\makeatletter
\@ifundefined{HCode}
{\documentclass[screen,CRMECA,Unicode,published]{cedram}
\newenvironment{noXML}{}{}%
\def\tsup#1{$^{{#1}}$}
\def\tsub#1{$_{{#1}}$}
\def\sfrac#1#2{{#1}/{#2}}
\def\thead{\noalign{\relax}\hline}
\def\endthead{\noalign{\relax}\hline}
\def\tbody{\noalign{\relax}\hline}
\def\xtbody{\noalign{\relax}}
\def\tabnote#1{\vskip4pt\parbox{.51\linewidth}{#1}}
\def\xxtabnote#1{\vskip4pt\parbox{.65\linewidth}{#1}}
\def\mathbi#1{\boldsymbol{#1}}
\def\xmathop#1{\mathop{\text{#1}}}
\def\rmDelta{\Delta}
\usepackage{etoolbox}
\usepackage{upgreek}
\csdef{Seqnsplit}{\\}
\def\botline{\\\hline}
\newcounter{runlevel}
\usepackage{multirow} 
\def\nrow#1{\@tempcnta #1\relax%
\advance\@tempcnta by 1\relax%
\xdef\lenrow{\the\@tempcnta}}
\def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}}
\def\xmorerows#1#2{#2}
\let\MakeYrStrItalic\relax
\def\jobid{crmeca20230392}
%\graphicspath{{/tmp/\jobid_figs/web/}} 
\graphicspath{{./figures/}} 
\def\0{\phantom{0}}
\def\rmmu{\upmu}
\def\mn{\phantom{$-$}}
\def\refinput#1{}
\def\back#1{}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\bgamma}{\boldsymbol{\gamma}}
\newcommand{\bdelta}{\boldsymbol{\delta}}
\newenvironment{inftab}{}{}
\def\Custref#1#2{\hyperlink{#2}{#1}}
\def\og{\guillemotleft}
\def\fg{\guillemotright}
\makeatletter
\g@addto@macro{\UrlBreaks}{\UrlOrds}
\gappto{\UrlBreaks}{\UrlOrds}
\makeatother
\DOI{10.5802/crmeca.233}
\datereceived{2023-04-18}
\daterevised{2023-10-02}
\dateaccepted{2023-10-17}
\ItHasTeXPublished
}
{\documentclass[crmeca]{article}
\usepackage[T1]{fontenc} 
\usepackage{upgreek}
\def\CDRdoi{10.5802/crmeca.233}
\def\newline{\unskip\break}
\def\xmorerows#1#2{\morerows{#1}{#2}}
\def\sfrac#1#2{{#1}/{#2}}
\def\xxtabnote#1{\tabnote{#1}}
\def\xmathop#1{\mathop{{#1}}}
\def\xtbody{\tbody}
\def\hypertarget#1#2{\label{#1}}
}
\makeatother

\newcommand{\bbR}{\mathbb{R}}
\newcommand{\bbC}{\mathbb{C}}
\newcommand{\bbZ}{\mathbb{Z}}

\newtheorem{remark}{Remark}

\dateposted{2023-11-23}
\begin{document}

\begin{noXML}


\TopicEF{Research article}{Article de recherche}

\title{Bounds for the blow-up time a class of integro-differential
problem of parabolic type with variable reaction term}

\alttitle{Bornes sur le temps d'explosion pour une \'equation
int\'egro-diff\'erentielle de type parabolique avec un terme r\'eactif
variable}

\author{\firstname{Rabil} \lastname{Ayazoglu}\CDRorcid{0000-0003-4493-2937}\IsCorresp}
\address{Faculty of Education, Bayburt University, Bayburt, Turkey}
\address{Institute of Mathematics and Mechanics of ANAS, Baku, Azerbaijan}
\email[R. Ayazoglu]{rabilmashiyev@gmail.com}

\author{\firstname{Ebubekir} \lastname{Akkoyunlu}\CDRorcid{0000-0003-2989-4151}}
\addressSameAs{1}{Faculty of Education, Bayburt University, Bayburt, Turkey}
\email[E. Akkoyunlu]{eakkoyunlu@bayburt.edu.tr}

\begin{abstract}
This paper is concerned with the blow-up time of the solutions to an
integro-differential problem of parabolic type with variable growth if
blow-up occurs. By using the differential inequality technique, we
obtain lower bounds for the blow-up time and some global existence
results under some conditions to variable exponent of reaction, memory
kernel, and initial value.
\end{abstract}

\begin{altabstract}
Cet article traite du temps d'explosion des solutions d'un probl\`eme
int\'egro-diff\'erentiel de type parabolique avec une croissance variable
si l'explosion se produit. En utilisant la technique de l'in\'egalit\'e
diff\'erentielle, nous obtenons des bornes inf\'erieures pour le temps
d'explosion et des r\'esultats d'existence globale sous certaines
conditions sur l'exposant variable de r\'eaction, le noyau de m\'emoire
et la valeur initiale.
\end{altabstract}

\keywords{\kwd{Integro-differential
problem}\kwd{Parabolic}\kwd{Variable reaction}\kwd{Global
existence}\kwd{Blow-up}\kwd{Lower bounds}}

\altkeywords{\kwd{Probl\`eme
int\'egro-diff\'erentiel}\kwd{Parabolique}\kwd{R\'eaction
variable}\kwd{Existence globale}\kwd{Explosion}\kwd{Limites
inf\'erieures}}

\maketitle

\end{noXML}

\section{Introduction}

In this paper, we study the lower bounds for the blow-up time of a class
integro-differential problem of parabolic type with  variable
reaction
{\begin{equation}
\begin{cases}
u_{t}-\Delta u+\displaystyle\int_{0}^{t}g\left( t-s\right) \Delta
u\left( x,s\right) \mathrm{d}s=\left\vert u\right\vert
^{p(x)-2}u,&\text{in }\Omega \times \left( 0,T\right) , \\
u(x,t)=0,&\text{on }\partial \Omega \times \left( 0,T\right) ,\\
u(x,0)=u_{0}(x),&\text{in }\Omega ,
\end{cases}
  \label{1.1}
\end{equation}}\unskip\noindent
where $\Omega $ is a bounded domain with smooth boundary in $
\mathbb{R}^{N} \left( N\geq 1\right) $, $T\in \left( 0,\infty \right]
$, the initial value $u_{0} \in H_{0}^{1} (\Omega )$, $p:\Omega
\rightarrow \left( 1,\infty \right) $ measurable function, $g:
\mathbb{R}^{+}\rightarrow \mathbb{R} ^{+}$ is a nonincreasing $C^{1}$
function (memory kernel)\ satisfying some additional conditions to be
specified later.

