Threshold phenomena for symmetric-decreasing radial solutions of reaction-diffusion equations

TL;DR

This paper investigates the long-term behavior of symmetric, decreasing radial solutions to reaction-diffusion equations with various nonlinearities, establishing threshold phenomena and propagation speeds for different initial data classes.

Contribution

It characterizes ignition behavior via energy analysis, establishes sharp thresholds for initial data, and links propagation speed to one-dimensional traveling waves.

Findings

Threshold behavior depends on initial data and nonlinearities.

Existence of sharp thresholds for ignition and propagation.

Propagation speed matches one-dimensional traveling wave speed.

Abstract

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial data that are radial and non-increasing as a function of the distance to the origin, we characterize the ignition behavior in terms of the long time behavior of the energy associated with the solution. We then use this characterization to establish existence of a sharp threshold for monotone families of initial data in the considered class under various assumptions on the nonlinearities and spatial dimension. We also prove that for more general initial data that are sufficiently localized the solutions that exhibit ignition behavior propagate in all directions with the asymptotic speed equal to that of the unique one-dimensional variational traveling wave.