Extinction for a singular diffusion equation with strong gradient absorption revisited

When $2N/(N+1)<p<2$ and $0<q<p/2$, non-negative solutions to the singular equation with gradient absorption \∂\ₜu-\Δ\ₚ u +|\∇ u|q=0 \\ \ in \\ (0,\∞)\×N vanish after a time. This phenomenon is usually referred to as finite time extinction takes place provided the initial condition u\₀ decays sufficiently as |x|\→\∞. On the one hand, the optimal decay of u\₀ at guaranteeing the occurence of finite time extinction is identified. On other hand, assuming further that $p-1<q<p/2$, optimal extinction rates the extinction time are derived.