In mathematical point of view, equations of the types 
(\ref{1.1}) with variable exponent (or variable\ source) 
are usually referred
to as equations with nonstandard growth conditions.\ Under certain
conditions on the initial data and certain ranges of exponents, the
existence, uniqueness, and other qualitative properties of solutions for
parabolic and hyperbolic equations with variable nonlinearity have been
studied by many authors (see \cite{Alaoui, Antontsev1, Autuori, Ayazoglu,
Baghaei2, Bokalo, Buhrii, Guo, Han, Kholyavka} and references therein).
 These types of problems appear in many modern physical and
engineering models such as electrorheological fluids (smart fluids), fluids
with temperature-dependent viscosity, nonlinear viscoelasticity, filtration
processes through a porous media, image processing, nuclear science,
chemical reactions, heat transfer, population dynamics, biological sciences,
etc., and have attracted a great deal of attention in the literature. More
applications and details on the subject can be found in \cite{Antontsev2,
Antontsev, Chen, Mashiyev, Ruzicka, Wang, Wu} and the references therein.

Problem (\ref{1.1}) arises from many important mathematical models in
engineering and physical sciences. For example, in the study of heat
conduction in materials with memory, the classical Fourier's law is replaced
by the following form (cf. \cite{Nohel})
{\begin{equation}
q=-d\nabla u-\int_{-\infty }^{t}g\left( x,t-s\right) u\left( x,s\right)
\mathrm{d}s, \label{1.2}
\end{equation}}\unskip\noindent
where $q$ is proportional to the temperature differences per unit length, $u$
is the temperature, $d$ is the diffusion coefficient and $g$ is a relaxation
kernel. Here (\ref{1.2}) means that $q$ is not linearly dependent on $
\nabla u$. Then substituting Fourier's law (\ref{1.2}) into the
conservation of heat law, we can deduce
{\begin{equation}
u_{t}-\Delta u+\int_{0}^{t}g\left( t-s\right) \Delta u\left( x,s\right)
\mathrm{d}s=|u|^{p-2}u,\quad\text{in }\Omega \times \left( 0,T\right) ,  \label{1.3}
\end{equation}}\unskip\noindent
which is the linear form of (\ref{1.1}) for $p(x)=p$ for all $x\in
\overline{\Omega }$. Results concerning the controllability of Equation~(\ref
{1.3}) are given in \cite{Barbu, Leugering}. In recent years, semilinear
parabolic problems 
{\begin{equation}
\begin{cases}
u_{t}-\Delta u+\displaystyle\int_{0}^{t}g\left( t-s\right) \Delta
u\left( x,s\right) \mathrm{d}s=\left\vert u\right\vert
^{p-2}u,&\text{in }\Omega \times \left( 0,T\right) , \\
u(x,t)=0,&\text{on }\partial \Omega \times \left( 0,T\right) , \\
u(x,0)=u_{0}(x),&\text{in }\Omega 
\end{cases}
  \label{1.4}
\end{equation}}\unskip\noindent
with a memory term associated with the Laplace operator and source term with
Dirichlet type condition in the different cases of the values of the memory
kernel, specifically when $g=0$\ or $g>0$, have been studied by many authors.
 By assuming suitable conditions on $g$, $p$, and $u_{0}$, using
some known theorems in the mathematical literature, the global existence in
time, blow-up in finite time, the asymptotic behavior, and a lower bound for
the blow-up time of the unique weak solution have been discussed. 
When $g=0$, problem~(\ref{1.4}) has been studied by various authors and
several results of global and nonglobal existence, have been established.
For instance, in the early 1970s, Levine \cite{Levine} introduced the
concavity method and proved that solutions with negative energy blow-up in
finite time. Later, this method was improved by Kalantarov and Ladyzhenskaya
\cite{Kalantarov} to accommodate more general situations. Ball
\cite{Ball} also studied (\ref{1.4}) with $f(u,\nabla u)$ instead of $
|u|^{p-2}u$ and established a nonglobal existence result in bounded domains.
Recently, for problem~(\ref{1.4}), Payne \etal~\cite{Payne1, Payne2}
obtained both the lower and the upper bounds for blow-up time when the
blow-up occurs.


When the viscoelastic term $g$\ is positive, in \cite{Messaoudi1},
 Messaoudi concerned with the finite-time blow-up of solutions for the
initial boundary value problem (\ref{1.4}), where $g:
\mathbb{R}
_{+}\rightarrow
\mathbb{R}
_{+}$\ is a bounded $C^{1}$\ function, $p>2$, and $\Omega $\ is a bounded
domain of $
\mathbb{R}
^{N}$\ $(N\geq 1)$, with a smooth boundary $\partial \Omega $. Under
suitable conditions on $g$\ and $p$, the author proved a blow-up result for
certain solutions with positive initial energy.

