A phase-field system, non-local in space and non-smooth in time, with heat flux
proportional to the gradient of the inverse temperature, is shown to admit
a unique strong thermodynamically consistent solution on the whole time axis.
The temperature remains globally bounded both from above and from below,
and its space gradient as well as the time derivative of the order parameter
asymptotically vanish in $L^2$-norm as time tends to infinity.