Traveling waves for a Fisher-type reaction-diffusion equation with a flux in divergence form

TL;DR

This paper investigates the propagation speeds of traveling waves in Fisher-type reaction-diffusion equations with divergence form flows, revealing a range of speeds and the existence of a singular minimal speed under various flow conditions.

Contribution

It introduces a comprehensive analysis of wave speeds in Fisher-type equations with non-elliptic flows, including the concept of a singular minimal speed and the construction of corresponding traveling wave profiles.

Findings

The wave speed range is an unbounded interval to the right.

The infimum of wave speeds may not be attained in non-elliptic flows.

A singular minimal speed exists, justified as a viscous limit.

Abstract

Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction-diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic traveling waves move is an interval unbounded to the right. Contrary to classic examples, the infimum may not be reached. When the flow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic traveling wave speed threshold is complemented by another value by analyzing an extension of the first order boundary value problem to which the classic case is reduced. This singular minimum speed can be justified as a viscous limit of classic minimal speeds in elliptic or over-elliptic flows. We construct a singular profile for each speed between the minimum singular speed and the speeds at which classic traveling waves move. Under additional assumptions, the constructed profile can be justified as that of a traveling wave of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine-Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.