# Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion

## TL;DR

This paper proves the nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis on a cylindrical domain, extending previous linear stability results to nonlinear stability for small domain perimeter.

## Contribution

It establishes the nonlinear stability of traveling wave solutions in a chemotaxis model with chemical diffusion on a cylindrical domain, addressing an open problem from prior research.

## Key findings

- Nonlinear stability holds for small domain perimeter.
- Extends linear stability results to nonlinear stability.
- Addresses an open problem in the stability analysis of tumor angiogenesis models.

## Abstract

We consider a simplified chemotaxis model of tumor angiogenesis, described by a Keller-Segel system on the two dimensional infinite cylindrical domain $(x, y) \in \mathbb{R} \times {\mathbf S^{\lambda}}$, where $ \mathbf S^{\lambda}$ is the circle of perimeter $\lambda>0$. The domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located. The system is known to allow planar traveling wave solutions of an invading type. In this paper, we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if $\lambda$ is sufficiently small. The same result for the corresponding system in one-dimension was obtained by Li-Li-Wang (2014) [16]. Our result solves the problem remained open in [3] at which only linear stability of the waves was obtained under certain artificial assumption.