Multiplicity of supercritical fronts for reaction-diffusion equations in cylinders

TL;DR

This paper investigates the existence and uniqueness of supercritical traveling front solutions in reaction-diffusion equations within cylinders, revealing that only certain solutions exist beyond a critical speed, with some additional solutions possible in specific speed ranges.

Contribution

It provides a variational framework to characterize and distinguish between unique and multiple traveling front solutions in reaction-diffusion models in cylinders.

Findings

Traveling fronts exist for all speeds above a critical value.

Uniqueness of solutions is established for sufficiently large speeds.

Additional solutions may exist within a specific speed interval.

Abstract

We study multiplicity of the supercritical traveling front solutions for scalar reaction-diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions connecting an unstable equilibrium to the closest stable equilibrium for all speeds exceeding a critical value. We show that these are, in fact, the only traveling front solutions in the considered problems for sufficiently large speeds. In addition, we show that other traveling fronts connecting to the unstable equilibrium may exist in a certain range of the wave speed. These results are obtained with the help of a variational characterization of such solutions.