# A nonlocal memory strange term arising in the critical scale homogenisation of a diffusion equation with a dynamic boundary condition

## TL;DR

This paper investigates the homogenization of a parabolic diffusion equation with nonlinear dynamic boundary conditions on a periodically perforated domain, revealing a nonlocal memory term in the limit problem.

## Contribution

It introduces a novel homogenization result showing the emergence of a nonlocal memory term as a strange limit in critical scale diffusion problems with dynamic boundaries.

## Key findings

- Weak convergence to a reaction-diffusion problem with a strange term
- The strange term is a nonlocal memory solving an ODE
- The resulting system satisfies a comparison principle

## Abstract

Our main interest in this paper is the study of homogenised limit of a parabolic equation with a nonlinear dynamic boundary condition of the micro-scale model set on a domain with periodically place particles. We focus on the case of particles (or holes) of critical diameter with respect to the period of the structure. Our main result proves the weak convergence of the sequence of solutions of the original problem to the solution of a reaction-diffusion parabolic problem containing a `strange term'. The novelty of our result is that this term is a nonlocal memory solving an ODE. We prove that the resulting system satisfies a comparison principle.