Existence and uniqueness of a global-in-time solution to a phase segregation problem of the Allen-Cahn type

Abstract. We study a model of phase segregation of the Allen-Cahn type, consisting in a system of two differential equations, one partial the other ordinary, respectively interpreted as balances of microforces and microenergy; the two unknowns are the order parameter entering the standard A-C equation and the chemical potential. We introduce a notion of maximal solution to the o.d.e., parameterized on the order-parameter field; and, by substitution in the p.d.e. of the so-obtained chemical potential field, we give the latter equation the form of an Allen-Cahn equation for the order parameter, with a memory term. Finally, we prove existence and uniqueness of global-in-time smooth solutions to this modified A-C equation, and we give a description of the relative omega limit set.