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.

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 
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     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},
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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://www.numdam.org/item/ASNSP_2010_5_9_2_385_0/

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