Multi-phase Stefan problems for a nonlinear 1-d model of cell-to-cell adhesion and diffusion

TL;DR

This paper develops an existence theory for multi-phase Stefan problems modeling cell adhesion and diffusion, addressing phase annihilation, solution continuation, and long-term stability with numerical support.

Contribution

It introduces a novel existence framework for nonlinear multi-phase Stefan problems with phase annihilation and long-term stability analysis.

Findings

Existence of solutions with phase annihilation

Long-time stability results for solutions

Numerical simulations supporting theoretical findings

Abstract

We consider a family of multi-phase Stefan problems for a certain 1-d model of cell-to-cell adhesion and diffusion, which takes the form of a nonlinear forward-backward parabolic equation. In each material phase the cell density stays either high or low, and phases are connected by jumps across an `unstable' interval. We develop an existence theory for such problems which allows for the annihilation of phases and the subsequent continuation of solutions. Stability results for the long-time behaviour of solutions are also obtained, and, where necessary, the analysis is complemented by numerical simulations.