Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
840593 | Nonlinear Analysis: Theory, Methods & Applications | 2012 | 21 Pages |
Abstract
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.
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Authors
Emmanuele DiBenedetto, Ugo Gianazza, Naian Liao,