A new short proof of regularity for local weak solutions for a certain class of singular parabolic equations

We shall establish the interior Holder continuity for locally bounded weak solutions to a class of parabolic singular equations whose prototypes are begin{equation} u_t= abla cdot bigg( | abla u|^{p-2} abla u bigg), quad text{ for } quad 1<p<2, end{equation} and begin{equation} u_{t}- abla cdot ( u^{m-1} | abla u |^{p-2} abla u ) =0 , quad text{for} quad m+p>3-frac{p}{N}, end{equation} via a new and simplified proof using recent techniques on expansion of positivity and $L^{1}$-Harnack estimates.