Space-time homogenization problems for porous medium equations with nonnegative initial data

TL;DR

This paper investigates the space-time homogenization of porous medium equations with nonnegative initial data, characterizing the homogenized matrix through cell problems and overcoming previous growth restrictions with new gradient estimates.

Contribution

It introduces a novel approach to handle growth restrictions in homogenization of porous medium equations by developing local uniform gradient estimates.

Findings

Homogenized matrix characterized via cell problems.

Overcomes growth restrictions in previous models.

Develops new gradient estimates for solutions.

Abstract

This paper concerns a space-time homogenization limit of nonnegative weak solutions to porous medium equations. In particular, the so-called homogenized matrix will be characterized in terms of solutions to cell problems, which drastically vary in a scaling parameter $r > 0$. A similar problem has already been studied in [1], where the growth of the power nonlinearity is strictly restricted due to some substantial obstacles. In the present paper, such obstacles will be overcome by developing local uniform estimates for the gradients of nonnegative weak solutions.