Infinitely many saturated travelling waves for a degenerate Fisher-KPP equation not in divergence form

TL;DR

This paper studies a degenerate Fisher-KPP equation arising from an epidemic model with distributed contacts, revealing infinitely many traveling wave solutions, including both smooth and sharp types, with diverse tail behaviors, contrasting previous models.

Contribution

It provides an exhaustive analysis of traveling waves for a degenerate Fisher-KPP equation not in divergence form, showing the existence of infinitely many saturated waves for each admissible speed.

Findings

Existence of a unique smooth wave for each speed.

Existence of infinitely many saturated (sharp) waves.

Diverse tail behaviors of the traveling waves.

Abstract

We consider an epidemic model with distributed-contacts. When the contact kernel concentrates, one formally reaches a very degenerate Fisher-KPP equation with a diffusion term that is not in divergence form. We make an exhaustive study of its travelling waves. For every admissible speed, there exist not only a unique non-saturated (smooth) wave but also infinitely many saturated (sharp) ones. Furthermore their tails may differ from what is usually expected. These results are thus in sharp contrast with their counterparts on related models.