Geodesic
Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations
DiBenedetto, Emmanuele1  ; Gianazza, Ugo2  ; Vespri, Vincenzo3 
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, Italia
3 Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni, 67/A, 50134 Firenze, Italia
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 2, pp. 385-422

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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

DiBenedetto, Emmanuele 1 ; Gianazza, Ugo 2 ; Vespri, Vincenzo 3

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, Italia
3 Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni, 67/A, 50134 Firenze, Italia 
@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},
     pages = {385--422},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {2},
     year = {2010},
     mrnumber = {2731161},
     zbl = {1206.35053},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_2_385_0/}
}
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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, Série 5, Tome 9 (2010) no. 2, pp. 385-422. http://geodesic.mathdoc.fr/item/ASNSP_2010_5_9_2_385_0/