The Cahn-Hilliard Equation with Singular Potentials and Dynamic Boundary Conditions

TL;DR

This paper investigates the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions, establishing existence, uniqueness, and long-term behavior of solutions through approximation and variational methods.

Contribution

It introduces a novel approach to handle singular potentials in the Cahn-Hilliard equation with dynamic boundary conditions, proving existence, uniqueness, and attractor properties.

Findings

Existence and uniqueness of solutions established.

Solutions are shown to separate from singularities.

Global and exponential attractors are proven to exist.

Abstract

Our aim in this paper is to study the Cahn-Hilliard equation with singular potentials and dynamic boundary conditions. In particular, we prove, owing to proper approximations of the singular potential and a suitable notion of variational solutions, the existence and uniqueness of solutions. We also discuss the separation of the solutions from the singularities of the potential. Finally, we prove the existence of global and exponential attractors.