Logarithmically singular parabolic equations as limits of the porous medium equation

Let {um}{um} be a local, weak solution to the porous medium equation um,t−Δwm=0where wm=umm−1m. It is shown that if {um}{um} is locally in Llocr for r>12N uniformly in mm and if wmwm is in Llocp for p>N+2p>N+2 in the space variables, uniformly in time, then {um}{um} contains a subsequence converging in Clocα,12α to a local, weak solution to the logarithmically singular equation ut=Δlnuut=Δlnu. The result is based on local upper and lower bounds on {um}{um}, uniform in mm. The uniform, local lower bounds are realized by a Harnack type inequality.