Threshold solutions for nonlocal reaction diffusion equations

TL;DR

This paper analyzes the long-term behavior of solutions to nonlocal reaction diffusion equations with bistable nonlinearity, identifying conditions for extinction or propagation and demonstrating a sharp transition between these states.

Contribution

It provides a rigorous analysis of threshold solutions and their asymptotic behaviors, including the sharp transition criterion, supported by numerical verification.

Findings

Solutions decay to 0 for small initial data

Solutions converge to 1 for large initial data

Transition between extinction and propagation is sharp

Abstract

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp. Numerical results are provided to verify the theoretical results.