In this paper we deal with the Cauchy problem associated to a class of nonlinear degenerate parabolic equations, whose prototype is the parabolic p-Laplacian (2>p>∞). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ‘Harnack chain’ method. In the linear case sharp estimates come by using Nash's approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In this paper, by using the DiBenedetto–DeGiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations. To obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions.

Moser–Nash kernel estimates for degenerate parabolic equations

Ragnedda, F.;Vernier Piro;
2017

Abstract

In this paper we deal with the Cauchy problem associated to a class of nonlinear degenerate parabolic equations, whose prototype is the parabolic p-Laplacian (2>p>∞). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ‘Harnack chain’ method. In the linear case sharp estimates come by using Nash's approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In this paper, by using the DiBenedetto–DeGiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations. To obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions.
File in questo prodotto:
File Dimensione Formato  
Moser-Nash kernel est.JFA.pdf

Solo gestori archivio

Descrizione: Articolo principale
Tipologia: versione post-print
Dimensione 1.36 MB
Formato Adobe PDF
1.36 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11584/233385
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 2
  • ???jsp.display-item.citation.isi??? 1
social impact