A New Approach to the Expansion of Positivity Set of Non-Negative Solutions toCertain Singular Parabolic Partial Differential Equations
Articolo
Data di Pubblicazione:
2010
Abstract:
{Let u be a non-negative solution to a
singular parabolic equation of p-Laplacian type
(1
for times comparable to $\rho$ and M, then it is bounded
below by $\sigma M$, for some $\sigma\in(0,1)$, on a larger ball,
say $B_{2\rho}$ for comparable times. This fact, stated
quantitatively in Proposition Theorem1.1,
is referred to as the ``spreading of positivity'' of solutions
of such singular equations, and is at the heart of any
form of Harnack inequality. The proof of such a ``spreading
of positivity'' effect, first given in a paper by Chen and DiBenedetto,
is rather involved and not intuitive. Here we give a new
proof which is more direct being based on geometrical ideas.
singular parabolic equation of p-Laplacian type
(1
for times comparable to $\rho$ and M, then it is bounded
below by $\sigma M$, for some $\sigma\in(0,1)$, on a larger ball,
say $B_{2\rho}$ for comparable times. This fact, stated
quantitatively in Proposition Theorem1.1,
is referred to as the ``spreading of positivity'' of solutions
of such singular equations, and is at the heart of any
form of Harnack inequality. The proof of such a ``spreading
of positivity'' effect, first given in a paper by Chen and DiBenedetto,
is rather involved and not intuitive. Here we give a new
proof which is more direct being based on geometrical ideas.
Tipologia CRIS:
1.1 Articolo in rivista
Keywords:
Singular; Parabolic Equations; Expansion of Positivity
Elenco autori:
Dibenedetto, Emmanuele; Gianazza, UGO PIETRO; Vespri, Vincenzo
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