Multi-phase Stefan problems for a non-linear one-dimensional model of cell-to-cell adhesion and diffusion

Abstract

We consider a family of multi-phase Stefan problems for a certain one-dimensional model of cell-to-cell adhesion and diffusion, which takes the form of a non-linear 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.