Homogenisation of Monotone Parabolic Problems with Several Temporal Scales: The Detailed arXiv e-Print Version

TL;DR

This paper develops a homogenisation framework for monotone parabolic equations with multiple spatial and temporal scales, identifying distinct local problems under scale separation assumptions.

Contribution

It introduces a detailed homogenisation approach for problems with several temporal and spatial scales, including self-similar cases, under well-separatedness conditions.

Findings

Existence of an H-limit characterized by up to four local problems.

Identification of different regimes based on temporal oscillation speed.

Framework applicable to complex multiscale parabolic problems.

Abstract

In this paper we homogenise monotone parabolic problems with two spatial scales and finitely many temporal scales. Under a certain well-separatedness assumption on the spatial and temporal scales as explained in the paper, we show that there is an H-limit defined by at most four distinct sets of local problems corresponding to slow temporal oscillations, slow resonant spatial and temporal oscillations (the "slow" self-similar case), rapid temporal oscillations, and rapid resonant spatial and temporal oscillations (the "rapid" self-similar case), respectively.