Existence and uniqueness of solutions to singular Cahn-Hilliard equations with nonlinear viscosity terms and dynamic boundary conditions

TL;DR

This paper establishes the global existence and uniqueness of solutions for a highly nonlinear Cahn-Hilliard system with dynamic boundary conditions, using approximation and monotonicity methods.

Contribution

It introduces a novel approach to prove existence and uniqueness for a complex nonlinear Cahn-Hilliard model with boundary dynamics.

Findings

Proved global existence and uniqueness of solutions.

Analyzed asymptotic behavior as boundary diffusion vanishes.

Developed an approximation scheme based on doubly nonlinear evolution theory.

Abstract

We prove global existence and uniqueness of solutions to a Cahn-Hilliard system with nonlinear viscosity terms and nonlinear dynamic boundary conditions. The problem is highly nonlinear, characterized by four nonlinearities and two separate diffusive terms, all acting in the interior of the domain or on its boundary. Through a suitable approximation of the problem based on abstract theory of doubly nonlinear evolution equations, existence and uniqueness of solutions are proved using compactness and monotonicity arguments. The asymptotic behaviour of the solutions as the the diffusion operator on the boundary vanishes is also shown.