\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 332, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/332\hfil Blow-up of solutions]
{Blow-up of solutions for viscoelastic equations of Kirchhoff
type with arbitrary positive \\ initial energy}

\author[Z. Yang, Z. Gong \hfil EJDE-2016/332\hfilneg]
{Zhifeng Yang, Zhaogang Gong}

\address{Zhifeng Yang \newline
 College of Mathematics and Statistics,
Hengyang Normal University,
Hengyang, Hunan, 421002, China}
\email{zhifeng\_yang@126.com}

\address{Zhaogang Gong (corresponding author) \newline
 College of Mathematics and Statistics,
Hengyang Normal University,
Hengyang, Hunan, 421002, China}
\email{zhaogang\_gong@126.com}

\thanks{Submitted June 2, 2016. Published December 28, 2016.}
\subjclass[2010]{35L05, 35L55, 35L70}
\keywords{Viscoelastic equation; blow-up; arbitrary positive initial energy}

\begin{abstract}
 We consider the viscoelastic equation
 $$
 u_{tt}(x,t)-M(\|\nabla u\|_2^2) \Delta u(x,t)+\int_0^t
 g(t-s)\Delta u(x,s)ds+u_t =|u|^{p-1}u
 $$
 with suitable initial data and boundary conditions.
 Under certain assumptions on the kernel $g$ and the initial data,
 we establish a new blow-up result for arbitrary positive initial energy, 
 by using simple analysis techniques.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks

\section{Introduction}

 The wave equation
\begin{equation}\label{eq1.1}
u_{tt}-\Delta u+h(u_t)=f(u)
\end{equation}
 with suitable initial data and boundary conditions has been extensively studied
and several results concerning existence and blow-up have been
established (see \cite{JB,VB-MI, SAM,LP-DS}).
Here $h$ represents the friction or damping, and $f$ the source.
To describe the nonlinear vibrations of an elastic string, the so-called
Kirchhoff equation
\begin{equation}\label{eq1.2}
u_{tt}-M(\|\nabla u\|_2^2)\Delta u+h(u_t)=f(u)
\end{equation}
was introduced \cite{GK}, where $M(s)=m_0+bs^\gamma$ is a positive
$C^1$-function ($m_0>0$, $b\geq 0$, $\gamma>0$, $s\geq 0$).
In this case the existence and blow-up of solutions have been discussed
by many authors (see \cite{MH-YY,RI,KO,KO2,STW-LYT}
and the references cited therein).

When we take the viscoelastic materials into consideration, the
 models \eqref{eq1.1} and \eqref{eq1.2} become
\begin{equation}\label{eq1.3}
u_{tt}-\Delta u+\int_0^t g(t-s)\Delta u(s)ds+h(u_t)=f(u)
\end{equation}
and
 \begin{equation}\label{eq1.4}
u_{tt}-M(\|\nabla u\|_2^2)\Delta u+\int_0^t g(t-s)\Delta u(s)ds+h(u_t)=f(u)
\end{equation}
respectively, where $g$ represents the kernel of the memory.

For \eqref{eq1.3}, many existence and blow-up results have been proved.
See in this regard \cite{MK-SAM,SAM2,SAM3,HTS-CKZ,HTS,YW}. For example,
Messaoudi \cite{SAM2} studied \eqref{eq1.3} with $h(u_t)=|u_t|^{m-2}u_t$
and $f(u)=|u|^{p-2}u$ and proved a blow-up result for solutions with
negative initial energy if $p>m\geq2$ and a global existence result for
$2\leq p\leq m$. This result has been improved by the same author
in \cite{SAM3} to the case of positive initial energy.
In \cite{HTS-CKZ}, Song and Zhang consider \eqref{eq1.3} with
$h(u_t)=-\Delta u_t$ and $f(u)=|u|^{p-2}u$ and prove a blow-up result
for solutions with positive initial energy by using potential well theory
introduced by Payne and Sattinger\cite{LP-DS}.
Later, Song \cite{HTS} obtained the blow-up result of \eqref{eq1.3}
in the case of $h(u_t)=|u_t|^{m-2}u_t$.

