On a generalized Cahn-Hilliardequation with biological applications

In this paper, we are interested in the study of the asymptotic behavior of a generalization of the Cahn-Hilliard equation with a proliferation term andendowed with Neumann boundary conditions. Such a model has, in particular,applications in biology. We show that either the average of the local density of cells is bounded, in which case we have a global in timesolution, or the solution blows up in finite time.We further prove that the relevant, from a biological point of view, solutions converge to $1$ as time goes to infinity.We finally give some numerical simulations which confirm the theoretical results.