1-DIMENSIONAL HARNACK ESTIMATES


DÜZGÜN F. G. , GIANAZZA U., Vespri V.

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S, vol.9, no.3, pp.675-685, 2016 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 3
  • Publication Date: 2016
  • Doi Number: 10.3934/dcdss.2016021
  • Title of Journal : DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
  • Page Numbers: pp.675-685

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.