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A hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions

A hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear... This paper studies the hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions and logistic source: $$u_{t}=-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+\mu u(1-u^k)$$ u t = - χ ∇ · ( u ∇ v ) + ξ ∇ · ( u ∇ w ) + μ u ( 1 - u k ) , $$0=\Delta v+\alpha u^q-\beta v$$ 0 = Δ v + α u q - β v , $$ 0=\Delta w+\gamma u^r-\delta w$$ 0 = Δ w + γ u r - δ w , in a bounded domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n\ge 1$$ n ≥ 1 , subject to the non-flux boundary condition. We at first establish the local existence of solutions (the so-called strong $$W^{1,p}$$ W 1 , p -solutions, satisfying the hyperbolic equation weakly and solving the elliptic ones classically) to the model via applying the viscosity vanishing method and then give criteria on global boundedness versus finite- time blowup for them. It is proved that if the attraction is dominated by the logistic source or the repulsion with $$\max \{r,k\}>q$$ max { r , k } > q , the solutions would be globally bounded; otherwise, the finite-time blowup of solutions may occur whenever $$\max \{r,k\}<q$$ max { r , k } < q . Under the balance situations with $$q=r=k$$ q = r = k , $$q=r>k$$ q = r > k or $$q=k>r$$ q = k > r , the boundedness or possible finite-time blowup would depend on the sizes of the coefficients involved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

A hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions

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References (37)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-018-0428-4
Publisher site
See Article on Publisher Site

Abstract

This paper studies the hyperbolic–elliptic–elliptic system of an attraction–repulsion chemotaxis model with nonlinear productions and logistic source: $$u_{t}=-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+\mu u(1-u^k)$$ u t = - χ ∇ · ( u ∇ v ) + ξ ∇ · ( u ∇ w ) + μ u ( 1 - u k ) , $$0=\Delta v+\alpha u^q-\beta v$$ 0 = Δ v + α u q - β v , $$ 0=\Delta w+\gamma u^r-\delta w$$ 0 = Δ w + γ u r - δ w , in a bounded domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , $$n\ge 1$$ n ≥ 1 , subject to the non-flux boundary condition. We at first establish the local existence of solutions (the so-called strong $$W^{1,p}$$ W 1 , p -solutions, satisfying the hyperbolic equation weakly and solving the elliptic ones classically) to the model via applying the viscosity vanishing method and then give criteria on global boundedness versus finite- time blowup for them. It is proved that if the attraction is dominated by the logistic source or the repulsion with $$\max \{r,k\}>q$$ max { r , k } > q , the solutions would be globally bounded; otherwise, the finite-time blowup of solutions may occur whenever $$\max \{r,k\}<q$$ max { r , k } < q . Under the balance situations with $$q=r=k$$ q = r = k , $$q=r>k$$ q = r > k or $$q=k>r$$ q = k > r , the boundedness or possible finite-time blowup would depend on the sizes of the coefficients involved.

Journal

Journal of Evolution Equations – Springer Journals

Published: Feb 28, 2018

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