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

A phase-field system which describes the evolution of both the absolute temperature θθ and the phase variable ff 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 G2′(f) appears in the energy balance in place of the (customarily constant) latent heat. As a consequence, the bounds 0≤f≤10≤f≤1 hold whenever θθ 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.