Asymptotic Behaviors for Nonlocal Diffusion Equations about the Dispersal Spread

TL;DR

This paper investigates how the dispersal range influences the long-term behavior of solutions to nonlocal diffusion equations, revealing different asymptotic patterns depending on the dispersal spread size and boundary conditions.

Contribution

It provides a comprehensive analysis of the asymptotic behaviors of eigenvalues and solutions in nonlocal diffusion equations under varying dispersal spreads and boundary conditions.

Findings

Large dispersal spread leads to unitary asymptotic behavior.

Small dispersal spread results in different asymptotics based on the cost parameter.

Boundary conditions significantly affect asymptotic properties for zero dispersal spread.

Abstract

This paper studies the effects of the dispersal spread, which characterizes the dispersal range, on nonlocal diffusion equations with the nonlocal dispersal operator $\frac{1}{\sigma^{m}}\int_{\Omega}J_{\sigma}(x-y)(u(y,t)-u(x,t))dy$ and Neumann boundary condition in the spatial heterogeneity environment. More precisely, we are mainly concerned with asymptotic behaviors of generalised principal eigenvalue to the nonlocal dispersal operator, positive stationary solutions and solutions to the nonlocal diffusion KPP equation in both large and small dispersal spread. For large dispersal spread, we show that their asymptotic behaviors are unitary with respect to the cost parameter $m\in[0,\infty)$. However, small dispersal spread can lead to different asymptotic behaviors as the cost parameter $m$ is in a different range. In particular, for the case $m=0$, we should point out that asymptotic properties for the nonlocal diffusion equation with Neumann boundary condition are different from those for the nonlocal diffusion equation with Dirichlet boundary condition.