The Hardy inequality and Nonlinear parabolic equations on Carnot groups

Abstract

In this paper we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equation:\[ \begin{cases} \frac{\partial u}{\partial t}= \Delta_{\mathbb{G},p}u+V(x)u^{p-1} & \text{in}\quad \Omega \times (0, T), \quad 1<p<2, u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T) \end{cases} \] where $ \Delta_{\mathbb{G},p}$ is the $p$-sub-Laplacian on Carnot group $ \mathbb{G}$ and $V\in L_{\text{loc}}^1(\Omega)$.