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

Emmanuele DiBenedetto, Ugo Gianazza, Naian Liao

For the logarithmically singular parabolic equation \[ u_t-Δ\ln u=0\qquad\text{weakly in}\ \ E\times(0,T], \] we establish a Harnack type estimate in the $L_{loc}^1$ topology, and…