  Phase-field models are a popular choice in computational physics to describe
complex dynamics of substances with multiple phases and are widely used in
various applications. We present nonlocal non-isothermal phase-field models of
Cahn-Hilliard and Allen-Cahn types involving a nonsmooth double-well obstacle
potential. Mathematically, in a weak form, the model translates to a system of
variational inequalities coupled to a temperature evolution equation. We
demonstrate that under certain conditions and with a careful choice of the
nonlocal operator one can obtain a model that allows for sharp interfaces in
the solution that evolve in time, which is a desirable property in many
applications. This can be contrasted to the diffuse-interface local models that
can not resolve sharp interfaces. We present the well-posedness analysis of the
models, discuss an appropriate numerical discretization scheme, and supplement
our findings with several numerical experiments.
