Asymptotic behavior of nonlocal bistable reaction-diffusion equations

TL;DR

This paper investigates the long-term behavior of solutions to nonlocal bistable reaction-diffusion equations, revealing conditions for propagation, extinction, or pinning, with a focus on pinned solutions and numerical analysis.

Contribution

It provides a detailed analysis of the asymptotic regimes of nonlocal bistable equations, especially the pinned state, including a case study with discontinuous ground states.

Findings

Solutions can propagate, go extinct, or remain pinned depending on parameters.

Pinned solutions are thoroughly characterized, including discontinuous ground states.

Numerical analysis supports theoretical findings.

Abstract

In this paper, we study the asymptotic behavior of the solutions of nonlocal bistable reaction-diffusion equations starting from compactly supported initial conditions. Depending on the relationship between the nonlinearity, the interaction kernel and the diffusion coefficient, we show that the solutions can either: propagate, go extinct or remain pinned. We especially focus on the latter regime where solutions are pinned by thoroughly studying discontinuous ground state solutions of the problem for a specific interaction kernel serving as a case study. We also present a detailed numerical analysis of the problem.