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

TL;DR

This paper establishes well-posedness for an Allen-Cahn equation with singular potentials and dynamic boundary conditions, modeling phase transitions with wall interactions, using regularization and compactness techniques.

Contribution

It provides the first rigorous proof of existence and uniqueness for this class of Allen-Cahn equations with dynamic boundary conditions and singular potentials.

Findings

Proved well-posedness for the problem

Developed regularization techniques for nonlinearities

Established a priori estimates and compactness arguments

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.