Traveling waves for the Keller-Segel-FKPP equation with strong chemotaxis

2023/04/20 by Christopher Henderson, Henderson, Christopher, Maximilian Rezek +1
Mathematics · Medicine · Biochemistry, Genetics and Molecular Biology · #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models #Evolution and Genetic Dynamics

paper · pdf · doi:10.48550/arxiv.2304.10024

Abstract

We show that there exist traveling wave solutions of the Keller-Segel-FKPP equation, which models a diffusing and logistically growing population subject to chemotaxis. In contrast to previous results, our result is in the strong aggregation regime; that is, we make no smallness assumption on the parameters. The lack of a smallness condition makes L^∞-estimates difficult to obtain as the comparison principle no longer gives them ``for free." Instead, our proof is based on suitable energy estimates in a carefully tailored uniformly local L^p-space. Interestingly, our uniformly local space involves a scaling parameter, the choice of which is a crux of the argument. Numerical experiments exploring the stability, qualitative properties, and speeds of these waves are presented as well.

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