Abstract approach of degenerate parabolic equations with dynamic boundary conditions

TL;DR

This paper develops an abstract framework to prove the existence of weak solutions for nonlinear diffusion equations with dynamic boundary conditions, emphasizing the role of mean-zero function spaces and total mass conservation.

Contribution

It introduces an abstract approach based on evolution equations governed by the subdifferential to handle degenerate parabolic equations with dynamic boundary conditions.

Findings

Existence of weak solutions is established.

The approach emphasizes the importance of mean-zero function spaces.

Total mass conservation is incorporated into the analysis.

Abstract

An initial boundary value problem of the nonlinear diffusion equation with a dynamic boundary condition is treated. The existence problem of the initial-boundary value problem is discussed. The main idea of the proof is an abstract approach from the evolution equation governed by the subdifferential. To apply this, the setting of suitable function spaces, more precisely the mean-zero function spaces, is important. In the case of a dynamic boundary condition, the total mass, which is the sum of volumes in the bulk and on the boundary, is a point of emphasis. The existence of a weak solution is proved on this basis.