Asymptotic Behavior of a Nonlocal KPP Equation with an Almost Periodic Nonlinearity

TL;DR

This paper studies the long-term behavior of a nonlocal KPP equation with almost periodic nonlinearity, showing solutions converge to stationary states separated by a front described by a Hamilton-Jacobi inequality.

Contribution

It introduces a homogenization approach to analyze the asymptotic behavior of a nonlocal KPP equation with almost periodic nonlinearity, linking it to Hamilton-Jacobi theory.

Findings

Solutions asymptotically converge to stationary states.

The front location is characterized by a Hamilton-Jacobi variational inequality.

Homogenization techniques are effectively applied to nonlocal equations.

Abstract

We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and an almost-periodic nonlinearity. By employing and adapting the theory of homogenization, we show that solutions of this equation asymptotically converge to its stationary states in regions of space separated by a front that is determined by a Hamilton-Jacobi variational inequality.