Front propagation on a general metric graph

Abstract

We consider a bistable reaction-diffusion equation on a metric graph that is a generalization of the so-called star graphs. More precisely, our graph $\hat{I}\text{c◯}$ consists of a bounded finite metric graph $D$ of arbitrary configuration and a finite number of branches $\hat{I}{\text{c◯}}_1,\dots ,\hat{I}{\text{c◯}}_N(N\ge 2)$ of infinite length emanating from some of the vertices of $D$. Each $\hat{I}{\text{c◯}}_i(i=1,\dots ,N)$ is called an ``outer path''. Our goal is to investigate the behavior of the front coming from infinity along a given outer path $\hat{I}{\text{c◯}}_i$ and to discuss whether or not the front propagates into other outer paths $\hat{I}{\text{c◯}}_j(j\ne i)$. Unlike the case of star graphs, where $D$ is a single vertex, the dynamics of solutions can be far more complex and may depend sensitively on the configuration of the center graph $D$. We first focus on general principles that hold regardless of the structure of the center graph $D$. Among other things, we introduce the notion ``limit profile'', which allows us to define ``propagation'' and ``blocking'' without ambiguity, then we prove transient properties, that is, propagation $\hat{I}{\text{c◯}}_i\to \hat{I}{\text{c◯}}_j$ and $\hat{I}{\text{c◯}}_j\to \hat{I}{\text{c◯}}_k$ imply propagation $\hat{I}{\text{c◯}}_i\to \hat{I}{\text{c◯}}_k$. Next we consider perturbations of the graph $D$ while fixing the outer paths $\hat{I}{\text{c◯}}_1,\dots ,\hat{I}{\text{c◯}}_N$ and prove that if, for a given choice of $i,j$, propagation $\hat{I}{\text{c◯}}_i\to \hat{I}{\text{c◯}}_j$ occurs for a graph $D$, then the same holds for any graph $D^{\prime }$ that is sufficiently close to $D$ (robustness under perturbation). We also consider several specific classes of graphs, such as those with a ``reservoir'' type subgraph, and study their intriguing properties.

47 pages, 17 figures