Entire solutions to reaction-diffusion equations in multiple half-lines with a junction

Abstract

There exists a traveling front wave to a bistable reaction-diffusion equation in a whole line under a certain condition of reaction term f(u). We deal with the bistable reaction-diffusion equation with the same f(u) in a domain Ω which is a graph of special type, that is, a union of half-lines starting at a common point, so the domain has a unique junction of the half-lines. The aim of our study is to show the existence of nontrivial entire solutions, which are classical solutions defined for all (x,t)∈Ω×R. We prove that there are entire solutions which converge to the front waves in some of half-lines and converge to zero in the remaining half-lines as t→−∞. We also give a condition under that the entire solutions exhibit the blocking of the front propagation. This blocking is caused by the emergence of stationary solutions. The stability/instability of the stationary solutions are proved.