# American Institute of Mathematical Sciences

April  2019, 39(4): 1923-1955. doi: 10.3934/dcds.2019081

## The Schnakenberg model with precursors

 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).

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].
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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