  The Cahn-Hilliard equation is a fundamental model that describes phase
separation processes of binary mixtures. In this paper we focus on the dynamics
of these binary media, when the underlying temperature is not constant. The aim
of this paper is twofold. We first derive two distinct models that extend the
classical Cahn-Hilliard equation with an evolutionary equation for the absolute
temperature. Secondly, we analyse the local well-posedness of classical
solutions for one of these systems. Our modelling introduces the systems of
PDEs by means of a general and unified formalism. This formalism couples
standard principles of mechanics together with the main laws of thermodynamics.
Our work highlights how certain assumptions on the transport of the temperature
effect the overall physics of the systems. The variety of these
thermodynamically consistent models opens the question of which one should be
more appropriate. Our analysis shows that one of the derived models might be
more desirable to the well-posedness theory of classical solutions, a property
that might be natural as a selection criteria.
