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

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  • 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.

    Mathematics Subject Classification: Primary: 35D40; Secondary: 35R02, 35F21, 65M06, 49L25.

    Citation:

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  • Figure 1.  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).

    Figure 2.  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).

    Figure 3.  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).

    Figure 4.  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).

    Figure 5.  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).

    Figure 6.  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).

    Figure 7.  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).

    Figure 8.  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).

    Figure 9.  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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