Existence of spiral strategies for blocking fire spreading

In this paper we address the problem for blocking fire by constructing a wall $\zeta$ whose shape is spiral-like. This is supposed to be the best strategy when a single firefighter is constructing the wall with a finite construction speed $\sigma$: the barriers which satisfy this bound on the construction speed are called admissible. We prove a sharp version of Bressan's Fire Conjecture in this case, i.e. when admissible barriers are spiral-like curves: namely, there exists a spiral-like barrier confining the fire in a bounded region of $\mathbb R^2$ if and only if the speed of construction of the barrier $\sigma$ is strictly larger than a critical speed $\bar \sigma = 2.614...$. The existence of confining spiral barriers for $\sigma > \bar \sigma$ is already known [Bressan A. et al., 2008, Klein R. et al., 2019], while we concentrate on the negative side, i.e. if $\sigma \leq \bar \sigma$ no admissible spiral blocks the fire. The proof of these results relies on: 1) the precise definition of spiral barrier and its representation; 2) the analysis of saturated spiral barriers as a Retarded Differential Equation (RDE) in the spirit of [Klein R. et al., 2019]; 3) the equivalent reformulation of the conjecture as a minimum problem for a prescribed functional; 4) the construction of the optimal closing spiral; 5) the analysis of a differentiable path of admissible spirals along which the functional is differentiable, and in particular increasing when moving from the optimal spiral to any other one (homotopy argument). Due to the complexity of the solution, the evaluation of the quantities needed to prove that the functional is increasing is performed numerically.