Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations

TL;DR

This paper provides the first sharp quantitative estimates for threshold phenomena in reaction-diffusion equations with compact initial data, supported by numerical conjectures and analysis of degenerate cases.

Contribution

It introduces the first precise estimates for threshold values in reaction-diffusion equations, including ignition and bistable cases, and explores degenerate nonlinearities.

Findings

First sharp quantitative threshold estimates for ignition and bistable cases

Numerical conjectures for refined threshold estimates

Analysis of degenerate monostable nonlinearities without the hair trigger effect

Abstract

We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic equations), 35B40 (Asymptotic behavior of solutions).