On the existence, uniqueness and regularity of solutions to the phase-field system with a general regular potential and a general class of nonlinear and non-homogeneous boundary conditions

This paper studies a Caginalp phase-field transition system endowed with a general regular potential, as well as a general class, in both unknown functions, of nonlinear and non-homogeneous (depending on time and space variables) boundary conditions. We first prove the existence, uniqueness and regularity of solutions to the Allen–Cahn equation, subject to the nonlinear and non-homogeneous dynamic boundary conditions. The existence, uniqueness and regularity of solutions to the Caginalp system in this new formulation are also proved. This extends previous works concerned with regular potential and nonlinear boundary conditions, allowing the present mathematical model to better approximate the real physical phenomena, especially phase transitions.