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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]  %Headings in Russian
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
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     {\flushleft\it Том 16, стр. 1--23 (2022) \hspace{65mm}{\rm\small УДК \udcs}} %Volume, pages in Russian,
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         {\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small MSC\ \ \mscs }
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\begin{document}
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\title[Blow-up analysis for a class of $p(x)-$Kirchhoff type inverse problem]{Blow-up analysis for a class of plate viscoelastic $p(x)-$Kirchhoff type inverse source problem with variable-exponent nonlinearities}
\author{{M. Shahrouzi}}%
\address{Mohammad Shahrouzi
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} Jahrom University, Jahrom,
\newline\hphantom{iii} P.O.Box: 74137-66171, Iran}%
\email{mshahrouzi@jahromu.ac.ir}%

\author{{J. Ferreira}}%
\address{Jorge Ferreira
\newline\hphantom{iii} Department of Exact Sciences,
\newline\hphantom{iii} Federal Fluminense University, Volta Redonda,
\newline\hphantom{iii} 27255-125, Brazil}%
\email{ferreirajorge2012@gmail.com}%

\author{{E. Pi\c{s}kin}}%
\address{Erhan Pi\c{s}kin
\newline\hphantom{iii} Department of Mathematics,
\newline\hphantom{iii} Dicle University, Diyarbak\i r,
\newline\hphantom{iii} TR-21280, Turkey}%
\email{episkin@dicle.edu.tr}%

\author{{N. Boumaza}}%
\address{Nouri Boumaza
\newline\hphantom{iii} Department of Mathematics and Computer Science,
\newline\hphantom{iii} Larbi Tebessi University, Tebessa,
\newline\hphantom{iii} Algeria}%
\email{nouri.boumaza@univ-tebessa.dz}%
%\thanks{\sc Ivanov, I.I.,
%On some problems of commutative algebra}
\thanks{Corresponding author: Mohammad Shahrouzi}
%\thanks{\rm The work is supported by RFFI (grant 03-01-11111)}
%\thanks{\it Received  January, 1, 2015, published  March, 1,  2015.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } In this work, we study the blow-up analysis for a class of plate viscoelastic $p(x)$-Kirchhoff type inverse source problem of the form:
\begin{align*}
  u_{tt}+\Delta^{2}u&-\left(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(x)}u-\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau \\ & +\beta|u_{t}|^{m(x)-2}u_{t}=\alpha|u|^{q(x)-2}u+f(t)\omega(x).
\end{align*}
Under suitable conditions on kernel of the memory, initial data and variable exponents, we prove the blow up of solutions in two cases: linear damping term ($m(x)\equiv2$) and nonlinear damping term ($m(x)>2$). Precisely, we show that the solutions with positive initial energy blow up in a finite time when $m(x)\equiv2$ and blow up at infinity if $m(x)>2$.\medskip

\noindent{\bf Keywords:} inverse source problem, blow-up, viscoelastic, p(x)-Kirchhoff type equation.
 \end{quote}
}


\section{Introduction}

In this paper, we consider the following plate viscoelastic $p(x)-$Kirchhoff type inverse source problem:
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber && u_{tt}+\Delta^{2}u-\left(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(x)}u-\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau \\
   && \quad +\beta|u_{t}|^{m(x)-2}u_{t}=\alpha|u|^{q(x)-2}u+f(t)\omega(x),\quad (x,t)\in\Omega\times(0,+\infty) \\
   && u(x,t)=\frac{\partial{u}}{\partial\nu}(x,t)=0,\qquad (x,t)\in\partial\Omega\times(0,+\infty) \\
   &&  u(x,0)=u_{0}(x),\ \ u_{t}(x,0)=u_{1}(x),\qquad x\in\Omega \\
   &&  \int_{\Omega}u(x,t)\omega(x)dx=\phi(t),\qquad t>0
\end{eqnarray}
while the pair of functions $\{u(x,t),f(t)\}$ are unknown. In this problem, $\Omega\subset {R}^{n}\ (n\geq 1)$ is a bounded domain with smooth boundary $\partial\Omega$ and a unit outer normal $\nu$. Here, $\Delta_{p(x)}$ is called $p(x)-$Laplace operator defined as
\begin{equation*}
  \Delta_{p(x)}u=div(|\nabla u|^{p(x)-2}\nabla u),
\end{equation*}
and $a,b>0$. Also, $\alpha$ and $\beta$ are positive constants and $g(t)$, $\omega(x)$ and $\phi(t)$ are real valued functions with specific conditions that will be enunciated later.\newline
In addition, $p(x)$, $m(x)$ and $q(x)$ are given continuous and measurable functions on $\overline{\Omega}$ such that
\begin{eqnarray}
\nonumber && 2<p^{-}\leq p(x)\leq p^{+}<\infty \\
&& 2\leq m^{-}\leq m(x)\leq m^{+}<\infty \label{5} \\
\nonumber && 2<q^{-}\leq q(x)\leq q^{+}<\infty ,
\end{eqnarray}
with
\begin{eqnarray}
\nonumber && p^{-}:= essinf_{x\in\overline{\Omega}} p(x),\ \ p^{+}:= esssup_{x\in\overline{\Omega}} p(x) \\
\nonumber && m^{-}:= essinf_{x\in\overline{\Omega}} m(x),\ \ m^{+}:= esssup_{x\in\overline{\Omega}} m(x) \\
\nonumber && q^{-}:= essinf_{x\in\overline{\Omega}} q(x),\ \ q^{+}:= esssup_{x\in\overline{\Omega}} q(x).
\end{eqnarray}
\newline
The inverse source problems in waves arise in many scientific and industrial areas such as antenna design and synthesis, biomedical imaging and photo-acoustic tomography \cite{0ref}. Solving the inverse problems are rather difficult, because they are nonlinear and improperly posed. It is known that there is no uniqueness for the inverse source problem at a fixed frequency due to the existence of non-radiating sources \cite{00ref}. Therefore, additional information is required for the source in order to obtain a unique solution such as (4) and
\begin{equation}
  \omega\in H_{0}^{2}(\Omega)\cap L^{p(.)}(\Omega)\cap L^{m(.)}(\Omega)\cap L^{q(.)}(\Omega),\ \ \int_{\Omega}\omega^{2}(x)dx=1.\label{6}
\end{equation}\par
To the best of our knowledge, the stability and blow up of solutions of inverse source problems with variable-exponent nonlinearities are less investigated area. In this paper, we are going to extend previous results in the inverse problems with constant-exponent nonlinearities to our inverse source problem (1)-(4) with variable-exponent nonlinearities. Thus, firstly we point out some previous results in the inverse problems with constant-exponent nonlinearities. For example, Eden and Kalantarov \cite{3ref} studied the following inverse problem
\begin{align*}
   & u_{t}-\Delta u+b(x,t,u,\nabla u)-|u|^{p}u=F(t)\omega(x),\ \   x\in\Omega,\ t>0 \\
   & u(x,t)=0,\ \ x\in\partial\Omega,\ t>0 \\
   & u(x,0)=u_{0}(x),\ \ x\in\Omega \\
   & \int_{\Omega}u(x,t)\omega(x)dx=\phi(t), \ \ t>0.
\end{align*}
They found conditions on data which guaranteed the global nonexistence of solutions when $\phi(t)\equiv1$. Also, authors established a stability result with the opposite sign on the power type nonlinearity and $b(x,t,u,\nabla u)\equiv0$. Next, Tahamtani and Shahrouzi \cite{4ref} extended previous results to a Petrovsky inverse source problem (see also \cite{5ref}). Shahrouzi in \cite{7ref} studied the following damped viscoelastic inverse problem
\begin{align*}
   & u_{tt}-\nabla[(a_{0}+a|\nabla u|^{m})\nabla u]+\int_{0}^{t}e^{\lambda(t-\tau)}g(t-\tau)\Delta u(\tau)d\tau+bu_{t} =h(x,t,u,\nabla u) \\
   & \quad\quad +|u|^{p}u+f(t)\omega(x),\ \ x\in\Omega,\ t>0 \\
   & u(x,t)=0,\ \ x\in\Gamma,\ t>0 \\
   & u(x,0)=u_{0}(x),\ \ x\in\Omega \\
   & \int_{\Omega}u(x,t)\omega(x)dx=1, \ \ t>0,
\end{align*}
and proved the blow up of solutions under sufficient conditions on initial functions by using the modified concavity argument. See \cite{6ref,8ref,88ref}.\par
On the other hand, it is known that modeling of some physical phenomena such as flows of electro-rheological fluids, nonlinear viscoelasticity and image processing give rise to equations with nonstandard growth conditions, i.e, equations with variable exponents of nonlinearities. In direct problems, equations with nonlinearities of variable-exponent type have largely been discussed by several authors. For instance, Antontsev \cite{9ref} considered the equation:
\begin{equation*}
  u_{tt}=div\big(a(x,t)|\nabla u|^{p(x,t)-2}\nabla u\big)+\alpha\Delta u_{t}+b(x,t)u|u|^{\sigma(x,t)-2}+f(x,t)
\end{equation*}
in $\Omega\subseteq R^{n}$, where $\alpha>0$ is a constant and $a,b,p,\sigma$ are given functions. For specific conditions on $a,b,p,\sigma$, the existence theorems for small and any finite time have been proved and blow up of solutions under some suitable conditions on data has been established. Messaoudi and Talahmeh \cite{10ref}, considered the following nonlinear equation with variable exponents:
\begin{equation}
  u_{tt}-div\big(|\nabla u|^{r(.)-2}\nabla u\big)+a|u_{t}|^{m(.)-2}u_{t}=b|u|^{p(.)-2}u.
