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Multi-spike Patterns for the Gierer-Meinhardt Model with Heterogeneity on Y-shaped Metric Graph

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Abstract

In this paper, we consider the existence and the linear stability of spiky stationary solutions for the Gierer-Meinhardt model with heterogeneity on the Y-shaped compact metric graph. The existence is shown by the Liapunov-Schmidt reduction method, and the stability is shown by investigating the associated linearized eigenvalue problem. In particular, it is revealed that the location, amplitude, and stability of spikes are decided by the interaction of the heterogeneity functions with the geometry of the graph, represented by the associated Green’s function. Moreover, by applying our abstract theorem to several concrete examples, we study the detailed effects of the geometry of the graph and the heterogeneity function on spiky solutions.

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Acknowledgements

This work was supported by JSPS KAKENHI Grant Number 21K20341. The author would like to thank the referees for the careful reading of the manuscript and many useful comments.

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  1. National Institute of Technology, Ibaraki College, 866 Nakane, Hitachinaka, Ibaraki, 312-8508, Japan

    Yuta Ishii

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  1. Yuta Ishii
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Correspondence to Yuta Ishii.

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Ishii, Y. Multi-spike Patterns for the Gierer-Meinhardt Model with Heterogeneity on Y-shaped Metric Graph. J Dyn Diff Equat 36, 833–869 (2024). https://doi.org/10.1007/s10884-022-10157-y

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