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 $\Omega$ consists of a bounded finite metric graph $D$ of arbitrary configuration and a finite number of branches $\Omega_1,\ldots,\Omega_N\,(N\geq 2)$ of infinite length emanating from some of the vertices of $D$. Each $\Omega_i\,(i=1,\ldots,N)$ is called an ``outer path''. Our goal is to investigate the behavior of the front coming from infinity along a given outer path $\Omega_i$ and to discuss whether or not the front propagates into other outer paths $\Omega_j\,(j\ne i)$. 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 $\Omega_i\to \Omega_j$ and $\Omega_j\to \Omega_k$ imply propagation $\Omega_i\to \Omega_k$. Next we consider perturbations of the graph $D$ while fixing the outer paths $\Omega_1,\ldots,\Omega_N$ and prove that if, for a given choice of $i,j$, propagation $\Omega_i\to \Omega_j$ occurs for a graph $D$, then the same holds for any graph $D'$ 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.