# American Institute of Mathematical Sciences

May  2019, 39(5): 2807-2875. doi: 10.3934/dcds.2019118

## Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity

 Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

* Corresponding author: Yuta Ishii

Received  June 2018 Revised  September 2018 Published  January 2019

Fund Project: The second author is supported by JSPS KAKENHI Grant Number 16K05240.

In this paper, we consider stationary solutions of the following one-dimensional Schnakenberg model with heterogeneity:
 $\begin{equation*} \begin{cases} u_t-\varepsilon ^2 u_{xx} = d\varepsilon -u+g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ \varepsilon v_t-Dv_{xx} = \frac{1}{2}-\frac{c}{\varepsilon}g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ u_x (\pm 1) = v_x (\pm 1) = 0 . \end{cases} \end{equation*}$
We concentrate on the case that
 $d, c, D>0$
are given constants,
 $g(x)$
is a given symmetric function, namely
 $g(x) = g(-x)$
, and
 $\varepsilon>0$
is sufficiently small and are interested in the effect of the heterogeneity
 $g(x)$
on the stability. For the case
 $g(x) = 1$
and
 $d = 0$
, Iron, Wei, and Winter (2004) studied the existence of
 $N-$
peaks symmetric stationary solutions and their stability. In this paper, first we construct symmetric one-peak stationary solutions
 $(u_{\varepsilon}, v_{\varepsilon})$
by using the contraction mapping principle. Furthermore, we give a linear stability analysis of the solutions
 $(u_{\varepsilon}, v_{\varepsilon})$
in details and reveal the effect of heterogeneity on the stability, which is a new phenomenon compared with the case
 $g(x) = 1$
.
Citation: Yuta Ishii, Kazuhiro Kurata. Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2807-2875. doi: 10.3934/dcds.2019118
##### References:

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##### References:
For $D = 0.6$, the solution is stable
For $D_1 > D = \frac{1}{24}-0.02$, the solution is stable. For $D_1 < D = \frac{1}{24}+0.02$, the solution is unstable
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