On an evolution equation in a cell motility model

TL;DR

This paper studies a complex evolution equation modeling cell membrane dynamics derived from reaction-diffusion systems, establishing existence of solutions for various initial conditions using regularization and maximum principle techniques.

Contribution

It proves the existence of solutions for a non-linear, non-local evolution equation in a cell motility model, including non-smooth initial data.

Findings

Existence of solutions for smooth initial data using regularization.

Existence of solutions for non-smooth initial data with maximum principle estimates.

Addresses analytical challenges of a non-linear, non-local evolution equation.

Abstract

This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth ($H^2$) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solution exists rely on a maximum principle type argument.