  Recently we considered a stochastic discrete model which describes fronts of
cells invading a wound \cite{KSS}. In the model cells can move, proliferate,
and experience cell-cell adhesion. In this work we focus on a continuum
description of this phenomenon by means of a generalized Cahn-Hilliard equation
(GCH) with a proliferation term. As in the discrete model, there are two
interesting regimes. For subcritical adhesion, there are propagating "pulled"
fronts, similarly to those of Fisher-Kolmogorov equation. The problem of front
velocity selection is examined, and our theoretical predictions are in a good
agreement with a numerical solution of the GCH equation. For supercritical
adhesion, there is a nontrivial transient behavior, where density profile
exhibits a secondary peak. To analyze this regime, we investigated relaxation
dynamics for the Cahn-Hilliard equation without proliferation. We found that
the relaxation process exhibits self-similar behavior. The results of continuum
and discrete models are in a good agreement with each other for the different
regimes we analyzed.