The model \eqref{eq1.4} states that the dynamic equilibrium of a body depends
not only on the present state of deformation, but also on the previous
history of the deformation\cite{JMR}. This model was first studied
by Torrej\'on and Young \cite{RT-JY}, who proved the existence of weakly
 asymptotic stable solution for a large analytical datum.
Later, Munoz Rivera \cite{JMR} showed the global existence for small datum
and the total energy decays to zero exponentially under some restrictions.
 In \cite{STW-LYT} and \cite{STW-LYT2}, Wu and Tsai studied the
model \eqref{eq1.4} with strong damping and nonlinear damping respectively
and proved the existence and blow-up of solutions.
In \cite{STW-LYT2}, a blow-up result of the model \eqref{eq1.4}
with $m_0=1$, $h(u_t)=a|u_t|^{\nu-2}u_t+a|u_t|^{m-2}u_t$ and
 $f(u)=|u|^{p-2}u$ is obtained under some assumptions on the kernel $g$,
the exponential $p$ and the initial data. But this result holds only
in the case $0\leq E(0)<E_1$, where $E(0)$ is the initial energy of the
solution and $E_1$ is some a positive constant.
Recently, by using concavity method, Liu and Liang \cite{JL-FL}
improved the results of \cite{STW-LYT2} to the case of arbitrary positive
initial energy. They considered the following initial-boundary value problem
\begin{equation}\label{eq1.5}
\begin{gathered}
u_{tt}-M(\|\nabla u\|_2^2)\Delta u+\int_0^t g(t-s)\Delta u(s)ds +u_t
=f(u), \\ (x,t)\in \Omega\times (0,T),\\
u(x,t)=0, \quad (x,t)\in \partial \Omega\times (0,T), \\
u(x,0)=u_{0}(x),\quad u_t(x,0)=u_{1}(x), \quad x\in \Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with a smooth boundary
$ \partial\Omega$. $u_0$ and $u_1$ are given initial data. $M$ and $g$
are two functions which stated as in \eqref{eq1.2} and \eqref{eq1.3}.
For this model, they obtained a blow-up result under some basic assumptions
on $f,g,M$ and the initial data $u_0,u_1$. (Readers can see 
\cite[Conditions A1--A4, (2.3) and (2.4)]{JL-FL}.)
However, we find that \cite[conditions (A4) and (2.4)]{JL-FL}
are inessential. Moreover, it is difficult to construct a concrete model
 according to all the assumptions in \cite{JL-FL}, especially for
 (A4) and (2.4). So, motivated by \cite{HTS,STW-LYT2,JL-FL},
we try to consider the blow-up properties of the model \eqref{eq1.5}
with $m_0=1$ and $f(u)=|u|^{p-2}u$. That is, we study the following problem
\begin{equation}\label{eq1.6}
\begin{gathered}
u_{tt}-M(\|\nabla u\|_2^2)\Delta u+\int_0^t g(t-s)\Delta u(s)ds +u_t
=|u|^{p-2}u, \\ (x,t)\in \Omega\times (0,T),\\
u(x,t)=0, \quad (x,t)\in \partial \Omega\times (0,T), \\
u(x,0)=u_{0}(x),\quad u_t(x,0)=u_{1}(x), \quad x\in \Omega,
\end{gathered}
\end{equation}
where $M(s)=1+bs^\gamma$($b\geq 0,\gamma>0, s\geq 0$) is a positive
$C^1$ -function.
We hope to get some more concise sufficient conditions.

\section{Preliminaries and statement of main result}

 In this article, $C$ denotes a generic positive
constant. It may be different from line to line. And we use the
standard Lebesgue space $L^p(\Omega)$ with their usual norms
$\|\cdot\|_p$. Moreover, we denote by $(\cdot,\cdot)$
the usual $L^2(\Omega)$ inner product.

