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This paper investigates a nonlocal version of a model for phase separation onan 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 anonlinearly coupled system of two partial differential equations governing theevolution of an order parameter and the chemical potential. Singularcontributions to the local free energy in the form of logarithmic ordouble-obstacle potentials are admitted. In contrast to the local model, whichwas studied by P. Podio-Guidugli and the present authors in a series of recentpublications, in the nonlocal case the equation governing the evolution of theorder parameter contains in place of the Laplacian a nonlocal expression thatoriginates from nonlocal contributions to the free energy and accounts forpossible long-range interactions between the atoms. It is shown that just as inthe local case the model equations are well posed, where the technique ofproving existence is entirely different: it is based on an application ofTikhonov's fixed point theorem in a rather unusual separable and reflexiveBanach space.

On an application of Tikhonov's fixed point theorem to a nonlocal Cahn-Hilliard type system modeling phase separation