# Longtime behavior for a generalized Cahn-Hilliard system with fractional operators

## TL;DR

This paper analyzes the long-term behavior of solutions to a fractional Cahn-Hilliard system, characterizing the omega-limit set based on the eigenvalues of involved operators, with implications for stationary solutions and chemical potential behavior.

## Contribution

It provides a complete characterization of the omega-limit set for the fractional Cahn-Hilliard system, depending on the first eigenvalue of the operator, extending previous well-posedness results.

## Key findings

- Omega-limit set characterized by eigenvalues
- Chemical potential vanishes when eigenvalue is positive
- Conditions for uniqueness and constancy of related functions

## Abstract

In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn-Hilliard system, with possibly singular potentials, that we have recently investigated in the paper `Well-posedness and regularity for a generalized fractional Cahn-Hilliard system' (see arXiv:1804.11290). More precisely, we study the omega-limit of the phase parameter and characterize it completely. Our characterization depends on the first eigenvalue of one of the operators involved: if it is positive, then the chemical potential vanishes at infinity and every element of the omega-limit is a stationary solution to the phase equation; if, instead, the first eigenvalue is 0, then every element of the omega-limit satisfies a problem containing a real function related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by an example. However, we give sufficient conditions in order that this function be uniquely determined and constant.