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 diffusion equation with gradient absorption $$\partial\_tu-\Delta\_p u + |\nabla u|^q=0 \ \text{ in }\ (0,\infty)\times\mathbb{R}^N$$ vanish after a finite time. This phenomenon is usually referred to as finite time extinction and takes place provided the initial condition $u\_0$ decays sufficiently rapidly as $|x|\to\infty$. On the one hand, the optimal decay of $u\_0$ at infinity guaranteeing the occurence of finite time extinction is identified. On the other hand, assuming further that $p-1<q<p/2$, optimal extinction rates near the extinction time are derived.