# American Institute of Mathematical Sciences

October  2018, 11(5): 825-843. doi: 10.3934/dcdss.2018051

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

 1 Dip. di Scienze di Base e Applicate per l'Ingegneria, "Sapienza" Università di Roma, via Scarpa 16, 00161 Roma, Italy 2 Dipartimento di Matematica, "Sapienza" Università di Roma, p.le A. Moro 5, 00185 Roma, Italy 3 Dip. di Ingegneria dell'Informazione, Università di Padova, via Gradenigo 6/B, 35131 Padova, Italy

* Corresponding author

Received  February 2017 Revised  August 2017 Published  June 2018

We consider the Cauchy problem
 $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+H(x,Du) = 0&(x,t)\in \Gamma \times (0,T) \\ u(x,0) = {{u}_{0}}(x)&x\in \Gamma \\\end{array} \right.$
where
 $\Gamma$
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.
Citation: Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A flame propagation model on a network with application to a blocking problem. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 825-843. doi: 10.3934/dcdss.2018051
##### References:

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##### References:
Test1. Graph structure where $R_0$ is represented by the circle marker and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29), (Top Right) and its 3D view (Bottom).
Test1. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square marker). 2D view (Left) and 3D view (Right).
Test1. Optimal blocking strategy $\sigma ^h_{opt}$ (square marker), preserved network region (cross marker) and minimum burnt network region (continuum line) starting from $R_0$ (circle marker).
Test2. Graph structure where $R_0$ is represented by the circle markers and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29) (Top Right), and its 3D view (Bottom).
Test2. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square markers). 2D view(Left) and 3D view (Right).
Test2. Optimal blocking strategy $\sigma ^h_{opt}$ (square markers), preserved network region (thin line) and minimum burnt network region (thick line) starting from $R_0$ (circle markers).
Test3. Graph structure where $R_0$ is represented by the circle markers and the vertices by the rhombus markers (Top Left). Color map of the time $u_h(x)$ at which a node $x$ get burnt, computed by (29), (Top Right) and its 3D view (Bottom).
Test3. Time to reach a point $x$ from the operation center $x_0$ (circle marker) and set of the admissible nodes $V^h_{ad}$ (square markers). 2D view(Left) and 3D view (Right).
Test3. Optimal blocking strategy $\sigma ^h_{opt}$ (square marker), preserved network region (thin line) and minimum burnt network region (thick line) starting from $R_0$ (circle marker).
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