\end{equation}
They proved a finite-time blow-up result for the solutions with negative initial energy and also certain solutions with positive energy in appropriate range of $m(.), r(.)$ and $p(.)$. In another study, Messaoudi \cite{11ref} studied equation (7) with $a=1,\ b=0$ in the presence of damping term $-\Delta u_{t}$. He proved several decay results depending on the range of variable exponents $m$ and $r$. Shahrouzi \cite{13ref} studied the behavior of solutions to the following initial-boundary value problem with variable-exponent nonlinearities
\begin{align*}
   & u_{tt}-\Delta u-div(|\nabla u|^{m(x)}\nabla u)+\int_{0}^{t}g(t-\tau)\Delta u(\tau)d\tau+ h(x,t,u,\nabla u)+\beta u_{t} \\
   & \quad\quad=|u|^{p(x)}u,\ in\ \Omega\times(0,+\infty) \\
   & \left\{\begin{array}{rl}
     u(x,t)=0,\ \ \quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ x\in\Gamma_{0},t>0\\
     \frac{\partial u}{\partial n}(x,t)=\int_{0}^{t}g(t-\tau)\frac{\partial u}{\partial n}(\tau)d\tau-|\nabla u|^{m(x)}\frac{\partial u}{\partial n}+\alpha u,\qquad x\in\Gamma_{1},t>0
     \end{array}\right. \\
   & u(x,0)=u_{0}(x), \ \ u_{t}(x,0)=u_{1}(x),\ \ \ in \ \ \Omega.
\end{align*}
Under appropriate conditions, he proved a general decay result associated to solution energy. Moreover, regarding arbitrary positive initial energy, blow up of solutions has been proved. Antontsev and Ferreira \cite{12ref}, studied a nonlinear class viscoelastic plate equation with a lower order by perturbation of $\vec{p}(x,t)$-Laplace operator of the form
\begin{equation*}
  u_{tt}+\Delta^{2}u-\Delta_{\vec{p}(x,t)}u+\int_{0}^{t}g(t-s)\Delta u(s)ds-\varepsilon\Delta u_{t}+f(u)=0,
\end{equation*}
associated with initial and Dirichlet-Neumann boundary conditions. Here, $\Delta_{\vec{p}(x,t)}$ is the $\vec{p}(x,t)$-Laplace operator which is defined as
\begin{equation*}
  \Delta_{\vec{p}(x,t)}u=\sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}\big(|\frac{\partial u}{\partial x_{i}}|^{p_{i}(x,t)-2}\frac{\partial u}{\partial x_{i}}\big),\ \vec{p}(x,t)=(p_{1},p_{2},\cdots,p_{n}).
\end{equation*}
They proved a blow up in finite time with negative initial energy under suitable conditions on $g,f$ and the variable exponent of the $\vec{p}(x,t)$-Laplace operator.
Recently, Antontsev et al. \cite{Antontsev.et.al2021} looked into the following nonlinear Timoshenko equation with
variable exponents:
$$
u_{tt}+\Delta^{2}u-M(\|\nabla u\|_{L^{2}(\Omega)}^{2})\Delta u+|u_{t}|^{p(x)-2}u_{t}=|u|^{q(x)-2}u,
$$
and demonstrated the local existence of the solution under suitable conditions. Moreover, nonexistence of solutions was proved with negative initial energy (see also \cite{Antontsev2021e}).\newline
Dai and Hao \cite{Dai2009} studied the following equation
$$
-M\big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big)div(|\nabla u|^{p(x)-2}\nabla u)=f(x,u),
$$
and by means of a direct variational approach and the theory of the variable-exponent Sobolev spaces, they established conditions
through which the existence and multiplicity of solutions for the problem were verified. In another study, Hamdani et al. \cite{Hamdani2020} investigated the following nonlocal $p(x)-$Kirchhoff type equation
$$
-(a-b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)div(|\nabla u|^{p(x)-2}\nabla u)=\lambda|u|^{p(x)-2}u+g(x,u),
$$
and obtained a nontrivial weak solution by using the Mountain Pass theorem. Related to the inverse problems with variable exponent nonlinearities, Shahrouzi in \cite{ShahrouziMMA2021} studied the general decay and blow up of solutions for the following Lam\'{e} system of inverse problem
\begin{align*}
   & u_{tt}-\Delta_{e}u-div\big(|\nabla u|^{r(x)-2}\nabla u\big)+\beta u_{t}+h(x,t,u,\nabla u)+a|u_{t}|^{m(x)-2}u_{t} \\
   & \quad\quad=b|u|^{p(x)-2}u+f(t)\omega(x),\ (x,t)\in \Omega\times(0,\infty) \\
   & u(x,t)=\frac{\partial u}{\partial\nu}(x,t)=0, \ (x,t)\in \partial\Omega\times(0,\infty) \\
   & u(x,0)=u_{0}(x),\ \ u_{t}(x,0)=u_{1}(x),\ x\in\Omega \\
   & \int_{\Omega}u(x,t)\omega(x)dx=\phi(t), \ \ t>0.
\end{align*}
The author proved the general decay of solutions when $b=0$, $h(x,t,u,\nabla u)\equiv0$ and the integral overdetermination tends to zero as time goes to infinity in appropriate range of variable exponents. Furthermore, in the absence of damping terms ($a=\beta=0$) and when $\phi(t)\equiv1$, blow up of solutions in a finite time has been proved. The relevant equations with variable-exponent nonlinearities have also been studied in \cite{Park2021,Piskin202037,14ref,15ref,Mahdi8,Ghegal2018,Park2018,16ref}.\par
Motivated by the aforementioned works, in the present paper, we study the blow-up analysis for a class of fourth-order viscoelastic $p(x)-$Kirchhoff type inverse source problem with variable-exponent nonlinearities. We mentioned before, existence of variable-exponent nonlinearities makes the study of inverse problems difficult. However, we try to extend and improve the previous results (\cite{6ref,8ref,ShahrouziMMA2021}) to a class of plate viscoelastic $p(x)-$Kirchhoff type inverse problems with variable-exponent nonlinearities. To the best of our knowledge, this is the first work dealing with the blow-up result for a plate viscoelastic $p(x)-$Kirchhoff type inverse source problem subject to the variable-exponent nonlinearities and various damping terms.\newline
The rest of the paper is organized as follows. In Section 2, we recall some definitions and Lemmas about the variable-exponent Lebesgue space, $L^{p(.)}(\Omega)$, the Sobolev space, $W^{1,p(.)}(\Omega)$ and additional conditions to be used for the main results. Section 3 includes two parts. First, we prove that the solutions of (1)-(4) blow-up in a finite time with suitable conditions on initial data and variable exponents when $m(x)\equiv 2$. Next, in the second part, we show that for $m(x)\geq m^{-}>2$ and under appropriate conditions on data, the solutions of (1)-(4) blow up at infinity.

\section{Preliminaries}
In this section, we recall some notations and functionals. We denote by $\|.\|_{q}$ the $L^{q}$-norm over $\Omega$ and in particular, the $L^{2}$-norm is denoted $\|.\|$ in $\Omega$. We shall assume that the functions $g(t),\ \omega(x)$ and those appearing in the data satisfy the following conditions:\newline
\begin{equation}
  g(t)\geq0,\ \ \ g^{\prime}(t)\leq 0,\ \
1-\int_{0}^{\infty}g(t)dt=l>0,
\end{equation}
and
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber && u_{0}\in H_{0}^{2}(\Omega)\cap L^{p(.)}(\Omega)\cap L^{m(.)}(\Omega),\qquad u_{1}\in L^{2}(\Omega)\cap L^{q(.)}(\Omega), \\
   && \qquad\qquad\qquad\qquad\int_{\Omega}u_{0}(x)\omega(x)dx=\phi(0).
\end{eqnarray}
In order to study problem (1)-(4), we need some hypotheses and theories about Lebesgue and Sobolev spaces with variable-exponents (for details, see \cite{Dien13,Edmu14,Edmu15,Xian16,Xian17}). Let $p(x)\geq1$ and measurable, we assume that
$$
C_{+}(\overline{\Omega})=\{h| h\in C(\overline{\Omega}),\ h(x)>1\ for\ any\ x\in \overline{\Omega}\},
$$
$$
h^{+}=\max_{\overline{\Omega}}h(x),\ \ h^{-}=\min_{\overline{\Omega}}h(x)\ \ \ for\ any\ h\in C(\overline{\Omega}),
$$
\begingroup\makeatletter\def\f@size{10}\check@mathfonts
\begin{equation*}
L^{p(x)}(\Omega)=\Big\{u|\ \text{u\ is\ a\ measurable\ real-valued\ function},\ \int_{\Omega}|u(x)|^{p(x)}dx<\infty\Big\}.
\end{equation*}
\endgroup
We equip the Lebesgue space with a variable exponent, $L^{p(x)}(\Omega)$, with the following Luxembourg-type norm
$$
\|u\|_{p(x)}:=\inf\Big\{\lambda>0\Big| \int_{\Omega}|\frac{u(x)}{\lambda}|^{p(x)}dx\leq 1 \Big\}
$$
\begin{lemma}\cite{Dien13,Xian17}
Let $\Omega$ be a bounded domain in $R^{n}$\newline
(i) the space $(L^{p(x)}(\Omega),\|.\|_{p(x)})$ is a Banach space, and its conjugate space is $L^{q(x)}(\Omega)$, where $\frac{1}{q(x)}+\frac{1}{p(x)}=1$. For any $u\in L^{p(x)}(\Omega)$ and $v\in L^{q(x)}(\Omega)$, we have
$$
\big|\int_{\Omega}uvdx\big|\leq\big(\frac{1}{p^{-}}+\frac{1}{q^{-}}\big)\|u\|_{p(x)}\|v\|_{q(x)};
$$
(ii) If $p,q\in C_{+}(\overline{\Omega})$, $q(x)\leq p(x)$ for any $x\in \overline{\Omega}$, then $L^{p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$, and the imbedding is continuous.
