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Global Boundedness in a Two-Competing-Species Chemotaxis System with Two Chemicals

Global Boundedness in a Two-Competing-Species Chemotaxis System with Two Chemicals This paper deals with a two-competing-species chemotaxis system with two different chemicals { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ v ) + μ 1 u ( 1 − u − a 1 w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ v t = Δ v − v + w , ( x , t ) ∈ Ω × ( 0 , ∞ ) , w t = Δ w − χ 2 ∇ ⋅ ( w ∇ z ) + μ 2 w ( 1 − a 2 u − w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ z t = Δ z − z + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n $\varOmega \subset \mathbb{R}^{n}$ ( n ≥ 1 ) $(n\geq 1)$ with the nonnegative initial data ( u 0 , τ v 0 , w 0 , τ z 0 ) ∈ C 0 ( Ω ‾ ) × W 1 , ∞ ( Ω ) × C 0 ( Ω ‾ ) × W 1 , ∞ ( Ω ) $(u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )$ , where τ ∈ { 0 , 1 } $\tau \in \{0,1\}$ and the parameters χ i , μ i , a i $\chi_{i},\mu_{i},a_{i}$ ( i = 1 , 2 $i=1,2$ ) are positive. When τ = 0 $\tau =0$ , based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if n = 2 $n=2$ . On the other hand, when τ = 1 $\tau =1$ , relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that n ≥ 1 $n\geq 1$ and there exists θ 0 > 0 $\theta_{0}>0$ such that χ 2 μ 1 < θ 0 $\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}$ and χ 1 μ 2 < θ 0 $\frac{\chi_{1}}{\mu_{2}}<\theta_{0}$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Global Boundedness in a Two-Competing-Species Chemotaxis System with Two Chemicals

Acta Applicandae Mathematicae , Volume 148 (1) – Nov 4, 2016

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References (61)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Springer Science+Business Media Dordrecht
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Classical Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-016-0083-0
Publisher site
See Article on Publisher Site

Abstract

This paper deals with a two-competing-species chemotaxis system with two different chemicals { u t = Δ u − χ 1 ∇ ⋅ ( u ∇ v ) + μ 1 u ( 1 − u − a 1 w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ v t = Δ v − v + w , ( x , t ) ∈ Ω × ( 0 , ∞ ) , w t = Δ w − χ 2 ∇ ⋅ ( w ∇ z ) + μ 2 w ( 1 − a 2 u − w ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) , τ z t = Δ z − z + u , ( x , t ) ∈ Ω × ( 0 , ∞ ) , $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla v)+\mu_{1} u(1-u-a _{1}w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau v_{t}=\Delta v-v+w, & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle w_{t}=\Delta w-\chi_{2}\nabla \cdot (w\nabla z)+\mu_{2}w(1-a_{2}u-w), & (x,t)\in \varOmega \times (0,\infty ), \\ \displaystyle \tau z_{t}=\Delta z-z+u, & (x,t)\in \varOmega \times (0,\infty ), \end{array}\displaystyle \right . \end{aligned}$$ under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ⊂ R n $\varOmega \subset \mathbb{R}^{n}$ ( n ≥ 1 ) $(n\geq 1)$ with the nonnegative initial data ( u 0 , τ v 0 , w 0 , τ z 0 ) ∈ C 0 ( Ω ‾ ) × W 1 , ∞ ( Ω ) × C 0 ( Ω ‾ ) × W 1 , ∞ ( Ω ) $(u_{0},\tau v_{0},w_{0},\tau z_{0})\in C^{0}(\overline{\varOmega }) \times W^{1,\infty }(\varOmega )\times C^{0}(\overline{\varOmega })\times W ^{1,\infty }(\varOmega )$ , where τ ∈ { 0 , 1 } $\tau \in \{0,1\}$ and the parameters χ i , μ i , a i $\chi_{i},\mu_{i},a_{i}$ ( i = 1 , 2 $i=1,2$ ) are positive. When τ = 0 $\tau =0$ , based on some a priori estimates and Moser-Alikakos iteration, it is shown that regardless of the size of initial data, the system possesses a unique globally bounded classical solution for any positive parameters if n = 2 $n=2$ . On the other hand, when τ = 1 $\tau =1$ , relying on the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that n ≥ 1 $n\geq 1$ and there exists θ 0 > 0 $\theta_{0}>0$ such that χ 2 μ 1 < θ 0 $\frac{\chi_{2}}{ \mu_{1}}<\theta_{0}$ and χ 1 μ 2 < θ 0 $\frac{\chi_{1}}{\mu_{2}}<\theta_{0}$ .

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Nov 4, 2016

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