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arXiv · 2610.36919

On the Minimum Speed for a Fire Blocking Problem

Alberto Bressan·Luca Talamini
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Abstract

We consider a dynamic blocking problem: a fire, starting on a bounded open set $R_0\subset\mathbb R^2$, propagates in the plane with unit speed in all directions. To block it, a barrier can be constructed in real time, at speed $λ>0$. A long standing conjecture by the first author states that the fire can be entirely surrounded by the barrier, in finite time, if and only if $λ>2$. The present paper improves on earlier lower bounds. Namely, if $λ\leq Λ=1.62621170228\ldots $ we show that the fire cannot be blocked. We also prove that, if $λ\leq 2$, the fire cannot be blocked by constructing a barrier which is a simple closed curve. More precisely, if the barrier does not contain purely delaying arcs, i.e.~arcs which are eventually surrounded by the fire on both sides, then for $λ<2$ the region burned by the fire at time $t>0$ has area $A(t)>(2-λ)t^2/4$.

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Alberto Bressan, Luca Talamini. 2026-10-27. On the Minimum Speed for a Fire Blocking Problem. https://arxiv.org/abs/2610.36919

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