Stability Analysis of Spike Solutions to the Schnakenberg Model with Heterogeneity on Metric Graphs
Stability Analysis of Spike Solutions to the Schnakenberg Model with Heterogeneity on Metric Graphs
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DOI：
10.1007/s00332-021-09762-w
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发表时间：
2021-12
影响因子：
3
通讯作者：
Yuta Ishii
中科院分区：
文献类型：
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作者：
Yuta Ishii
本文研究了具有异质性的Schnakenberg模型的尖峰平稳解在紧度量图上的线性稳定性。作者和仓田在作品(石井和仓田2021)中证明了尖峰解的存在。通过研究相关的线性化特征值问题，建立了一般紧度量图解的稳定性的抽象定理。特别地，伴随的格林函数在计算本征值中起着重要的作用，我们揭示了一般图上格林函数的几个必要条件。为了证明稳定性，我们分别计算了O(1)阶和O(1)阶的两个本征值。利用非局部特征值问题的魏和温特引理，证明了O(1)阶本征值的稳定性。O(1)阶本征值的稳定性是由非均质性与格林函数的相互作用决定的。此外，基于抽象定理，我们给出了无异质性函数的解在他们形图和H形图上关于扩散常数的精确稳定性阈值。特别地，与一维区间情形相比，通过这些具体图的几何效应，我们得到了关于双峰解稳定性的新现象。此外，我们还通过一个典型的例子展示了异质性的影响。
In this paper, we consider the linear stability of spiky stationary solutions on compact metric graphs for the Schnakenberg model with heterogeneity. The existence of spiky solutions has been shown by the author and Kurata in the work (Ishii and Kurata 2021). By studying the associated linearized eigenvalue problem, we establish the abstract theorem on the stability of the solutions for general compact metric graphs. In particular, the associated Green’s function plays an important role in calculating eigenvalues, and we reveal the several needed conditions for Green’s function on general graphs. To show the stability, we calculate two eigenvalues of orderO(1) and of ordero(1), respectively. The stability of eigenvalues of orderO(1) is shown by using the lemma of Wei and Winter for non-local eigenvalue problem. The stability of eigenvalues of ordero(1) is determined by the interaction of the heterogeneity with Green’s function. Moreover, based on the abstract theorem, we give precise stability thresholds with respect to diffusion constants for the solutions without heterogeneity function on theY-shaped graph and theH-shaped graph. In particular, compared with the one-dimensional interval case, we obtain new phenomena on the stability of two-peak solutions by the effect of the geometry of these concrete graphs. In addition, we also present the effect of heterogeneity by using a typical example.