 \documentclass[10pt]{studiamnew}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage[T1]{fontenc}
\usepackage{amssymb}


\sloppy
\setcounter{page}{1} \setcounter{firstpage}{1} \setcounter{lastpage}{14} %
\renewcommand{\currentvolume}{} \renewcommand{\currentyear}{2019} %
\renewcommand{\currentissue}{}
\newtheorem{theorem}{Theorem} \newtheorem{example}[theorem]{Example} %
\newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} %
\newtheorem{corollary}[theorem]{Corollary} %
\newtheorem{acknowledgement}[theorem]{Acknowledgement} %
\newtheorem{proposition}[theorem]{Proposition} %
\newtheorem{claim}[theorem]{Claim}
\newcommand{\QATOP}[2]{\genfrac{}{}{0pt}{}{#1}{#2}}
\newcommand{\func}[1]{\operatorname{#1}}
\newtheorem{notation}[theorem]{Note}
 \begin{document}

\title[Bounds of the blow-up time ]{Bounds for blow-up time in a semilinear
parabolic problem with variable exponents}
\author{Abita Rahmoune}
\address{Laboratory of Pure and Applied Mathematics, University of Laghouat, P.O. Box 37G,
Laghouat 03000, Algeria}
\email{abitarahmoune@yahoo.fr}


\subjclass{Primary:35K55; 35K60; 35B44; Secondary: 74G45 }
\keywords{Parabolic equation; Variable nonlinearity; Bounds of the blow-up
time}

\begin{abstract}
This report deals with a blow-up of the solutions to a class of semilinear
parabolic equations with variable exponents nonlinearities. Under some
appropriate assumptions on the given data, more general lower bounds for a
blow-up time are gained if the solutions blow up. This result extends a
recent results by Baghaei Khadijeh et al. \cite{Baghaei}, which confirms the
Lower bounds for the blow-up time of solutions with initial data $\varphi
\left( 0\right) =\int_{\Omega }u_{0}{}^{k}dx$, $k$=constant.
\end{abstract}

\maketitle

\section{Introduction}

In this paper, we are concerned with the following semilinear parabolic
equation%
\begin{equation}
\left\{
\begin{array}{c}
u_{t}-\Delta u=u^{p\left( x\right) },\text{\ \ }x\in \Omega ,\text{ }t>0 \\
u=0\text{ on }\Gamma \text{, \ }t\geq 0 \\
u(x,0)=u_{0}(x)\geq 0\text{,\ }x\in \Omega ,%
\end{array}%
\right.  \label{1}
\end{equation}%
where $\Omega $ be a bounded domain in $\mathbb{R}^{n}$, with a smooth
boundary $\Gamma =\partial \Omega $, $T\in \left( 0,+\infty \right] $, and
the initial value $u_{0}\in H_{0}^{1}\left( \Omega \right) ,$ the exponent $%
p\left( .\right) $ is given measurable function on $\overline{\Omega }$ such
that:%
\begin{equation}
1<p_{1}=\underset{x\in \Omega }{ess\text{ }\inf }p\left( x\right) \leq
p\left( x\right) \leq p_{2}=\underset{x\in \Omega }{ess\text{ }\sup }p\left(
x\right) <\infty ,  \label{01}
\end{equation}%
and satisfy the following Zhikov-Fan uniform local continuity condition:%
\begin{equation}
\left\vert p\left( x\right) -p\left( y\right) \right\vert \leq \frac{M}{%
\left\vert \log \left\vert x-y\right\vert \right\vert }\text{, for all }x,%
\text{ }y\text{ in }\Omega \text{ with }\left\vert x-y\right\vert <\frac{1}{2%
},\text{ }M>0\text{.}  \label{03}
\end{equation}%
The problem (\ref{1}) arises from many important mathematical models in
engineering and physical sciences. For example, nuclear science, chemical
reactions, heat transfer, population dynamics, biological sciences, etc.,
and have interested a great deal of attention in the research, see \cite%
{Acerbi, Antonsev20, Ferreira} and the references therein. For problem (\ref%
{1}), Hua Wang et al. \cite{HuaWang} established a blow-up result with
positive initial energy under some suitable assumptions on the parameters $%
p(.)$ and $u_{0}$.

