Existence of a traveling wave solution in a free interface problem with fractional order kinetics

TL;DR

This paper proves the existence of traveling wave solutions in a reaction-diffusion system modeling fractional order thermal combustion, revealing finite settling times and contrasting behaviors with classical cases.

Contribution

It introduces a novel approach to analyze free interface problems with fractional kinetics, using topological methods and fixed-point techniques to establish wave existence.

Findings

Traveling wave solutions exist for fractional order kinetics.

The settling time to reach equilibrium is finite.

The behavior differs from classical cases with integer reaction order.

Abstract

In this paper we consider a system of two reaction-diffusion equations that models diffusional-thermal combustion with stepwise ignition-temperature kinetics and fractional reaction order 0 < $\alpha$ < 1. We turn the free interface problem into a scalar free boundary problem coupled with an integral equation. The main intermediary step is to reduce the scalar problem to the study of a non-C 1 vector field in dimension 2. The latter is treated by qualitative topo-logical methods based on the Poincar{\'e}-Bendixson Theorem. The phase portrait is determined and the existence of a stable manifold at the origin is proved. A significant result is that the settling time to reach the origin is finite, meaning that the trailing interface is finite in contrast to the case $\alpha$ = 1, but in accordance with $\alpha$ = 0. Finally, the integro-differential system is solved via a fixed-point method.