# Bifurcation of traveling waves in a Keller-Segel type free boundary model of cell motility

## TL;DR

This paper analyzes a complex free boundary model of cell motility, revealing bifurcations of traveling waves and steady states driven by nonlinear coupled PDEs with curvature effects.

## Contribution

It introduces a novel bifurcation analysis of traveling waves in a Keller-Segel type free boundary model of cell motility, including existence proofs for non-radial steady states.

## Key findings

- Existence of bifurcating traveling wave solutions.
- Presence of non-radial steady states.
- Application of Leray-Schauder degree theory in free boundary problems.

## Abstract

We study a two-dimensional free boundary problem that models motility of eukaryotic cells on substrates. This problem consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. The two key properties of this problem are (i) presence of the cross diffusion as in the classical Keller-Segel problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from a family of radially symmetric steady states. The traveling waves describe persistent motion without external cues or stimuli which is a signature of cell motility. We also prove existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a free boundary setting.