Homogenization of parabolic problems for non-local convolution type operators under non-diffusive scaling of coefficients

TL;DR

This paper investigates the homogenization of non-autonomous parabolic equations with non-symmetric convolution operators under non-diffusive scaling, revealing decoupling of space and time in the asymptotic limit.

Contribution

It introduces a homogenization framework for non-local convolution operators with non-symmetric kernels under non-diffusive scaling, showing solution decoupling in a moving frame.

Findings

Solutions exhibit decoupling of space and time asymptotically.

Homogenization holds in a moving frame despite non-symmetric kernels.

Non-diffusive scaling leads to distinct asymptotic behavior.

Abstract

We study homogenization problem for non-autonomous parabolic equations of the form $\partial_t u=L(t)u$ with an integral convolution type operator $L(t)$ that has a non-symmetric jump kernel which is periodic in spatial variables and in time. It is assumed that the space-time scaling of the environment is not diffusive. We show that asymptotically the spatial and temporal evolutions of the solutions are getting decoupled, and the homogenization result holds in a moving frame.