Front Profile in Time Backward for the Bistable Reaction-Diffusion Equation on Metric Graphs

In the bistable reaction-diffusion equation on a half-line there exists an entire solution exhibiting the front propagation from the infinity of the line. We glue an arbitrary number of half-lines at a single point (junction) to build a metric graph and examine the front propagation on given half-lines (upstream branches). The aim of the article is to show the existence of an entire solution which converges to the profile of the traveling wave with an arbitrarily assigned phase-shift in each upstream branch as t -> -infinity. The proof is carried out by constructing an appropriate pair of sub-and super-solutions.