We first state the general assumptions on $g$ and $p$ as
follows:
\begin{itemize}
\item[(A1)] $g\in C^1([0,\infty))$ is a non-negative and non-increasing
function satisfying
\begin{equation}\label{eq2.1}
0<k:=\int_0^\infty g(s)ds<1.
\end{equation}

\item[(A2)] If the space dimension $n=1,2$, then $2(\gamma+1)<p<\infty$; If
$n\geq 3$, then $$2(\gamma+1)<p\leq \frac{2(n-1)}{n-2}.$$
\end{itemize}
 To simplify the notation, we set
$$
(\phi\circ\psi)(t):=\int_0^t \phi(t-s)\int_\Omega |\psi(t)-\psi(s)|^2dx ds,
$$
where $\psi$ may be a scalar, or a vector valued function. A direct
computation shows that, for any $g\in C^1(\mathbb{R})$ and
$u \in H^2(0,T,L^2(\Omega))$, the following identity holds:
\begin{equation}\label{eq2.2}
\begin{aligned}
&\int_0^t g(t-s)\big(\nabla u(s),\nabla u_t(t)\big)ds \\
&= \frac{1}{2}(g'\circ \nabla u)(t) -\frac{1}{2}g(t)\|\nabla u(t)\|_2^2\\
&\quad -\frac{1}{2}\frac{d}{dt}\Big\{(g\circ \nabla
u)(t)-\Big(\int_0^t g(s)ds\Big)\|\nabla u(t)\|_2^2\Big\}.
\end{aligned}
\end{equation}

 Now, we state a local existence theorem that can be established by adopting
the arguments of \cite{STW-LYT2}.

\begin{theorem}[Local solution] \label{thm2.1}
Assume that {\rm (A1)} and {\rm (A2)} hold. Let
 $u_0\in H_0^2(\Omega)$ and $u_1 \in H_0^1(\Omega)$ be given.
Then, there exists a unique weak solution $u(t)$ of \eqref{eq1.5} such that
 \begin{equation}\label{eq2.3}
u\in C([0,T]; H_0^2(\Omega)) \cap C^1([0,T]; L^2(\Omega)),\quad
u_t \in L^2([0,T]; H_0^1(\Omega)).
\end{equation}
for a small enough $T>0$.
\end{theorem}

The energy functional of the solution $u$ of \eqref{eq1.5} is defined
as
 \begin{equation}\label{eq2.4}
\begin{aligned}
E(t)&:=\frac{1}{2}\|u_t\|_2^2+\frac{1}{2}\Big(1-\int_0^t
g(s)ds\Big)\|\nabla u\|_2^2
 +\frac{b}{2(\gamma+1)}\|\nabla u\|_2^{2(\gamma+1)}\\
&\quad +\frac{1}{2}(g\circ \nabla u)(t)
-\frac{1}{p}\|u\|_{p}^{p}.
\end{aligned}
\end{equation}
By \eqref{eq2.2} and assumption (A1), direct computations
yield
\begin{equation}\label{eq2.5}
E'(t)=\frac{1}{2}(g'\circ \nabla u)(t)
-\frac{1}{2}g(t)\|\nabla u\|_{2}^{2}-\|u_t\|_2^2 \leq -\|u_t\|_2^2\leq 0.
\end{equation}

According to \cite{STW-LYT2}, we can obtain the following blow-up
with negative initial energy:

\begin{theorem} \label{thm2.2}
Assume that {\rm (A1), (A2)} and $k<\frac{2(p-2)}{2p-3}$ hold.
if $E(0)<0$, then for all the initial data $u_0\in H_0^2(\Omega)$ and
$u_1 \in H_0^1(\Omega)$, the corresponding solution $u(x,t)$
of the problem \eqref{eq1.5} blows up in finite time.
\end{theorem}

Our main result is a blow-up with positive initial energy that reads
as follows.

\begin{theorem} \label{thm2.3}
Assume that {\rm (A1), (A2)} and $k<\frac{p(p-2)}{(p-1)^2}$ hold.
Moreover, $E(0)>0$ (maybe large enough) is a given initial energy state.
If we choose initial data $u_0\in H_0^2(\Omega)$ and $u_1 \in H_0^1(\Omega)$
satisfying
\begin{equation}\label{eq2.6}
\int_\Omega u_0u_1dx>\beta E(0),
\end{equation}
where $\beta=\frac{1}{2\varepsilon_0},\varepsilon_0\in(0,1)$
is a positive constant, then the corresponding solution $u(x,t)$
of the problem \eqref{eq1.5} blows up in finite time.
\end{theorem}

In \cite{JL-FL}, the kernel $g$ must be the so-called positive type
function. But, we do not need that assumption.
Moreover, our kernel function space is bigger than the one in
\cite{STW-LYT2} since $\frac{p(p-2)}{(p-1)^2}>\frac{2(p-2)}{2p-3}$.

