Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, p. 385-422

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the $p$-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold.

Classification:  35K65,  35B65,  35B45
@article{ASNSP_2010_5_9_2_385_0,
author = {DiBenedetto, Emmanuele and Gianazza, Ugo and Vespri, Vincenzo},
title = {Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola Normale Superiore, Pisa},
volume = {Ser. 5, 9},
number = {2},
year = {2010},
pages = {385-422},
zbl = {1206.35053},
mrnumber = {2731161},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2010_5_9_2_385_0}
}

Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 2, pp. 385-422. http://www.numdam.org/item/ASNSP_2010_5_9_2_385_0/

[1] Y. Z. Chen and E. Dibenedetto, Hölder estimates of solutions of singular parabolic equations with measurable coefficients, Arch. Ration. Mech. Anal. 118 (1992), 257–271. | MR 1158938 | Zbl 0836.35029

[2] E. De Giorgi, Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari, Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3 (1957), 25–43. | MR 93649 | Zbl 0084.31901

[3] E. Dibenedetto, Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations, Arch. Ration. Mech. Anal. 100 (1988), 129–147. | MR 913961 | Zbl 0708.35017

[4] E. Dibenedetto and Y. C. Kwong, Intrinsic Harnack estimates and extinction profile for certain singular parabolic equations, Trans. Amer. Math. Soc. 330 (1992), 783–811. | Zbl 0772.35006

[5] E. Dibenedetto, Y. C. Kwong and V. Vespri, Local space analiticity of solutions of certain singular parabolic equations, Indiana Univ. Math. J. 40 (1991), 741–765. | MR 1119195 | Zbl 0784.35055

[6] E. Dibenedetto, “Degenerate Parabolic Equations”, Springer Verlag, Series Universitext, New York, 1993. | MR 1230384 | Zbl 0794.35090

[7] E. Dibenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equations, Acta Math. 200 (2008), 181–209. | MR 2413134 | Zbl 1221.35213

[8] J. Hadamard, Extension à l’équation de la chaleur d’un théorème de A. Harnack, Rend. Circ. Mat. Palermo 3 (1954), 337–346. | MR 68713 | Zbl 0058.32201

[9] B. H. Gilding, On a class of similarity solutions of the porous media equation III, J. Math. Anal. Appl. 77 (1980), 381–402. | MR 593221 | Zbl 0454.35053

[10] J. Moser, On Harnack’s theorem for elliptic differential equations, Comm. Pure Appl. Math. 14 (1961), 577–591. | MR 159138 | Zbl 0111.09302

[11] J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. | MR 159139 | Zbl 0149.06902

[12] J. Moser, On a pointwise estimate for parabolic differential equations, Comm. Pure Appl. Math. 24 (1971), 727–740. | MR 288405 | Zbl 0227.35016

[13] M. A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, Differential Integral Equations 8 (1995), 2045–2064. | MR 1348964 | Zbl 0845.35057

[14] B. Pini, Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico, Rend. Sem. Mat. Univ. Padova 23 (1954), 422–434. | Numdam | MR 65794 | Zbl 0057.32801