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

TL;DR

This paper proves regularity and analytic properties of solutions to a logarithmically singular diffusion equation, showing stability of estimates from porous medium approximations as the parameter approaches zero.

Contribution

It establishes a Harnack estimate and regularity results for solutions with high integrability, bridging porous medium equations and logarithmic singularities.

Findings

Solutions are locally analytic in space and differentiable in time.

Harnack estimates hold under high integrability conditions.

Estimates from porous medium equations are stable as the parameter tends to zero.

Abstract

For the logarithmically singular parabolic equation \[ u_t-\Delta\ln u=0\qquad\text{weakly in}\ \ E\times(0,T], \] 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, namely \begin{equation*} \ln u\in L^\infty_{loc}\big(0,T;L^p_{loc}(E)\big) \quad\text{for some} p\ge1. \end{equation*} These two properties are known for solutions of singular porous medium type equations ($0<m<1$), 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.