Let {u_m} be a local, weak solution to the porous medium equation u_m,t − \Delta w_m = 0 where w_m = (u_m − 1)/m. It is shown that if {u_m} is locally in L^r_{loc} for r > 1/2 N uniformly in m and if w_m is in L^p_{loc} for p > N + 2 in the space variables, uniformly in time, then {u_m} contains a subsequence converging in C_{loc}^{\alpha,1/2 \alpha} to a local, weak solution to the logarithmically singular equation u_t = \Delta ln u. The result is based on local upper and lower bounds on {u_m}, uniform in m. The uniform, local lower bounds are realized by a Harnack type inequality.
Logarithmically Singular Parabolic Equations as Limits of the Porous Medium Equation
GIANAZZA, UGO PIETRO;
2012-01-01
Abstract
Let {u_m} be a local, weak solution to the porous medium equation u_m,t − \Delta w_m = 0 where w_m = (u_m − 1)/m. It is shown that if {u_m} is locally in L^r_{loc} for r > 1/2 N uniformly in m and if w_m is in L^p_{loc} for p > N + 2 in the space variables, uniformly in time, then {u_m} contains a subsequence converging in C_{loc}^{\alpha,1/2 \alpha} to a local, weak solution to the logarithmically singular equation u_t = \Delta ln u. The result is based on local upper and lower bounds on {u_m}, uniform in m. The uniform, local lower bounds are realized by a Harnack type inequality.File in questo prodotto:
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