On the Minimum Speed for a Fire Blocking Problem

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 $A(t)>(2-λ)t^2/4$.