\section{Proof of main result}

Assume $u$ is a global solution of problem \eqref{eq1.6}. Let
 $$
Q(t)=\int_\Omega uu_t dx.
$$
 Multiplying the first equation of \eqref{eq1.6} by $u$ and integrating over
$\Omega$, we get
$$
\int_\Omega uu_{tt} dx+M(\|\nabla u\|_2^2)\|\nabla u\|_2^2
-\int_\Omega \Big(\int_0^t
g(t-s)\Delta u(s)ds\Big)u dx
+\int_\Omega uu_t dx=\|u\|_{p}^{p}.
$$
Then, we easily obtain
\begin{equation}\label{eq3.1}
\begin{aligned}
Q'(t)&=\| u_t \|_{2}^{2}-M(\|\nabla u\|_2^2)\|\nabla u\|_2^2
 +\|u\|_{p}^{p} \\
&\quad -\int_\Omega \left(\int_0^tg(t-s)\Delta u(s)ds\right)u dx
 -\int_\Omega uu_t dx.
\end{aligned}
\end{equation}
For the last term on the right side of \eqref{eq3.1}, using Cauchy inequality,
we deduce that
\begin{equation}\label{eq3.2}
\begin{aligned}
&-\int_\Omega \Big(\int_0^t g(t-s)\Delta u(s)ds\Big)u dx \\
&= \int_0^t g(t-s)\int_\Omega \nabla u(s) \nabla u(t)dx ds \\
&= \int_0^t g(t-s)\int_\Omega \nabla u(t)(\nabla u(s)-\nabla u(t))dx ds
 + \int_0^t g(s)ds \|\nabla u\|_2^2 \\
&\geq -\frac{p(1-\varepsilon)}{2}(g\circ \nabla u)(t)
+\big(1-\frac{1}{2p(1-\varepsilon)}\big)\int_0^t g(s)ds \|\nabla u\|_2^2
\end{aligned}
\end{equation}
for all $\varepsilon \in (0,1)$.
By \eqref{eq3.2} and \eqref{eq2.4}, we have
\begin{equation}\label{eq3.3}
\begin{aligned}
Q' (t)
&\geq \| u_t \|_{2}^{2}-\big(1-\int_0^t g(s)ds\big)
 \|\nabla u\|_2^2-b\|\nabla u\|_2^{2(\gamma+1)}+\|u\|_{p}^{p}-\int_\Omega uu_t dx \\
&\quad -\frac{p(1-\varepsilon)}{2}(g\circ \nabla u)(t)
 -\frac{1}{2p(1-\varepsilon)} \int_0^t g(s)ds \|\nabla u\|_2^2 \\
&=\big(\frac{p(1-\varepsilon)}{2}+1\big) \| u_t \|_{2}^{2}
 +\big(\frac{p(1-\varepsilon)}{2}-1\big)\Big(1-\int_0^t g(s)ds\Big)
 \|\nabla u\|_2^2 \\
&\quad -\frac{1}{2p(1-\varepsilon)} \int_0^t g(s)ds \|\nabla u\|_2^2
 - p(1-\varepsilon)E(t)+\varepsilon \|u\|_{p}^{p}-\int_\Omega uu_t dx \\
&\quad +\Big(\frac{bp(1-\varepsilon)}{2(\gamma+1)}-b\Big)
 \|\nabla u\|_2^{2(\gamma+1)}.
\end{aligned}
\end{equation}
Now, by assumption (A2), we select $\varepsilon$ small enough to ensure that
$$
\frac{bp(1-\varepsilon)}{2(\gamma+1)}-b>0.
$$
Moreover, using H\"older inequality and Young inequality, we can get
$$
\big| \int_\Omega uu_t dx\big|
\leq \|u\|_2\|u_t\|_2\leq \frac{\varepsilon}{2}\|u\|_2^2
+\frac{1}{2\varepsilon}\|u_t\|_2^2.
$$
Then, by assumption (A1), \eqref{eq2.5} and Poincar\'e's inequality, we have
\begin{equation}\label{eq3.4}
\begin{aligned}
\Big(Q(t)-\frac{ E(t)}{2\varepsilon}\Big)'
&\geq Q'(t)+\frac{1}{2\varepsilon}\| u_t \|_{2}^{2}\\
&\geq \Big(\frac{p(1-\varepsilon)}{2}+1\Big)
 \|u_t\|_{2}^{2}- p(1-\varepsilon)E(t)-\frac{\varepsilon}{2}\|u\|_2^2 \\
&\quad +\Big(\Big(\frac{p(1-\varepsilon)}{2}-1\Big)(1-k)
 -\frac{k}{2p(1-\varepsilon)}\Big)\|\nabla u\|_2^2 \\
