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Boundedness and Stabilization in a Two-Species and Two-Stimuli Chemotaxis System with Signaling Loop

Boundedness and Stabilization in a Two-Species and Two-Stimuli Chemotaxis System with Signaling Loop This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop {ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)−χ3∇⋅(w∇v)+μ2w(1−w−a2u),x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\chi _{1}\nabla \cdot (u\nabla v)+\mu _{1} u(1-u-a_{1}w), \quad &x\in \Omega ,\quad t>0, \\ v_{t}=\Delta v-v+w,\quad &x\in \Omega ,\quad t>0, \\ w_{t}=\Delta w-\chi _{2}\nabla \cdot (w\nabla z)-\chi _{3}\nabla \cdot (w\nabla v)+\mu _{2} w(1-w-a_{2}u),\quad &x\in \Omega ,\quad t>0, \\ z_{t}=\Delta z-z+u,\quad &x\in \Omega ,\quad t>0 \end{array}\displaystyle \right . \end{aligned}$$ \end{document} in a bounded domain Ω⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Omega \subset \mathbb{R}^{n}$\end{document} with n≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\geq 1$\end{document}, where χ1,χ2,χ3>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\chi _{1},\chi _{2},\chi _{3}>0$\end{document}, a1,a2>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$a_{1},a_{2}>0$\end{document} and μ1,μ2>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu _{1},\mu _{2}>0$\end{document}. The system models the communication between macrophages and breast tumor cells.It will be proved that if n≤2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\leq 2$\end{document}, then for all appropriately regular nonnegative initial data u0,v0,w0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u_{0}, v_{0}, w_{0}$\end{document} and z0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$z_{0}$\end{document}, the solution to the corresponding Neumann initial-boundary value problem is global and bounded. Moreover, the asymptotic stabilization of arbitrary global bounded solutions for any n≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\geq 1$\end{document} under some explicit conditions will be investigated. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Boundedness and Stabilization in a Two-Species and Two-Stimuli Chemotaxis System with Signaling Loop

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References (63)

Publisher
Springer Journals
Copyright
Copyright © The Author(s), under exclusive licence to Springer Nature B.V. 2021
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-021-00447-w
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Abstract

This paper deals with the following competitive two-species and two-stimuli chemotaxis system with chemical signalling loop {ut=Δu−χ1∇⋅(u∇v)+μ1u(1−u−a1w),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−χ2∇⋅(w∇z)−χ3∇⋅(w∇v)+μ2w(1−w−a2u),x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} u_{t}=\Delta u-\chi _{1}\nabla \cdot (u\nabla v)+\mu _{1} u(1-u-a_{1}w), \quad &x\in \Omega ,\quad t>0, \\ v_{t}=\Delta v-v+w,\quad &x\in \Omega ,\quad t>0, \\ w_{t}=\Delta w-\chi _{2}\nabla \cdot (w\nabla z)-\chi _{3}\nabla \cdot (w\nabla v)+\mu _{2} w(1-w-a_{2}u),\quad &x\in \Omega ,\quad t>0, \\ z_{t}=\Delta z-z+u,\quad &x\in \Omega ,\quad t>0 \end{array}\displaystyle \right . \end{aligned}$$ \end{document} in a bounded domain Ω⊂Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\Omega \subset \mathbb{R}^{n}$\end{document} with n≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\geq 1$\end{document}, where χ1,χ2,χ3>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\chi _{1},\chi _{2},\chi _{3}>0$\end{document}, a1,a2>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$a_{1},a_{2}>0$\end{document} and μ1,μ2>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$\mu _{1},\mu _{2}>0$\end{document}. The system models the communication between macrophages and breast tumor cells.It will be proved that if n≤2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\leq 2$\end{document}, then for all appropriately regular nonnegative initial data u0,v0,w0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$u_{0}, v_{0}, w_{0}$\end{document} and z0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$z_{0}$\end{document}, the solution to the corresponding Neumann initial-boundary value problem is global and bounded. Moreover, the asymptotic stabilization of arbitrary global bounded solutions for any n≥1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$n\geq 1$\end{document} under some explicit conditions will be investigated.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: Oct 1, 2021

Keywords: Two-species chemotaxis; Global boundedness; Asymptotic stability; 92C17; 35K35; 35A01; 35B35

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