Binary differential equations with symmetries
Asymptotic expansion of the mean-field approximation
April 2019, 39(4): 1923-1955. doi: 10.3934/dcds.2019081
The Schnakenberg model with precursors
Weiwei Ao and Chao Liu
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Received January 2018 Revised March 2018 Published January 2019
Fund Project: The research of the first author is supported by NSFC (No. 11801421 and No. 11631011).
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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*}$
$ D_{1}>0 $
$ B>0 $
We establish the existence and stability of
$ N- $
peaked steady-states in terms of the precursor
and the diffusion coefficients
$ D_{1} $
. It is shown that
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].
Keywords: Schnakenberg model, precursors, $ N- $peaked solutions, existence and stability.
Mathematics Subject Classification: Primary: 35J47; Secondary: 35J60.
Citation: Weiwei Ao, Chao Liu. The Schnakenberg model with precursors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1923-1955. doi: 10.3934/dcds.2019081
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Weiwei Ao Chao Liu