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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