On the Minimum Speed for a Fire Blocking Problem
Abstract
We consider a dynamic blocking problem: a fire, starting on a bounded open set R0⊂ℝ2, 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≤ 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)t2/4.
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BibTeX
@misc{bressan2026minimum,
title = {{On the Minimum Speed for a Fire Blocking Problem}},
author = {Alberto Bressan and Luca Talamini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36919},
eprint = {2610.36919},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36919},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Minimum Speed for a Fire Blocking Problem
AU - Alberto Bressan
AU - Luca Talamini
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36919
UR - https://arxiv.org/abs/2610.36919
AB - We consider a dynamic blocking problem: a fire, starting on a bounded open set R_0⊂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≤ 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.
ER -