137/16M1082329 doi: 10.1137/16M1082329
[7]	X. Li, Y. Chen, C. Chen, An improved two-grid technique for the nonlinear time-fractional parabolic equation based on the block-centered finite difference method, J. Comput. Math., 40 (2022), 455–473. https://doi.org/10.4208/jcm.2011-m2020-0124 doi: 10.4208/jcm.2011-m2020-0124
[8]	X. Peng, D. Xu, W. Qiu, Pointwise error estimates of compact difference scheme for mixed-type time-fractional Burgers' equation, Math. Comput. Simulat., 208 (2023, ) 702–726. https://doi.org/10.1016/j.matcom.2023.02.004
[9]	H. Wang, Y. Chen, Y. Huang, W. Mao, A posteriori error estimates of the Galerkin spectral methods for space-time fractional diffusion equations, Adv. Appl. Math. Mech., 12 (2020), 87–100.
[10]	X. Li, C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal., 47 (2009), 2108–2131. https://doi.org/10.1137/080718942 doi: 10.1137/080718942
[11]	F. Liu, P. Zhuang, I. Turner, K. Burrage, V. Anh, A new fractional finite volume method for solving the fractional diffusion equation, Appl. Math. Model., 38 (2014), 3871–3878. https://doi.org/10.1016/j.apm.2013.10.007 doi: 10.1016/j.apm.2013.10.007
[12]	C. Huang, M. Stynes, Superconvergence of a finite element method for the multi-term time-fractional diffusion problem, J. Sci. Comput., 82 (2020), 10. https://doi.org/10.1007/s10915-019-01115-w doi: 10.1007/s10915-019-01115-w
[13]	B. Tang, Y. Chen, X. Lin, A posteriori error error estimates of spectral Galerkin methods for multi-term time fractional diffusion equations, Appl. Math. Lett., 120 (2021), 107259. https://doi.org/10.1016/j.aml.2021.107259 doi: 10.1016/j.aml.2021.107259
[14]	H. Liu, X. Zheng, C. Chen, H. Wang, A characteristic finite element method for the time-fractional mobile/immobile advection diffusion model, Adv. Comput. Math., 47 (2021), 41. https://doi.org/10.1007/s10444-021-09867-6 doi: 10.1007/s10444-021-09867-6
[15]	S. Toprakseven, A weak Galerkin finite element method for time fractional reaction-diffusion-convection problems with variable coefficients, Appl. Numer. Math., 168 (2021), 1–12. https://doi.org/10.1016/j.apnum.2021.05.021 doi: 10.1016/j.apnum.2021.05.021
[16]	Y. Zhao, P. Chen, W. Bu, X. Liu, Y. Tang, Two mixed finite element methods for time-fractional diffusion equations, J. Sci. Comput., 70 (2017), 407–428. https://doi.org/10.1007/s10915-015-0152-y doi: 10.1007/s10915-015-0152-y
[17]	Z. Shi, Y. Zhao, F. Liu, Y. Tang, F. Wang, Y. Shi, High accuracy analysis of an $H^1$ -Galerkin mixed finite element method for two-dimensional time fractional diffusion equations, Comput. Math. Appl., 74 (2017), 1903–1914. https://doi.org/10.1016/j.camwa.2017.06.057 doi: 10.1016/j.camwa.2017.06.057
[18]	M. Abbaszadeh, M. Dehghan, Analysis of mixed finite element method (MFEM) for solving the generalized fractional reaction-diffusion equation on nonrectangular domains, Comput. Math. Appl., 78 (2019), 1531–1547. https://doi.org/10.1016/j.camwa.2019.03.040 doi: 10.1016/j.camwa.2019.03.040
[19]	X. Li, Y. Tang, Interpolated coefficient mixed finite elements for semilinear time fractional diffusion equations, Fractal Fract., 7 (2023), 482. https://doi.org/10.3390/fractalfract7060482 doi: 10.3390/fractalfract7060482
[20]	M. Li, J. Zhao, C. Huang, S. Chen, Nonconforming virtual element method for the time fractional reaction-subdiffusion equation with non-smooth data, J. Sci. Comput., 81 (2019), 1823–1859. https://doi.org/10.1007/s10915-019-01064-4 doi: 10.1007/s10915-019-01064-4
[21]	S. Jiang, J. Zhang, Q. Zhang, Z. Zhang, Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys., 21 (2017), 650–678. https://doi.org/10.4208/cicp.OA-2016-0136 doi: 10.4208/cicp.OA-2016-0136
[22]	J. Shen, Z. Sun, R. Du, Fast finite difference schemes for time-fractional diffusion equations with a weak singularity at initial time, East Asian J. Appl. Math., 8 (2018), 834–858.
