A reducing mechanism on wave speed for chemotaxis systems with degenerate diffusion

TL;DR

This paper investigates traveling wave solutions in chemotaxis models with degenerate diffusion, revealing that chemotaxis reduces wave speed, a novel finding supported by fixed point and variational methods.

Contribution

It establishes the existence of semi-finite traveling waves in chemotaxis systems with porous medium diffusion and shows chemotaxis slows wave propagation, a new insight in the field.

Findings

Chemotaxis decreases wave speed compared to pure porous medium diffusion.

Existence of sharp and $C^1$ semi-finite traveling waves is proven.

Wave speed is estimated using variational methods.

Abstract

This paper is concerned with traveling wave solutions for a chemotaxis model with degenerate diffusion of porous medium type. We establish the existence of semi-finite traveling waves, including the sharp type and $C^1$ type semi-finite waves. Our results indicate that chemotaxis slows down the wave speed of semi-finite traveling wave, that is, the traveling wave speed for chemotaxis with porous medium (degenerate) diffusion is smaller than that for the porous medium equation without chemotaxis. As we know, this is a new result not shown in the existing literature. The result appears to be a little surprising since chemotaxis is a connective force. We prove our results by the Schauder's fixed point theorem and estimate the wave speed by a variational approach.