A flame propagation model on a network with application to a blocking problem

TL;DR

This paper develops a mathematical model for flame propagation on a network, proving the uniqueness of solutions and exploring strategies to block fire spread, supported by numerical simulations.

Contribution

It introduces a novel flame propagation model on networks with a proven solution framework and analyzes optimal fire blocking strategies.

Findings

Hopf-Lax formula provides unique viscosity solutions

Optimal fire blocking strategies are identified

Numerical simulations demonstrate model effectiveness

Abstract

We consider the Cauchy problem \[\partial_t u+H(x,Du)=0 \quad (x,t)\in\Gamma\times (0,T),\quad u(x,0)=u_0(x) \quad x\in\Gamma\] where $\Gamma$ is a network and $H$ is a convex and positive homogeneous Hamiltonian which may change from edge to edge. In the former part of the paper, we prove that the Hopf-Lax type formula gives the (unique) viscosity solution of the problem. In the latter part of the paper we study a flame propagation model in a network and an optimal strategy to block a fire breaking up in some part of a pipeline; some numerical simulations are provided.