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Volume 35, Issue 4
Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields

Junqiang Han

DOI: 10.4208/jpde.v35.n4.1

J. Part. Diff. Eq., 35 (2022), pp. 307-319.

Published online: 2022-10

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  • 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 $p^*\leqslant  m+p<3$.

  • Keywords

Doubly nonlinear degenerate parabolic equations, Greiner vector fields, positive solutions, nonexistence, Hardy inequality.

  • AMS Subject Headings

35K55, 35K65

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

southhan@163.com (Junqiang Han)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-35-307, author = {Han , Junqiang}, title = {Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields}, journal = {Journal of Partial Differential Equations}, year = {2022}, volume = {35}, number = {4}, pages = {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 $p^*\leqslant  m+p<3$.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v35.n4.1}, url = {http://global-sci.org/intro/article_detail/jpde/21050.html} }
TY - JOUR T1 - Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields AU - Han , Junqiang JO - Journal of Partial Differential Equations VL - 4 SP - 307 EP - 319 PY - 2022 DA - 2022/10 SN - 35 DO - http://doi.org/10.4208/jpde.v35.n4.1 UR - https://global-sci.org/intro/article_detail/jpde/21050.html KW - Doubly nonlinear degenerate parabolic equations, Greiner vector fields, positive solutions, nonexistence, Hardy inequality. AB -

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 $p^*\leqslant  m+p<3$.

Han , Junqiang. (2022). Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields. Journal of Partial Differential Equations. 35 (4). 307-319. doi:10.4208/jpde.v35.n4.1
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