Uniqueness and nonuniqueness of fronts for degenerate diffusion-convection reaction equations

TL;DR

This paper investigates traveling-wave solutions for a one-dimensional scalar parabolic equation with degenerate diffusion, establishing existence, minimal speed estimates, regularity improvements, and the presence of multiple semi-wavefronts with identical speeds.

Contribution

It provides new existence results, sharp minimal speed estimates, and demonstrates the multiplicity of semi-wavefronts for degenerate diffusion-convection reaction equations.

Findings

Existence of traveling-wave solutions with specific properties.

Sharp estimates for the minimal wave speed.

Existence of infinitely many semi-wavefronts at the same speed.

Abstract

We consider a scalar parabolic equation in one spatial dimension. The equation is constituted by a convective term, a reaction term with one or two equilibria, and a positive diffusivity which can however vanish. We prove the existence and several properties of traveling-wave solutions to such an equation. In particular, we provide a sharp estimate for the minimal speed of the profiles and improve previous results about the regularity of wavefronts. Moreover, we show the existence of an infinite number of semi-wavefronts with the same speed.