&\geq \Big(\frac{p(1-\varepsilon)}{2}+1\Big) \|u_t\|_{2}^{2}
 - p(1-\varepsilon)E(t) \\
&\quad +\big(f(\varepsilon)\lambda_1-\frac{\varepsilon}{2}\big)\|u\|_2^2.
\end{aligned}
\end{equation}
where $\lambda_1$ is the first eigenvalue of $-\Delta$ and
\begin{equation}\label{eq3.5}
f(\varepsilon)=\Big(\frac{p(1-\varepsilon)}{2}-1\big)(1-k)
-\frac{k}{2p(1-\varepsilon)}.
\end{equation}
Since $k<\frac{p(p-2)}{(p-1)^2}$ and $p>2$, we deduce that
$1-k>\frac{1}{(p-1)^2}$ and
$$
\theta:=(p-2)(1-k)-\frac{k}{p}>0.
$$
Moreover, we note that $f(\varepsilon)\to \frac{\theta}{2}$ as
$\varepsilon\to 0^+$.
So, we can select $\varepsilon$ small enough such that
$f(\varepsilon)\lambda_1-\frac{\varepsilon}{2}>0$. Then, using Cauchy
inequality to \eqref{eq3.4}, we have
\begin{equation}\label{eq3.6}
\begin{aligned}
\Big(Q(t)-\frac{ E(t)}{2\varepsilon}\Big)'
&\geq h(\varepsilon)Q(t) - p(1-\varepsilon)E(t) \\
&= h(\varepsilon)\Big(Q(t)-\frac{p(1-\varepsilon)}{h(\varepsilon)}E(t)\Big),
\end{aligned}
\end{equation}
where
$$
h(\varepsilon)=2\sqrt{\big(\frac{p(1-\varepsilon)}{2}+1\big)
\big(f(\varepsilon)\lambda_1-\frac{\varepsilon}{2}\big)}.
$$
Denote
$$
\varphi(\varepsilon)
=\Big(\frac{p(1-\varepsilon)}{2}+1\Big)
\Big(f(\varepsilon)\lambda_1-\frac{\varepsilon}{2}\Big).
$$
It is easy to see that
\begin{gather*}
f(\varepsilon)\lambda_1-\frac{\varepsilon}{2}\to \frac{\theta\lambda_1}{2},\quad
 \varphi(\varepsilon)\to \theta\lambda_1(p+2),\quad\text{as } \varepsilon\to 0^+,
\\
f(\varepsilon)\to -\infty,\quad f(\varepsilon)\lambda_1-\frac{\varepsilon}{2}
\to -\infty,\quad \varphi(\varepsilon)\to -\infty \quad
\text{as } \varepsilon\to 1^-.
\end{gather*}
Hence, by the continuity of $\varphi(\varepsilon)$, there exists
$\tilde{\varepsilon}\in(0,1)$ such that $\varphi(\tilde{\varepsilon})=0$
and $\varphi(\varepsilon)>0$ for all $\varepsilon\in(0,\tilde{\varepsilon})$.
So, we have $h(\tilde{\varepsilon})=2\sqrt{\varphi(\tilde{\varepsilon})}=0$
and $h(\varepsilon)=2\sqrt{\varphi(\varepsilon)}>0$ for all
$\varepsilon\in(0,\tilde{\varepsilon})$.
And then, we easily deduce that
\begin{gather*}
\frac{p(1-\varepsilon)}{h(\varepsilon)}\to \frac{p}{\sqrt{\theta\lambda_1(p+2)}},\quad
\frac{1}{2\varepsilon}\to +\infty,\quad\text{as } \varepsilon\to 0^+,\\
\frac{p(1-\varepsilon)}{h(\varepsilon)}\to +\infty,\quad
\frac{1}{2\varepsilon}\to \frac{1}{2\tilde{\varepsilon}},\quad\text{as }
\varepsilon\to \tilde{\varepsilon}^{-}.
\end{gather*}
Thus, using the continuity in $\varepsilon$ of
$\frac{p(1-\varepsilon)}{h(\varepsilon)}$ and $\frac{1}{2\varepsilon}$,
there exists $\varepsilon_0\in(0, \tilde{\varepsilon})\subset (0,1)$ such that
$$
\frac{1}{2\varepsilon_0}=\frac{p(1-\varepsilon_0)}{h(\varepsilon_0)}.
$$

