Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions

We prove well-posedness results for the solution to an initial and boundary-value problem for an Allen–Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain. The dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls in Fischer (1997) [13], Kenzler (2001) [14]. We show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments.