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

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels

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. 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.

Keywords


Fractional operators, Cahn--Hilliard systems, longtime behavior

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DOI: https://doi.org/10.1478/AAPP.98S2A4

Copyright (c) 2020 Pierluigi Colli, Giovanni Gilardi, Jürgen Sprekels

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