\end{lemma}
The variable-exponent Lebesgue Sobolev space $W^{1,p(x)}(\Omega)$ is defined by
$$
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega)| \nabla u\ exists\ and\ |\nabla u|\in L^{p(x)}(\Omega)\}.
$$
This space is a Banach space with respect to the norm $\|u\|_{W^{1,p(x)}(\Omega)}=\|u\|_{p(x)}+\|\nabla u\|_{p(x)}$. Furthermore, let $W^{1,p(x)}_{0}(\Omega)$ be the closure of $C_{0}^{\infty}(\Omega)$ in $W^{1,p(x)}(\Omega)$. The dual of $W^{1,p(x)}_{0}(\Omega)$ is defined as $W^{-1,p^{\prime}(x)}(\Omega)$, by the same way as the usual Sobolev spaces, where $\frac{1}{p(x)}+\frac{1}{p^{\prime}(x)}=1$. \newline
If we define
$$
p^{*}(x)=\left\{\begin{array}{rl}
\frac{Np(x)}{N-p(x)},\qquad p^{+}<N\\
\infty,\qquad\ \ \ \ p^{+}\geq N,
\end{array}\right.
$$
then we have
\begin{lemma}\cite{Dien13,Xian17}
Let $\Omega$ be a bounded domain in $R^{n}$. Then for any measurable bounded exponent $p(x)$ we have \newline
  (i) $W^{1,p(x)}(\Omega)$ and $W^{1,p(x)}_{0}(\Omega)$ are separable Banach spaces;\newline
  (ii) if $q\in C_{+}(\overline{\Omega})$ and $q(x)<p^{*}(x)$ for any $x\in\overline{\Omega}$, then the imbedding $W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is compact and continuous;\newline
  (iii) if  $p(x)$ is uniformly continuous in $\Omega$, then there exists a constant $C>0$, such that
  $$
  \|u\|_{p(x)}\leq C\|\nabla u\|_{p(x)}\ \ \ \ \forall u\in W^{1,p(x)}_{0}(\Omega).
  $$
\end{lemma}
By (iii) of Lemma 2, we know that the space $W^{1,p(x)}_{0}(\Omega)$ has an equivalent norm given by $\|u\|_{W^{1,p(x)}(\Omega)}=\|\nabla u\|_{p(x)}$.\newline
We recall the Young's inequality
\begin{equation}
ab\leq\theta a^{q(x)}+ C(\theta,q(x))b^{q^{\prime}(x)},\ \ a,b\geq0,\ \ \beta>0,\ \ \frac{1}{q(x)}+\frac{1}{q^{\prime}(x)}=1,
\end{equation}
where $C(\theta,q(x))=\frac{1}{q^{\prime}(x)}(\theta q(x))^{-\frac{q^{\prime}(x)}{q(x)}}$. In special case when $\theta=\frac{1}{q(x)}$, we have from (10)
\begin{equation}
  ab\leq \frac{a^{q(x)}}{q(x)}+\frac{b^{q^{\prime}(x)}}{q^{\prime}(x)}.
\end{equation}
Adapting the conditions (6) and integral over-determination (4), by multiplying equation (1) in $\omega(x)$, the key observation is that the
problem (1)-(4) is equivalent to the following direct problem
\begin{align}
   & u_{tt}+\Delta^{2}u-\left(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\right)\Delta_{p(x)}u-\int_{0}^{t}g(t-\tau)\Delta^{2}u(\tau)d\tau \notag \\
   & \quad +\beta|u_{t}|^{m(x)-2}u_{t}=\alpha|u|^{q(x)-2}u+f(t)\omega(x),\ (x,t)\in\Omega\times(0,+\infty) \\
   & u(x,t)=\frac{\partial{u}}{\partial\nu}(x,t)=0,\ (x,t)\in\partial\Omega\times(0,+\infty) \\
   & u(x,0)=u_{0}(x),\ \ u_{t}(x,0)=u_{1}(x),\ x\in\Omega
\end{align}
in which the unknown function $f(t)$ is replaced by
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber f(t) &=& \phi^{\prime\prime}(t)+\int_{\Omega}\Delta u\Delta\omega(x) dx+\beta\int_{\Omega}|u_{t}|^{m(x)-1}\omega(x)dx \\
  \nonumber && +(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\int_{\Omega}|\nabla u|^{p(x)-1}\nabla\omega(x) dx \\
  \nonumber && -\int_{0}^{t}g(t-\tau)\int_{\Omega}(\Delta u(\tau)-\Delta u)\Delta\omega(x) dxd\tau \\
   && -\int_{0}^{t}g(t-\tau)\int_{\Omega}\Delta u\Delta\omega(x) dxd\tau-\alpha\int_{\Omega}|u|^{q(x)-1}\omega(x)dx.
\end{eqnarray}
At this point, we state the local existence of solutions for the problem (12)-(14), that can be established employing the Galerkin method as in \cite{9ref}.
\begin{theorem}
  (Local existence) Let $u_{0}\in W_{0}^{1,p(.)}(\Omega)$, $u_{1}\in L^{2}(\Omega)$ and assume that (6), (8) and (9) be satisfied. Then problem (12)-(14) has a unique weak solution such that
  \begin{align*}
     & u\in L^{\infty}\Big((0,T),W_{0}^{1,p(.)}(\Omega)\Big)\cap L^{q(.)}((0,T),\Omega), \\
     & u_{t}\in L^{\infty}\Big((0,T),L^{2}(\Omega)\Big)\cap L^{m(.)}((0,T),\Omega), \\
     & u_{tt}\in L^{\infty}\Big((0,T),W_{0}^{-1,p^{\prime}(.)}(\Omega)\Big),
  \end{align*}
  for any $T>0$ and $\frac{1}{p(.)}+\frac{1}{p^{\prime}(.)}=1$.
\end{theorem}
\bigskip
\section{Blow-up}

In this section, we are going to prove the blow-up result for certain solutions with
positive initial energy. At first, by using concavity method \cite{3ref,7ref,14ref}, we prove that the solutions of (1)-(4) blow-up in a finite time with suitable conditions on initial data and variable exponents when $m(x)\equiv 2$. Next, in the second part, by using modified method inspired by \cite{Yang2020}, we show that for $m(x)\geq m^{-}>2$ and under appropriate conditions on data, the solutions of (1)-(4) blow up at infinity.

\subsection{Blow-up result with $m(x)\equiv2$}
 In order to prove the blow up of solutions with $m(x)\equiv 2$, we use the following change variable
\begin{equation}
  v(x,t)=e^{-\lambda t}u(x,t).
\end{equation}
A direct computation by substituting (16) into the problem (1)-(4) yields
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber && v_{tt}+(2\lambda+\beta)v_{t}+\lambda(\lambda+\beta)v+\Delta^{2}v-\int_{0}^{t}g_{1}(t-\tau)\Delta^{2}v(\tau)d\tau \\
  \nonumber && -(a+b\int_{\Omega}\frac{e^{\lambda p(x)t}}{p(x)}|\nabla v|^{p(x)}dx)div\big(e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)-2}\nabla v\big) \\
   && =\alpha e^{\lambda(q(x)-2)t}|v|^{q(x)-2}v+e^{-\lambda t}f(t)\omega(x),\ (x,t)\in \Omega\times(0,\infty)
\end{eqnarray}
\begin{equation}
v(x,t)=\frac{\partial v}{\partial\nu}=0,  \ (x,t)\in \partial\Omega\times(0,\infty)
\end{equation}
\begin{equation}
v(x,0)=u_{0}(x),\ \ v_{t}(x,0)=u_{1}(x)-\lambda u_{0}(x),\qquad x\in \Omega,
\end{equation}
\begin{equation}
\int_{\Omega }v(x,t)\omega(x)dx=e^{-\lambda t}\phi(t),\qquad t>0,
\end{equation}
where $g_{1}(s)=e^{-\lambda s}g(s)$ and the value of the parameter $\lambda$ will be prescribed later.\newline
Similarly, adapting to the condition (6) and integral overdetermination, the inverse problem (17)-(20) is equivalent to the direct problem (17)-(19) when the unknown function $f(t)$ is replaced by
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber f(t) &=& \phi^{\prime\prime}(t)+\beta\phi^{\prime}(t)+e^{\lambda t}\int_{\Omega}\Delta v\Delta\omega(x) dx \\
  \nonumber && +a\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)-1}\nabla\omega(x) dx \\
  \nonumber && +b(\int_{\Omega}\frac{e^{\lambda p(x)t}}{p(x)}|\nabla v|^{p(x)}dx)(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)-1}\nabla\omega(x) dx) \\
  \nonumber && -e^{\lambda t}\int_{0}^{t}g_{1}(t-\tau)\int_{\Omega}\Delta v\Delta\omega(x) dxd\tau \\
  \nonumber && -e^{\lambda t}\int_{0}^{t}g_{1}(t-\tau)\int_{\Omega}(\Delta v(\tau)-\Delta v)\Delta\omega(x) dxd\tau \\
  && -\alpha\int_{\Omega}e^{\lambda(q(x)-1)}|v|^{q(x)-1}\omega(x)dx.