In \cite{Ferreira}, the authors proved that there are non-negative solutions
with a blow-up in finite time if and only if $p_{2}>1$. The authors in \cite%
{BinGao} obtained the solution of problem (\ref{1}) blows up in finite time
when the initial energy is positive.

In \cite{Baghaei}, the authors based exactly on the idea on the one in \cite%
{Aigu}, derived the lower bounds for the time of blow-up, if the solutions
blow up.

In order to declare the main results of this paper, we need to add the
following energy functional corresponding to the problem (\ref{1})%
\begin{equation}
E\left( t\right) =\frac{1}{2}\int_{\Omega }\left\vert \nabla u\left(
t\right) \right\vert ^{2}dx-\int_{\Omega }\frac{1}{p\left( x\right) +1}%
u\left( x,t\right) ^{p\left( x\right) +1}dx.  \label{8}
\end{equation}

\section{Lower bounds of the blow-up time}

In this section, we investigate the lower bound for the blow-up time $T$ in
some suitable measure. The idea of the proof of the following theorem is
inspired by on the one in \cite{Aigu}. For this goal, we start by the
following lemma concerning the energy of the solution.

\begin{lemma}
\label{Lem1}Let $u(x,t)$ be a weak solution of (\ref{1}), then $E(t)$ is a
nonincreasing function on $[0,T]$, that is%
\begin{equation}
\frac{dE\left( t\right) }{dt}=-\int_{\Omega }u_{t}^{2}\left( x,t\right)
dx\leq 0  \label{9}
\end{equation}%
and the inequality $E(t)\leq E(0)$ is satisfied.
\end{lemma}
We consider the following partition of $\Omega ,$%
\begin{equation*}
\Omega ^{-}=\left\{ x\in \Omega \mid 1>\left( k\left( x\right) -1\right) \ln
\left\vert u\right\vert \right\} ,\text{ \ }\Omega ^{+}=\left\{ x\in \Omega
\mid 1\leq \left( k\left( x\right) -1\right) \ln \left\vert u\right\vert
\right\} ,\text{ }\forall t>0
\end{equation*}%
where each $\Omega ^{\pm }$ depends on $t$, and setting%
\begin{equation*}
\widetilde{E}\left( 0\right) =\frac{1}{2}\int_{\Omega ^{-}}\left\vert \nabla
u_{0}\right\vert ^{2}dx-\int_{\Omega ^{-}}\frac{1}{p\left( x\right) +1}%
u_{0}^{p\left( x\right) +1}dx.
\end{equation*}%
Now, we are in a position to affirm our principal theorem results.

\begin{theorem}
\label{Theorem1}Assume $u_{0}\in L^{k\left( .\right) }\left( \Omega \right) $%
, and the nonnegative weak solution $u(x,t)$ of problem (\ref{1}) blows up
in finite time $T$, then $T$ has a lower bound by:%
\begin{equation}
\int_{\varphi \left( 0\right) }^{+\infty }\frac{d\gamma }{C_{1}+C_{2}\gamma
^{\frac{3n-6}{3n-8}}},  \label{07}
\end{equation}%
where%
\begin{equation}
\varphi \left( 0\right) =\int_{\Omega }\frac{1}{k\left( x\right) \left(
k\left( x\right) -1\right) }u_{0}{}^{k\left( x\right) }dx,  \label{08}
\end{equation}%
where $k\left( .\right) $ is a measurable function on $\overline{\Omega }$
such that%
\begin{equation}
\max \left( 1,2\left( n-2\right) \left( p_{2}-1\right) \right) <k_{1}=%
\underset{x\in \Omega }{ess\text{ }\inf }k\left( x\right) \leq k\left(
x\right) \leq k_{2}=\underset{x\in \Omega }{ess\text{ }\sup }k\left(
x\right) <\infty ,  \label{21}
\end{equation}%
and
\begin{equation}
\sqrt{C_{k}}\underset{x\in \overline{\Omega }}{=\sup }\left\vert \nabla
k\left( x\right) \right\vert \in L^{2}\left( \Omega \right) ,\text{ }C_{k}>0
\label{021}
\end{equation}%
and $C_{i}$ $\left( i=1,\text{ }2\right) $ are positive constants will be
described later.
\end{theorem}

