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