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 y_ω of the ω-limit is a stationary solution to the phase equation; if instead λ₁=0, then every element y_ω 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.