Spreading speeds of nonlocal KPP equations in heterogeneous media

TL;DR

This paper establishes the spreading speeds of nonlocal KPP equations in heterogeneous media, using advanced mathematical tools to handle almost periodic and periodic environments with different diffusion properties.

Contribution

It introduces a new theoretical framework combining generalized principal eigenvalues, homogenization, and Harnack inequalities for nonlocal diffusion in complex media.

Findings

Proved existence of spreading speeds in almost periodic media with continuous kernels.

Established spreading speeds in periodic media with weak irreducible diffusion.

Developed new mathematical tools for analyzing nonlocal diffusion in heterogeneous environments.

Abstract

In this paper, we prove the existence of the spreading speed of nonlocal KPP equations in two cases: 1. The media is almost periodic and the kernel of diffusion is continuous; 2. The media is periodic and the diffusion is not continuous but weak irreducible. To do this, we develop the theory of generalized principal eigenvalues, the method homogenization and special Harnack's inequalities of nonlocal diffusion equations.