[23]	X. Gu, H. Sun, Y. Zhang, Y. Zhao, Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian, Math. Methods Appl. Sci., 44 (2021), 441–463. https://doi.org/10.1002/mma.6746 doi: 10.1002/mma.6746
[24]	G. Gao, Z. Sun, H. Zhang, A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys., 259 (2014), 33–50. https://doi.org/10.1016/j.jcp.2013.11.017 doi: 10.1016/j.jcp.2013.11.017
[25]	A. Alikhanov, C. Huang, A high-order $L2$ type difference scheme for the time-fractional diffusion equation, Appl. Meth. Comput., 411 (2021), 126545. https://doi.org/10.1016/j.amc.2021.126545 doi: 10.1016/j.amc.2021.126545
[26]	J. Ren, H. Liao, J. Zhang, Z. Zhang, Sharp $H^1$ -norm error estimates of two time-stepping schemes for reaction-subdiffusion problems, J. Comput. Appl. Math., 389 (2021), 113352. https://doi.org/10.1016/j.cam.2020.113352 doi: 10.1016/j.cam.2020.113352
[27]	R. Feng, Y. Liu, Y. Hou, H. Li, Z. Fang, Mixed element algorithm based on a second-order time approximation scheme for a two-dimensional nonlinear time fractional coupled sub-diffusion model, Eng. Comput., 38 (2022), 51–68. https://doi.org/10.1007/s00366-020-01032-9 doi: 10.1007/s00366-020-01032-9
[28]	X. Zheng, H. Wang, A hidden-memory variable-order time-fractional optimal control model: Analysis and approximation, SIAM J. Control Optim., 59 (2021), 1851–1880. https://doi.org/10.1137/20M1344962 doi: 10.1137/20M1344962
[29]	C. Li, Z. Zhao, Y. Chen, Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion, Comput. Math. Appl., 62 (2011), 855–875. https://doi.org/10.1016/j.camwa.2011.02.045 doi: 10.1016/j.camwa.2011.02.045
[30]	D. Li, C. Wu, Z. Zhang, Linearized Galerkin fems for nonlinear time fractional parabolic problems with non-smooth solutions in time direction, J. Sci. Comput., 80 (2019), 403–419. https://doi.org/10.1007/s10915-019-00943-0 doi: 10.1007/s10915-019-00943-0
[31]	J. Xu, Two-grid discretization techniques for linear and nonlinear PDEs, SIAM J. Numer. Anal., 33 (1996), 1759–1777. https://doi.org/10.1137/S0036142992232949 doi: 10.1137/S0036142992232949
[32]	Q. Li, Y. Chen, Y. Huang, Y. Wang, Two-grid methods for semilinear time fractional reaction diffusion equations by expanded mixed finite element method, Appl. Numer. Math., 157 (2020), 38–54. https://doi.org/10.1016/j.apnum.2020.05.024 doi: 10.1016/j.apnum.2020.05.024
[33]	Y. Zeng, Z. Tan, Two-grid finite element methods for nonlinear time fractional variable coefficient diffusion equations, Appl. Math. Comput., 434 (2022), 127408. https://doi.org/10.1016/j.amc.2022.127408 doi: 10.1016/j.amc.2022.127408
[34]	H. Fu, B. Zhang, X. Zheng, A high-order two-grid difference method for nonlinear time-fractional Biharmonic problems and its unconditional $\alpha$ -robust error estimates, J. Sci. Comput., 96 (2023), 54. https://doi.org/10.1007/s10915-023-02282-7 doi: 10.1007/s10915-023-02282-7
[35]	W. Qiu, D. Xu, J. Guo, J. Zhou, A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model, Numer. Algor., 85 (2020), 39–58. https://doi.org/10.1007/s11075-019-00801-y doi: 10.1007/s11075-019-00801-y
[36]	A. Pehlivanov, G. Carey, R. Lazarov, Least-squares mixed finite elements for second-order elliptic problems, SIAM J. Numer. Anal., 31 (1994), 1368–1377. https://doi.org/10.1137/0731071 doi: 10.1137/0731071
[37]	A. Pani, An $H^1$ -Galerkin mixed finite element methods for parabolic partial differential equations, SIAM J. Numer. Anal., 35 (1998), 712–727. https://doi.org/10.1137/S0036142995280808 doi: 10.1137/S0036142995280808
[38]	D. Yang, A splitting positive definite mixed finite element method for miscible displacement of compressible flow in porous media, Numer. Methods Partial Differ. Equation, 17 (2001), 229–249. https://doi.org/10.1002/num.3 doi: 10.1002/num.3
[39]	Y. Liu, Y. Du, H. Li, J. Wang, An $H^1$ -Galerkin mixed finite element method for time fractional reaction-diffusion equation, J. Appl. Math. Comput., 47 (2015), 103–117. https://doi.org/10.1007/s12190-014-0764-7 doi: 10.1007/s12190-014-0764-7
[40]	J. Wang, T. Liu, H. Li, Y. Liu, S. He, Second-order approximation scheme combined with $H^1$ -Galerkin MFE method for nonlinear time fractional convection-diffusion equation, Comput. Math. Appl., 73 (2017), 1182–1196. https://doi.org/10.1016/j.camwa.2016.07.037 doi: 10.1016/j.camwa.2016.07.037
[41]	T. Hou, C. Liu, C. Dai, L. Chen, Y. Yang, Two-grid algorithm of $H^1$ -Galerkin mixed finite element methods for semilinear parabolic integro-differential equations, J. Comput. Math., 40 (2022), 667–685. https://doi.org/10.4208/jcm.2101-m2019-0159 doi: 10.4208/jcm.2101-m2019-0159
[42]	M. Tripathy, R. Sinha, Superconvergence of $H^1$ -Galerkin mixed finite element methods for parabolic problems, Appl. Anal., 88 (2009), 1213–1231. https://doi.org/10.1080/00036810903208163 doi: 10.1080/00036810903208163
[43]	F. Brezzi, M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991. https:doi.org//10.1007/978-1-4612-3172-1
[44]	R. Ewing, M. Liu, J. Wang, Superconvergence of mixed finite element approximations over quadrilaterals, SIAM J. Numer. Anal., 36 (1999), 772–787. https://doi.org/10.1137/S0036142997322801 doi: 10.1137/S0036142997322801
[45]	C. Huang, M. Stynes, Optimal $H^1$ sptial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation, Adv. Comput. Math., 46 (2020), 63. https://doi.org/10.1007/s10444-020-09805-y doi: 10.1007/s10444-020-09805-y
[46]	R. Li, W. Liu, The AFEPack Handbook, 2006. Available from: http://dsec.pku.edu.cn/rli/software.php

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