Phase Fronts in Reaction-Diffusion Problems

Reaction-diffusion equations occur in the description of chemical reactions of reactants which are distributed in space. The motion of a surface along its normal, with a velocity proportional to its mean curvature, is called "flow by curvature" by geometers. Their results on such flows can be applied to the motion of fronts. Those results are primarily for motions in unbounded domains, and the strongest results are for curves embedded in the plane. One part of that lecture concerned the stability of crystals that grow or evaporate by step propagation. The step mechanism of crystal growth was proposed by W. K. Burton, N. Cabrera and F. C. Frank in 1951. It leads to a Stefan-like problem for the motion of a step on a crystal surface. This problem has solutions in which a single step moves at constant velocity, and other solutions in which an infinite number of equally spaced parallel steps move at constant velocity.