Journal of Statistical Physics, Vol. 95, Nos. 5/6, 1999. p. 1119-1139.
We study the existence and stability of stationary solutions of an integro-differential model for phase transitions, which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity. As such, this model is a nonlocal extension of the Allen-Cahn equation, which incorporates long-range interactions. We find that the set of stationary solutions for this model is much larger than that of the Allen-Cahn equation.