Blow-down and immediate onset of positivity in the Neumann problem for (super-)fast diffusion equations

For the degenerate parabolic equation $u_t=\nabla\cdot (u^{m-1}\nabla u)$, the homogeneous Neumann problem in smoothly bounded domains $Ω\subset {\mathbb{R}}^n$ is considered in the (super-fast) diffusion range when $m\in (-\infty,1)$. Upper and lower estimates for solutions are inferred by means of