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

• Main Authors: Pierluigi Colli, Giovanni Gilardi, Jürgen Sprekels
• Format: Article
• Language: English
• Published: Accademia Peloritana dei Pericolanti 2020-12-01
• Series: Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali
• Online Access: http://dx.doi.org/10.1478/AAPP.98S2A4

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. More precisely, we study the ω-limit of the phase parameter y and characterize it completely. Our characterization depends on the first eigenvalues λ1 ≥ 0 of one of the operators involved: if λ1 > 0, then the chemical potential μ vanishes at infinity and every element yω of the ω-limit is a stationary solution to the phase equation; if instead λ1 = 0, then every element yω of the ω-limit satisfies a problem containing a real function μ∞ related to the chemical potential μ. Such a function μ∞ is nonunique and time dependent, in general, as we show by an example. However, we give sufficient conditions for μ∞ to be uniquely determined and constant.