Harnack estimates at large: Sharp pointwise estimates for nonnegative solutions to a class of singular parabolic equations

In this paper we deal with quasilinear singular parabolic equations with $L^\infty$ coefficients,whose prototypes are the p-Laplacian ($\frac{ 2N}{N+1} < p < 2$) equations. In this range of the parameters, we are in the so called fast diffusion case. Extending a recent result (Ragnedda et al. 2013), we are able to prove Harnack estimates at large, i.e. starting from the value attained in a point by the solution, we are able to give explicit and sharp pointwise estimates, from below by using the Barenblatt solutions. In the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i.e. $(\frac{N−2}{N})_+ < m < 1$.