The Hardy inequality and Nonlinear parabolic equations on Carnot groups
Ismail Kombe
TL;DR
The paper studies the nonexistence of positive solutions for a nonlinear parabolic equation driven by the p-sub-Laplacian on Carnot groups, in bounded domains with Dirichlet boundary and nonnegative initial data. It develops a framework combining a sharp $L^p$ Hardy inequality on polarizable Carnot groups with spectral conditions on the potential $V$ to derive nonexistence results, depending on the range of $p$ and the geometry (via the homogeneous dimension $Q$). The authors first establish a sharp $L^p$ Hardy inequality on polarizable Carnot groups, then apply it to obtain nonexistence results for problem (1.1) under conditions like $ rac{2Q}{Q+1}\le p<2$ and $igsigma_{ ext{inf}}^p((1- frac{}{} ext{e})Vig)=- ablaigotimesig$, including cases with singular or oscillatory potentials, yielding corollaries in terms of potentials of the form $V(x)= rac{ abla_{ an G}N}{N^p}$ with explicit thresholds. The results generalize known Euclidean and Heisenberg-group outcomes to the broader class of Carnot groups, highlighting the role of Hardy-type bounds and group geometry in the behavior of nonlinear parabolic equations. These findings have implications for understanding how geometric structure of the underlying space governs the existence of positive solutions for nonlinear diffusion-type 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 $V\in L_{\text{loc}}^1(\Omega)$.