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

We consider the Cauchy problem\[\partial_t u+H(x,Du)=0 \quad (x,t)\in\Gamma \times (0,T),\quad u(x,0)=u_0(x)\; x\in\Gamma \]where $\Gamma$ is a network and $H$ is a convex and positive homogeneous Hamiltonian. We introduce a definition of viscosity solution and we prove that the unique viscosity solution of the problem is given by a Hopf-Lax type formula. In the second part of the paper we study flame propagation in a network and we seek an optimal strategy to block a fire breaking up in some part of a pipeline.