Weak solutions to a phase-field model with non-constant thermal conductivity

We investigate the existence of weak solutions to a phase-field model when the thermal conductivity vanishes for some values of the order parameter. We obtain weak solutions for a general class of free energies, including non-differentiable ones. We also study the omega-limit set of these weak solutions, and investigate their convergence to a solution of a degenerate Cahn-Hilliard equation.