Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

TL;DR

This paper investigates how symmetric decreasing initial data influence the long-term behavior of solutions to reaction-diffusion equations, establishing a precise link between energy limits and solution outcomes, and identifying sharp thresholds for propagation versus extinction.

Contribution

It introduces a novel relation connecting the long-time behavior of solutions to their energy limits for symmetric decreasing initial data in reaction-diffusion equations.

Findings

Established a one-to-one relation between solution behavior and energy limits.

Derived sharp thresholds distinguishing propagation from extinction.

Applied results to monotone families of initial data.

Abstract

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in $L^2$ under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.