\end{eqnarray}
The energy function related with problem (17)-(19) is given by
\begin{equation}
  E_{\lambda}(t)=\alpha\int_{\Omega}\frac{e^{\lambda(q(x)-2)t}}{q(x)}|v|^{q(x)}dx-a\int_{\Omega}\frac{e^{\lambda(p(x)-2)t}}{p(x)}|\nabla v|^{p(x)}dx-\frac{1}{2}I(t),
\end{equation}
where
$$
I(t)=\|v_{t}\|^{2}+\lambda(\lambda+\beta)\|v\|^{2}+(1-\int_{0}^{t}g_{1}(s)ds)\|\Delta v\|^{2}+(g_{1}\odot\Delta v)(t)
$$
$$
+b\big(\int_{\Omega}\frac{e^{\lambda(p(x)-1)t}}{p(x)}|\nabla v|^{p(x)}dx\big)^{2},
$$
and $(g_{1}\odot\Delta v)(t)=\int_{0}^{t}g_{1}(t-\tau)\|\Delta v(\tau)-\Delta v\|^{2}d\tau$.\newline
Now we are in a position to state our blow-up result as follows:
\begin{theorem}
Let the conditions (5), (6) and (8), (9) be satisfied and suppose that the functions $\phi^{\prime\prime}(t), \phi^{\prime}(t)$ and $\phi(t)$ are continuous and bounded such that for constants $M_{1}$ and $M_{2}(\lambda)$:
$$|\phi^{\prime\prime}(t)+\beta\phi^{\prime}(t)|\leq M_{1}\quad {and}\quad |\phi^{\prime}(t)-\lambda\phi(t)|\leq M_{2}(\lambda).$$ Moreover, assume that
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
   && q^{-}>\max\{4p^{+}-2,\ 3+\frac{2(p^{+})^{2}(p^{+}-1)}{(p^{-})^{2}},\ \frac{4p^{+}(p^{-}+2)}{(p^{-})^{2}}\}, \\
   && l=1-\int_{0}^{\infty}g(s)ds\geq \frac{6}{q^{-}+2},\quad \alpha\geq \frac{2(q^{+}-1)}{q^{+}} \\
   && E_{\lambda}(0)\geq\frac{D_{1}}{\lambda(q^{-}-3)}+\frac{2D_{2}}{q^{-}},
\end{eqnarray}
where
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber D_{1} &=& M_{1}M_{2}(\lambda)+M_{2}^{2}(\lambda)\big[\frac{1}{2(1-l)}+\frac{1-l}{2\lambda(q^{-}-3)}+\frac{1-l}{2}\big]\|\Delta\omega\|^{2} \\
  \nonumber && +a\int_{\Omega}\frac{M_{2}^{p(x)}(\lambda)}{p(x)\lambda^{p(x)-1}}|\nabla\omega|^{p(x)}dx+\alpha\int_{\Omega}\frac{M_{2}^{q(x)}(\lambda)}{q(x)}|\omega|^{q(x)}dx \\
  \nonumber && +\frac{b}{4}\big(\int_{\Omega}\frac{M_{2}^{p(x)}(\lambda)}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega|^{p(x)}dx\big)^{2},
\end{eqnarray}
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber D_{2} &=& M_{1}+(\frac{l^{2}-2l+3}{2(1-l)})\|\Delta\omega\|^{2}+a\int_{\Omega}\frac{|\nabla\omega|^{p(x)}}{p(x)}dx +\int_{\Omega}\frac{(\alpha|\omega(x)|)^{q(x)}}{q(x)}dx\\
  \nonumber && +\frac{b}{4}\big(\int_{\Omega}\frac{(p(x)-1)^{p(x)-1}}{p(x)}|\nabla\omega|^{p(x)}dx\big)^{2}.
\end{eqnarray}
Then, for sufficiently large $\lambda$, there exists a finite time $t^{*}$ such that the solution of the problem (1)-(4) blows up in a finite time, that is
\begin{equation}
  \|u(t)\|\rightarrow +\infty\ \ as\ \ t\rightarrow t^{*}.
\end{equation}
\end{theorem}
To prove the blow-up result in this case, we need the following Lemmas.
\begin{lemma}
Under the conditions of Theorem 2, the unknown function $f(t)$, defined by (21), satisfies
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber e^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|f(t) &\leq & \frac{a\lambda(p^{+}-1)}{p^{+}}\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx\\
  \nonumber && +(1-l)\|\Delta v\|^{2}+\frac{\lambda(q^{-}-3)}{2}(g_{1}\odot\Delta v)(t) \\
  \nonumber && +\frac{2b}{(p^{-})^{2}}\big(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx\big)^{2} \\
  \nonumber && +\frac{\alpha(q^{+}-1)}{q^{+}}\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx+e^{-2\lambda t}D_{1}. \\
   &&
\end{eqnarray}
\end{lemma}
\begin{proof}
By using (21), we have
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber && e^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|f(t) \\
  \nonumber &=& e^{-2\lambda t}(\phi^{\prime\prime}(t)+\beta\phi^{\prime}(t))|\phi^{\prime}(t)-\lambda\phi(t)|+e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{\Omega}\Delta v\Delta\omega(x) dx \\
  \nonumber && +ae^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{\Omega}|\nabla (e^{\lambda t}v)|^{p(x)-1}\nabla\omega(x) dx \\
  \nonumber && +be^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|(\int_{\Omega}\frac{|\nabla (e^{\lambda t}v)|^{p(x)}}{p(x)}dx)(\int_{\Omega}|\nabla (e^{\lambda t}v)|^{p(x)-1}\nabla\omega(x) dx) \\
  \nonumber && -e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{0}^{t}g_{1}(t-\tau)\int_{\Omega}\Delta v(\tau)\Delta\omega(x) dxd\tau \\
   && -\alpha e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)-1}\omega(x) dx.
\end{eqnarray}
At this point, by using the Young's inequality (10) and Cauchy-Schwarz inequality and (5), we estimate the terms on the right-hand side of (28) as follows\newline
\begin{equation}
 e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|.|\int_{\Omega}\Delta v\Delta\omega dx| \leq \frac{1-l}{2}\|\Delta v\|^{2}+\frac{e^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|^{2}}{2(1-l)}\|\Delta\omega\|^{2}.
\end{equation}
$$
e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|.|\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)-1}\nabla\omega(x) dx|\qquad\qquad\qquad\qquad\qquad
$$
\begin{eqnarray}
 \nonumber &\leq & e^{-2\lambda t}\int_{\Omega}\frac{\lambda(p(x)-1)}{p(x)}e^{\lambda p(x)t}|\nabla v|^{p(x)}dx \\
 \nonumber && +e^{-2\lambda t}\int_{\Omega}\frac{|\phi^{\prime}(t)-\lambda\phi(t)|^{p(x)}}{p(x)\lambda^{p(x)-1}}|\nabla\omega(x)|^{p(x)}dx \\
 \nonumber &\leq & \frac{\lambda(p^{+}-1)}{p^{+}}\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
  && +e^{-2\lambda t}\int_{\Omega}\frac{|\phi^{\prime}(t)-\lambda\phi(t)|^{p(x)}}{p(x)\lambda^{p(x)-1}}|\nabla\omega(x)|^{p(x)}dx.
\end{eqnarray}
$$
e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|.(\int_{\Omega}\frac{e^{\lambda(p(x)-1)t}}{p(x)}|\nabla v|^{p(x)}dx)(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)-1}\nabla\omega(x) dx)\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\leq & e^{-2\lambda t}(\int_{\Omega}\frac{e^{\lambda p(x)t}}{p(x)}|\nabla v|^{p(x)}dx)(\int_{\Omega}\frac{1}{p(x)}e^{\lambda p(x)t}|\nabla v|^{p(x)}dx \\
  \nonumber && +\int_{\Omega}\frac{|\phi^{\prime}(t)-\lambda\phi(t)|^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx) \\
  \nonumber &\leq & \frac{2}{(p^{-})^{2}}\big(\int_{\Omega}e^{\lambda (p(x)-1)t}|\nabla v|^{p(x)}dx\big)^{2} \\
  && +\frac{e^{-2\lambda t}}{4}\big(\int_{\Omega}\frac{|\phi^{\prime}(t)-\lambda\phi(t)|^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx\big)^{2}.
\end{eqnarray}
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber && e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{0}^{t}g_{1}(t-\tau)\int_{\Omega}\Delta v(\tau)\Delta\omega(x) dxd\tau \\
  \nonumber &=& e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{0}^{t}g_{1}(t-\tau)(\int_{\Omega}(\Delta v(\tau)-\Delta v)\Delta\omega(x) dx \\
  \nonumber && +\int_{\Omega}\Delta v\Delta\omega(x) dx)d\tau \\
  \nonumber &\leq & \frac{\lambda(q^{-}-3)}{2}(g_{1}\odot\Delta v)(t)+\frac{1-l}{2}\|\Delta v\|^{2} \\
  && +e^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|^{2}(\frac{1-l}{2}+\frac{1-l}{2\lambda(q^{-}-3)})\|\Delta\omega\|^{2},
\end{eqnarray}
where the fact $\int_{0}^{\infty}g_{1}(s)ds<\int_{0}^{\infty}g(s)ds=1-l$ has been used.
$$
e^{-\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)-1}\omega(x) dx\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\leq & e^{-2\lambda t}\big(\int_{\Omega}\frac{q(x)-1}{q(x)}e^{\lambda q(x)t}|v|^{q(x)}dx+\int_{\Omega}\frac{|\phi^{\prime}(t)-\lambda\phi(t)|^{q(x)}}{q(x)}|\omega(x)|^{q(x)}dx\big) \\
  \nonumber &\leq & \frac{q^{+}-1}{q^{+}}\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx+e^{-2\lambda t}\int_{\Omega}\frac{|\phi^{\prime}(t)-\lambda\phi(t)|^{q(x)}}{q(x)}|\omega(x)|^{q(x)}dx. \\
\end{eqnarray}
Combining estimations (29)-(33) with (28) and by using hypotheses of Theorem 2 about $\phi^{\prime\prime}(t), \phi^{\prime}(t)$ and $\phi(t)$, we derive inequality (27) and proof of Lemma 3 is completed.
