A new dual approach for a class of phase transitions with memory: existence and long-time behaviour of solutions

In this paper we describe a class of phase transitions with
thermal memory using a dual approach with respect to the
temperature (and the free energy). More precisely, we use as state
variables the phase parameter, the entropy (in place of the
absolute temperature), and the history contribution of the entropy
flux. The equations are recovered from a generalization of the
principle of virtual power (to describe the evolution of the
phases), including the effects of micro-motions responsible for
the phase transition, and a rescalation of the internal energy
balance (to describe the evolution of the entropy). We discuss
thermodynamical
consistency in terms of the properties of the
involved energy functionals: the internal energy (in place of the
free energy) and the pseudo-potential of dissipation. Hence, we
prove existence of a solution (in a proper functional framework)
for the resulting nonlinear integrodifferential PDE system.
Finally, we discuss the long-time behaviour of solutions, characterizing the w-limit of
trajectories.