Standing Waves of Reaction-Diffusion Equations on an Unbounded Graph with Two Vertices

We deal with bistable reaction-diffusion equations in a domain of a metric graph with two vertices, that is, a domain of multiple half-lines with two junctions connected by a line segment. We prove that there exist two types of standing waves: standing front waves and unimodal waves, if the line segment is long enough. We also numerically show the exact number of standing wave solutions for a cubic nonlinearity and for a piecewise linear case. The stability and instability of the standing wave solutions are also investigated. Standing waves play a role in blocking the front propagation.