Logarithmically Singular Parabolic Equations as Limits of the Porous Medium Equation

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.