Nonlinear degenerate parabolic equations with singular coefficients for Greiner vector fields

Abstract

In this work, we consider the following nonlinear parabolic partial differential equations:

in a cylinder Ω × (0, T) with initial condition u(w, 0) = u ₀(w) ≥ 0 and vanishing on the boundary ∂Ω × (0, T), where Ω is a Carnot Carathéodory metric ball in ℝ²ⁿ⁺¹ with smooth boundary and the time-dependent potential function is , 0 < m < q and 1 < p < q + 1. We investigate the nonexistence of positive solutions of these two problems and present our results on nonexistence.

Keywords:

• nonlinear parabolic equations
• Greiner vector fields
• positive solutions
• nonexistence

AMS Subject Classifications: :

• 35K55
• 35H10
• 35K65
• 35R05

Acknowledgement

I would like to thank the referees for their constructive helps leading to improvements to this article.