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

## TL;DR

This paper investigates a complex doubly nonlinear Cahn-Hilliard system with internal constraints and potential, establishing existence, uniqueness, and asymptotic behavior of bounded solutions with novel methods and regularization limits.

## Contribution

It introduces a new approach to prove existence and uniqueness of solutions for a doubly nonlinear Cahn-Hilliard system with regularizations and singular potentials.

## Key findings

- Proved existence and uniqueness of bounded solutions.
- Established convergence of solutions as regularization parameters vanish.
- Improved upon previous methods for doubly nonlinear Cahn-Hilliard equations.

## Abstract

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 viscosity 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.