Time periodic solutions of {C}ahn--{H}illiard system with dynamic boundary conditions

TL;DR

This paper proves the existence of time-periodic weak solutions for the Cahn--Hilliard system with dynamic boundary conditions using an abstract evolution equation approach, fixed point theorem, and monotone operator theory.

Contribution

It introduces a novel application of viscosity and fixed point methods to establish existence results for time-periodic solutions of the Cahn--Hilliard system with dynamic boundary conditions.

Findings

Existence of weak solutions under certain conditions.

Application of Schauder fixed point theorem in this context.

Use of maximal monotone graph assumptions for the analysis.

Abstract

The existence problem for {C}ahn--{H}illiard system with dynamic boundary conditions and time periodic conditions is discussed. We apply the abstract theory of evolution equations by using viscosity approach and the Schauder fixed point theorem in the level of approximate ploblem. One of the key point is the assumption for maximal monotone graphs with respect to their domains. Thanks to this, we obtain the existence result of the weak solution by using the passage to the limit.