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

Abstract

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. 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.