Mathematical Models for Vegetation Fire PropagationExistence and Uniqueness Analysis

This work presents a rigorous mathematical study of a vegetation fire propagation model specifically adapted to the Ivorian environmental context, where climatic conditions and vegetation types significantly influence fire dynamics. The model aims to describe the thermal behavior of vegetation during combustion and the spatial spread of fire fronts. We first analyze a nonlinear ordinary differential equation (ODE) governing the temporal evolution of temperature, which captures the essential mechanisms of heat production and dissipation during combustion. Using classical results from the theory of differential equations, we establish the existence and uniqueness of solutions under suitable assumptions on the model parameters and initial conditions. The study is then extended to a transport-reaction partial differential equation (PDE) that incorporates spatial effects and allows the description of fire propagation in a heterogeneous domain. This PDE model accounts for both the advective transport of heat and the local reaction terms associated with combustion processes. The analysis relies on tools from functional analysis, including appropriate function spaces and a priori estimates, combined with the method of characteristics to handle the transport component of the equation. Under physically relevant assumptions, we prove the existence and uniqueness of weak solutions to the PDE model. The proposed mathematical framework provides a solid theoretical foundation for vegetation fire modeling in West African environments. Beyond its theoretical interest, this work contributes to a better understanding of fire dynamics and offers a basis for future numerical simulations and risk assessment tools aimed at improving fire prevention and management strategies in Côte d'Ivoire and similar regions.