A variational approach to a Cahn-Hillliard model in a domain with nonpermeable walls.

This paper deals with the well-posedness and the long time behavior of a Cahn-Hilliard model with a singular bulk potential and suitable dynamic boundary conditions. We assume here that the system is confined in a vessel with non-permeable walls and that the total mass, in the bulk and on the boundary, is conserved. As a result, the well-posedness in the sense of distributions may not hold and new notions of solutions are required. The same problem has been analyzed in \cite{Gold}, relying on duality techniques, under weak assumptions on the nonlinearities. However, the regularity of solutions and the study of the asymptotic behavior of the system required growth restrictions on the bulk nonlinearity which exclude the thermodynamically relevant logarithmic potentials. Our aim in this paper is to improve these results by introducing instead, in the spirit of [23], a variational formulation of the problem, based on a proper variational inequality.