Tian \cite{Tian} considered the problem (\ref{1.4}) with $g:
\mathbb{R}
_{+}\rightarrow
\mathbb{R}
_{+}$ is a bounded $C^{1}$ function, $p>2$, and $\Omega \subset 
\mathbb{R}
^{N} (N\geq 1)$ is a bounded domain with smooth boundary, $T\in \left(
0,+\infty \right] $, the initial value $u_{0}\in H_{0}^{1}\left( \Omega
\right) $. For $N\geq 1$, by using the differential
inequality technique, the author obtained a lower bound for blow-up time of
the solution if blow-up occurs. Also, established a new blow-up criterion
and gave an upper bound for blow-up time of the solution under some
conditions on $p,g,u_{0}$.

In \cite{Rahmoune}, the author generalized the results obtained by Tian in
\cite{Tian} using $|u|^{p\left( x\right) -2}u$\ instead of $|u|^{p-2}u$.
Using energy methods, the author obtained a lower bound for blow-up time of
the solution if blow-up occurs. Furthermore, assuming the initial energy is
negative established a new blow-up criterion and gave an upper bound for
blow-up time of the solution to the problem~(\ref{1.1}).


In our this paper, we study some global existence results and give
sufficient conditions on the variable reaction $p(\cdot )$, memory term $g$
 for the lower bounds at the time blow-up in $L^{2}$\ of the problem 
(\ref{1.1}). We extend the value range of the $p$ in
the conditions given in \cite{Tian} and \cite{Rahmoune}.

\section{{Main results and proofs}}


In this section, we list and recall some well-known results and facts from
the theory of Sobolev spaces with a variable exponent (for details, see
\cite{Diening, Kovacik}).

Let\ $\ p:\Omega \rightarrow \left( 1,\infty \right) $ be a measurable
function. We introduce $p^{-}$ and $p^{+}$ such that
{\begin{equation}
1<p^{-}\underset{x\in \Omega }{:=\mathrm{ess}\inf p(x)}\leq p(x)\leq p^{+}
\underset{x\in \Omega }{:=\mathrm{ess}\sup p(x)}<\infty .  \label{2.1}
\end{equation}}\unskip\noindent

Throughout this paper, $\Omega $ is considered to be a bounded domain of $
\mathbb{R}
^{N}$, with a smooth boundary $\partial \Omega $, assuming that $p(\cdot )$
is a measured function on $\overline{\Omega }$ and satisfies the following
logarithmic continuity condition (see \cite{Diening}):
{\begin{equation}
\left\vert p\left( x\right) -p\left( y\right) \right\vert \leq \frac{C_{\log
}}{\left\vert \log \left\vert x-y\right\vert \right\vert }\quad\text{for all }x,y
\ \in \ \Omega \text{, }\left\vert x-y\right\vert <\frac{1}{2},
\ C_{\log }>0.  \label{2.2}
\end{equation}}\unskip\noindent
We denote $L^{p(\cdot )}(\Omega )$ the set of measurable real-valued
functions $u$ on $\Omega $ such that
{\begin{equation*}
\rho _{p(\cdot )}\left( u\right) =\int_{\Omega }\left\vert u\left( x\right)
\right\vert ^{p\left( x\right) }\mathrm{d}x<\infty .
\end{equation*}}\unskip\noindent
The variable exponent space $L^{p(\cdot )}(\Omega )$ equipped with the
Luxemburg norm
{\begin{equation*}
\left\Vert u\right\Vert _{p(\cdot )}=\inf \left\{ a>0,\rho _{p(\cdot
)}\left( \frac{u}{a}\right) \leq 1\right\} ,
\end{equation*}}\unskip\noindent
is a Banach space and it is called variable exponent Lebesgue space.


Next, we denote $L^{p}(\Omega )$ by $L^{p}$\ and $H_{0}^{1}(\Omega )$\ by $
H_{0}^{1}$, the usual Sobolev space. The norm and inner of $L^{p}(\Omega )$
are denoted by $\left\Vert u\right\Vert _{L^{p}(
\Omega
)}:=\left\Vert u\right\Vert _{p} \left( 1\leq p\leq \infty \right) $ and $
H_{0}^{1}(\Omega )$ is the closure of $C_{0}^{\infty }(\Omega )$ with
respect to the equivalent norm $\left\Vert u\right\Vert
_{H_{0}^{1}}=\left\Vert \nabla u\right\Vert _{2}$.



\subsection{{Global existence}}


In this section, we first show the global existence result for $N\geq 1$.
The main idea of this section is the comparison principle. Let $\varphi (x)$
satisfies the following elliptic problem:
{\begin{equation*}
-\Delta \varphi =1,\ x\in \Omega ;\quad \varphi (x)=1,\ x\in
\partial \Omega .
\end{equation*}}\unskip\noindent

By using the result in \cite{Zhou}, we can see that the above nonlinear
problem has an unique solution, and the following inequalities hold:
{\begin{equation*}
M:=\max_{x\in \overline{\Omega }\ }\varphi (x)<+\infty ,\quad \varphi
(x)>1,\ x\in \Omega ;\quad \nabla \varphi \cdot \nu <0,x\in \partial
\Omega .
\end{equation*}}\unskip\noindent
We assume that:


