Traveling waves to a logarithmic chemotaxis model with fast diffusion and singularities

TL;DR

This paper investigates a chemotaxis model with logarithmic sensitivity and fast diffusion, establishing the existence and asymptotic stability of traveling waves despite strong singularities, supported by novel weighted energy techniques and numerical simulations.

Contribution

It introduces new methods to handle singularities in chemotaxis models and demonstrates the stability and structure of traveling waves with fast diffusion effects.

Findings

Existence of traveling waves connecting singular zero-end states.

Asymptotic stability of these traveling waves.

Fast diffusion causes traveling waves to become steeper, resembling shock waves.

Abstract

This paper is concerned with a chemotaxis model with logarithmic sensitivity and fast diffusion, which possesses strong singularities for the sensitivity at zero-concentration of chemical signal, and for the diffusion at zero-population of cells, respectively. The main purpose is to show the existence of traveling waves connecting the singular zero-end-state, and particularly, to show the asymptotic stability of these traveling waves. The challenge of the problem is the interaction of two kinds of singularities involved in the model: one is the logarithmic singularity of the sensitivity; and the other is the power-law singularity of the diffusivity. To overcome the singularities for the wave stability, some new techniques of weighted energy method are introduced artfully. Numerical simulations are also carried out, which further confirm our theoretical stability results, in particular, the numerical results indicate that the effect of fast diffusion to the structure of traveling waves is essential, which causes the traveling waves much steeper like shock waves. This new phenomenon is a first observation.