Boundedness in a two-species chemotaxis parabolic system with two chemicals

Xie Li; Yilong Wang

doi:10.3934/dcdsb.2017132

Article Contents

2017, Volume 22, Issue 7: 2717-2729. Doi: 10.3934/dcdsb.2017132

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Boundedness in a two-species chemotaxis parabolic system with two chemicals

Xie Li^1, , and
Yilong Wang^2,

1.
College of Mathematic & Information, China West Normal University, Nanchong, 637002, China
2.
School of Sciences, Southwest Petroleum University, Chengdu, 610500, China

* Corresponding author: Xie Li
* Corresponding author: Xie Li

Received: August 2016

Revised: February 2017

Published: October 2017

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Abstract

This paper is devoted to the chemotaxis system

$\left\{\begin{aligned}&u_t=Δ u-χ\nabla·(u\nabla v), &x∈Ω,\,t>0,\\& τ v_t=Δ v-v+w, &x∈Ω,\,t>0,\\&w_t=Δ w-ξ\nabla·(w\nabla z), &x∈Ω,\,t>0,\\& τ z_t=Δ z-z+u, &x∈Ω,\,t>0,\end{aligned}\right.$

which models the interaction between two species in presence of two chemicals, where $χ, \, ξ∈\mathbb{R}$, $Ω\subset\mathbb{R}^2$ are bounded domains with smooth boundary. It is shown that under the homogeneous Neumann boundary conditions the system possesses a unique global classical solution which is bounded whenever both $\int_Ω u_0dx$ and $\int_Ω w_0dx$ are appropriately small. In particular, we extend the recent results obtained by Tao and Winkler (2015, Disc. Cont. Dyn. Syst. B) to the fully parabolic case, i.e., the case of $τ=1$.

Keywords:

Chemotaxis,
attraction,
repulsion,
blow-up.

Mathematics Subject Classification: Primary:35A01, 35B44, 35K57, 35Q92;Secondary:92C17.

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