  We focus on the (sharp) threshold phenomena arising in some
reaction-diffusion equations supplemented with some compactly supported initial
data. In the so-called ignition and bistable cases, we prove the first sharp
quantitative estimate on the (sharp) threshold values. Furthermore, numerical
explorations allow to conjecture some refined estimates. Last we provide
related results in the case of a degenerate monostable nonlinearity "not
enjoying the hair trigger effect". AMS Subject Classifications: 35K57
(Reaction-diffusion equations), 35K15 (Initial value problems for second-order
parabolic equations), 35B40 (Asymptotic behavior of solutions).
