Year: 2022
Author: Junqiang Han
Journal of Partial Differential Equations, Vol. 35 (2022), Iss. 4 : pp. 307–319
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, $V\in L_{loc}^{1} (\Omega )$, $k\in \mathbb{N}$, $m\in \mathbb{R}$, $1<p<2n+2k$ and $m+p-2>0$. The better lower bound $p^*$ for $m + p_{ }$ is found and the nonexistence results are proved for $p^*\leqslant m+p<3$.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v35.n4.1
Journal of Partial Differential Equations, Vol. 35 (2022), Iss. 4 : pp. 307–319
Published online: 2022-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Doubly nonlinear degenerate parabolic equations Greiner vector fields positive solutions nonexistence Hardy inequality.