The Schnakenberg model with precursors

Weiwei Ao; Chao Liu

doi:10.3934/dcds.2019081

Article Contents

2019, Volume 39, Issue 4: 1923-1955. Doi: 10.3934/dcds.2019081

This issue Previous Article Asymptotic expansion of the mean-field approximation Next Article Binary differential equations with symmetries

The Schnakenberg model with precursors

Weiwei Ao and
Chao Liu

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received: December 31, 2017

Revised: February 28, 2018

Published: January 2019

The research of the first author is supported by NSFC (No. 11801421 and No. 11631011)

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Abstract

In this paper, we mainly consider the following Schnakenberg model with a precursor $ \mu(x) $ on the interval $ (-1,1) $:

$ \begin{equation*}\left\{\begin{array}{l}u_{t} = D_{1}u''-\mu(x)u+vu^{2} \hspace{1.64cm} \text{in }(-1,1),\\v_{t} = D_{2}v''+B-vu^{2}\hspace{2.2cm} \;\;\;\; \text{in } (-1,1),\\u'(\pm1) = v'(\pm1) = 0,\end{array}\right.\end{equation*}$

where $ D_{1}>0 $, $ D_{2}>0 $, $ B>0 $.

We establish the existence and stability of $ N- $peaked steady-states in terms of the precursor $ \mu(x) $ and the diffusion coefficients $ D_{1} $ and $ D_{2} $. It is shown that $ \mu(x) $ plays an essential role for both existence and stability of the above pattern. Similar result has been obtained for the Gierer-Meinhardt system by Wei and Winter [21].

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Mathematics Subject Classification: Primary: 35J47; Secondary: 35J60.

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References

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