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

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  • 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.

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  • 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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