\end{proof}
\begin{lemma}
Under the conditions of Theorem 2, the energy functional $E_{\lambda}(t)$, defined by (22), satisfies
\begin{equation}
E_{\lambda}(t)\geq E_{\lambda}(0)-\frac{D_{1}}{\lambda(q^{-}-3)}\qquad\qquad\forall t\in R^{+},
\end{equation}
\end{lemma}
\begin{proof}
A multiplication of equation (17) by $v_{t}$ and integrating over $\Omega$ give
\begin{eqnarray}
\nonumber E_{\lambda}^{\prime}(t)&=&(2\lambda+\beta)\|v_{t}\|^{2}-a\int_{\Omega}\frac{\lambda(p(x)-2)}{p(x)}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
\nonumber && -b\lambda\big(\int_{\Omega}\frac{e^{\lambda(p(x)-1)t}}{p(x)}|\nabla v|^{p(x)}dx\big)^{2}-\frac{1}{2}(g_{1}^{\prime}\odot\Delta v)(t)+\frac{1}{2}g_{1}(t)\|\Delta v\|^{2}\\
\nonumber && -b(\int_{\Omega}\frac{e^{\lambda(p(x)-1)t}}{p(x)}|\nabla v|^{p(x)}dx)(\int_{\Omega}\frac{\lambda(p(x)-2)}{p(x)}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx) \\
\nonumber && +\alpha\int_{\Omega}\frac{\lambda(q(x)-2)}{q(x)}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx-e^{-2\lambda t}(\phi^{\prime}(t)-\lambda\phi(t))f(t) \\
\nonumber &\geq & (2\lambda+\beta)\|v_{t}\|^{2}-\frac{a\lambda(p^{+}-2)}{p^{+}}\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
\nonumber && -\frac{b\lambda(p^{+}-1)}{(p^{-})^{2}}\big(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx\big)^{2} \\
\nonumber && +\frac{\alpha(q^{-}-2)}{q^{-}}\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx \\
 && -e^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|f(t),
\end{eqnarray}
where conditions (5) and (8) have been used.\newline
Employing (22), we obtain from (35) the following inequality for some $\varepsilon>0$
$$
E_{\lambda}^{\prime}(t)-\varepsilon E_{\lambda}(t) \geq \frac{\alpha[\lambda(q^{-}-2)-\varepsilon]}{q^{-}}\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx+\frac{\varepsilon\lambda(\lambda+\beta)}{2}\|v\|^{2}
$$
\begin{eqnarray}
\nonumber && +(2\lambda+\beta+\frac{\varepsilon}{2})\|v_{t}\|^{2}+\frac{\varepsilon}{2}(1-\int_{0}^{t}g_{1}(s)ds)\|\Delta v\|^{2}+\frac{\varepsilon}{2}(g_{1}\odot\Delta v)(t) \\
\nonumber && +\frac{a[\varepsilon-\lambda(p^{+}-2)]}{p^{+}}\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
\nonumber && +b\big[\frac{\varepsilon}{2(p^{+})^{2}}-\frac{\lambda(p^{+}-1)}{(p^{-})^{2}})\big]\big(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx\big)^{2} \\
&& -e^{-2\lambda t}|\phi^{\prime}(t)-\lambda\phi(t)|f(t),
\end{eqnarray}
where (5) has been used.\newline
Thanks to the Lemma 3 and taking into account (27) and set $\varepsilon:=\lambda(q^{-}-3)$, then we get
$$
E_{\lambda}^{\prime}(t)-\lambda(q^{-}-3) E_{\lambda}(t) \geq \frac{\alpha}{q^{-}}[\lambda-\frac{q^{-}(q^{+}-1)}{q^{+}}]\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx\qquad\qquad
$$
\begin{eqnarray}
\nonumber && +[\frac{\lambda(q^{-}-3)}{2}(1-\int_{0}^{t}g_{1}(s)ds)+l-1]\|\Delta v\|^{2} \\
\nonumber && +\frac{\lambda a}{p^{+}}(q^{-}-2p^{+})\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
\nonumber && +b\big[\frac{\lambda(q^{-}-3)}{2(p^{+})^{2}}-\frac{\lambda(p^{+}-1)}{(p^{-})^{2}}-\frac{2}{(p^{-})^{2}}\big]\big(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx\big)^{2} \\
&& -e^{-2\lambda t}D_{1},
\end{eqnarray}
where $D_{1}$ is satisfied in Theorem 2.\newline
By using (23) and for sufficiently large $\lambda$, we deduce from (37)
\begin{equation*}
 E_{\lambda}^{\prime}(t)-\lambda(q^{-}-3) E_{\lambda}(t) \geq -e^{-2\lambda t}D_{1} \geq -D_{1}.
\end{equation*}
Integrating over $(0,t)$, we observe that
\begin{equation*}
E_{\lambda}(t)\geq E_{\lambda}(0)-\frac{D_{1}}{\lambda(q^{-}-3)},\ \ \ \forall
t\geq0,
\end{equation*}
and proof of Lemma 4 is complete.
\end{proof}
Now, we are in a position to prove the Theorem 2 by using Lemma 3 and Lemma 4.\newline
\textbf{Proof of Theorem 2.} For obtaining the blow-up result, we apply concavity method by defining the following functional
\begin{equation}
\psi(t)=\|v(t)\|^{2},
\end{equation}
then%
\begin{equation}
\psi^{\prime}(t)=2\int_{\Omega}vv_{t}dx,
\end{equation}
\begin{equation}
\psi^{\prime\prime}(t)=2\int_{\Omega}vv_{tt}dx+2\|v_{t}\|^{2}.
\end{equation}
A multiplication of equation (17) by $v$ and integrating over
$\Omega$ give
\begin{eqnarray}
\nonumber \int_{\Omega}vv_{tt}dx &=& -(2\lambda+\beta)\int_{\Omega}vv_{t}dx-\lambda(\lambda+\beta)\|v\|^{2}-(1-\int_{0}^{t}g_{1}(s)ds)\|\Delta v\|^{2} \\
\nonumber && -a\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx+\alpha\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx \\
\nonumber && -b(\int_{\Omega}\frac{e^{\lambda p(x)t}}{p(x)}|\nabla v|^{p(x)}dx)(\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx) \\
&& +\int_{0}^{t}g_{1}(t-\tau)\int_{\Omega}(\Delta v(\tau)-\Delta v)\Delta vdxd\tau+e^{-2\lambda t}f(t).
\end{eqnarray}
By virtue of the Young's inequality (10) with $\theta=\frac{1-l}{2},\ q(x)=q^{\prime}(x)=2$, we obtain
$$
|\int_{\Omega}\Delta v\int_{0}^{t}g_{1}(t-\tau)(\Delta v(\tau)-\Delta u)d\tau dx|\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\leq & \frac{1-l}{2}\|\Delta v\|^{2}+\frac{1}{2(1-l)}\int_{\Omega}\Big(\int_{0}^{t}g_{1}(t-\tau)|\Delta v(\tau)-\Delta v|d\tau\Big)^{2}dx \\
  \nonumber &= & \frac{1-l}{2}\|\Delta v\|^{2}+\frac{1}{2(1-l)}\int_{\Omega}\Big(\int_{0}^{t}\frac{g_{1}(t-\tau)}{\sqrt{g_{1}(t-\tau)}}\sqrt{g_{1}(t-\tau)}|\Delta v(\tau)-\Delta v|d\tau\Big)^{2}dx \\
  \nonumber &\leq & \frac{1-l}{2}\|\Delta v\|^{2}+\frac{1}{2(1-l)}\big(\int_{0}^{t}g_{1}(s)ds\big)\int_{\Omega}\int_{0}^{t}g_{1}(t-\tau)|\Delta v(\tau)-\Delta v|^{2}d\tau dx \\
 \nonumber &\leq & \frac{1-l}{2}\|\Delta v\|^{2}+\frac{1}{2}(g_{1}\odot\Delta v)(t), \\
\end{eqnarray}
where $\int_{0}^{t}g_{1}(s)ds<\int_{0}^{\infty}g_{1}(s)ds<\int_{0}^{\infty}g(s)ds=1-l$.\newline
Combining (42) with (41) and by using (5), we deduce
\begin{eqnarray}
\nonumber \int_{\Omega}vv_{tt}dx &\geq & -(2\lambda+\beta)\int_{\Omega}vv_{t}dx-\lambda(\lambda+\beta)\|v\|^{2} \\
\nonumber && -[(1-\int_{0}^{t}g_{1}(s)ds)+\frac{1-l}{2}]\|\Delta v\|^{2} \\
\nonumber && -\frac{1}{2}(g_{1}\odot\Delta v)(t)-a\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
\nonumber && -\frac{b}{p^{-}}(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx)^{2} \\
 && +\alpha\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx+e^{-2\lambda t}f(t).
