  In this paper we prove a local-in-time existence theorem for an
initial-boundary value problem related to a model of temperature-dependent
phase segregation that generalizes the standard Allen-Cahn's model. The problem
is ruled by a system of two differential equations, one partial the other
ordinary, interpreted as balances, respectively, of microforces and of
microenergy, complemented by a transcendental condition on the three unknowns,
that are: the order parameter entering the standard A-C equation, the chemical
potential, and the absolute temperature. The results obtained by the authors in
a recent paper and dealing with the isothermal case serve as a starting point
for our existence proof, which relies on a fixed-point argument involving the
Tychonoff-Schauder theorem.
