Traveling waves for a boundary reaction-diffusion equation

TL;DR

This paper proves the existence of traveling wave solutions for a boundary reaction-diffusion equation with combustion nonlinearity, using explicit formulas from a free boundary problem to analyze decay properties.

Contribution

It introduces an explicit formula for traveling waves in a boundary reaction-diffusion context, linking it to a free boundary problem and analyzing decay rates.

Findings

Existence of traveling wave solutions established.

Explicit formula for solutions derived from a free boundary problem.

Traveling waves decay faster than exponential at infinity.

Abstract

We prove the existence of a traveling wave solution for a boundary reaction diffusion equation when the reaction term is the combustion nonlinearity with ignition temperature. A key role in the proof is plaid by an explicit formula for traveling wave solutions of a free boundary problem obtained as singular limit for the reaction-diffusion equation (the so-called high energy activation energy limit). This explicit formula, which is interesting in itself, also allows us to get an estimate on the decay at infinity of the traveling wave (which turns out to be faster than the usual exponential decay).