$(G)$ The memory kernel $g:
\mathbb{R}
^{+}\rightarrow
\mathbb{R}
^{+}$ is a $C^{1}\left(
\mathbb{R}
^{+}\right) $ function satisfying
{\begin{equation*}
g\left( t\right) \geq 0,g^{\prime }\left( t\right) \leq 0\quad\text{and}\quad
1-\int_{0}^{\infty }g\left( s\right) \mathrm{d}s=l>0.
\end{equation*}}\unskip\noindent

\begin{theorem}
\label{Theorem 2.1.} {Let }$u${\ be a nonnegative solution of
the problem }(\ref{1.1}) {, memory kernel }$g$ {satisfies} $(G)$, {and function} $p$ {satisfies the
conditions} (\ref{2.1}) and ( \ref{2.2}). If $1<p\left(
x\right) <\infty $ for all $x\in \overline{\Omega }$, $u$\ {remains
globally bounded.}
\end{theorem}

\begin{proof}
\bigskip Let $1<p\left( x\right) <\infty $ for all $x\in 
\overline{\Omega }$. Define
{\begin{equation*}
\overline{u}=A\varphi (x),
\end{equation*}}\unskip\noindent
where
{\begin{equation*}
A\geq \max \left\{ \left( \frac{l}{M^{p^{+}}}\right) ^{\frac{1}{p^{+}-1}},
\ \max_{x\in \overline{\Omega }}u_{0}(x),1\right\} .
\end{equation*}}\unskip\noindent
It can be checked that
{\begin{eqnarray*}
\Delta \overline{u}+\overline{u}^{p(x)}-\int_{0}^{t}g\left( t-s\right)
\Delta \overline{u}\left( x,s\right) \mathrm{d}s 
&\leq& -A+M^{p^{+}}A^{p^{+}}+A\int_{0}^{\infty }g\left( \xi \right) \mathrm{d}\xi \\
&\leq &-A+M^{p^{+}}A^{p^{+}}+A\left( 1-l\right) =-lA+M^{p^{+}}A^{p^{+}} \\
&\leq &\overline{u}_{t}=0,
\end{eqnarray*}}\unskip\noindent
in $\Omega \times \left( 0,T\right) $ and $\overline{u}\geq  0$ on $
\partial \Omega \times (0,T)$, and $\overline{u}(x,0)\geq  u_{0}(x)$ in $
\Omega $. By the comparison principle, $\overline{u}(x)$ is a bounded
supersolution of (\ref{1.1}). The proof of Theorem~\ref{Theorem 2.1.} is
completed.
\end{proof}

\subsection{Blow-up in finite time for any initial data}


In this section, we derive a lower bound for $T$ if the weak solution $
u(x,t) $ of (\ref{1.1}) blows up in finite time $T$. We start with a local
existence result for the problem (\ref{1.1}), which is a direct result of
the existence theorem by \cite{Akagi, Lions}.


\begin{theorem}
\label{Theorem 2}{For all }$u_{0} \in H_{0}^{1}(\Omega )$, 
problem~(\ref{1.1}) {possesses a weak solution} $u$
 {on }$[0,T_{0}]$ {satisfying:}
{\begin{equation*}
u\in C_{\omega }\left( [0,T_{0}];W^{1,2}\left( \Omega \right) \right) \cap
C\left( [0,T_{0}];L^{p(\cdot )}\left( \Omega \right) \right) \cap
W^{1,2}\left( [0,T_{0}];L^{2}\left( \Omega \right) \right) ,
\end{equation*}}\unskip\noindent
{where} $T_{0}\in (0,T]$ {is a suitable number and }$
C_{\omega }\left( X\right) $ the set of all real functions defined on $X$
such that their growths are tempered by the modulus of continuity $\omega $
with $\omega \left( 0\right) =0$, $\omega \left( \varepsilon \right) >0$ for
$\varepsilon >0.$
\end{theorem}

The energy functional corresponding to problem (\ref{1.1}) is
{\begin{equation}
E\left( t\right) =\frac{1}{2}\left( g\circ \nabla u\right) \left( t\right) +
\frac{1}{2}\left( 1-\int_{0}^{t}g\left( s\right) \mathrm{d}s\right) \left\Vert \nabla
u\right\Vert _{2}^{2}-\int_{\Omega }\frac{1}{p(x)}\left\vert u\right\vert
^{p(x)}\mathrm{d}x,  \label{E}
\end{equation}}\unskip\noindent
where
{\begin{eqnarray}
\left( g\circ \nabla u\right) \left( t\right) &=&\int_{0}^{t}g\left(
t-s\right) \int_{\Omega }\left\vert \nabla u\left( t\right) -\nabla u\left(
s\right) \right\vert ^{2}\mathrm{d}x\mathrm{d}s  \nonumber \\
&=&\int_{0}^{t}g\left( t-s\right) \left\Vert \nabla u\left( t\right) -\nabla
u\left( s\right) \right\Vert _{2}^{2}\mathrm{d}s.  \label{G}
\end{eqnarray}}\unskip\noindent

\begin{lemma}
\label{Lemma 2.2}{Let} $u(x,t)$ {be a nonnegative solution of
the problem }(\ref{1.1}), {then} $E$ {is a nonincreasing
function on }$[0,T]$.
\end{lemma}


