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 ∂ t u - Δ u m = 0 for m ≥ 2 in a space-time cylinder Ω T = Ω × ( 0 , T ] . Fix a point ( x o , t o ) ∈ Ω T : if the average then the quantity | ∇ 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 Hölder continuous with exponent α = 1 m - 1 , which is known to be optimal. Moreover, u presents a sort of instantaneous regularization, which we discuss.