Kinks and solitons in linear and nonlinear-diffusion Keller-Segel type models with logarithmic sensitivity

TL;DR

This paper explores traveling wave patterns in Keller-Segel models with logarithmic sensitivity, comparing linear and nonlinear diffusion mechanisms, and rigorously establishing the existence of compact support waves.

Contribution

It provides a detailed analysis of wave existence in both linear and nonlinear diffusion regimes, highlighting qualitative differences due to diffusion types.

Findings

Existence of traveling waves with compact support established.

Differences between linear and nonlinear diffusion effects analyzed.

Traveling wave solutions depend on the diffusion mechanism used.

Abstract

This paper investigates the existence of traveling--wave--type patterns in the Keller--Segel model with logarithmic sensitivity. We consider both the linear diffusion case and the nonlinear, flux-saturated diffusion of relativistic heat--equation type, providing a detailed comparison between the two regimes. Particular attention is devoted to traveling waves exhibiting compact support or support restricted to a half-line. We rigorously establish the existence of such patterns and highlight the qualitative differences arising from the choice of diffusion mechanism.