\begin{proof}
We define as $f\left( x,s\right) =\left\vert s\right\vert ^{p(x)-2}s$ and $
F(x,s)=\int_{0}^{s} f\left( x,\zeta \right) \mathrm{d}\zeta $ for all $x\in \Omega $
. Multiplying $u_{t}(x,t)$ on both sides of (\ref{1.1}) and integrating
over $\Omega $, we can obtain
{\begin{eqnarray*}
\int_{\Omega }u_{t}^{2}\left( x,t\right) \mathrm{d}x 
&=&\int_{\Omega }\Delta u\left( x,t\right) u_{t}\left( x,t\right)
dx-\int_{\Omega }u_{t}\left( x,t\right) \left( \int_{0}^{t}g\left(
t-s\right) \Delta u\left( x,s\right) \mathrm{d}s\right) \mathrm{d}x \\
&&+\,\int_{\Omega }u_{t}\left( x,t\right) f\left( x,u\right) \mathrm{d}x \\
&=&\int_{\Omega }f(x,u)u_{t}\mathrm{d}x-\int_{\Omega }\nabla u\nabla
u_{t}\mathrm{d}x+\int_{\Omega }\left( \int_{0}^{t}g\left( t-s\right) \nabla u\left(
s\right) \nabla u_{t}\left( t\right) \mathrm{d}s\right) \mathrm{d}x \\
&=&-\frac{\mathrm{d}}{\mathrm{d}t}\left( \frac{1}{2}\int_{\Omega }\left\vert \nabla
u\right\vert ^{2}\mathrm{d}x\right) +\frac{\mathrm{d}}{\mathrm{d}t}F(x,u)\mathrm{d}x \\
&&+\,\frac{\mathrm{d}}{\mathrm{d}t}\left( \int_{0}^{t}g\left( t-s\right) \int_{\Omega }\nabla
u_{t}\left( t\right) \nabla u\left( s\right) \mathrm{d}x\mathrm{d}s\right).
\end{eqnarray*}}\unskip\noindent
The last term of the right-hand side of the above equality can be rewritten
as
{\begin{eqnarray*}
\int_{0}^{t}g\left( t-s\right) \int_{\Omega }\nabla u_{t}\left( t\right)
\nabla u\left( s\right) \mathrm{d}x\mathrm{d}s 
&=&-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( \int_{0}^{t}g\left( t-s\right)
\int_{\Omega }\left\vert \nabla u\left( t\right) -\nabla u\left( s\right)
\right\vert ^{2}\mathrm{d}x\mathrm{d}s\right)  \\
&&+\,\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( \int_{0}^{t}g\left( s\right) \mathrm{d}s\int_{\Omega
}\left\vert \nabla u\left( t\right) \right\vert ^{2}\mathrm{d}x\right)  \\
&&+\,\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left( \int_{0}^{t}g^{\prime }\left( t-s\right)
\int_{\Omega }\left\vert \nabla u\left( t\right) -\nabla u\left( s\right)
\right\vert ^{2}\mathrm{d}x\mathrm{d}s\right)  \\
&&-\,\frac{1}{2}g\left( t\right) \int_{\Omega }\left\vert \nabla u\left(
t\right) \right\vert ^{2}\mathrm{d}x.
\end{eqnarray*}}\unskip\noindent
Combining the above two equalities with (\ref{E}) 
and~(\ref{G}), we see that
{\begin{equation*}
E^{\prime }\left( t\right) =-\frac{1}{2}g\left( t\right) \left\Vert \nabla
u\left( t\right) \right\Vert ^{2}+\frac{1}{2}\left( g^{\prime }\circ \nabla
u\right) \left( t\right) -\int_{\Omega }u_{t}^{2}\mathrm{d}x\leq 0,
\end{equation*}}\unskip\noindent
for smooth solutions. Thus $E(t)$ is a nonincreasing function on $[0,T]$,
which implies that $E(t)\leq E(0)$. The same result can be established for
strong solutions for almost every $t$, by a standard density argument. The
proof of the Lemma~\ref{Lemma 2.2} is completed.
\end{proof}

Now, we give a sufficient condition for the nonnegative solution of problem 
(\ref{1.1})  to blow-up in $L^{2}$-norm and establish a lower
bound for the blow-up time.


\begin{theorem}
\textbf{\label{Theorem 2.2.}}Let $u(x,t)$\ be a nonnegative solution of
problem (\ref{1.1}) in a bounded domain\ $\Omega \subset
\mathbb{R}
^{N}$ and $u_{0}\in H_{0}^{1}(\Omega )$ such that $\left\Vert
u_{0}\right\Vert _{2} \neq 0$, function $p$\ satisfies the condition 
(\ref{2.1}) and $g$\ satisfies condition $\left( G\right) $. We define $\varphi
\left( t\right) $\ as following
{\begin{equation*}
\varphi \left( t\right) =\int_{\Omega }u^{2}\mathrm{d}x.
\end{equation*}}\unskip\noindent

(i) If $N\geq 3$ and
{\begin{equation*}
1<p^{-},\quad p^{+}<\frac{2N-3}{N-2},
\end{equation*}}\unskip\par\pagebreak\noindent
then a lower bound for the time of blow-up for any solution which blows up
in $L^{2}$\ norm\ is given by
{\begin{equation}
\int_{\left\Vert u_{0}\right\Vert ^{2}}^{+\infty }
\frac{\mathrm{d}\xi }{K_{4}\xi ^{
\frac{3\left( N-2\right) }{3N-8}}+K_{3}}\leq T,  \label{2.5}
\end{equation}}\unskip\noindent
where $\left\Vert u_{0}\right\Vert ^{2}=\int_{\Omega
}u_{0}^{2}\mathrm{d}x$\ and $ K_{3}$,\ $K_{4}$\ are positive constants
which will be stated later. 

(ii) If $N=1,~2$ and
{\begin{equation*}
1<p^{-},\quad p^{+}<4,
\end{equation*}}\unskip\noindent
then a lower bound for the time of blow-up for any solution which blows up
in $L^{2}$\ norm\ is given by
{\begin{equation*}
\int_{\left\Vert u_{0}\right\Vert ^{2}}^{+\infty }\frac{\mathrm{d}\xi }{K_{5}\xi ^{
\frac{2}{4-p^{+}}}+K_{3}}\leq T,
\end{equation*}}\unskip\noindent
where $K_{3}$\ is the constant in (\ref{2.5}) and $K_{5}$\ is a positive
constant which will be stated later.
\end{theorem}

\setcounter{remark}{4}
\begin{remark}
\label{Remark 1}
\upshape
Let $g$ satisfies condition $\left( G\right) $. Parameter $
p$ had given as
{\begin{equation*}
2<p<2+\frac{4}{N},\quad  N\geq 3\text{ or }2<p<4,\ N=1,2
\end{equation*}}\unskip\noindent
in \cite{Tian} and function $p(\cdot )$ had given as
{\begin{equation*}
2<p^{-}\leq p^{+}<2+\frac{4}{N},\quad  N\geq 3\text{ or }2<p^{-}\leq
p^{+}<4,\ N=1,2
\end{equation*}}\unskip\noindent
in \cite{Rahmoune} to obtain blow-up in finite time $T$ for problem 
(\ref{1.4})  and (\ref{1.1})  respectively.
\end{remark}

