Longtime behavior for a generalized Cahn-Hilliard system with fractional operators
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 pr...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Accademia Peloritana dei Pericolanti
2020-12-01
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| Series: | Atti della Accademia Peloritana dei Pericolanti : Classe di Scienze Fisiche, Matematiche e Naturali |
| Online Access: |
http://dx.doi.org/10.1478/AAPP.98S2A4
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| Summary: | 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. |
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| ISSN: | 0365-0359 1825-1242 |