\begin{notation}
We note that the presence of the variable-exponent nonlinearities in (\ref%
{02}) below, makes analysis in the paper somewhat harder than that in the
related ones, we will establish and give a precise estimate for the lifespan
$T$ of the solution in this case. The method used here is the differential
inequality technique. However, our argument is considerably different and it
is more abbreviated.
\end{notation}

\begin{proof}[Proof of Theorem \protect\ref{Theorem1}]
Set%
\begin{equation}
\varphi \left( t\right) =\int_{\Omega }\frac{1}{k\left( x\right) \left(
k\left( x\right) -1\right) }u\left( x,t\right) ^{k\left( x\right) }dx.
\label{02}
\end{equation}%
Multiplying the equation Eq. (\ref{1}) by $u$ and integrating by parts, we
see%
\begin{gather*}
\varphi ^{\prime }\left( t\right) =\int_{\Omega }\frac{1}{k\left( x\right) -1%
}u^{k\left( x\right) -1}u_{t}dx=\int_{\Omega }\frac{1}{k\left( x\right) -1}%
u^{k\left( x\right) -1}\left( \Delta u+u^{p\left( x\right) }\right) dx \\
=\int_{\Omega }\frac{1}{k\left( x\right) -1}u^{k\left( x\right) -1}\Delta
udx+\int_{\Omega }\frac{1}{k\left( x\right) -1}u^{k\left( x\right) +p\left(
x\right) -1}dx \\
=-\int_{\Omega }\nabla \left( \frac{1}{k\left( x\right) -1}u^{k\left(
x\right) -1}\right) .\nabla udx+\int_{\Omega }\frac{1}{k\left( x\right) -1}%
\left\vert u\right\vert ^{k\left( x\right) +p\left( x\right) -1}dx
\end{gather*}%
where we have used the divergence theorem, the boundary condition on $u.$