Our in this paper, we extend the value range of the $p$ in the conditions
given in \cite{Tian} and \cite{Rahmoune}.


\begin{proof}[{Proof of Theorem \protect\ref{Theorem 2.2.}}]

(i) Let $N\geq 3$. By multiplying the equation in (\ref{1.1})
 by $u(x,t)$, integrating by parts over $\Omega $, one obtains that
{\begin{eqnarray}
\varphi ^{\prime }(t)  
&=&2\int_{\Omega }u\left( x,t\right) u_{t}\left( x,t\right) \mathrm{d}x  \nonumber \\
&=&2\int_{\Omega }u\left( \Delta u-\int_{0}^{t}g\left( t-s\right) \Delta
u\left( x,s\right) \mathrm{d}s+\left\vert u\right\vert ^{p(x)-2}u\right) \mathrm{d}x  \nonumber \\
&=&-2\left( 1-\int_{0}^{t}g\left( s\right) \mathrm{d}s\right) \left\Vert \nabla
u\left( t\right) \right\Vert _{2}^{2}+2\int_{\Omega }\left\vert u\right\vert
^{p(x)}\mathrm{d}x+I,  \label{2.10}
\end{eqnarray}}\unskip\noindent
where
{\begin{equation*}
I=2\int_{0}^{t}g\left( t-s\right) \int_{\Omega }\nabla u\left( t\right)
\left( \nabla u\left( s\right) -\nabla u\left( t\right) \right) \mathrm{d}x\mathrm{d}s.
\end{equation*}}\unskip\noindent
Furthermore, by the condition (\ref{2.1}), we derive
{\begin{eqnarray}
\int_{\Omega }u^{p(x)}\mathrm{d}x &=&\int_{\Omega \cap \left\{ x:u\geq 1\right\}
}u^{p(x)}\mathrm{d}x+\int_{\Omega \cap \left\{ x:u<1\right\} }u^{p(x)}\mathrm{d}x  \nonumber \\
&\leq &\int_{\Omega }u^{p^{+}}\mathrm{d}x+\left\vert \Omega \right\vert.
\label{2.rr}
\end{eqnarray}}\unskip\noindent
Then by (\ref{2.10}) and (\ref{2.rr}), we
obtain
{\begin{eqnarray*}
\varphi ^{\prime }(t) 
&\leq& -2\left( 1-\int_{0}^{t}g\left( s\right) \mathrm{d}s\right) \left\Vert \nabla
u\left( t\right) \right\Vert _{2}^{2}+2\int_{\Omega }u^{p^{+}}\mathrm{d}x+I+K_{0} \\
&\leq &-2l\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+2\int_{\Omega }u^{p^{+}}\mathrm{d}x+I+K_{0},
\end{eqnarray*}}\unskip\noindent
where
{\begin{equation*}
K_{0}=2\left\vert \Omega \right\vert >0.
\end{equation*}}\unskip\noindent
By H\"{o}lder's inequality and the condition $(G)$, we know
{\begin{eqnarray*}
\left\vert I\right\vert &\leq &2l\int_{0}^{t}g\left( s\right) \mathrm{d}s\left\Vert
\nabla u\left( t\right) \right\Vert _{2}^{2}+\frac{1}{2l}\left( g\circ
\nabla u\right) \\
&\leq &2l\left( 1-l\right) \left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+\frac{1}{2l}\left( g\circ \nabla u\right) .
\end{eqnarray*}}\unskip\noindent
From (\ref{E}), (\ref{G}) and Lemma \ref{Lemma 2.2}, we have
{\begin{eqnarray}
\frac{1}{2l}\left( g\circ \nabla u\right)  
&=&\frac{1}{l}\left[ E(t)-\frac{1}{2}\left\Vert \nabla u\left( t\right)
\right\Vert _{2}^{2}\left( 1-\int_{0}^{t}g\left( s\right) \mathrm{d}s\right)
+\int_{\Omega }\frac{1}{p(x)}u^{p(x)}\mathrm{d}x\right]  \nonumber \\
&\leq &\frac{1}{l}E(0)-\frac{1}{2}\left\Vert \nabla u\left( t\right)
\right\Vert _{2}^{2}+\frac{1}{lp^{-}}\int_{\Omega }u^{p^{+}}\mathrm{d}x+K_{1},
\label{A3}
\end{eqnarray}}\unskip\noindent
where
{\begin{equation*}
K_{1}=\frac{\left\vert \Omega \right\vert }{lp^{-}}>0.
\end{equation*}}\unskip\noindent
From (\ref{2.10}) and (\ref{A3}), we obtain
{\begin{eqnarray}
\varphi ^{\prime }(t)  
&\leq& -2l\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+2\int_{\Omega }u^{p^{+}}\mathrm{d}x+2l\left( 1-l\right) \left\Vert \nabla
u\left( t\right) \right\Vert _{2}^{2}  \nonumber \\
&&+\,\frac{1}{l}E(0)-\frac{1}{2}\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+\frac{1}{lp^{-}}\int_{\Omega }u^{p^{+}}\mathrm{d}x+K_{2}  \nonumber \\
&\leq &-2l\left\Vert \nabla u\left( t\right) \right\Vert _{2}^{2}+2l\left(
1-l\right) \left\Vert \nabla u\left( t\right) \right\Vert _{2}^{2}+\frac{1}{l
}E(0)  \nonumber \\
&&-\,\frac{1}{2}\left\Vert \nabla u\left( t\right) \right\Vert _{2}^{2}+\frac{
2lp^{-}+1}{lp^{-}}\int_{\Omega }u^{p^{+}}\mathrm{d}x+K_{2}  \nonumber \\
&=&-\frac{4l^{2}+1}{2}\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+K\int_{\Omega }u^{p^{+}}\mathrm{d}x+K_{3},  \label{2.AA}
\end{eqnarray}}\unskip\noindent
where
{\begin{eqnarray*}
&\displaystyle K=\frac{2lp^{-}+1}{lp^{-}}>0, \Seqnsplit
&\displaystyle K_{2}=K_{0}+K_{1},
\end{eqnarray*}}\unskip\noindent
and
{\begin{equation*}
K_{3}=K_{2}+\frac{1}{l}E(0)\quad\text{with} \ \max \left\{ \frac{1}{l}
E(0),0\right\} \geq 0.
\end{equation*}}\unskip\noindent
By using the H\"{o}lder and Young inequalities we get
{\begin{eqnarray}