\end{eqnarray}
At this point, similar to Lemma 3.1 (when $|\phi^{\prime}(t)-\lambda\phi(t)|:=1$), one can observe the following estimation of the last term on the right-hand side of (43):
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\nonumber e^{-2\lambda t}f(t) &\leq & (1-l)\|\Delta v\|^{2}+\frac{a(p^{+}-1)}{p^{+}}\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
  \nonumber && +\frac{1}{2}(g_{1}\odot\Delta v)(t)+\frac{2b}{(p^{-})^{2}}\big(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx\big)^{2} \\
   && +\frac{q^{+}-1}{q^{+}}\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx+e^{-2\lambda t}D_{2},
\end{eqnarray}
where $D_{2}$ satisfies Theorem 2.\newline
Therefore by utilizing (44) and (22) into (43), we obtain for $\delta>0$
\begin{eqnarray}
\nonumber \int_{\Omega}vv_{tt}dx &\geq & \delta E_{\lambda}(t) -(2\lambda+\beta)\int_{\Omega}vv_{t}dx+(\frac{\delta}{2}-1)\lambda(\lambda+\beta)\|v\|^{2} \\
\nonumber && +[(\frac{\delta}{2}-1)(1-\int_{0}^{t}g_{1}(s)ds)-\frac{3(1-l)}{2}]\|\Delta v\|^{2} \\
\nonumber && +\frac{\delta}{2}\|v_{t}\|^{2}+(\frac{\delta}{2}-1)(g_{1}\odot\Delta v)(t) \\
\nonumber && +a(\frac{\delta}{p^{+}}-\frac{p^{+}-1}{p^{+}}-1)\int_{\Omega}e^{\lambda(p(x)-2)t}|\nabla v|^{p(x)}dx \\
 \nonumber && +b(\frac{\delta}{2(p^{+})^{2}}-\frac{2}{(p^{-})^{2}}-\frac{1}{p^{-}})(\int_{\Omega}e^{\lambda(p(x)-1)t}|\nabla v|^{p(x)}dx)^{2} \\
 \nonumber && +[\alpha(1-\frac{\delta}{q^{-}})-\frac{q^{+}-1}{q^{+}}]\int_{\Omega}e^{\lambda(q(x)-2)t}|v|^{q(x)}dx-e^{-2\lambda t}D_{2}.\\
 &&
\end{eqnarray}
Thus, by using the fact that
$$
1-\int_{0}^{t}g_{s}ds\geq 1-\int_{0}^{\infty}g_{1}(s)ds\geq 1-\int_{0}^{\infty}g(s)ds=l,
$$
we choose $\delta:=\frac{q^{-}}{2}$ and apply the conditions of Theorem 2 to obtain from (45)
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber \int_{\Omega}vv_{tt}dx &\geq & \frac{q^{-}}{2}E_{\lambda}(t)+\frac{q^{-}}{4}\|v_{t}\|^{2}+\frac{(q^{-}-4)}{4}\lambda(\lambda+\beta)\|v\|^{2} \\
   && -(2\lambda+\beta)\int_{\Omega}vv_{t}dx-D_{2}.
\end{eqnarray}
Now, by using Lemma 4 and (25), we get from (46)
\begin{equation}
\int_{\Omega}vv_{tt}dx\geq \frac{q^{-}}{4}\|v_{t}\|^{2}-(2\lambda+\beta)\int_{\Omega}vv_{t}dx.
\end{equation}
By substituting (38)-(40) in (47) we get
\begin{equation*}
\psi^{\prime\prime}(t)\geq \frac{(q^{-}+4)}{2}\|v_{t}\|^{2}-(2\lambda+\beta)\psi^{\prime}(t),
\end{equation*}
thus
\begin{equation}
\psi(t)\psi^{\prime\prime}(t)\geq \frac{(q^{-}+4)}{8}(\psi^{\prime}(t))^{2}-(2\lambda+\beta)\psi(t)\psi^{\prime}(t),
\end{equation}
where inequality $(\psi^{\prime}(t))^{2}\leq 4\|v_{t}\|^{2}\|v\|^{2}$ has been used.\newline
Hence, the concavity argument (see \cite{3ref}) gives us
\begin{equation*}
 \lim_{t\rightarrow t^{*}} \psi(t)=\infty,
\end{equation*}
which yields solutions of problem (17)-(19) blow up in a finite time $t^{*}$. Since this system is equivalent to (1)-(4), the proof of Theorem 2 is complete.\par
\textbf{Remark.}
Under the conditions of Theorem 2, if we choose initial data appropriately such that
$$
\psi^{\prime}(0)-\frac{8(2\lambda+\beta)}{q^{-}-4}\psi(0)>0,
$$
then we obtain an upper bound for the lifetime of the solutions as
$$
t^{*}<\frac{1}{2\lambda+\beta}\ln\frac{(q^{-}-4)\psi^{\prime}(0)}{(q^{-}-4)\psi^{\prime}(0)-8(2\lambda+\beta)\psi(0)}.
$$

\bigskip
\subsection{Blow-up result with $m(x)>2$}
In this part, we suppose that $2<m^{-}\leq m(x)\leq m^{+}<+\infty$ and we shall prove that the solutions of problem (1)-(4) blow up at infinity. By constructing a proper auxiliary functional and using modified method inspired by \cite{Yang2020}, blow up at infinity has been proved when the variable exponents and initial data satisfy appropriate conditions and the initial energy is positive.\newline
Firstly, we define
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber E(t) &=& \frac{1}{2}\|u_{t}\|^{2}+\frac{1}{2}(1-\int_{0}^{t}g(s)ds)\|\Delta u\|^{2}+a\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx \\
  \nonumber && +\frac{b}{2}\big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big)^{2}+\frac{1}{2}(g\odot\Delta u)(t) \\
  && -\alpha\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx.
\end{eqnarray}
By definition of $E(t)$ and using (8), we deduce
\begin{equation}
  E^{\prime}(t)\leq -\beta\int_{\Omega}|u_{t}|^{m(x)}dx+f(t)\phi^{\prime}(t).
\end{equation}
We are in a position that state blow-up result as follows:
\begin{theorem}
Let the conditions (5) (with $m^{-}>2$), (6) and (8), (9) be satisfied and suppose that the functions $\phi^{\prime\prime}(t), \phi^{\prime}(t)$ and $\phi(t)$ are continuous and bounded such that there exist constants $M_{3}$ and $M_{4}$
$$
|\phi^{\prime\prime}(t)|\leq M_{3}\quad {and}\quad |\phi^{\prime}(t)-m^{+}\phi(t)|\leq M_{4}.
$$
Moreover, Assume that
\begin{equation}
\begin{aligned}
   & \max\{2p^{+}-1,\ \frac{3-l}{l},\ \frac{2(p^{+})^{2}(p^{-}+2)}{(p^{-})^{2}}\}<m^{-}<q^{-}, \\
   & m^{+}>\frac{m^{-}}{\sqrt{2(m^{-}+2)}},
\end{aligned}
\end{equation}
and suppose that $E(0)>0$ is a given initial energy level. If we choose initial data $u_{0}$ and $u_{1}$ such that satisfying
\begin{equation*}
  \int_{\Omega}u_{0}u_{1}dx>m^{+}E(0)+\frac{m^{+}D_{3}}{m^{-}},
\end{equation*}
where $D_{3}$ will be enunciate in Lemma 5. Then, for sufficiently large $\alpha$ and sufficiently small $\beta$, the solution of the problem (1)-(4) blows up at infinity, that is
\begin{equation}
  \|u(t)\|\rightarrow +\infty\ \ as\ \ t\rightarrow +\infty.
\end{equation}
\end{theorem}
Before going to prove of Theorem 3, we state and prove the following Lemma which will be used later:
\begin{lemma}
  Under the conditions of Theorem 3, for any $\varepsilon>0$ the unknown function $f(t)$, defined by (21), satisfies
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
\nonumber |\phi(t)-\varepsilon\phi^{\prime}(t)|f(t) &\leq & \frac{l(m^{-}-2)}{6}\|\Delta u\|^{2}+\frac{a(p^{+}-1)}{p^{+}}\int_{\Omega}|\nabla u|^{p(x)}dx \\
\nonumber && +\frac{2b}{(p^{-})^{2}}\big(\int_{\Omega}|\nabla u|^{p(x)}dx\big)^{2} \\
\nonumber && +\frac{3(1-l)}{2l}(g\odot\Delta u)(t)+\frac{\beta(m^{+}-1)}{m^{+}}\int_{\Omega}|u_{t}|^{m(x)}dx \\
   && +\frac{(q^{+}-1)}{q^{+}}\int_{\Omega}|u|^{q(x)}dx+D_{3},
\end{eqnarray}
where
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber D_{3} &=& M_{3}M_{4}+M_{4}^{2}\big(\frac{3(l^{2}-2l+2)}{l(m^{-}-2)}+\frac{l}{6}\big)\|\Delta\omega\|^{2} \\
  \nonumber && +\beta\int_{\Omega}\frac{M_{4}^{m(x)}}{m(x)}|\omega(x)|^{m(x)}dx+\int_{\Omega}\frac{(\alpha M_{4})^{q(x)}}{q(x)}|\omega(x)|^{q(x)}dx \\
  \nonumber && +\frac{b}{4}\big(\int_{\Omega}\frac{M_{4}^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx\big)^{2}.
