Bounded solutions and their asymptotics for a doubly nonlinear Cahn–Hilliard system

In this paper we deal with a doubly nonlinear Cahn–Hilliard system, where both an internal constraint on the time derivative of the concentration and a potential for the concentration are introduced. The definition of the chemical potential includes two regularizations: a viscous and a diffusive term. First of all, we prove existence and uniqueness of a bounded solution to the system using a nonstandard maximum-principle argument for time-discretizations of doubly nonlinear equations. Possibly including singular potentials, this novel result brings improvements over previous approaches to this problem. Secondly, under suitable assumptions on the data, we show the convergence of solutions to the respective limit problems once either of the two regularization parameters vanishes.