K\int_{\Omega }u^{p^{+}}\mathrm{d}x &\leq &K\left( \int_{\Omega }u^{\frac{2N-3}{N-2}
}dx\right) ^{a_{1}}\left\vert \Omega \right\vert ^{a_{2}}  \nonumber \\
&\leq &a_{1}K\int_{\Omega }u^{\frac{2N-3}{N-2}}\mathrm{d}x+a_{2}\left\vert \Omega
\right\vert ,  \label{2.18i}
\end{eqnarray}}\unskip\noindent
where
{\begin{equation*}
a_{1}=\frac{(N-2)p^{+}}{2N-3},\quad a_{2}=1-\frac{2(N-2)p^{+}}{2N-3}.
\end{equation*}}\unskip\noindent
We now make use of Schwarz's inequality to the first term on the right hand
side of (\ref{2.18i}) as follows:
{\begin{eqnarray}
a_{1}K\int_{\Omega }u^{\frac{2N-3}{N-2}}\mathrm{d}x 
&\leq& a_{1}K\left( \int_{\Omega }u^{2}\mathrm{d}x\right) ^{\frac{1}{2}}\left(
\int_{\Omega }u^{\frac{2\left( N-1\right) }{N-2}}\mathrm{d}x\right) ^{\frac{1}{2}}
\nonumber \\
&\leq &a_{1}K\left( \int_{\Omega }u^{2}\mathrm{d}x\right) ^{\frac{3}{4}}\left(
\int_{\Omega }u^{2^{\ast }}\mathrm{d}x\right) ^{\frac{1}{4}},  \label{2.19}
\end{eqnarray}}\unskip\noindent
where $2^{\ast }=({2N}/{N-2})$. Next, by using the Sobolev inequality (see
\cite{Talenti}), for $N\geq 3$, we get
{\begin{equation}
\left\Vert u\right\Vert _{2^{\ast }}\leq B\left\Vert \nabla u\right\Vert
_{2},\quad \forall u\in H_{0}^{1}(\Omega ),  \label{2.20}
\end{equation}}\unskip\noindent
where
{\begin{equation*}
B=\left( \frac{1}{N(N-2)\uppi }\right) ^{1/2}\left( \frac{N!}{2\Gamma \left(
\frac{N}{2}+1\right) }\right) ^{1/N}>0,
\end{equation*}}\unskip\noindent
is the best constant in the Sobolev inequality. By inserting 
the (\ref{2.20}) to the last inequality in (\ref{2.19}), we have
{\begin{equation}
a_{1}K\int_{\Omega }u^{\frac{2N-3}{N-2}}\mathrm{d}x\leq a_{1}KB^{\frac{N}{2\left(
N-2\right) }}\varphi ^{\frac{3}{4}}(t)\left\Vert \nabla u\left( t\right)
\right\Vert _{2}^{\frac{N}{2\left( N-2\right) }}.  \label{rrr}
\end{equation}}\unskip\noindent
Now, we can use the Young inequality to inequality (\ref{rrr})
and we get
{\begin{eqnarray}
a_{1}K\int_{\Omega }u^{\frac{2N-3}{N-2}}\mathrm{d}x  
&\leq& \frac{a_{1}K\left( 3N-8\right) }{4\left( N-2\right) }B^{\frac{N}{
2\left( N-2\right) }}\left( \frac{4\varepsilon \left( N-2\right) }{3N-8}
\right) ^{-\frac{N}{3N-8}}\varphi ^{\frac{3\left( N-2\right) }{3N-8}}(t)
\nonumber \\
&&+\,a_{1}K\varepsilon \left\Vert \nabla u\left( t\right) \right\Vert _{2}^{2}.
\label{2.2a}
\end{eqnarray}}\unskip\noindent
Combining (\ref{2.AA}), (\ref{2.18i}) and 
(\ref{2.2a}), we have $\varepsilon $\ is a positive constant
to be determined later:
{\begin{eqnarray}
\varphi ^{\prime }(t)  
&\leq& -\frac{4l^{2}+1}{2}\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+\frac{a_{1}\left( 3N-8\right) K}{4\left( N-2\right) }B^{\frac{N}{
2\left( N-2\right) }}\left( \frac{4\varepsilon \left( N-2\right) }{3N-8}
\right) ^{-\frac{N}{3N-8}}\varphi ^{\frac{3\left( N-2\right) }{3N-8}}(t)
\nonumber \\
&&+\,a_{1}K\varepsilon \left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+K_{3}.  \label{2.2b}
\end{eqnarray}}\unskip\noindent
If we choose $\varepsilon >0$ stated in (\ref{2.2b}) such that
{\begin{equation*}
\varepsilon =\frac{1+4l^{2}}{2a_{1}K},
\end{equation*}}\unskip\noindent
then we obtain the ordinary differential inequality
{\begin{equation}
\varphi ^{\prime }(t)\leq K_{4}\varphi ^{\frac{3\left( N-2\right) }{3N-8}
}(t)+K_{3},  \label{DD1}
\end{equation}}\unskip\noindent
where
{\begin{equation*}
K_{4}=\frac{a_{1}\left( 3N-8\right) K}{4\left( N-2\right) }B^{\frac{N}{
2\left( N-2\right) }}\left( \frac{2\left( 1+4l^{2}\right) \left( N-2\right)
}{a_{1}K\left( 3N-8\right) }\right) ^{-\frac{N}{3N-8}}>0.
\end{equation*}}\unskip\noindent
An integration of the differential inequality (\ref{DD1})
from $0$ to $t$, we obtain the following inequality
{\begin{equation*}
\int_{\varphi (0)}^{\varphi \left( t\right) }\frac{\mathrm{d}\xi }{K_{4}\xi ^{\frac{
3\left( N-2\right) }{3N-8}}+K_{3}}\leq t,
\end{equation*}}\unskip\noindent
which with $\lim {}_{t\rightarrow T^{-}}\varphi (t)=+\infty $ implies that
{\begin{equation*}
\int_{\varphi (0)}^{+\infty }\frac{\mathrm{d}\xi }{K_{4}\xi ^{\frac{3\left(
N-2\right) }{3N-8}}+K_{3}}\leq T,
\end{equation*}}\unskip\noindent
where $\varphi (0)=\int_{\Omega }u_{0}^{2}\left( x\right) \mathrm{d}x$. Note that $
{3\left( N-2\right) }/({3N-8})>1 \left( N\geq 3\right) $, hence the
left-hand side of the above inequality is finite.

