Wavefronts for degenerate diffusion-convection reaction equations with sign-changing diffusivity

TL;DR

This paper studies traveling wave solutions for a one-dimensional degenerate diffusion-convection reaction equation with sign-changing diffusivity, revealing existence, profile properties, and sharp behaviors at degeneracy points.

Contribution

It introduces new results on the existence and properties of traveling waves in equations with sign-changing diffusivity, including novel sharp behaviors at degeneracy points.

Findings

Existence of globally defined traveling waves connecting equilibria

Profiles exhibit monotony and sharp behaviors at degeneracy points

New types of sharp behaviors are identified at interior degeneracy points

Abstract

We consider in this paper a diffusion-convection reaction equation in one space dimension. The main assumptions are about the reaction term, which is monostable, and the diffusivity, which changes sign once or twice; then, we deal with a forward-backward parabolic equation. Our main results concern the existence of globally defined traveling waves, which connect two equilibria and cross both regions where the diffusivity is positive and regions where it is negative. We also investigate the monotony of the profiles and show the appearance of sharp behaviours at the points where the diffusivity degenerates. In particular, if such points are interior points, then the sharp behaviours are new and unusual.