The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model
The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model
复制标题
异质性对 Schnakenberg 模型单峰平稳解的影响
DOI：
10.1016/j.jde.2021.03.007
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发表时间：
2021
影响因子：
2.4
通讯作者：
Ishii Yuta
中科院分区：
文献类型：
--
作者：
Kitamura Takashi;Nakata Hirotaka;Takahashi Daisuke;Toshima Kazunobu;Ishii Yuta
本文考虑区间(−1，1)上具有异质性的Schnakenberg模型。我们首先利用Liapunov-Schmidt约化方法构造出集中在适当点的定常解。此外，通过研究相关的线性化特征值问题，我们建立了上述解的线性稳定性。Iron，魏和温特(2004)建立了非均质情形下多峰对称定态解的存在性和稳定性。在他们的工作中，单峰解总是稳定的。对于对称异质情形，Ishii和Kurata(2019)给出了单峰对称解的详细分析，揭示了异质性的失稳效应。在这篇文章中，我们揭示了集中点的位置和相对于o(1)阶本征值的稳定性是由非均质性与相应的格林函数相互作用决定的。特别是，我们并不认为这种异质性是对称的。此外，通过一个典型的例子，我们进行了几个数值模拟来说明我们的主要结果。
In this paper, we consider the Schnakenberg model with heterogeneity on the interval (− 1, 1). We first construct stationary solutions which concentrate at a suitable point by using the Liapunov-Schmidt reduction method. Moreover, by investigating the associated linearized eigenvalue problem, we establish the linear stability of the solutions above. Iron, Wei, and Winter (2004) established the existence and stability of multi-peak symmetric stationary solutions in non-heterogeneity case. In their work, the one-peak solution is always stable. For the symmetric heterogeneity case, Ishii and Kurata (2019) gave the analysis of one-peak symmetric solutions in details and revealed a destabilization effect of the heterogeneity. In this paper, we reveal that the mechanism which the location of the concentration point and the stability with respect to eigenvalues of order o (1) are determined by the interaction of the heterogeneity with the associated Green's function. In particular, we not suppose that the heterogeneity is symmetric. Also, by a typical example, we performed several numerical simulations to illustrate our main results.