Asymptotic Behavior of a Nonlocal KPP Equation with a Stationary Ergodic Nonlinearity

TL;DR

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

Contribution

It introduces a stochastic homogenization approach to analyze the asymptotic behavior of nonlocal KPP equations with ergodic nonlinearities.

Findings

Solutions asymptotically approach stationary states

The front location is characterized by a Hamilton-Jacobi inequality

The approach extends homogenization techniques to nonlocal equations

Abstract

We consider a space-inhomogeneous Kolmogorov-Petrovskii-Piskunov (KPP) equation with a nonlocal diffusion and a stationary ergodic nonlinearity. By employing and adapting the theory of stochastic 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.