Shock wavefronts for parabolic equations with sign-changing diffusivity

TL;DR

This paper studies reaction-diffusion equations with sign-changing diffusivity, proving the existence of shock wavefronts with discontinuous profiles and analyzing their properties and speeds.

Contribution

It introduces the concept of shock wavefronts in models with negative diffusivity regions, expanding understanding beyond continuous wavefront solutions.

Findings

Existence of shock wavefronts with jump discontinuities.

Profiles and speeds of these shock wavefronts are characterized.

Application to population movement models with mixed individual states.

Abstract

We consider a reaction-diffusion equation in a one-dimensional space, where the diffusion coefficient changes sign from positive to negative and back to positive. The reaction term is bistable, with its interior zero located in the region where the diffusivity is negative. The model does not admit continuous wavefronts, i.e., continuous traveling waves that connect the steady states $0$ and $1$. We prove the existence of a family of shock wavefronts, that is, wavefronts with profiles exhibiting a jump discontinuity. We investigate the properties of these profiles and their propagation speeds. Finally, we apply the results to a recently proposed model describing the movement of a population composed of both isolated and grouped individuals.