Extinction for a singular diffusion equation with strong gradient absorption revisited

When $2N\mathrm{/}(N+1)<p<2$ and $0<q<p\mathrm{/}2$, non-negative solutions to the singular diffusion equation with gradient absorption $$\mathrm{\partial }\mathrm{\_}tu-\mathrm{\Delta }\mathrm{\_}pu+\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q=0\text{ in }(0,\mathrm{\infty })\times {\mathbb{R}}^N$$ vanish after a finite time. This phenomenon is usually referred to as finite time extinction and takes place provided the initial condition $u\mathrm{\_}0$ decays sufficiently rapidly as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. On the one hand, the optimal decay of $u\mathrm{\_}0$ at infinity guaranteeing the occurence of finite time extinction is identified. On the other hand, assuming further that $p-1<q<p\mathrm{/}2$, optimal extinction rates near the extinction time are derived.