Root colonization dynamics of alginate encapsulated rhizobacteria: implications for <i>Arabidopsis thaliana</i> root growth and durum wheat performance

Amel Balla; Allaoua Silini; Hafsa Cherif-Silini; Francesca Mapelli; Sara Borin; Amel Balla; Allaoua Silini; Hafsa Cherif-Silini; Francesca Mapelli; Sara Borin

doi:10.3934/microbiol.2025006

AIMS Microbiology

2025, Volume 11, Issue 1: 87-125. doi: 10.3934/microbiol.2025006

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Research article

Root colonization dynamics of alginate encapsulated rhizobacteria: implications for Arabidopsis thaliana root growth and durum wheat performance

1.
Laboratory of Applied Microbiology, Department of Microbiology, Faculty of Natural and Life Sciences, University Ferhat Abbas of Setif -1, 19000 Setif, Algeria
2.
Department of Food, Environmental and Nutritional Sciences (DeFENS), University of Milan, 20133 Milan, Italy

Received: 04 September 2024 Revised: 11 January 2025 Accepted: 20 January 2025 Published: 05 February 2025

Bioencapsulation in alginate capsules offers an interesting opportunity for the efficient delivery of microbial inoculants for agricultural purposes. The present study evaluated the ionic gelation technique to prepare beads loaded with two plant growth-promoting bacteria (PGPB), Bacillus thuringiensis strain B25 and Pantoea agglomerans strain Pa in 1% alginate supplemented with 5mM proline as an osmoprotectant. Capsule morphology, survival rate, encapsulation efficiency, and viability during 24 months of storage as well as the stability of PGP activities were studied. Our results indicate that more than 99% of bacteria were effectively trapped in the alginate beads, which successfully released live bacteria after 60 days of storage at room temperature. A considerable survival of B. thuringiensis B25 throughout the storage period was detected, while the inoculated concentration of 8.72 × 10⁹ (±0.04 ×10⁹) CFU/mL was reduced to 99.9% for P. agglomerans Pa after 24 months of storage. Notably, a higher survival of individually encapsulated bacteria was observed compared to their co-inoculation. The colonization capacity of model plant Arabidopsis thaliana roots by free and encapsulated bacteria was detected by the triphenyltetrazolium chloride test. Moreover, both strains effectively colonized the rhizosphere, rhizoplane, and endosphere of durum wheat plants and exerted a remarkable improvement in plant growth, estimated as a significant increase in the quantities of total proteins, sugars, and chlorophyll pigments, besides roots and shoots length. This study demonstrated that alginate-encapsulated B. thuringiensis B25 and P. agglomerans Pa could be used as inoculants in agriculture, as their encapsulation ensures robust protection, maintenance of viability and PGP activity, and controlled bacterial biostimulant release into the rhizosphere.

Keywords:

Citation: Amel Balla, Allaoua Silini, Hafsa Cherif-Silini, Francesca Mapelli, Sara Borin. Root colonization dynamics of alginate encapsulated rhizobacteria: implications for Arabidopsis thaliana root growth and durum wheat performance[J]. AIMS Microbiology, 2025, 11(1): 87-125. doi: 10.3934/microbiol.2025006

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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.

Conflict of interest

The authors declare no conflicts of interest.

Author contributions

Conceptualization, A.B., HCS.,and A.S.; methodology, A.S., and H.C-S.; software, A.B.; validation, A.S., HCS., F.M. and S.B.; formal analysis, F.M., and S.B.; investigation, A.B., and F.M.; data curation, A.B., and A.S.; writing—original draft preparation, A.B.; writing—review and editing, A.S., HCS., F.M., and S.B.; visualization, A.B.; supervision, A.S., and H.C-S.; All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

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