Existence, uniqueness and homogenization of nonlinear parabolic problems with dynamical boundary conditions in perforated media

TL;DR

This paper studies a nonlinear parabolic PDE with dynamic boundary conditions in perforated media, proving existence, uniqueness, and deriving a homogenized limit as the perforations vanish.

Contribution

It introduces a nonlinear model with boundary nonlinearity and rigorously derives the homogenized limit problem as perforations disappear.

Findings

Existence and uniqueness of solutions established.

Homogenized nonlinear parabolic problem derived.

New boundary influence terms identified in the limit.

Abstract

We consider a nonlinear parabolic problem with nonlinear dynamical boundary conditions of pure-reactive type in a media perforated by periodically distributed holes of size $\varepsilon$. The novelty of our work is to consider a nonlinear model where the nonlinearity also appears in the boundary. The existence and uniqueness of solution is analyzed. Moreover, passing to the limit when $\varepsilon$ goes to zero, a new nonlinear parabolic problem defined on a unified domain without holes with zero Dirichlet boundary condition and with extra-terms coming from the influence of the nonlinear dynamical boundary conditions is rigorously derived.