Long-time extinction of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equation of the following type t_∂u−Δu+a₀(x)uq=0 where , d₀>0, 1>q>0, and ω is a positive continuous radial 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.