On the Cahn-Hilliard equation with mass source for biological applications

On a Generalized Cahn-hilliard Equation 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 and endowed 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 time solution, or the solution blows...

The Cahn-hilliard Equation

1. Steady states. There are many two component systems in which phase separation can be induced by rapidly cooling the system. Thus, if a two component system, which is spatially uniform at temperature T1, is rapidly cooled to a second sufficiently lower temperature T2, then the cooled system will separate into regions of higher and lower concentration. A phenomenological description of the beh...

The Cahn–Hilliard Equation

The convective Cahn-Hilliard equation

We consider the convective Cahn–Hilliard equation with periodic boundary conditions as an infinite dimensional dynamical system and establish the existence of a compact attractor and a finite dimensional inertial manifold that contains it. Moreover, Gevrey regularity of solutions on the attractor is established and used to prove that four nodes are determining for each solution on the attractor...

The critical mass constraint in the Cahn-Hilliard equation

When the mass constraint of the Cahn-Hilliard equation in two dimensions is lowered to the order of ǫ, where ǫ is the interface thickness parameter, the existence of droplet solutions becomes conditional. For interior single droplet solutions, there is a critical value for the mass constraint such that above this value two interior single droplet solutions exist, and below this value interior s...