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
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", UniversitÓ 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.

References

[1] M. Bonforte and J. L. Vázquez, Positivity, local smoothing, and Harnack inequalities for very fast diffusion equations, Adv. Math. 223 (2010), no. 2, 529-578.
[2] E. DiBenedetto, U. Gianazza and N. Liao, On the local behavior of non-negative solutions to a logarithmically singular equation, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), no. 6, 1841-1858.
[3] E. DiBenedetto, U. Gianazza and N. Liao, Logarithmically singular parabolic equations as limits of the porous medium equation, Nonlinear Anal. 75 (2012), no. 12, 4513-4533.
[4] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations, Manuscripta Math. 131 (2010), no. 1-2, 231-245.
[5] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's inequality for degenerate and singular parabolic equations, Springer Monographs in Mathematics, Springer, New York 2012.
[6] E. DiBenedetto, Y. C. Kwong and V. Vespri, Local space-analyticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J. 40 (1991), no. 2, 741-765.
[7] S. Fornaro and V. Vespri, Harnack estimates for non-negative weak solutions of a class of singular parabolic equations, Manuscripta Math. 141 (2013), no. 1-2, 85-103.
[8] M. A. Herrero and M. Pierre, The Cauchy problem for \(u_t = \Delta u^m\) when \(0 < m < 1\), Trans. Amer. Math. Soc. 291 (1985), no. 1, 145-158.
[9] D. Kinderlehrer and L. Nirenberg, Analyticity at the boundary of solutions of nonlinear second-order parabolic equations, Comm. Pure Appl. Math. 31 (1978), no. 3, 283-338.
[10] O. A. Ladyzenskaja, N. A. Solonnikov and N. N. Ural'ceva, Linear and quasi-linear equations of parabolic type, Translated from the Russian, Translations of Mathematical Monographs, 23, American Mathematical Society, Providence, RI 1968.


Home Riv.Mat.Univ.Parma