# American Institute of Mathematical Sciences

June  2020, 19(6): 2965-3031. doi: 10.3934/cpaa.2020130

## Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

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

Received  February 2019 Revised  October 2019 Published  March 2020

In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity:
 $\begin{equation*} \begin{cases} u_t-\varepsilon ^2 u_{xx} = d\varepsilon -u+g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ \varepsilon v_t-Dv_{xx} = \frac{1}{2}-\frac{c}{\varepsilon}g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ u_x (\pm 1) = v_x (\pm 1) = 0 .\end{cases} \end{equation*}$
where
 $d,c,D>0$
are given constants,
 $\varepsilon >0$
is sufficiently small, and
 $g(x)$
is a given positive function. Let
 $N \ge 1$
be an arbitrary natural number. We assume that
 $g(x)$
is a periodic and symmetric function, namely
 $g(x) = g(-x)$
and
 $g(x) = g(x+2N^{-1})$
. We study the stability of
 $N$
-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity
 $g(x)$
above on their stability. For the standard Schnakenberg model, namely the case of
 $g(x) = 1$
, with
 $d = 0$
, the stability of
 $N$
-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of
 $N$
-peak solution. In particular, we investigate how
 $N$
-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case
 $g(x) = 1$
.
Citation: Yuta Ishii. Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2965-3031. doi: 10.3934/cpaa.2020130
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##### References:
For $D_2^{2,+}(\xi_2) > D = (3+\sqrt{17})/16-0.1$, two-peak solution is stable. For $D_2^{2,+}(\xi_2) < D = (3+\sqrt{17})/16+0.1$, two-peak solution is unstable
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