Pattern formation of a coupled two-cell Schnakenberg model

Guanqi Liu; Yuwen Wang

doi:10.3934/dcdss.2017056

Article Contents

2017, Volume 10, Issue 5: 1051-1062. Doi: 10.3934/dcdss.2017056

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Pattern formation of a coupled two-cell Schnakenberg model

Guanqi Liu and
Yuwen Wang^1,2, ,

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.
Y. Y. Tseng Functional Analysis Research Center and School of Mathematical Sciences, Harbin Normal University, Harbin, Heilongjiang 150025, China

* Corresponding author: Yuwen Wang
* Corresponding author: Yuwen Wang

Received: June 30, 2016

Revised: December 31, 2016

Published: October 2017

Partially supported by NSFC grant 11571086,11471091 and Science Research Funds for Over-seas Returned Chinese Scholars of Heilongjiang Province LC2013C01

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Abstract

In this paper, we study a coupled two-cell Schnakenberg model with homogenous Neumann boundary condition, i.e.,

$\left\{ \begin{gathered} -d_1Δ u=a-u+u^2v+c(w-u),&\text{ in } Ω, \\-d_2Δ v=b-u^2v,&\text{ in } Ω , \\-d_1Δ w=a-w+w^2z+c(u-w),&\text{ in } Ω, \\-d_2Δ z=b-w^2z,&\text{ in } Ω, \\\dfrac{\partial u}{\partial ν}=\dfrac{\partial v}{\partial ν}=\dfrac{\partial w}{\partial ν}=\dfrac{\partial z}{\partial ν}=0, &\text{ on } \partialΩ.\end{gathered} \right.$

We give a priori estimate to the positive solution. Meanwhile, we obtain the non-existence and existence of positive non-constant solution as parameters $ d_1, d_2, a$ and b changes.

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Mathematics Subject Classification: 92C40, 92C15, 35B45, 35B35, 35A01.

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