**Emmanuele DiBenedetto** ^{[1]}, **Ugo Gianazza** ^{[2]} and **Naian Liao** ^{[1]}

*
Two remarks on the local behavior of solutions to logarithmically singular diffusion equations and its porous-medium type approximations
*

**Pages:** 139-182

**Received:** 30 April 2013

**Accepted :** 18 July 2013

**Mathematics Subject Classification (2010):** Primary 35K65, 35B65; Secondary 35B45.

**Keywords:** Singular parabolic equations, \(L^{1}_{loc}\)-Harnack estimates, analyticity.

**Authors addresses:**

[1] : Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN 37240, USA

[2] : Dipartimento di Matematica "F. Casorati", Università di Pavia, via Ferrata 1, 27100 Pavia, Italy

**Abstract:**
For the logarithmically singular parabolic
equation \((1.1)\) below, 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
(see \((1.3)\) for a precise statement). These
two properties are known for solutions
to
singular porous medium type equations \((0 < m < 1)\),
which formally approximate the logarithmically
singular equation (1.1) below.
However, the corresponding
estimates deteriorate as \(m \to 0\). It is
shown that these estimates become stable and
carry to the limit as \(m \to 0\),
provided the indicated
sufficiently high order of integrability is in force.
The latter then appears as the discriminating
assumption between solutions
to parabolic
equations with power-like singularities and
logarithmic singularities to insure such
solutions to be regular.

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