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 λ 1 ≥ 0 of one of the operators involved: if λ 1 > 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 λ 1 = 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.

Asymptotic analysis of a tumor growth model with fractional operators

“…Notice also that in the case P ≡ 0 the equation (1.3) decouples from the other two equations (1.1), (1.2); the latter system of equations has for the case α = 0 recently been the subject of a series of investigations by the present authors (cf. the papers [15][16][17][18]).…”

Well-posedness and regularity for a fractional tumor growth model

“…For a review of some related work, let us refer the interested reader to our recent papers [14,15] and [8], which offer a recapitulation of various contributions. In our approach here, which follows closely the setting used in [8,[14][15][16], we deal with fractional operators defined via spectral theory. Then we can easily consider powers of a second-order elliptic operator with either Dirichlet or Neumann or Robin homogeneous boundary conditions, as well as other operators like, e.g., fourth-order ones or systems involving the Stokes operator.…”