Longtime behavior for a generalized Cahn-Hilliard system with fractional operators
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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PDFDOI: https://doi.org/10.1478/AAPP.98S2A4
Copyright (c) 2020 Pierluigi Colli, Giovanni Gilardi, Jürgen Sprekels

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