Two Remarks on the Local Behavior of Solutions to Logarithmically Singular Diffusion Equations and its Porous-Medium Type Approximations

For the logarithmically singular parabolic equation u_t−\Delta \ln u=0, we establish a Harnack type estimate in the L^1_{loc} topology, and we show that the solutions are locally analytic in the space variables and differentiable in time. The main assumption is that ln u possesses a sufficiently high degree of integrability. These two properties are known for solutions of singular porous medium type equations (m\in(0,1)), which formally approximate the logarithmically singular equation. However, the corresponding estimates deteriorate as m\to0. It is shown that these estimates become stable and carry to the limit as m\to0, provided the indicated sufficiently high order of integrability is in force. The latter then appears as the discriminating assumption between solutions of parabolic equations with power-like singularities and logarithmic singularities to insure such solutions to be regular.