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\begin{document}

\title{Nonexistence of Global Solutions of Systems of Time Fractional
Differential equations posed on the Heisenberg group}



\author[1]{Mokhtar Kirane}%
\author[1]{alrazi abdeljabbar}%
\affil[1]{Khalifa University of Science and Technology}%


\vspace{-1em}



  
  \date{May 22, 2024}


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\begin{abstract}
We first consider the nonlinear time fractional diffusion equation
D\^{}\{1+\selectlanguage{greek}α\selectlanguage{english}\}u+D\^{}\selectlanguage{greek}β \selectlanguage{english}u-[?]\_\{H\} u=\textbar{}u\textbar{}\^{}p posed on
the Heisenberg group H, where 1 \textless{} p is a positive real nimber
to be specified later; D\^{}\selectlanguage{greek}δ\selectlanguage{english}\_\{0\textbar{}t\} is the Liouville-Caputo
derivative of order \selectlanguage{greek}δ. \selectlanguage{english}For 0 \textless{} \selectlanguage{greek}α \selectlanguage{english}\textless{} 1,0 \textless{} \selectlanguage{greek}β
\selectlanguage{english}[?] \selectlanguage{english}1. This equation interpolates the heat equation and the wave equation
with the linear damping D\^{}\selectlanguage{greek}β\selectlanguage{english}\_\{0\textbar{}t\}u. We present the Fujita
exponent for blow-up. Then establish sufficient conditions ensuring
non-existence of local solutions. We extend the analysis to the case of
the system D\^{}\{1+\selectlanguage{greek}α\selectlanguage{english}\}u+D\^{}\selectlanguage{greek}β \selectlanguage{english}u-[?]\_\{H\} u=\textbar{}v\textbar{}\^{}q
D\^{}\{1+\selectlanguage{greek}δ\selectlanguage{english}\}v+D\^{}\selectlanguage{greek}γ \selectlanguage{english}v-[?]\_\{H\} v=\textbar{}u\textbar{}\^{}p. Our method
of proof is based on the nonlinear capacity method.%
\end{abstract}%



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