Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity
Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity
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具有周期异质性的Schnakenberg模型多峰对称平稳解的稳定性
DOI：
10.3934/cpaa.2020130
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发表时间：
2020
影响因子：
1
通讯作者：
Yuta Ishii
中科院分区：
文献类型：
--
作者：
Yuta Ishii
本文考虑具有周期异质性的一维Schnakenberg模型：Begin{Document}$Begin{方程*}\Begin{Case}u_t-varepsilon^2u_(Xx)=d\varepsilon-u+g(X)u^2v，&x\in(-1，1)，t>；0，\varepsilon v_t-dv_{xx}=\frac{1}{2}-\frac{c}{\varepsilon}g(X)u^2 v，&x\in(-1，1)，\；t>；0，\u_x(\pm 1)=v_x(\pm 1)=0。\end{case}\end{等式*}$\end{Document}其中\Begin{Document}$d，c，D>；0$\end{Document}是给定的常量，\Begin{Document}$\varepsilon>；0$\end{Document}足够小，且\Begin{Document}$g(X)$\end{Document}是给定的正函数。设\Begin{Document}$N\ge 1$\end{Document}为任意自然数。我们假设\Begin{Document}$g(X)$\end{Document}是一个周期对称函数，即\Begin{Document}$g(X)=g(-x)$\end{Document}和\Begin{Document}$g(X)=g(x+2N^{-1})$\end{Document}。研究了Begin{Document}$N$end{Document}-峰平稳对称解的稳定性。特别是，我们感兴趣的是上面的周期异质性对它们稳定性的影响。对于标准的Schnakenberg模型，即Begin{Document}$g(X)=1$\end{Document}，其中Begin{Document}$d=0$\end{Document}，Iron，魏和温特在2004年建立了Begin{Document}$N$\end{Document}-Peak解的稳定性。在本文中，我们严格地给出了线性稳定性分析，揭示了周期非均质性对峰解的稳定性的影响。特别地，与Begin{Document}$g(X)=1$\end{Document}情形相比，我们研究了周期异质性的影响如何使Begin{Document}$N\end{Document}-峰解稳定或不稳定。
In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity: \begin{document}$ \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*} $\end{document} where \begin{document}$ d,c,D>0 $\end{document} are given constants, \begin{document}$ \varepsilon >0 $\end{document} is sufficiently small, and \begin{document}$ g(x) $\end{document} is a given positive function. Let \begin{document}$ N \ge 1 $\end{document} be an arbitrary natural number. We assume that \begin{document}$ g(x) $\end{document} is a periodic and symmetric function, namely \begin{document}$ g(x) = g(-x) $\end{document} and \begin{document}$ g(x) = g(x+2N^{-1}) $\end{document} . We study the stability of \begin{document}$ N $\end{document} -peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity \begin{document}$ g(x) $\end{document} above on their stability. For the standard Schnakenberg model, namely the case of \begin{document}$ g(x) = 1 $\end{document} , with \begin{document}$ d = 0 $\end{document} , the stability of \begin{document}$ N $\end{document} -peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of \begin{document}$ N $\end{document} -peak solution. In particular, we investigate how \begin{document}$ N $\end{document} -peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case \begin{document}$ g(x) = 1 $\end{document} .
异质性Schnakenberg模型单峰对称平稳解的构造及稳定性分析
DOI：
--
发表时间：
2020
期刊：
RIMS Kokyuroku
影响因子：
--
作者：
J. Yamada;S. Matsuyama;S. Yasuda;Y. Sano;Y. Kohmura;M. Yabashi;T. Ishikawa;and K. Yamauchi;Simon Schnyder;Yuta Ishii and Kazuhiro Kurata
通讯作者：
Yuta Ishii and Kazuhiro Kurata