Accelerated front propagation for monostable equations with nonlocal diffusion: Multidimensional case

TL;DR

This paper investigates how solutions to certain nonlinear equations with nonlocal diffusion in multiple dimensions accelerate their front propagation, especially when initial conditions or kernels have heavy tails, providing sharp estimates and generalizing previous one-dimensional results.

Contribution

It extends previous one-dimensional results to multidimensional cases, analyzing acceleration phenomena with heavy-tailed initial conditions or kernels in nonlocal monostable equations.

Findings

Acceleration occurs with heavy-tailed initial conditions or kernels.

Sharp estimates for the propagation zones are provided.

Various propagation rates, from slightly faster than linear to exponential, are characterized.

Abstract

We describe acceleration of the front propagation for solutions to a class of monostable nonlinear equations with a nonlocal diffusion in $\mathbb{R}^d$, $d\geq1$. We show that the acceleration takes place if either the diffusion kernel or the initial condition has 'regular' heavy tails in $\mathbb{R}^d$ (in particular, decays slower than exponentially). Under general assumptions which can be verified for particular models, we present sharp estimates for the time-space zone which separates the region of convergence to the unstable zero solution with the region of convergence to the stable positive constant solution. We show the variety of different possible rates of the propagation starting from a little bit faster than a linear one up to the exponential rate. The paper generalizes to the case $d>1$ our results for the case $d=1$ obtained early in https://dx.doi.org/10.1080/00036811.2017.1400537 .