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\begin{document}

\title{A blow-up result for the wave equation with localized initial data: the
scale-invariant damping and mass term with combined nonlinearities}



\author[1]{Makram Hamouda}%
\author[1]{M.Ali Hamza}%
\affil[1]{Imam Abdulrahman Bin Faisal University}%


\vspace{-1em}



  
  \date{May 24, 2024}


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\begin{abstract}
We are interested in this article in studying the damped wave equation
with localized initial data, in the
\textbackslash{}textit\{scale-invariant case\} with mass term and two
combined nonlinearities. More precisely, we consider the following
equation: \textbackslash{}begin\{displaymath\} \textbackslash{}d (E)
\textbackslash{}hspace\{1cm\} u\_\{tt\}-\textbackslash{}Delta
u+\textbackslash{}frac\{\textbackslash{}mu\}\{1+t\}u\_t+\textbackslash{}frac\{\textbackslash{}nu\^{}2\}\{(1+t)\^{}2\}u=\textbar{}u\_t\textbar{}\^{}p+\textbar{}u\textbar{}\^{}q,
\textbackslash{}quad \textbackslash{}mbox\{in\}\textbackslash{}
\textbackslash{}R\^{}N\textbackslash{}times{[}0,\textbackslash{}infty),
\textbackslash{}end\{displaymath\} with small initial data. Under some
assumptions on the mass and damping coefficients, \$\textbackslash{}nu\$
and \$\textbackslash{}mu\textgreater{}0\$, respectively, we show that
blow-up region and the lifespan bound of the solution of \$(E)\$ remain
the same as the ones obtained in (missing citation) in the case of a
mass-free wave equation, \{\textbackslash{}it i.e.\} \$(E)\$ with
\$\textbackslash{}nu=0\$. Furthermore, using in part the computations
done for \$(E)\$, we enhance the result in (missing citation) on the
Glassey conjecture for the solution of \$(E)\$ with omitting the
nonlinear term \$\textbar{}u\textbar{}\^{}q\$. Indeed, the blow-up
region is extended from \$p \textbackslash{}in (1,
p\_G(N+\textbackslash{}sigma){]}\$, where \$\textbackslash{}sigma\$ is
given by \textbackslash{}eqref\{sigma\} below, to \$p \textbackslash{}in
(1, p\_G(N+\textbackslash{}mu){]}\$ yielding, hence, a better estimate
of the lifespan when
\$(\textbackslash{}mu-1)\^{}2-4\textbackslash{}nu\^{}2\textless{}1\$.
Otherwise, the two results coincide. Finally, we may conclude that the
mass term \{\textbackslash{}it has no influence\} on the dynamics of
\$(E)\$ (resp. \$(E)\$ without the nonlinear term
\$\textbar{}u\textbar{}\^{}q\$), and the conjecture we made in
(missing citation) on the threshold between the blow-up and the global
existence regions obtained holds true here.%
\end{abstract}%



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\section*{References}\sloppy
\end{document}

