  A nonisothermal phase field system of Cahn-Hilliard type is introduced and
analyzed mathematically. The system constitutes an extension of the classical
Caginalp model for nonisothermal phase transitions with a conserved order
parameter. It couples a Cahn-Hilliard type equation with source term for the
order parameter with the universal balance law of internal energy. In place of
the standard Fourier form, the constitutive law of the heat flux is assumed in
the form given by the theory developed by Green and Naghdi, which accounts for
a possible thermal memory of the evolution. This has the consequence that the
balance law of internal energy becomes a second-order in time equation for the
thermal displacement or freezing index, that is, a primitive with respect to
time of the temperature. Another particular feature of our system is the
presence of the source term in the equation for the order parameter, which
entails additional mathematical difficulties because the mass conservation of
the order parameter is lost. We provide several mathematical results under
general assumptions on the source term and the double-well nonlinearity
governing the evolution: existence and continuous dependence results are shown
for weak and strong solutions to the corresponding initial-boundary value
problem.
