Local bounds of the gradient of weak solutions to the porous medium equation

Let $u$ be a nonnegative, local, weak solution to the porous medium equation
\[
\partial_t u-\Delta u^m=0
\]
for $m\ge2$ in a space-time cylinder $\Om_T=\Om\times(0,T]$. Fix a point $\pto\in\Om_T$: if the average
\[
a\df=\dashint_{B_r(x_o)}u(x,t_o)\,dx>0,
\]
then the quantity $|\nabla u^{m-1}|$ is locally bounded in a proper cylinder, whose center lies at time $t_o+a^{1-m}r^2$. {This implies that in the same cylinder the solution $u$ is H\"older continuous with exponent $\al=\frac1{m-1}$, which is known to be optimal}. Moreover, $u$ presents a sort of instantaneous regularization, which we discuss.