  We study properties of the solutions of a family of second order
integro-differential equations, which describe the large scale dynamics of a
class of microscopic phase segregation models with particle conserving
dynamics. We first establish existence and uniqueness as well as some
properties of the instantonic solutions. Then we concentrate on formal
asymptotic (sharp interface) limits. We argue that the obtained interface
evolution laws (a Stefan-like problem and the Mullins-Sekerka solidification
model) coincide with the ones which can be obtained in the analogous limits
from the Cahn-Hilliard equation, the fourth order PDE which is the standard
macroscopic model for phase segregation with one conservation law.