Now, let
\begin{equation}\label{eq3.7}
\beta=\frac{1}{2\varepsilon_0}\quad\text{and}\quad
H(t)=Q(t)-\beta E(t).
\end{equation}
By using \eqref{eq2.6}, \eqref{eq2.5} and \eqref{eq3.6}, we deduce that
\begin{gather*}
H(0)=Q(0)-\beta E(0)>0, \\
H'(t)\geq Q'(t)\geq h(\varepsilon_0)H(t).
\end{gather*}
Then, we have
$$
H(t)\geq e^{h(\varepsilon_0)t}H(0).
$$
Since $u$ is global, by \eqref{eq2.5} and Theorem \ref{thm2.2}, the energy $E(t)$
remains nonnegative, i.e., $0\leq E(t)\leq E(0)$ for all $t\in [0,+\infty)$.
So, we deduce that
$Q(t)\geq e^{h(\varepsilon_0)t}H(0)$
and
\begin{equation}\label{eq3.8}
\begin{aligned}
\|u(t)\|_2^2
&= \|u(0)\|_2^2 +2\int_0^t Q(s)ds \\
&\geq \|u(0)\|_2^2 +2\int_0^t e^{h(\varepsilon_0)s}H(0)ds \\
&= \|u(0)\|_2^2 +\frac{2H(0)}{h(\varepsilon_0)}\big(e^{h(\varepsilon_0)t}-1\big).
\end{aligned}
\end{equation}

By \eqref{eq2.5}, Theorem \ref{thm2.2}, and H\"older inequality, we obtain
\begin{equation}\label{eq3.9}
\begin{aligned}
\|u(t)\|_2
&\leq \|u(0)\|_2 +\int_0^t \|u_s(s)\|_2ds \\
&\leq \|u(0)\|_2 +t^{1/2}\Big(\int_0^t \|u_s(s)\|_2^2 ds\Big)^{1/2} \\
&\leq \|u(0)\|_2 +t^{1/2}\left(E(0)-E(t)\right)^{1/2} \\
&\leq \|u(0)\|_2 +t^{1/2}(E(0))^{1/2}
\end{aligned}
\end{equation}
which contradicts \eqref{eq3.8}.
\hfill\qed

As a simple example, we consider a one-dimension model with
 $M(s)=1+s,\Omega=[0,2\pi]$ and $p=5$. Let
$$
u_0=\xi \sin(\eta x),\quad u_1=\xi\eta^2\sin(\eta x),
$$
where $\xi>0$ and $\eta$ is a positive integer. Then, we have
 $Q(0)=(u_0,u_1)=\xi^2\eta^2 \pi$ and
\begin{align*}
E(0)
&= \frac{1}{2}\|u_1\|_2^2 +\frac{1}{2}\|\nabla u_0\|_2^2
 +\frac{1}{4}\|\nabla u_0\|_2^4-\frac{1}{5}\|u_0\|_5^5 \\
&= \int_0^{2\pi}|\xi\eta^2\sin(\eta x)|^2 dx
 -\frac{1}{5}\int_0^{2\pi}|\xi \sin(\eta x)|^5 dx \\
&= \xi^2\eta^4 \pi-\frac{32}{75}\xi^5.
\end{align*}
Now, we choose $\eta>\sqrt{1/(2\beta)}$ and
$\xi=\sqrt[3]{\frac{75}{32}\eta^2\pi(\eta^2-\frac{1}{2\beta})}$.
Then, we can deduce that
$$
Q(0)=2\beta E(0)>\beta E(0).
$$
According Theorem \ref{thm2.3}, the corresponding solution blows up
in finite time.

