Existence and Instability of Traveling Pulses of Keller-Segel System with Nonlinear Chemical Gradients and Small Diffusions

TL;DR

This paper investigates the existence and instability of traveling pulse solutions in a generalized Keller-Segel model with nonlinear chemical gradients and small cell diffusion, using geometric singular perturbation and spectral analysis.

Contribution

It establishes the existence of traveling pulses via geometric singular perturbation and analyzes their linear instability through spectral methods, extending understanding of Keller-Segel dynamics.

Findings

Traveling pulses exist under certain conditions.

Necessary conditions involve slow flows on the critical manifold.

Traveling pulses are linearly unstable.

Abstract

In this paper, we consider a generalized model of $2\times 2$ Keller-Segel system with nonlinear chemical gradient and small cell diffusion. The existence of the traveling pulses for such equations is established by the methods of geometric singular perturbation (GSP in short) and trapping regions from dynamical systems theory. By the technique of GSP, we show that the necessary condition for the existence of traveling pulses is that their limiting profiles with vanishing diffusion can only consist of the slow flows on the critical manifold of the corresponding algebraic-differential system. We also consider the linear instability of these pulses by the spectral analysis of the linearized operators.