Let u be a non-negative super-solution to a 1-dimensional singular parabolic equation of p-Laplacian type (1 < p < 2). If u is bounded below on a time-segment {y} x (0, T] by a positive number M, then it has a power like decay of order 2 Pp with respect to the space variable x in R x [T/2, T]. This fact, stated quantitatively in Proposition 1.2, is a "sidewise spreading of positivity" of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.
$1$-dimensional Harnack estimates
Düzgün, Fatma Gamze;Vespri, Vincenzo
2016-01-01
Abstract
Let u be a non-negative super-solution to a 1-dimensional singular parabolic equation of p-Laplacian type (1 < p < 2). If u is bounded below on a time-segment {y} x (0, T] by a positive number M, then it has a power like decay of order 2 Pp with respect to the space variable x in R x [T/2, T]. This fact, stated quantitatively in Proposition 1.2, is a "sidewise spreading of positivity" of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.
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