We study the long-time behavior of solutions of semilinear parabolic equation of the following type ∂t u − ∆u + a0(x)u^q = 0 where a0(x) ≥ d0 exp(− [ω(|x|)]/|x|2 ), d0 > 0, 1 > q > 0, and ω is a positive continuousradial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.