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 -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.
@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}
}
DiBenedetto, Emmanuele; Gianazza, Ugo; Vespri, Vincenzo. 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/
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