Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

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
Accepted : 18 July 2013
Mathematics Subject Classification (2010): Primary 35K65, 35B65; Secondary 35B45.

Keywords: Singular parabolic equations, $$L^{1}_{loc}$$-Harnack estimates, analyticity.
[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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