Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs
Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs
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DOI：
10.3934/cpaa.2021035
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发表时间：
2021
影响因子：
1
通讯作者：
Yuta Ishii;K. Kurata
中科院分区：
文献类型：
--
作者：
Yuta Ishii;K. Kurata
本文研究了紧致度量图上具有异质性的Schnakenberg模型的尖峰平稳解的存在性。这些解是用Liapunov-Schmidt约化方法构造的，并采取与[ 14，11 ]相同的策略。首先，我们给出了一般紧度量图的多峰解的存在性的抽象定理，并对相应的绿色函数作了若干假设。特别是，我们揭示了如何集中点的位置和尖峰的解决方案的振幅是由相互作用的异质性与几何的紧凑度量图，由绿色的功能表示。其次，我们将我们的抽象定理应用于非异质情况下的\开始{document}$ Y $\end{document}形度量图和\开始{document}$ H $\end{document}形度量图。特别是，我们展示了这些紧凑的图形的几何形状的精确效果，这些具体的图的集中点的位置，分别。
In this paper, we study the existence of spiky stationary solutions of the Schnakenberg model with heterogeneity on compact metric graphs. These solutions are constructed by using the Liapunov–Schmidt reduction method and taking the same strategy as that in [ 14 , 11 ]. First, we give the abstract theorem on the existence of multi-peak solutions for general compact metric graphs under several assumptions for the associated Green's function. In particular, we reveal that how locations of concentration points and amplitudes of spiky solutions are determined by the interaction of the heterogeneity with the geometry of the compact metric graph, represented by Green's function. Second, we apply our abstract theorem to the \begin{document}$ Y $\end{document} -shaped metric graph and the \begin{document}$ H $\end{document} -shaped metric graph in non-heterogeneity case. In particular, we show the precise effect of the geometry of those compact graphs to the locations of concentration points for these concrete graphs, respectively.