Cauchy problem and periodic homogenization for nonlocal Hamilton-Jacobi equations with coercive gradient terms

TL;DR

This paper studies the homogenization of nonlocal Hamilton-Jacobi equations with superlinear gradient growth, revealing different limit behaviors based on the nonlocal operator's order and introducing a new comparison principle for viscosity solutions.

Contribution

It introduces a novel homogenization analysis for nonlocal Hamilton-Jacobi equations with superlinear growth, including a new comparison principle for viscosity solutions.

Findings

Different limit problems depending on the nonlocal operator's order

Convergence to the effective equation's solution is established

A new comparison principle for viscosity semi-solutions is developed

Abstract

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton-Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different limit problems. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integro-differential equations that can be of independent interest.