It is straightforward to check that%
\begin{equation*}
\nabla \left( \frac{1}{k\left( x\right) -1}u^{k\left( x\right) -1}\right)
=u^{k\left( x\right) }\left\vert u\right\vert ^{-2}\nabla u+\frac{\nabla
k\left( x\right) }{k\left( x\right) -1}u^{k\left( x\right) -1}\left( \ln
\left\vert u\right\vert -\frac{1}{k\left( x\right) -1}\right)
\end{equation*}%
then, we get%
\begin{equation}
\varphi ^{\prime }\left( t\right) =-\int_{\Omega }u^{k\left( x\right)
}\left\vert u\right\vert ^{-2}\left\vert \nabla u\right\vert
^{2}dx+\int_{\Omega }\frac{1}{k\left( x\right) -1}u^{k\left( x\right)
+p\left( x\right) -1}dx+\mathcal{Q}  \label{012}
\end{equation}%
where%
\begin{equation*}
\mathcal{Q=}\int_{\Omega }u^{k\left( x\right) -1}\left( \frac{1}{\left(
k\left( x\right) -1\right) ^{2}}-\frac{1}{\left( k\left( x\right) -1\right) }%
\ln \left\vert u\right\vert \right) \nabla k\left( x\right) .\nabla udx
\end{equation*}%
Considering the following properties of the function $\mathcal{G}$,
\begin{gather*}
\mathcal{G}\left( \lambda \right) =\frac{\lambda ^{\gamma }}{\gamma ^{2}}%
\left( 1-\gamma \ln \lambda \right) ,\text{ \ }0\leq \lambda \leq e^{\frac{1%
}{\gamma }}; \\
\mathcal{G}\left( 0\right) =\mathcal{G}\left( e^{\frac{1}{\gamma }}\right)
=0,\text{ }\mathcal{G}^{\prime }\left( \lambda \right) =-\lambda ^{\gamma
-1}\ln \lambda ,\text{ }\underset{0\leq \lambda \leq e^{\frac{1}{\gamma }}}{%
\max }\mathcal{G}\left( \lambda \right) =\mathcal{G}\left( 1\right) =\frac{1%
}{\gamma ^{2}},
\end{gather*}
and using the fact that
\begin{equation*}
\int_{\Omega ^{-}}\left\vert \nabla u\right\vert ^{2}dx\leq 2\widetilde{E}%
\left( 0\right) +2\int_{\Omega ^{-}}\frac{1}{p\left( x\right) +1}u\left(
x,t\right) ^{p\left( x\right) +1}dx,\text{ (by (\ref{8}) and (\ref{9}))}
\end{equation*}%
applying the H\"{o}lder, Young inequalities and (\ref{021}), $\mathcal{Q}$
is evaluated as follows:%
\begin{gather}
\mathcal{Q}=\int_{\Omega }u^{k\left( x\right) -1}\left( \frac{1}{\left(
k\left( x\right) -1\right) ^{2}}-\frac{1}{k\left( x\right) -1}\ln \left\vert
u\right\vert \right) \nabla k\left( x\right) .\nabla udx  \notag \\
=\int_{\Omega \cap \left( 1>\left( k\left( x\right) -1\right) \ln \left\vert
u\left( x,t\right) \right\vert \right) }u^{k\left( x\right) -1}\left( \frac{1%
}{\left( k\left( x\right) -1\right) ^{2}}-\frac{1}{k\left( x\right) -1}\ln
\left\vert u\right\vert \right) \nabla k\left( x\right) .\nabla udx  \notag
\\
\int_{\Omega \cap \left( 1\leq \left( k\left( x\right) -1\right) \ln
\left\vert u\left( x,t\right) \right\vert \right) }u^{k\left( x\right)
-1}\left( \frac{1}{\left( k\left( x\right) -1\right) ^{2}}-\frac{1}{k\left(
x\right) -1}\ln \left\vert u\right\vert \right) \nabla k\left( x\right)
.\nabla udx  \notag \\
\leq \int_{\Omega ^{-}}\frac{1}{\left( k\left( x\right) -1\right) ^{2}}%
\left\vert u\right\vert ^{k\left( x\right) -1}\left( 1-\left( k\left(
x\right) -1\right) \ln \left\vert u\right\vert \right) \left\vert \nabla
u\right\vert \left\vert \nabla k\left( x\right) \right\vert dx  \notag \\
\leq \int_{\Omega ^{-}}\frac{1}{\left( k\left( x\right) -1\right) ^{2}}%
\left\vert \nabla k\left( x\right) \right\vert \left\vert \nabla
u\right\vert dx\leq \frac{1}{2\left( k_{1}-1\right) ^{2}}\left(
