A Lower Bound on the Blob

RESUMO

A blob grows on the plane (as in films called "The Blob" of 1958 and 1988). Suppose that it grows in all directions at unit rate, and can be stopped only by a certain kind of fence that can be manufactured at rate $\lambda$. What is the critical value $\lambda_C$ for this rate, above which the blob can be surrounded and contained, and below which we are all doomed? This problem, introduced by Bressan in 2007 and independently by Barghi and Winkler in 2013, was in both cases motivated by consideration of strategies in fighting forest fires. In particular, does it pay to place "gambit" barriers which will subsequently be overwhelmed by the fire, in order to slow it down enabling later containment? A non-gambit strategy, which will be described, succeeds when $\lambda > 2$ and is widely believed to be optimal; but up until now no one has succeeded in obtaining a lower bound greater than 1 for $\lambda_C$. We introduce a novel weakened version of the blob which is used to raise the lower bound to 1.5.