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

Source: [Existence of a traveling wave solution in a free interface problem with fractional order kinetics](<https://www.sota2.com/research/paper/68fa577a1e209131ce9a5de1>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T14:45:29.807Z

Authors: Claude-Michel Brauner, Robert Roussarie, Peipei Shang, Linwan Zhang

Published: 2020-10-29T15:28:44.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2010.15685>)

[PDF](<https://arxiv.org/pdf/2010.15685.pdf>)