Harnack type inequalities for some doubly nonlinear singular parabolic equations

Simona Fornaro; Maria Sosio; Vincenzo  Vespri

doi:10.3934/dcds.2015.35.5909

Article Contents

2015, Volume 35, Issue 12: 5909-5926. Doi: 10.3934/dcds.2015.35.5909

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Harnack type inequalities for some doubly nonlinear singular parabolic equations

1.
Università degli Studi di Pavia, Dipartimento di Matematica “F. Casorati”, via Ferrata 1, 27100 Pavia
2.
Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia
3.
Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, viale Morgagni, 67/A, 50134, Firenze

Received: March 2014

Published: December 2015

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Abstract

We prove Harnack type inequalities for a wide class of parabolic doubly nonlinear equations including $u_t=$ ${ div}(|u|^{m-1}|Du|^{p-2}Du)$. We will distinguish between the supercritical range $3 - \frac {p} {N} < p+m < 3$ and the subcritical $2 < p+m \le 3 - \frac {p} {N}$ range. Our results extend similar estimates holding for general equations having the same structure as the parabolic $p$-Laplace or the porous medium equation and recently collected in [6].

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Mathematics Subject Classification: Primary: 35B65, 35K67; Secondary: 35K55.

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