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Existence-uniqueness theory and small-data decay for a reaction-diffusion model of wildfire spread

Part of: Thermodynamics and heat transfer Parabolic equations and systems

Published online by Cambridge University Press:  08 July 2025

A. George Morgan Open the ORCID record for A. George Morgan [Opens in a new window]
A. George Morgan*
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George St., Room 6290, Toronto, ON, M5S 2E4, Canada
*
e-mail: agmorgan@my.yorku.ca
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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 spatial $L^2$-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 traveling combustion waves.

Keywords

Wildfire spread modelssolid-fuel combustionreaction-diffusion equations in combustion theoryexistence and uniqueness of solutions to reaction-diffusion equationstime decay of solutions to reaction-diffusion equations

MSC classification

Primary: 35K57: Reaction-diffusion equations 80A25: Combustion

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Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 12
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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