On the existence, uniqueness and regularity of solutions to the phase-field transition system with non-homogeneous Cauchy–Neumann and nonlinear dynamic boundary conditions

This paper is devoted to the study of a Caginalp phase-field system endowed with non-homogeneous Cauchy–Neumann and nonlinear dynamic boundary conditions. We first prove the existence, uniqueness and regularity of solutions to an Allen–Cahn equation. Our approach allows to consider in the dynamic boundary conditions a nonlinearity of higher order than in the known results. The existence, uniqueness and regularity of solutions to the Caginalp system in this new formulation is also proven.