On an application of Tikhonov's fixed point theorem to a nonlocal Cahn--Hilliard type system modeling phase separation
- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 35K40 35K86 45K05 47H10 80A22
- Cahn-Hilliard system, nonlocal energy, phase separation, singular potentials, initial-boundary value problem, Tikhonov's fixed point theorem
This paper investigates a nonlocal version of a model for phase separation on an atomic lattice that was introduced by P. Podio-Guidugli in Ric. Mat. 55 (2006) 105-118. The model consists of an initial-boundary value problem for a nonlinearly coupled system of two partial differential equations governing the evolution of an order parameter ρ and the chemical potential μ. Singular contributions to the local free energy in the form of logarithmic or double-obstacle potentials are admitted. In contrast to the local model, which was studied by P. Podio-Guidugli and the present authors in a series of recent publications, in the nonlocal case the equation governing the evolution of the order parameter contains in place of the Laplacian a nonlocal expression that originates from nonlocal contributions to the free energy and accounts for possible long-range interactions between the atoms. It is shown that just as in the local case the model equations are well posed, where the technique of proving existence is entirely different: it is based on an application of Tikhonov's fixed point theorem in a rather unusual separable and reflexive Banach space.
- J. Differential Equations, 260 (2016), pp. 7940--7964.