Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields

Abstract

The purpose of this paper is to investigate the nonexistence of positive solutions of the following doubly nonlinear degenerate parabolic equations: \begin{align*}\begin{cases}  {\dfrac{\partial u}{\partial t}=\nabla_{k} \cdot \left( {u^{m-1}\left| {\nabla_{k} u} \right|^{p-2}\nabla_{k} u} \right)+V(w)u^{m+p-2}},\qquad & {\mbox{in}\  \Omega \times (0,T),} \\  {u(w,0)=u_{0} (w)\geqslant 0}, & {\mbox{in}\  \Omega ,} \\  {u(w,t)=0}, & {\mbox{on}\  \partial \Omega \times (0,T),}  \end{cases} \end{align*} where $\Omega$ is a Carnot-Carathéodory metric ball in $\mathbb{R}^{2n+1}$ generated by Greiner vector fields, $V\in L_{loc}^{1} (\Omega )$, $k\in \mathbb{N}$, $m\in \mathbb{R}$, $10$. The better lower bound $p^*$ for $m + p_{ }$ is found and the nonexistence results are proved for $p^*\leqslant  m+p<3$.