(ii) Let $N=1,2$. Now, recall the Sobolev embedding
$H_{0}^{1}(\Omega )\hookrightarrow L^{\infty }(\Omega )$ which provide the
inequality
{\begin{equation}
\left\Vert u\right\Vert _{\infty }\leq B\left\Vert \nabla u\right\Vert
_{2},\quad \forall u\in H_{0}^{1}(\Omega ),  \label{2.27}
\end{equation}}\unskip\noindent
where $B>0$ is the best constant of the Sobolev embedding. Using 
(\ref{2.27}) and H\"{o}lder's inequality to $\left\Vert u\right\Vert _{p^{+}}^{p^{+}}$
which is in (\ref{2.AA}), we\ show that
{\begin{eqnarray}
\left\Vert u\right\Vert _{p^{+}}^{p^{+}} &=&\left\Vert u\right\Vert
^{2}\left\Vert u\right\Vert _{\infty }^{p^{+}-2}=\varphi (t)\left\Vert
u\right\Vert _{\infty }^{p^{+}-2}  \nonumber \\
&\leq &\varphi (t)B^{p^{+}-2}\left\Vert \nabla u\right\Vert _{2}^{p^{+}-2}.
\label{23}
\end{eqnarray}}\unskip\noindent
Now, by using Young's inequality to (\ref{23}), we have for all $
\varepsilon >0$,
{\begin{equation}
\left\Vert u\right\Vert _{p^{+}}^{p^{+}}\leq \frac{4-p^{+}}{2}\left( \frac{
2\varepsilon }{p^{+}-2}\right) ^{-\frac{p^{+}-2}{4-p^{+}}}\varphi (t)^{\frac{
2}{4-p^{+}}}+\varepsilon \left\Vert \nabla u\right\Vert _{2}^{2}.  \label{24}
\end{equation}}\unskip\noindent
From (\ref{2.AA})  and (\ref{24}), we have
{\begin{eqnarray*}
\varphi ^{\prime }(t)
&\leq& -\frac{4l^{2}+1}{2}\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+K\int_{\Omega }u^{p^{+}}\mathrm{d}x+K_{3} \\
&=&-\frac{4l^{2}+1}{2}\left\Vert \nabla u\left( t\right) \right\Vert
_{2}^{2}+\frac{\left( 4-p^{+}\right) K}{2}\left( \frac{2\varepsilon }{p^{+}-2
}\right) ^{-\frac{p^{+}-2}{4-p^{+}}}\varphi (t)^{\frac{2}{4-p^{+}}} \\
&&+\,\varepsilon K\left\Vert \nabla u\right\Vert _{2}^{2}+K_{3}.
\end{eqnarray*}}\unskip\noindent
Let us choose $\varepsilon >0$ such that
{\begin{equation*}
\varepsilon =\frac{4l^{2}+1}{2K}.
\end{equation*}}\unskip\noindent
Then we obtain the ordinary differential inequality
{\begin{equation}
\varphi ^{\prime }(t)\leq K_{5}\varphi (t)^{\frac{2}{4-p^{+}}}+K_{3},
\label{2.121}
\end{equation}}\unskip\noindent
where
{\begin{equation*}
K_{5}=\frac{\left( 4-p^{+}\right) K}{2}\left( \frac{4l^{2}+1}{K\left(
p^{+}-2\right) }\right) ^{-\frac{p^{+}-2}{4-p^{+}}}>0.
\end{equation*}}\unskip\noindent
By integration of the differential inequality (\ref{2.121})
from $0$ to $t$, we obtain the following inequality
{\begin{equation*}
\int_{\varphi (0)}^{\varphi \left( t\right) }\frac{\mathrm{d}\xi }{K_{5}\zeta ^{\frac{
2}{4-p^{+}}}+K_{3}}\leq t,
\end{equation*}}\unskip\noindent
which with $\lim {}_{t\rightarrow T^{-}}\varphi (t)=+\infty $ implies
{\begin{equation*}
\int_{\varphi (0)}^{+\infty }\frac{\mathrm{d}\zeta }{K_{5}\zeta ^{\frac{2}{4-p^{+}}
}+K_{3}}\leq T
\end{equation*}}\unskip\noindent
with $p^{+}\in \left( 2,4\right) $ and $N=1,~2$. Thus the Theorem \ref{Theorem 2.2.} is proved.
\end{proof}

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\back{}

\bibliographystyle{crunsrt}
\bibliography{crmeca20230392}
\refinput{crmeca20230392-reference.tex}

\end{document}