C_{k}+\int_{\Omega ^{-}}\left\vert \nabla u\right\vert ^{2}dx\right)  \notag
\\
\leq \frac{1}{2\left( k_{1}-1\right) ^{2}}\left( C_{k}+2E\left( 0\right)
+2\int_{\Omega ^{-}}\frac{1}{p\left( x\right) +1}u\left( x,t\right)
^{p\left( x\right) +1}dx\right)  \notag \\
\leq \frac{1}{2\left( k_{1}-1\right) ^{2}}\left( C_{k}+2E\left( 0\right) +%
\frac{2}{p_{1}+1}\max \left( \int_{\Omega ^{-}}\left\vert u\right\vert
^{p_{2}+1}dx,\int_{\Omega ^{-}}\left\vert u\right\vert ^{p_{1}+1}dx\right)
\right)  \notag \\
\leq \frac{1}{\left( k_{1}-1\right) ^{2}}\left( \frac{1}{2}C_{k}+E\left(
0\right) +\frac{1}{p_{1}+1}e^{\frac{p_{2}+1}{k_{1}-1}}\left\vert \Omega
\right\vert \right) .  \label{22}
\end{gather}
Because in $\Omega ^{+}$, we have%
\begin{equation*}
\int\limits_{\Omega ^{+}}\left\vert u\right\vert ^{k\left( x\right)
-1}\left( \frac{1}{\left( k\left( x\right) -1\right) ^{2}}-\frac{1}{k\left(
x\right) -1}\ln \left\vert u\right\vert \right) \left\vert \nabla k\left(
x\right) \right\vert dx\leq 0,
\end{equation*}%
while that of the first term in the right-hand side of (\ref{012}) was
estimated as follows%
\begin{equation*}
-\int_{\Omega }\left\vert u\right\vert ^{k\left( x\right) -2}\left\vert
\nabla u\right\vert ^{2}dx\leq -\min \left( \int_{\Omega }\left\vert
u\right\vert ^{k_{2}-2}\left\vert \nabla u\right\vert ^{2}dx,\int_{\Omega
}\left\vert u\right\vert ^{k_{1}-2}\left\vert \nabla u\right\vert
^{2}dx\right) .
\end{equation*}%
Using the fact%
\begin{equation*}
\left\vert \nabla u^{\gamma }\right\vert =\gamma u^{\gamma -1}\left\vert
\nabla u\right\vert
\end{equation*}%
to get%
\begin{equation}
-\int_{\Omega }\left\vert u\right\vert ^{k\left( x\right) -2}\left\vert
\nabla u\right\vert ^{2}dx\leq -\min \left( \frac{4}{\left( k_{2}\right) ^{2}%
}\int_{\Omega }\left\vert \nabla u^{\frac{k_{2}}{2}}\right\vert ^{2}dx,\frac{%
4}{\left( k_{1}\right) ^{2}}\int_{\Omega }\left\vert \nabla u^{\frac{k_{1}}{2%
}}\right\vert ^{2}dx\right)  \label{23}
\end{equation}%
Plugging this estimate (\ref{22}) and (\ref{23}) into (\ref{012}), we obtain
\begin{gather}
\varphi ^{\prime }\left( t\right) \leq \min \left( \frac{-4}{\left(
k_{2}\right) ^{2}}\int_{\Omega }\left\vert \nabla u^{\frac{k_{2}}{2}%
}\right\vert ^{2}dx,\frac{-4}{\left( k_{1}\right) ^{2}}\int_{\Omega
}\left\vert \nabla u^{\frac{k_{1}}{2}}\right\vert ^{2}dx\right)  \notag \\
+\frac{1}{k_{1}-1}\int_{\Omega }u^{k\left( x\right) +p_{2}-1}dx+\frac{1}{%
k_{1}-1}\int_{\Omega }u^{k\left( x\right) +p_{1}-1}dx  \notag \\
+\frac{1}{\left( k_{1}-1\right) ^{2}}\left( \frac{1}{2}C_{k}+E\left(
0\right) +\frac{1}{p_{1}+1}e^{\frac{p_{2}+1}{k_{1}-1}}\left\vert \Omega
\right\vert \right)  \label{24}
\end{gather}
By using (\ref{21}), we can apply the H\"{o}lder and Young inequalities to
get%
\begin{eqnarray}
\int_{\Omega }u^{k\left( x\right) +p_{2}-1}dx &\leq &\int_{\Omega }1.\alpha
_{1}dx+\int_{\Omega }\alpha _{2}.u^{\frac{k\left( x\right) \left(
2n-3\right) }{2\left( n-2\right) }}dx  \label{25} \\
&\leq &\left( \sup \alpha _{1}\right) \left\vert \Omega \right\vert +\left(
\sup \alpha _{2}\right) \left( \int_{\Omega }u^{\frac{k\left( x\right)
\left( 2n-3\right) }{2\left( n-2\right) }}dx\right) ,  \notag
\end{eqnarray}%
and%
\begin{eqnarray}
