  In this paper, we prove the existence and global boundedness from above for a
solution to an integrodifferential model for nonisothermal multi-phase
transitions under nonhomogeneous third type boundary conditions. The system
couples a quasilinear internal energy balance ruling the evolution of the
absolute temperature with a vectorial integro-differential inclusion governing
the (vectorial) phase-parameter dynamics. The specific heat and the heat
conductivity are allowed to depend both on the order parameter and on the
absolute temperature of the system, and the convex component of the free energy
may or may not be singular. Uniqueness and continuous data dependence are also
proved under additional assumptions.
