Existence and Nonexistence of Traveling Wave Solutions for a Bistable Reaction-Diffusion Equation in an Infinite Cylinder Whose Diameter is Suddenly Increased

Abstract

ABSTRACT We consider a reaction-diffusion equation of bistable type in a square cylinder whose diameter varies with Neumann boundary conditions in dimension 2 and 3. We prove the nonexistence of generalized traveling wave solution of this equation when the diameter is suddenly strongly increased. At the same time, we prove that the solution of the equation with an exponentially decreasing initial condition cannot pass over a certain threshold far enough in the direction of propagation. The proof is divided in two steps. First, we extend the solution in the cylinder to a solution of the same equation in the half space. Then we overestimate it using Green's functions.