Extinction of solutions for some nonlinear parabolic equations

We are dealing with the first vanishing time for solutions of the Cauchy–Neumann problem for the semilinear parabolic equation $\partial_t u-\Delta u+a(x)u^q=0$, where $a(x)\ge d_0\exp(-\omega(|x|)/|x|^2)$, $d_0>0$, $1>q>0$, and $\omega$ is a positive continuous radial function. We give a Dini-like condition on the function $\omega$ which implies that any solution of the above equation vanishes in finite time. The proof is derived from semi-classical limits of some Schrödinger operators.