# Stabilization of the Moving Front Solution of the Reaction-Diffusion-Advection Problem

^{*}

## Abstract

**:**

## 1. Introduction

- a mathematically justified solution with an internal transition layer approximation;
- the conditions for the existence of its stable stationary solution;
- the domain of attraction of a stable stationary solution.

## 2. Problem Statement

**Proposition**

**1.**

#### 2.1. Associated Systems

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

## 3. Study Algorithm

## 4. Asymptotic Approximation of Moving Front Solution

## 5. The Moving Front Upper and Lower Solutions

## 6. Stationary Solution Asymptotic Approximation

## 7. The Upper and Lower Solutions of the Stationary Problem

**Theorem**

**1.**

## 8. Stabilization of the Solution of the Initial-Boundary Value Problem

**Theorem**

**2.**

- Step 1. First we will estimate the time interval ${T}_{0}$ such that for $t\ge {T}_{0}$ it is performed ${\widehat{x}}_{\alpha}\left(t\right)-{x}_{s}=O\left({\epsilon}^{2}\right)$ and ${x}_{s}-{\widehat{x}}_{\beta}\left(t\right)=O\left({\epsilon}^{2}\right)$.
- Step 2. Then we will prove that for some t$(t>{T}_{0})$ the inequalities are valid$${\alpha}_{s2}(x,{\underline{x}}_{s},\epsilon )\le {\widehat{\alpha}}_{2}(x,t,\underline{x},\underline{W},\epsilon )\le {\widehat{\beta}}_{2}(x,t,\overline{x},\overline{W},\epsilon )\le {\beta}_{s2}(x,{\overline{x}}_{s},\epsilon ),\phantom{\rule{0.277778em}{0ex}}x\in [0,1].$$

#### 8.1. Step 1

#### 8.2. The Estimates for Q-Functions

#### 8.3. The Estimate for $\overline{\mathrm{\Delta}}\left(t\right)$

#### 8.4. Step 2

## 9. Examples of Solution Formation

**Example**

**1.**

**Example**

**2.**

## 10. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**–

**g**): the numerical solution of problem (51) (solid black bold line) and location of the upper and lower solutions (solid gray bold lines) at different moments of time; (

**h**) the illustration of solution stabilization over time. Thin oblique straight lines denote the reduced equation solutions ${\phi}^{(-)}=x-1.5$ and ${\phi}^{(+)}=x$.

**Figure 2.**The numerical solution of problems ((52) I) (

**a**–

**c**); ((52) II) (

**e**–

**g**); ((52) III) (

**i**–

**k**) at various moments of time (solid black bold line) and location of the upper and lower solutions (solid gray bold lines); the illustration of the stabilization over time of the solution to problem ((52) I) (

**d**); ((52) II) (

**h**); ((52) III) (

**l**). Thin thin black curves denote the reduced equation solutions ${\phi}^{(-)}=-{(x+2)}^{-1}$ and ${\phi}^{(+)}=-{(x-3)}^{-1}$.

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**MDPI and ACS Style**

Nefedov, N.; Polezhaeva, E.; Levashova, N.
Stabilization of the Moving Front Solution of the Reaction-Diffusion-Advection Problem. *Axioms* **2023**, *12*, 253.
https://doi.org/10.3390/axioms12030253

**AMA Style**

Nefedov N, Polezhaeva E, Levashova N.
Stabilization of the Moving Front Solution of the Reaction-Diffusion-Advection Problem. *Axioms*. 2023; 12(3):253.
https://doi.org/10.3390/axioms12030253

**Chicago/Turabian Style**

Nefedov, Nikolay, Elena Polezhaeva, and Natalia Levashova.
2023. "Stabilization of the Moving Front Solution of the Reaction-Diffusion-Advection Problem" *Axioms* 12, no. 3: 253.
https://doi.org/10.3390/axioms12030253