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 ∂ _t u − Δu + a(x)u ^q = 0, where a(x) \geqslant d₀exp( - \fracw( | x | )| x |² ) a(x) \geqslant {d_0}\exp \left( { - \frac{{\omega \left( {\left| x \right|} \right)}}{{{{\left| x \right|}^2}}}} \right) , d ₀ > 0, 1 > q > 0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω which implies that any solution of the above equation vanishes in finite time. The proof is derived from semi-classical limits of some Schr¨odinger operators.