WIAS Preprint No. 2588, (2019)

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


  • Colli, Pierluigi
  • Gilardi, Gianni
  • Sprekels, Jürgen

2010 Mathematics Subject Classification

  • 35K45 35K90 35R11 35B40


  • Fractional operators, Cahn--Hilliard systems, longtime behaviour




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, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue 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 it is zero instead, then every element of the Omega-limit solves a problem containing a real function which is related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by means of an example; however, we give sufficient conditions for it to be uniquely determined and constant.

Appeared in

  • Atti Accad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur., 98 (2020), pp. A4/1--A4/18, DOI 10.1478/AAPP.98S2A4 .

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