In this article, we establish the phenomenon of existence and nonexistence of positive weak solutions of parabolic quasi-linear equations perturbed by a singular Hardy potential on the whole Euclidean space depending on the controllability of the given singular potential. To control the singular potential we use a weighted Hardy inequality with an optimal constant, which was recently discovered in Hauer and Rhandi (2013). Our results in this paper extend the ones in Goldstein et al. (2012) concerning a linear Kolmogorov operator significantly in several ways: firstly, by establishing existence of positive global solutions of singular parabolic equations involving nonlinear operators of p-Laplace type with a nonlinear convection term for 1 < p < ∞, and secondly, by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions.

Existence and nonexistence of positive solutions of p-Kolmogorov equations perturbed by a Hardy potential

RHANDI, Abdelaziz
2016

Abstract

In this article, we establish the phenomenon of existence and nonexistence of positive weak solutions of parabolic quasi-linear equations perturbed by a singular Hardy potential on the whole Euclidean space depending on the controllability of the given singular potential. To control the singular potential we use a weighted Hardy inequality with an optimal constant, which was recently discovered in Hauer and Rhandi (2013). Our results in this paper extend the ones in Goldstein et al. (2012) concerning a linear Kolmogorov operator significantly in several ways: firstly, by establishing existence of positive global solutions of singular parabolic equations involving nonlinear operators of p-Laplace type with a nonlinear convection term for 1 < p < ∞, and secondly, by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4649899
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