  In this paper we prove the existence of weak solutions for a
thermodynamically consistent phase-field model introduced in [26] in two and
three dimensions of space. We use a notion of solution inspired by [18], where
the pointwise internal energy balance is replaced by the total energy
inequality complemented with a weak form of the entropy inequality. Moreover,
we prove existence of local-in-time strong solutions and, finally, we show
weak-strong uniqueness of solutions, meaning that every weak solution coincides
with a local strong solution emanating from the same initial data, as long as
the latter exists.
