Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain

TL;DR

This paper investigates the stability of planar traveling waves in a Keller-Segel model on a two-dimensional domain, establishing nonlinear stability without diffusion and linear stability with diffusion, using transformations and inequalities.

Contribution

It extends stability analysis of Keller-Segel equations to a 2D domain with small perimeter, providing new nonlinear stability results for invading traveling waves.

Findings

Nonlinear stability of traveling waves without chemical diffusion.

Linear stability when diffusion is present.

Solutions tend to become planar over time under certain conditions.

Abstract

A simplified model of the tumor angiogenesis can be described by a Keller-Segel equation \cite{FrTe,Le,Pe}. The stability of traveling waves for the one dimensional system has recently been known by \cite{JinLiWa,LiWa}. In this paper we consider the equation on the two dimensional domain $ (x, y) \in \mathbf R \times {\mathbf S^{\lambda}}$ for a small parameter $\lambda>0$ where $ \mathbf S^{\lambda}$ is the circle of perimeter $\lambda$. Then the equation allows a planar traveling wave solution of invading types. We establish the nonlinear stability of the traveling wave solution if the initial perturbation is sufficiently small in a weighted Sobolev space without a chemical diffusion. When the diffusion is present, we show a linear stability. Lastly, we prove that any solution with our front conditions eventually becomes planar under certain regularity conditions. The key ideas are to use the Cole-Hopf transformation and to apply the Poincar\'{e} inequality to handle with the two dimensional structure.