Non-admissibility of spiral-like strategies in Bressan's Fire conjecture

In this talk we will introduce Bressan's Fire Conjecture: it is concerned with the model of wild fire spreading in a region of the plane and the possibility to block it using barriers constructed in real time.

The fire starts spreading at time $t=0$ from the unit ball $B_1(0)$ in every direction with speed $1$, while the length of the barrier constructed within the time $t$ has to be lower than $\sigma t$, where $\sigma>0$ is a positive constant (construction speed). In 2007 Bressan conjectured that if $\sigma\in[1,2]$ no barrier can block the spreading of the fire. In this talk we will prove Bressan's Fire Conjecture in the case barriers are spirals. Spirals are thought to be the best strategies a firefighter can do in order to confine the fire for $\sigma\leq 2$. We will introduce the new concept of family of generalized barriers and we will prove that, if there exists such a family satisfying a diverging condition, then no spiral can confine the fire. This is a joint work with Stefano Bianchini.