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

We consider the Cauchy problem ∂ t u + H(x, Du) = 0 (x, t) ∈ Γ × (0, T ) u(x, 0) = u 0 (x) x ∈ Γ where Γ is a network and H is a positive homogeneous Hamiltonian which may change from edge to edge. In the first 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.