Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

TL;DR

This paper develops integral variational principles to determine bounds on the spreading speed of reaction-diffusion fronts with Stefan boundary conditions, applicable to various reaction types, including monostable and arbitrary reactions.

Contribution

It introduces two new variational principles for reaction-diffusion front speeds with Stefan conditions, providing a unified framework for bounds and asymptotic estimates.

Findings

Established variational principles for front speed bounds.

Constructed generalized lower bounds similar to Zeldovich-Frank-Kamenetskii.

Derived asymptotically exact lower bounds matching perturbation theory results.

Abstract

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich-Frank-Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.