Front propagation on a general metric graph

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 $\mathrm{\Omega }$ consists of a bounded finite metric graph $D$ of arbitrary configuration and a finite number of branches ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_N(N\ge 2)$ of infinite length emanating from some of the vertices of $D$. Each ${\mathrm{\Omega }}_i(i=1,\dots ,N)$ is called an outer path''. Our goal is to investigate the behavior of the front coming from infinity along a given outer path $Ω_i$ and to discuss whether or not the front propagates into other outer paths $Ω_j\,(j\ne i)$. 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 ${\mathrm{\Omega }}_i\to {\mathrm{\Omega }}_j$ and ${\mathrm{\Omega }}_j\to {\mathrm{\Omega }}_k$ imply propagation ${\mathrm{\Omega }}_i\to {\mathrm{\Omega }}_k$. Next we consider perturbations of the graph $D$ while fixing the outer paths ${\mathrm{\Omega }}_1,\dots ,{\mathrm{\Omega }}_N$ and prove that if, for a given choice of $i,j$, propagation ${\mathrm{\Omega }}_i\to {\mathrm{\Omega }}_j$ occurs for a graph $D$, then the same holds for any graph $D^{\mathrm{\prime }}$ 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.