Long-time extinction of solutions of some semilinear parabolic equations

We study the long time behaviour of solutions of semi-linear parabolic equation of the following type ${\mathrm{\partial }}_{t}u-\mathrm{\Delta }u+a_0(x)u^q=0$ where $a_0(x)\ge d_0\exp (\frac{\omega (\mathrm{\mid }x\mathrm{\mid })}{\mathrm{\mid }x{\mathrm{\mid }}^2})$, $d_0>0$, $1>q>0$ and $\omega$ a positive continuous radial function. We give a Dini-like condition on the function $\omega$ by two different method 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 Schr"odinger operators.