Existence-Uniqueness Theory and Small-Data Decay for a Reaction-Diffusion Model of Wildfire Spread

TL;DR

This paper analyzes a reaction-diffusion wildfire model, proving global existence, uniqueness, and decay of solutions, and showing the model's stability against thermal blow-up with small initial data.

Contribution

It establishes the global well-posedness and decay properties of solutions for a nonlinear wildfire spread model, highlighting conditions preventing thermal blow-up.

Findings

Global existence and uniqueness of solutions

Solutions decay to zero with small initial data

Model prevents thermal blow-up under certain conditions

Abstract

I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the $L^2$-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into travelling combustion waves.