\end{eqnarray}
\end{lemma}
\begin{proof}
  Recalling (15), by virtue of Cauchy and Yang inequalities, we estimate the terms on the RHS of (15) as follows:
$$
|(\phi(t)-\varepsilon\phi^{\prime}(t))\int_{\Omega}\Delta u\Delta\omega dx|\leq \frac{l(m^{-}-2)}{12}\|\Delta u\|^{2}+\frac{3|\phi(t)-\varepsilon\phi^{\prime}(t)|^{2}}{l(m^{-}-2)}\|\Delta\omega\|^{2}.
$$
\begin{equation}
\end{equation}
$$
|(\phi(t)-\varepsilon\phi^{\prime}(t))(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\int_{\Omega}|\nabla u|^{p(x)-1}\nabla\omega(x) dx|\qquad\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &=& |(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\int_{\Omega}|\nabla u|^{p(x)-1}(\phi(t)-\varepsilon\phi^{\prime}(t))\nabla\omega(x) dx| \\
  \nonumber &\leq & a\underbrace{|\int_{\Omega}|\nabla u|^{p(x)-1}(\phi(t)-\varepsilon\phi^{\prime}(t))\nabla\omega(x) dx|}_{I_{1}} \\
   && +b\underbrace{\big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big)|\int_{\Omega}|\nabla u|^{p(x)-1}(\phi(t)-\varepsilon\phi^{\prime}(t))\nabla\omega(x) dx|}_{I_{2}}.
\end{eqnarray}
For $I_{1}$ and $I_{2}$ we have
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber I_{1} &\leq & \int_{\Omega}\frac{p(x)-1}{p(x)}|\nabla u|^{p(x)}dx+\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{p(x)}}{p(x)}|\nabla\omega(x)|^{p(x)}dx \\
   &\leq & \frac{p^{+}-1}{p^{+}}\int_{\Omega}|\nabla u|^{p(x)}dx+\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{p(x)}}{p(x)}|\nabla\omega(x)|^{p(x)}dx.
\end{eqnarray}
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber I_{2} &\leq & \big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big)\big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx \\
  \nonumber && +\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx\big) \\
  \nonumber &\leq & \frac{1}{(p^{-})^{2}}\big(\int_{\Omega}|\nabla u|^{p(x)}dx\big)^{2} \\
  \nonumber && +\big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\big)\big(\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx\big) \\
  \nonumber &\leq & \frac{2}{(p^{-})^{2}}\big(\int_{\Omega}|\nabla u|^{p(x)}dx\big)^{2} \\
   && +\frac{1}{4}\big(\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx\big)^{2}.
\end{eqnarray}
By combining (56) and (57), we get
$$
|(\phi(t)-\varepsilon\phi^{\prime}(t))(a+b\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)\int_{\Omega}|\nabla u|^{p(x)-1}\nabla\omega(x) dx|\qquad\qquad\qquad\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\leq & \frac{p^{+}-1}{p^{+}}\int_{\Omega}|\nabla u|^{p(x)}dx+\frac{2}{(p^{-})^{2}}\big(\int_{\Omega}|\nabla u|^{p(x)}dx\big)^{2} \\
   && +\frac{1}{4}\big(\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{p(x)}}{p(x)}(p(x)-1)^{p(x)-1}|\nabla\omega(x)|^{p(x)}dx\big)^{2}.
\end{eqnarray}
$$
|\int_{\Omega}|u_{t}|^{m(x)-1}|\phi(t)-\varepsilon\phi^{\prime}(t)|\omega(x)dx|\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\leq & \int_{\Omega}\frac{m(x)-1}{m(x)}|u_{t}|^{m(x)}dx+\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{m(x)}}{m(x)}|\omega(x)|^{m(x)}dx \\
   &\leq & \frac{m^{+}-1}{m^{+}}\int_{\Omega}|u_{t}|^{m(x)}dx+\int_{\Omega}\frac{|\phi(t)-\varepsilon\phi^{\prime}(t)|^{m(x)}}{m(x)}|\omega(x)|^{m(x)}dx.
\end{eqnarray}
$$
|\int_{0}^{t}g(t-\tau)\int_{\Omega}(\Delta u(\tau)-\Delta u)|\phi(t)-\varepsilon\phi^{\prime}(t)|\Delta\omega(x) dxd\tau|\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\leq \frac{3(1-l)}{2l}(g\odot\Delta u)(t)+\frac{1}{6}|\phi(t)-\varepsilon\phi^{\prime}(t)|^{2}\|\Delta\omega\|^{2}.
\end{equation}
$$
  |\int_{0}^{t}g(t-\tau)\int_{\Omega}\Delta u|\phi(t)-\varepsilon\phi^{\prime}(t)|\Delta\omega(x) dxd\tau|\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{equation}
\leq \frac{l(m^{-}-2)}{12}\|\Delta u\|^{2}+\frac{3(1-l)^{2}|\phi(t)-\varepsilon\phi^{\prime}(t)|^{2}}{l(m^{-}-2)}\|\Delta\omega\|^{2}.
\end{equation}
$$
|\alpha\int_{\Omega}|u|^{q(x)-1}|\phi(t)-\varepsilon\phi^{\prime}(t)|\omega(x)dx|\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\leq & \int_{\Omega}\frac{q(x)-1}{q(x)}|u|^{q(x)}dx+\int_{\Omega}\frac{(\alpha|\phi(t)-\varepsilon\phi^{\prime}(t)|)^{q(x)}}{q(x)}|\omega(x)|^{q(x)}dx \\
   &\leq & \frac{(q^{+}-1)}{q^{+}}\int_{\Omega}|u|^{q(x)}dx+\int_{\Omega}\frac{(\alpha|\phi(t)-\varepsilon\phi^{\prime}(t)|)^{q(x)}}{q(x)}|\omega(x)|^{q(x)}dx.
\end{eqnarray}
Finally, utilizing (54) and (58)-(62) into (15) and using conditions of Theorem 3 about $\phi^{\prime\prime}(t), \phi^{\prime}(t)$ and $\phi(t)$, we get inequality (53) and proof of Lemma 5 is completed.
\end{proof}
\textbf{Proof of Theorem 3.}
Multiplying equation (1) by $u$ and integrating over $\Omega$ yield
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber \frac{d}{dt}\int_{\Omega}uu_{t}dx &=& \|u_{t}\|^{2}+\int_{\Omega}uu_{tt}dx \\
  \nonumber &=& \|u_{t}\|^{2}-(1-\int_{0}^{t}g(s)ds)\|\Delta u\|^{2}-a\int_{\Omega}|\nabla u|^{p(x)}dx \\
  \nonumber && -b(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx)(\int_{\Omega}|\nabla u|^{p(x)}dx) \\
  \nonumber && +\int_{0}^{t}g(t-\tau)\int_{\Omega}\Delta u(\Delta u(\tau)-\Delta u)dxd\tau \\
   && -\beta\int_{\Omega}u|u_{t}|^{m(x)-2}u_{t}dx+\alpha\int_{\Omega}|u|^{q(x)}dx+f(t)\phi(t).
\end{eqnarray}
Similar to (42) and using (8), we have
$$
 |\int_{0}^{t}g(t-\tau)\int_{\Omega}\Delta u(\Delta u(\tau)-\Delta u)dxd\tau|\leq \frac{l(m^{-}-2)}{12}\|\Delta u\|^{2}+\frac{3(1-l)}{l(m^{-}-2)}(g\odot\Delta u)(t).
$$
\begin{equation}
\end{equation}
Utilizing (64) in (63) and using (5), we obtain
\begin{align}
% \nonumber % Remove numbering (before each equation)
  \nonumber \frac{d}{dt}\int_{\Omega}uu_{t}dx \geq & \|u_{t}\|^{2}-[(1-\int_{0}^{t}g(s)ds)+\frac{l(m^{-}-2)}{12}]\|\Delta u\|^{2}-a\int_{\Omega}|\nabla u|^{p(x)}dx \\
  \nonumber & -\frac{b}{p^{-}}(\int_{\Omega}|\nabla u|^{p(x)}dx)^{2}-\frac{3(1-l)}{l(m^{-}-2)}(g\odot\Delta u)(t) \\
   & -\beta\int_{\Omega}u|u_{t}|^{m(x)-2}u_{t}dx+\alpha\int_{\Omega}|u|^{q(x)}dx+f(t)\phi(t).
\end{align}
For any $\delta>0$ and using definition of $E(t)$, we have
\begin{align}
% \nonumber % Remove numbering (before each equation)
  \nonumber \frac{d}{dt}\int_{\Omega}uu_{t}dx \geq & -\delta E(t)+(1+\frac{\delta}{2})\|u_{t}\|^{2}+[(\frac{\delta}{2}-1)(1-\int_{0}^{t}g(s)ds)-\frac{l(m^{-}-2)}{12}]\|\Delta u\|^{2} \\
  \nonumber & +a(\frac{\delta}{p^{+}}-1)\int_{\Omega}|\nabla u|^{p(x)}dx+b(\frac{\delta}{2(p^{+})^{2}}-\frac{1}{p^{-}})(\int_{\Omega}|\nabla u|^{p(x)}dx)^{2} \\
  \nonumber & +(\frac{\delta}{2}-\frac{3(1-l)}{l(m^{-}-2)})(g\odot\Delta u)(t)+\alpha(1-\frac{\delta}{q^{-}})\int_{\Omega}|u|^{q(x)}dx \\
   & -\beta\int_{\Omega}u|u_{t}|^{m(x)-2}u_{t}dx+f(t)\phi(t),
\end{align}
where condition (5) has been used.\newline
Also, for any $\varepsilon>0$ and using (50), we have
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber && \frac{d}{dt}\Big(\int_{\Omega}uu_{t}dx-\varepsilon E(t)\Big) \geq -\delta E(t)+\varepsilon\beta\int_{\Omega}|u_{t}|^{m(x)}dx+(1+\frac{\delta}{2})\|u_{t}\|^{2} \\
  \nonumber && +[(\frac{\delta}{2}-1)(1-\int_{0}^{t}g(s)ds)-\frac{l(m^{-}-2)}{12}]\|\Delta u\|^{2} \\
  \nonumber && +a(\frac{\delta}{p^{+}}-1)\int_{\Omega}|\nabla u|^{p(x)}dx+b(\frac{\delta}{2(p^{+})^{2}}-\frac{1}{p^{-}})(\int_{\Omega}|\nabla u|^{p(x)}dx)^{2} \\
  \nonumber && +(\frac{\delta}{2}-\frac{3(1-l)}{l(m^{-}-2)})(g\odot\Delta u)(t)+\alpha(1-\frac{\delta}{q^{-}})\int_{\Omega}|u|^{q(x)}dx \\
   && -\beta\int_{\Omega}u|u_{t}|^{m(x)-2}u_{t}dx+f(t)|\phi(t)-\varepsilon\phi^{\prime}(t)|.
