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We consider, in a $D-$dimensional cylinder, a non--local evolution equationthat describes the evolution of the local magnetization in a continuum limit ofan Ising spin system with Kawasaki dynamics and Kac potentials.  We consider sub--critical temperatures, for which there are two localspatially homogeneous equilibria, and show a local nonlinear stability resultfor the minimum free energy profiles for the magnetization at the interfacebetween regions of these two different local equilibrium; i.e., the planarfronts: We show that an initial perturbation of a front that is sufficientlysmall in $L^2$ norm, and sufficiently localized yields a solution that relaxesto another front, selected by a conservation law, in the $L^1$ norm at analgebraic rate that we explicitly estimate. We also obtain rates for therelaxation in the $L^2$ norm and the rate of decrease of the excess freeenergy.

Stability of planar fronts for a non--local phase kinetics equation with a conservation law in $D \le 3$