Global asymptotic stability in a parabolic–elliptic chemotaxis system with competitive kinetics and loop,Applicable Analysis

当前位置： X-MOL 学术 › Appl. Anal. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

Global asymptotic stability in a parabolic–elliptic chemotaxis system with competitive kinetics and loop
Applicable Analysis ( IF 1.1 ) Pub Date : 2020-06-23 , DOI: 10.1080/00036811.2020.1783536
Xinyu Tu ₁ , Chunlai Mu ₂ , Shuyan Qiu ₂

Affiliation

ABSTRACT

This paper deals with the initial-boundary value problem for the two-species chemotaxis-competition system with two signals $\{\begin{cases} \partial_{t} u_{1} = Δ u_{1} - χ_{11} \nabla \cdot (u_{1} \nabla v_{1}) - χ_{12} \nabla \cdot (u_{1} \nabla v_{2}) + μ_{1} u_{1} (1 - u_{1} - a_{1} u_{2}), \\ \partial_{t} u_{2} = Δ u_{2} - χ_{21} \nabla \cdot (u_{2} \nabla v_{1}) - χ_{22} \nabla \cdot (u_{2} \nabla v_{2}) + μ_{2} u_{2} (1 - u_{2} - a_{2} u_{1}), \\ 0 = Δ v_{1} - λ_{1} v_{1} + α_{11} u_{1} + α_{12} u_{2}, \\ 0 = Δ v_{2} - λ_{2} v_{2} + α_{21} u_{1} + α_{22} u_{2}, \end{cases}$ under the homogeneous Neumann boundary condition, where $x \in Ω, t > 0$ , $χ_{i j} > 0$ , $μ_{i} > 0$ , $a_{i} > 0$ , $α_{i j} > 0$ , $λ_{i} > 0$ $(i, j = 1, 2)$ , and $Ω \subset R^{n} (n \geq 2)$ is a smooth bounded domain. If $χ_{11} / μ_{1}$ , $χ_{12} / μ_{1}$ , $χ_{21} / μ_{2}$ and $χ_{22} / μ_{2}$ are sufficiently small, then the system possesses a globally bounded classical solution for any suitably regular initial data $u_{10}, u_{20}$ . Furthermore, by constructing some appropriate functionals, it is shown that

For the weak competition case, if $μ_{1}, μ_{2}$ are sufficiently large, then the solution $(u_{1}, u_{2}, v_{1}, v_{2})$ converges to $\begin{aligned} (\frac{1 - a_{1}}{1 - a_{1} a_{2}}, \frac{1 - a_{2}}{1 - a_{1} a_{2}}, \frac{α_{11} (1 - a_{1}) + α_{12} (1 - a_{2})}{λ_{1} (1 - a_{1} a_{2})}, \\ \frac{α_{21} (1 - a_{1}) + α_{22} (1 - a_{2})}{λ_{2} (1 - a_{1} a_{2})}) \end{aligned}$ exponentially as $t \to \infty$ .
For the strong-weak competition case, if $μ_{2}$ is sufficiently large, then the solution $(u_{1}, u_{2}, v_{1}, v_{2})$ converges to $(0, 1, α_{12} / λ_{1}, α_{22} / λ_{2})$ with exponential decay when $a_{1} > 1$ , and with algebraic decay when $a_{1} = 1$ .

中文翻译：

具有竞争动力学和循环的抛物线-椭圆趋化系统的全局渐近稳定性

摘要

本文研究了具有两个信号的双物种趋化竞争系统的初始边值问题 $\{\begin{cases} \partial_{吨} 你_{1} = Δ 你_{1} - χ_{11} \nabla \cdot (你_{1} \nabla v_{1}) - χ_{12} \nabla \cdot (你_{1} \nabla v_{2}) + μ_{1} 你_{1} (1 - 你_{1} - {一种}_{1} 你_{2}), \\ \partial_{吨} 你_{2} = Δ 你_{2} - χ_{21} \nabla \cdot (你_{2} \nabla v_{1}) - χ_{22} \nabla \cdot (你_{2} \nabla v_{2}) + μ_{2} 你_{2} (1 - 你_{2} - {一种}_{2} 你_{1}), \\ 0 = Δ v_{1} - λ_{1} v_{1} + α_{11} 你_{1} + α_{12} 你_{2}, \\ 0 = Δ v_{2} - λ_{2} v_{2} + α_{21} 你_{1} + α_{22} 你_{2}, \end{cases}$ 在齐次 Neumann 边界条件下，其中 $X \in Ω, 吨 > 0$ , $χ_{一世 j} > 0$ , $μ_{一世} > 0$ , ${一种}_{一世} > 0$ , $α_{一世 j} > 0$ , $λ_{一世} > 0$ $(一世, j = 1, 2)$ ，和 $Ω \subset R^{n} (n \geq 2)$ 是一个光滑的有界域。如果 $χ_{11} / μ_{1}$ , $χ_{12} / μ_{1}$ , $χ_{21} / μ_{2}$ 和 $χ_{22} / μ_{2}$ 足够小，则系统对任何适当规则的初始数据都具有全局有界经典解 $你_{10}, 你_{20}$ . 此外，通过构造一些适当的泛函，表明

对于弱竞争情况，如果 $μ_{1}, μ_{2}$ 足够大，那么解决方案 $(你_{1}, 你_{2}, v_{1}, v_{2})$ 收敛到 $\begin{aligned} (\frac{1 - {一种}_{1}}{1 - {一种}_{1} {一种}_{2}}, \frac{1 - {一种}_{2}}{1 - {一种}_{1} {一种}_{2}}, \frac{α_{11} (1 - {一种}_{1}) + α_{12} (1 - {一种}_{2})}{λ_{1} (1 - {一种}_{1} {一种}_{2})}, \\ \frac{α_{21} (1 - {一种}_{1}) + α_{22} (1 - {一种}_{2})}{λ_{2} (1 - {一种}_{1} {一种}_{2})}) \end{aligned}$ 指数地为 $吨 \to \infty$ .
对于强弱竞争情况，如果 $μ_{2}$ 足够大，则解 $(你_{1}, 你_{2}, v_{1}, v_{2})$ 收敛到 $(0, 1, α_{12} / λ_{1}, α_{22} / λ_{2})$ 指数衰减时 ${一种}_{1} > 1$ ，并且当代数衰减时 ${一种}_{1} = 1$ .

更新日期：2020-06-23

点击分享查看原文

点击收藏

阅读更多本刊最新论文本刊介绍/投稿指南

全部期刊列表>>