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Nonlocal reaction-diffusion equations and nucleation. (English) Zbl 0763.35051
Summary: A nonlocal reaction-diffusion equation is presented and analysed using matched asymptotic expansions and multiple timescales. The problem models a binary mixture undergoing phase separation. The particular form of the equation is motivated by arguments from the calculus of variations, with the nonlocality arising from an enforcement of conservation of mass. It is shown that the evolution of the solution can be characterized by tracking the motion of fronts separating phases. The propagation of the interfaces is found to be a coarsening process which depends in a nonlocal fashion on mean curvature. Several special features of the equations of motion for the fronts are studied, and the relation of this evolution to Cahn-Hilliard theory and nucleation is discussed.

MSC:
35K57 Reaction-diffusion equations
35A15 Variational methods applied to PDEs
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