Selection of the ground state on a compact metric graph

2025/08/08 by Robert Marangell, Marangell, Robert, Dmitry E. Pelinovsky +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2508.05986

openalex publication_date 2025/08/08 · openalex created_date 2025/10/15 · openalex updated_date 2026/07/28

Abstract

We show that the ground state in the Fisher--KPP model on a compact metric graph with Dirichlet conditions on boundary vertices is either trivial (zero) or nontrivial and strictly positive. For positive initial data, we prove that the trivial ground state is globally asymptotically stable if the edges of the metric graph are uniformly short and the nontrivial ground state is globally asymptotically stable if at least one edge is sufficiently long. For the intermediate case, we find a sharp criterion for the existence, uniqueness and global asymptotic stability of the trivial versus nontrivial ground state. Besides standard methods based on the comparison theory, energy minimizers, and the lowest eigenvalue of the graph Laplacian, we develop a novel method based on the period function for differential equations to fully characterize the nontrivial ground state in the particular case of flower graphs.

Citations

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