# Homogenization of non-autonomous operators of convolution type in periodic media

## TL;DR

This paper establishes homogenization results for a class of non-autonomous convolution operators with rapidly oscillating periodic coefficients, showing convergence to a second order differential parabolic operator with constant coefficients.

## Contribution

It introduces a homogenization framework for convolution-type operators with oscillating coefficients in both space and time, under diffusive scaling, and proves convergence to a constant-coefficient parabolic operator.

## Key findings

- Homogenization holds for convolution operators with oscillating periodic coefficients.
- The limit operator is a second order differential parabolic operator with constant coefficients.
- The convolution kernel has a finite second moment and the operator is symmetric in spatial variables.

## Abstract

The paper deals with periodic homogenization problem for a para\-bo\-lic equation whose elliptic part is a convolution type operator with rapidly oscillating coefficients. It is assumed that the coefficients are rapidly oscillating periodic functions both in spatial and temporal variables and that the scaling is diffusive that is the scaling factor of the temporal variable is equal to the square of the scaling factor of the spatial variable. Under the assumption that the convolution kernel has a finite second moment and that the operator is symmetric in spatial variables we show that the studied equation admits homogenization and prove that the limit operator is a second order differential parabolic operator with constant coefficients.