Front propagation in the bistable reaction-diffusion equation on tree-like graphs

Abstract

We deal with the bistable reaction-diffusion equation on an unbounded metric graph consisting of half-lines and triple junctions. It is known that there exists an entire solution which asymptotically converges to the traveling front profile as t→−∞ in the infinity of a half-line. Analyzing the behavior of the entire solution, we examine the condition that the front can propagate beyond the junctions of the graph having a tree-structure and a symmetry. As a result, the condition for blocking of the propagation depends not only a condition at the junctions but also the length between the neighboring junctions, specifically, in a certain case long segments between the two junctions can prevent blocking of the propagation.