\int_{\Omega }u^{k\left( x\right) +p_{1}-1}dx &\leq &\int_{\Omega }1.\alpha
_{3}dx+\int_{\Omega }\alpha _{4}.u^{\frac{k\left( x\right) \left(
2n-3\right) }{2\left( n-2\right) }}dx  \label{26} \\
&\leq &\left( \underset{\Omega }{\sup }\alpha _{3}\right) \left\vert \Omega
\right\vert +\left( \underset{\Omega }{\sup }\alpha _{4}\right) \left(
\int_{\Omega }u^{\frac{k\left( x\right) \left( 2n-3\right) }{2\left(
n-2\right) }}dx\right) ,  \notag
\end{eqnarray}%
where%
\begin{gather*}
\alpha _{1}=1-\frac{2\left( n-2\right) \left( k\left( x\right)
+p_{2}-1\right) }{\left( 2n-3\right) k\left( x\right) },\text{ \ \ }\alpha
_{2}=\frac{2\left( n-2\right) \left( k\left( x\right) +p_{2}-1\right) }{%
\left( 2n-3\right) k\left( x\right) }, \\
\alpha _{3}=1-\frac{2\left( n-2\right) \left( k\left( x\right)
+p_{1}-1\right) }{\left( 2n-3\right) k\left( x\right) },\text{ \ \ }\alpha
_{4}=\frac{2\left( n-2\right) \left( k\left( x\right) +p_{1}-1\right) }{%
\left( 2n-3\right) k\left( x\right) }; \\
\text{observe that }\alpha _{2}\geq \alpha _{4}\text{ and }\alpha _{1}\leq
\alpha _{3}.
\end{gather*}%
Combining (\ref{25}) and (\ref{26}) with (\ref{24}) give%
\begin{gather}
\varphi ^{\prime }\left( t\right) \leq \frac{-1}{2}\frac{4}{\left(
k_{2}\right) ^{2}}\left( \int_{\Omega }\left\vert \nabla u^{\frac{k_{2}}{2}%
}\right\vert ^{2}dx+\int_{\Omega }\left\vert \nabla u^{\frac{k_{1}}{2}%
}\right\vert ^{2}dx\right)  \notag \\
+\frac{2}{k_{1}-1}\left( \underset{\Omega }{\sup }\alpha _{2}\right)
\int_{\Omega }u^{\frac{k\left( x\right) \left( 2n-3\right) }{2\left(
n-2\right) }}dx  \notag \\
+\frac{1}{\left( k_{1}-1\right) ^{2}}\left( \frac{1}{2}C_{k}+E\left(
0\right) +\frac{1}{p_{1}+1}e^{\frac{p_{2}+1}{k_{1}-1}}\left\vert \Omega
\right\vert \right) +\frac{\left\vert \Omega \right\vert }{k_{1}-1}\underset{%
\Omega }{\sup }\left( \alpha _{3}+\alpha _{1}\right)  \label{27}
\end{gather}%
We now make use of Schwarz's inequality to the second term on the right-hand
side of (\ref{27}) as follows%
\begin{gather}
\int_{\Omega }u^{\frac{k\left( x\right) \left( 2n-3\right) }{2\left(
n-2\right) }}dx\leq \left( \int_{\Omega }u^{k\left( x\right) }dx\right) ^{%
\frac{1}{2}}\left( \int_{\Omega }u^{\frac{k\left( x\right) \left( n-1\right)
}{n-2}}dx\right) ^{\frac{1}{2}}  \label{28} \\
\leq \left( \int_{\Omega }u^{k\left( x\right) }dx\right) ^{\frac{3}{4}%
}\left( \int_{\Omega }\left( u^{^{\frac{k\left( x\right) }{2}}}\right) ^{%
\frac{2n}{n-2}}dx\right) ^{\frac{1}{4}},  \notag
\end{gather}%
Next, by using the Sobolev inequality (see \cite{Adams}), for $n\geq 3$, we
get%
\begin{eqnarray}
\left\Vert u^{^{\frac{k\left( x\right) }{2}}}\right\Vert _{\frac{2n}{n-2}}^{%
\frac{n}{2\left( n-2\right) }} &\leq &B^{\frac{n}{2\left( n-2\right) }}\max
\left( \left\Vert \nabla u^{^{\frac{k_{2}}{2}}}\right\Vert _{2}^{\frac{n}{%
2\left( n-2\right) }},\left\Vert \nabla u^{^{\frac{k_{1}}{2}}}\right\Vert
_{2}^{\frac{n}{2\left( n-2\right) }}\right)  \label{29} \\
&\leq &B^{\frac{n}{2\left( n-2\right) }}\left( \left\Vert \nabla u^{^{\frac{%
k_{2}}{2}}}\right\Vert _{2}^{\frac{n}{2\left( n-2\right) }}+\left\Vert
\nabla u^{^{\frac{k_{1}}{2}}}\right\Vert _{2}^{\frac{n}{2\left( n-2\right) }%
}\right) ,  \notag
\end{eqnarray}%
where $B$ is the best constant in the Sobolev inequality.

