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

• 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 $g(x) = 1$.

Mathematics Subject Classification: Primary: 35K57, 35J66; Secondary: 35B35, 35Q92.

 Citation: • • Figure 1.  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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