\end{eqnarray}
Thanks to the Lemma 5, we get
\begin{align}
% \nonumber % Remove numbering (before each equation)
  \nonumber \frac{d}{dt}\Big(\int_{\Omega}uu_{t}dx-\varepsilon E(t)\Big) \geq & -\delta E(t)+\beta(\varepsilon-\frac{m^{+}-1}{m^{+}})\int_{\Omega}|u_{t}|^{m(x)}dx +(1+\frac{\delta}{2})\|u_{t}\|^{2}\\
  \nonumber & +[(\frac{\delta}{2}-1)(1-\int_{0}^{t}g(s)ds)-\frac{l(m^{-}-2)}{4}]\|\Delta u\|^{2} \\
  \nonumber & +a(\frac{\delta+1}{p^{+}}-2)\int_{\Omega}|\nabla u|^{p(x)}dx \\
  \nonumber & +b(\frac{\delta}{2(p^{+})^{2}}-\frac{1}{p^{-}}-\frac{2}{(p^{-})^{2}})(\int_{\Omega}|\nabla u|^{p(x)}dx)^{2} \\
  \nonumber & +(\frac{\delta}{2}-\frac{3(1-l)}{l(m^{-}-2)}-\frac{3(1-l)}{2l})(g\odot\Delta u)(t) \\
  \nonumber & +[\alpha(1-\frac{\delta}{q^{-}})-\frac{q^{+}-1}{q^{+}}]\int_{\Omega}|u|^{q(x)}dx \\
   & -\beta\int_{\Omega}u|u_{t}|^{m(x)-2}u_{t}dx-D_{3}.
\end{align}
Again by using Young's inequality (11), we obtain
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber |\int_{\Omega}u|u_{t}|^{m(x)-1}dx| &\leq & \int_{\Omega}\frac{1}{m(x)}|u|^{m(x)}dx+\int_{\Omega}\frac{m(x)-1}{m(x)}|u_{t}|^{m(x)}dx \\
  &\leq & \frac{1}{m^{-}}\int_{\Omega}|u|^{m(x)}dx+\frac{m^{+}-1}{m^{+}}\int_{\Omega}|u_{t}|^{m(x)}dx.
\end{eqnarray}
On the other hand, let $c^{*}$ be the best constant of embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{m(.)}(\Omega)$. Then we have
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber \int_{\Omega}|u|^{m(x)}dx &\leq & \max\{\|u\|_{m(x)}^{m^{-}}, \|u\|_{m(x)}^{m^{+}}\} \\
  \nonumber &\leq & \max\{(c^{*})^{m^{-}}\|\Delta u\|^{m^{-}}, (c^{*})^{m^{+}}\|\Delta u\|^{m^{+}}\} \\
  \nonumber &\leq & \max\{(c^{*})^{m^{-}}\|\Delta u\|^{m^{-}-2}, (c^{*})^{m^{+}}\|\Delta u\|^{m^{+}-2}\}\|\Delta u\|^{2} \\
   &\leq & \overline{C}\|\Delta u\|^{2}.
\end{eqnarray}
Combining (69) with (70), we get
$$
|\int_{\Omega}u|u_{t}|^{m(x)-1}dx| \leq \frac{\overline{C}}{m^{-}}\|\Delta u\|^{2}+\frac{m^{+}-1}{m^{+}}\int_{\Omega}|u_{t}|^{m(x)}dx.
$$
Substituting last inequality into (68) and set $\varepsilon:=m^{+},\ \delta:=m^{-}$, we obtain
\begin{align}
% \nonumber % Remove numbering (before each equation)
  \nonumber \frac{d}{dt}\Big(\int_{\Omega}uu_{t}dx-m^{+}E(t)\Big) \geq & -m^{-} E(t)+\beta(m^{+}-\frac{2(m^{+}-1)}{m^{+}})\int_{\Omega}|u_{t}|^{m(x)}dx \\
  \nonumber & +(1+\frac{m^{-}}{2})\|u_{t}\|^{2}+[\frac{l(m^{-}-2)}{4}-\frac{\beta\overline{C}}{m^{-}}]\|\Delta u\|^{2} \\
  \nonumber & +a(\frac{m^{-}+1}{p^{+}}-2)\int_{\Omega}|\nabla u|^{p(x)}dx \\
  \nonumber & +b(\frac{m^{-}}{2(p^{+})^{2}}-\frac{1}{p^{-}}-\frac{2}{(p^{-})^{2}})(\int_{\Omega}|\nabla u|^{p(x)}dx)^{2} \\
  \nonumber & +(\frac{m^{-}}{2}-\frac{3(1-l)}{l(m^{-}-2)}-\frac{3(1-l)}{2l})(g\odot\Delta u)(t) \\
   & +[\alpha(1-\frac{m^{-}}{q^{-}})-\frac{q^{+}-1}{q^{+}}]\int_{\Omega}|u|^{q(x)}dx-D_{3},
\end{align}
where $1-\int_{0}^{t}g(s)ds>1-\int_{0}^{\infty}g(s)ds=l$ has been used.\newline
Using the conditions of Theorem 3, if $\alpha$ is large enough and $\beta$ sufficiently small and (51) satisfied, then we have
$$
\frac{d}{dt}\Big(\int_{\Omega}uu_{t}dx-m^{+}E(t)\Big)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber &\geq & -m^{-} E(t)+(1+\frac{m^{-}}{2})\|u_{t}\|^{2}+[\frac{l(m^{-}-2)}{4}-\frac{\beta\overline{C}}{m^{-}}]\|\Delta u\|^{2}-D_{3} \\
  \nonumber &\geq & -m^{-} E(t)+(1+\frac{m^{-}}{2})\|u_{t}\|^{2}+\frac{1}{B^{2}}[\frac{l(m^{-}-2)}{4}-\frac{\beta\overline{C}}{m^{-}}]\|u\|^{2}-D_{3}, \\
  &&
\end{eqnarray}
where $B$ is the best constant of embedding $H_{0}^{2}(\Omega)\hookrightarrow L^{2}(\Omega)$.\newline
By virtue of the Young's inequality and condition (51) i.e. $m^{+}>\frac{m^{-}}{\sqrt{2(m^{-}+2)}}$ and for sufficiently small $\beta$, it is easy to see that
\begin{eqnarray}
% \nonumber % Remove numbering (before each equation)
  \nonumber \frac{m^{-}}{m^{+}}\int_{\Omega}uu_{t}dx &\leq & \|u\|^{2}+\big(\frac{m^{-}}{2m^{+}}\big)^{2}\|u_{t}\|^{2} \\
   &\leq & \frac{1}{B^{2}}[\frac{l(m^{-}-2)}{4}-\frac{\beta\overline{C}}{m^{-}}]\|u\|^{2}+\frac{(m^{-}+2)}{2}\|u_{t}\|^{2}.
\end{eqnarray}
Thus using inequality (73) into (72) yields
\begin{equation}
  \frac{d}{dt}\Big(\int_{\Omega}uu_{t}dx-m^{+}E(t)\Big) \geq \frac{m^{-}}{m^{+}}\Big(\int_{\Omega}uu_{t}dx-m^{+}E(t)\Big)-D_{3}.
\end{equation}
Let define
\begin{equation*}
  H(t)=\int_{\Omega}uu_{t}dx-m^{+}E(t),
\end{equation*}
and therefore
\begin{equation*}
  H^{\prime}(t)\geq \frac{m^{-}}{m^{+}}H(t)-D_{3},
\end{equation*}
integrating over $(0,t)$ to get
\begin{equation}
  H(t)\geq e^{\frac{m^{-}t}{m^{+}}}(H(0)-\frac{m^{+}D_{3}}{m^{-}})+\frac{m^{+}D_{3}}{m^{-}}, \qquad \forall t>0
\end{equation}
where by the assumption of Theorem 3, $H(0)>\frac{m^{+}D_{3}}{m^{-}}$. \newline
Finally, inequality (75) shows that $H(t)$ tends to infinity when time goes to infinity and thus the proof of Theorem 3 is completed.

\section{Conclusion}
In this paper, we studied blow up of solutions for a class of plate viscoelastic $p(x)-$Kirchhoff type inverse source problem with variable-exponent nonlinearities. We obtained blow up of solutions for the inverse problem (1)-(4) in a finite time when $m(x)\equiv 2$. Moreover, if $2<m^{-}\leq m(x)$, then we proved blow-up at infinity of solutions for the inverse problem (1)-(4). Therefore, in the case of $2<m^{-}\leq m(x)$, blow-up of solutions in a finite time is an open problem for the inverse problem (1)-(4).
\bigskip

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