By inserting the last inequality in (\ref{28}) and (\ref{29}), we have%
\begin{equation*}
\int_{\Omega }u^{\frac{k\left( x\right) \left( 2n-3\right) }{2\left(
n-2\right) }}dx\leq B^{\frac{n}{2\left( n-2\right) }}\left( \int_{\Omega
}u^{k\left( x\right) }dx\right) ^{\frac{3}{4}}\left( \left( \int_{\Omega
}\left\vert \nabla u^{^{\frac{k_{2}}{2}}}\right\vert ^{2}dx\right) ^{\frac{n%
}{4\left( n-2\right) }}+\left( \int_{\Omega }\left\vert \nabla u^{^{\frac{%
k_{1}}{2}}}\right\vert ^{2}dx\right) ^{\frac{n}{4\left( n-2\right) }}\right)
,
\end{equation*}%
Now, we can use the Young inequality to get%
\begin{gather}
\int_{\Omega }u^{\frac{k\left( x\right) \left( 2n-3\right) }{2\left(
n-2\right) }}dx\leq 2B^{\frac{2n}{3n-8}}\frac{3n-8}{4\left( n-2\right)
\varepsilon ^{\frac{n}{3n-8}}}\left( \int_{\Omega }u^{k\left( x\right)
}dx\right) ^{\frac{3\left( n-2\right) }{3n-8}}  \label{30} \\
+\frac{\varepsilon n}{4\left( n-2\right) }\left( \int_{\Omega }\left\vert
\nabla u^{^{\frac{k_{1}}{2}}}\right\vert ^{2}dx+\int_{\Omega }\left\vert
\nabla u^{^{\frac{k_{2}}{2}}}\right\vert ^{2}dx\right)  \notag
\end{gather}%
where $\varepsilon $ is a positive constant to be determined later.
Combining (\ref{30}) with (\ref{27}), we obtain%
\begin{equation*}
\varphi ^{\prime }\left( t\right) \leq C_{1}+C_{2}\varphi \left( t\right) ^{%
\frac{3\left( n-2\right) }{3n-8}}+C_{3}\left( \int_{\Omega }\left\vert
\nabla u^{\frac{k_{2}}{2}}\right\vert ^{2}dx+\int_{\Omega }\left\vert \nabla
u^{\frac{k_{1}}{2}}\right\vert ^{2}dx\right) ,
\end{equation*}%
where%
\begin{gather*}
C_{1}=\frac{1}{\left( k_{1}-1\right) ^{2}}\left( \frac{1}{2}C_{k}+E\left(
0\right) +\frac{1}{p_{1}+1}e^{\frac{p_{2}+1}{k_{1}-1}}\left\vert \Omega
\right\vert \right) +\frac{\left\vert \Omega \right\vert }{k_{1}-1}\underset{%
\Omega }{\sup }\left( \alpha _{3}+\alpha _{1}\right) \\
C_{2}=\frac{4}{k_{1}-1}\left( \underset{\Omega }{\sup }\alpha _{2}\right) B^{%
\frac{2n}{3n-8}}\frac{3n-8}{4\left( n-2\right) \varepsilon ^{\frac{n}{3n-8}}}%
, \\
C_{3}=\frac{2}{k_{1}-1}\frac{\varepsilon n}{4\left( n-2\right) }\left(
\underset{\Omega }{\sup }\alpha _{2}\right) -\frac{2}{\left( k_{2}\right)
^{2}}
\end{gather*}%
If we choose $\varepsilon >0$ such that%
\begin{equation*}
0<\varepsilon \leq \frac{4\left( n-2\right) \left( k_{1}-1\right) }{\left(
\underset{\Omega }{\sup }\alpha _{2}\right) n\left( k_{2}\right) ^{2}}
\end{equation*}%
then, we obtain the differential inequality%
\begin{equation}
\varphi ^{\prime }\left( t\right) \leq C_{1}+C_{2}\varphi \left( t\right) ^{%
\frac{3\left( n-2\right) }{3n-8}}  \label{31}
\end{equation}%
Integration of the differential inequality (\ref{31}) from $0$ to $t$ leads
to%
\begin{equation}
\int_{\varphi \left( 0\right) }^{\varphi \left( t\right) }\frac{d\gamma }{%
C_{1}+C_{2}\gamma ^{\frac{3\left( n-2\right) }{3n-8}}}\leq t  \label{32}
\end{equation}%
In fact, let $t\rightarrow T^{-},$ (\ref{32}) leads to%
\begin{equation*}
\int_{\varphi \left( 0\right) }^{+\infty }\frac{d\gamma }{C_{1}+C_{2}\gamma
^{\frac{3\left( n-2\right) }{3n-8}}}\leq T.
\end{equation*}%
where%
\begin{equation*}
\varphi \left( 0\right) =\int_{\Omega }\frac{1}{k\left( x\right) \left(
k\left( x\right) -1\right) }u_{0}{}^{k\left( x\right) }dx.
\end{equation*}%
Thus, the proof is achieved.
\end{proof}

