Diffusion-convection reaction equations with sign-changing diffusivity and bistable reaction term

TL;DR

This paper investigates wavefront solutions in a reaction-diffusion equation with sign-changing diffusivity and bistable reaction, revealing new phenomena such as multiple wave speeds, loss of uniqueness, and singular profiles.

Contribution

It introduces analysis of wavefronts in equations with sign-changing diffusivity and bistable reactions, highlighting novel behaviors like multiple speeds and singularities.

Findings

Profiles can occur for a single speed or a bounded interval of speeds

Uniqueness of wavefronts is lost in this setting

Profiles can have singularities where diffusion vanishes

Abstract

We consider a reaction-diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.