Analysing the existence and uniqueness solution of a wildfire model with diffusion and convection of moisture

Abstract

Wildfire spread modeling is governed by a complex system of nonlinear partial differential equations (PDEs) that capture the intricate
dynamics of wildfire behavior, including heat transfer and moisture interaction. A comprehensive understanding of these dynamics is critical for developing effective management, mitigation, and intervention strategies. In this study, temperature-dependent diffusion and  convection terms are incorporated into the volume fraction of moisture, enriching the model framework and improving its accuracy in  representing wildfire spread. To ensure the mathematical robustness of the model, the non-linear PDE system is transformed into a  dimensionless form using appropriate dimensionless variables, facilitating the analysis of the equations. The model equations describe  the dynamics of combustible forest material (CFM) in terms of the volume fractions of dry organic matter, moisture, coke, heat, and  oxygen. The conditions for the existence and uniqueness of solutions to the model equations are rigorously established using the  Lipschitz continuity criterion. The results confirm that unique solutions exist when the Lipschitz conditions are satisfied.