\subsection*{Acknowledgements.}

The author is highly grateful to the anonymous referee for his/her valuable
comments and suggestions for the improvement of the paper.

\begin{thebibliography}{9}
\bibitem{Acerbi} Acerbi,  E., Mingione, G.,~ \emph{Regularity results for
stationary electrorheological fluids}, Arch. Ration. Mech. Anal \textbf{164}
(2002), 213--259.

\bibitem{Antonsev20} Antontsev, S., Shmarev, S.,  \emph{Blow-up of solutions to
parabolic equations with nonstandard growth conditions}, J. Comput. Appl.
Math \textbf{234} (2010), 2633--2645.

\bibitem{Aigu} Aiguo, B.,  Xianfa, S., \emph{Bounds for the blowup time of
the solutions to quasi-linear parabolic problems}, Zeitschrift f\"{u}r
Angewandte Mathematik und Physik (ZAMP) \textbf{65} (2014).

\bibitem{Baghaei} Baghaei, K., Ghaemi,~M. B., Hesaaraki, M., \emph{Lower
bounds for the blow-up time in a semilinear parabolic problem involving a
variable source}, Applied Mathematics Letters \textbf{27} (2014).

\bibitem{Ferreira} Ferreira, R., de Pablo, A., P�rez-LLanos, M., Rossi, J. D.,
 \emph{Critical exponents for a semilinear parabolic equation with
variable reaction}, Proceedings of the Royal Society of Edinburgh Section A
Mathematics \textbf{142} (2012).

\bibitem{Adams} Adams, R.A., \emph{Sobolev spaces}, AP, 1975.

\bibitem{HuaWang} Wang, H., He, Y.,  \emph{On blow-up of solutions for a
semilinear parabolic equation involving variable source and positive initial
energy}, Applied Mathematics Letters \textbf{26} (2013), no.~10, 1008 --1012.

\bibitem{BinGao} Wu, X., Guo, B., Gao, W.,  \emph{Blow-up of solutions for a
semilinear parabolic equation involving variable source and positive initial
energy}, Applied Mathematics Letters \textbf{26} (2013).

\bibitem{Rahmoune2020} Abita, R., \emph{Upper bound estimate for the blow-up 
 time of a class of integrodifferential equation of parabolic type involving 
 variable source}, Comptes Rendus Math�matique \textbf{357}, (2020), p. 23-32.



\end{thebibliography}

\end{document}
