Well-posedness for solid-liquid phase transitions with a forth-order nonlinearity

A phase-field system which describes the evolution of both the absolute temperature $\theta$ and the phase variable $f$ during first-order transitions in thermal insulators is considered. A thermodynamic approach is developed by regarding the order parameter as a phase field and its evolution equation as a balance law. By virtue of the special form of the internal energy, a third-order nonlinearity $G_2^\prime(f)$ appears into the energy balance in place of the (customary constant) latent-heat. As a consequence, the bounds $0\le f\le1$ hold true whenever $\theta$ is positive valued. In addition, a nonlinear Fourier law with conductivity proportional to temperature is assumed. Well-posedness for the resulting initial and boundary value problem are then established in a suitable setting.