Ismail Kombe
The Hardy inequality and Nonlinear parabolic equations on Carnot groups
(399K, PS)

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 $V\in L_{\text{loc}}^1(\Omega)$.