Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields (Journal of Partial Differential Equations, 2022)
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}= abla_{k} \cdot \left( {u^{m-1}\left| { abla_{k} u} \right|^{p-2} abla_{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, $V\in L_{loc}^{1} (\Omega )$, $k\in \mathbb{N}$, $m\in \mathbb{R}$, $1
Keywords: Doubly nonlinear degenerate parabolic equations; Greiner vector fields; positive solutions; nonexistence; Hardy inequality.