Superabundance of stationary solutions for the discrete Allen-Cahn equation (Q2731401)

The author considers a discrete analogue of the Allen-Cahn equation, a parabolic partial differential equation proposed as a simple model for phase separation in materials. The paper illustrates that, in some sense, the solutions of the discrete equation display a richer variety of behaviors than the corresponding solutions of the continuous model. In particular, the number of stationary solutions of the two equations may not agree, even when the lattice of the discrete equation is extremely fine. The author studies this phenomenon and its implications in areas such as dynamical metastability.