\subsection*{Acknowledgments}
The author would like to thank the anonymous referees for their invaluable
comments and suggestions. This research was supported by the Natural
Science Foundation of China (11671128), the Science and Technology Plan
Project of Hunan Province (2016TP1020), the Key Construction Disciplines
of Hunan Province and the Starting Project of Hengyang Normal University(16D01).

\begin{thebibliography}{00}

\bibitem{JB} J. Ball;
Remarks on blow up and nonexistence theorems for nonlinear evolutions equations.
 \emph{Q. J. Math.}, 1977, \textbf{28(2)}: 473-486.

\bibitem{VB-MI}  V. Barbu, M. Iannelli;
 Controllability of the heat equation with memory. \emph{Differ. Integral Equ.},
2000, \textbf{13}: 1393-1412.

\bibitem{CG-GG}  C. Giorgi, G. Gentili;
 Thermodynamic properties and stability for the heat flux equation with
linear memory. \emph{Q. Appl. Math.}, 1993, \textbf{51}: 343-362.

\bibitem{AH-EZ}  A. Haraux, E. Zuazua;
 Decay estimates for some semilinear damped hyperbolic problems.
 \emph{Arch. Ration. Mech. Anal.}, 1988, \textbf{150}: 191-206.

\bibitem{MH-YY}  M. Hosoya, Y. Yamada;
 On some nonlinear wave equations II: global existence and energy decay of solutions.
 \emph{J. Fac. Sci., Univ. Tokyo, Sect. IA, Math.}, 1991, \textbf{38}: 239-250.

\bibitem{RI}  R. Ikehata;
A note on the global solvability of solutions to some nonlinear wave equations
with dissipative terms. \emph{Differ. Integral Equ.}, 1995, \textbf{8}: 607-616.

\bibitem{MK-SAM}  M. Kafini, S. A. Messaoudi;
 A blow-up result in a Cauchy viscoelastic problem. \emph{Appl. Math. Lett.},
2008, \textbf{21}: 549-553.

\bibitem{GK}  G. Kirchhoff;
\emph{Vorlesungen \"uber Mechanik}. Teubner, Leipzig, 1883.

\bibitem{JL-FL}  J. Liu, F. Liang;
 Blow-up of solution for an integro-differential equation with arbitrary
positive initial energy. \emph{Boundary Value Problems}, 2015, \textbf{2015(96)}:
 1-10.

\bibitem{SAM}  S. A. Messaoudi;
 Blow up in a nonlinearly damped wave equation. \emph{Math. Nachr.}, 2001, 
\textbf{231}: 1-7.

\bibitem{SAM2}  S. A. Messaoudi;
 Blow up and global existence in a nonlinear viscoelastic wave equation.
\emph{Math. Nachr.}, 2003, \textbf{260}: 58-66.

\bibitem{SAM3}  S. A. Messaoudi;
 Blow up of positive-initial-energy solutions of a nonlinear viscoelastic
hyperbolic equation. \emph{J. Math. Anal. Appl.}, 2006, \textbf{320}: 902-915.

\bibitem{JMR}  J. Munoz Rivera;
 Global solution on a quasilinear wave equation with memory.
\emph{Boll. Unione Mat. Ital., B}, 1994, \textbf{7(8)}: 289-303.

\bibitem{KO}  K. Ono;
 On global existence, asymptotic stability and blowing up of solutions for 
some degenerate nonlinear wave equations of Kirchhoff type with a strong dissipation.
 \emph{Math. Methods Appl. Sci.}, 1997, \textbf{20}: 151-177.

\bibitem{KO2}  K. Ono;
On global solutions and blow-up solutions of nonlinear Kirchhoff strings 
with nonlinear dissipation. \emph{J. Math. Anal. Appl.}, 1997, \textbf{216}: 321-342.

\bibitem{LP-DS}  L. Payne, D. Sattinger;
 Saddle points and instability on nonlinear hyperbolic equations.
\emph{Isr. J. Math.}, 1975, \textbf{22}: 273-303.

\bibitem{HTS-CKZ}  H. Song, C. Zhong;
 Blow-up of solutions of a nonlinear viscoelastic wave equation. 
\emph{Nonlinear Anal.}, 2010, \textbf{11}: 3877-3883.

\bibitem{HTS} H. Song;
 Blow-up of arbitrarily positive initial energy solutions for a viscoelastic 
wave equation. \emph{Nonlinear Anal.: RWA}. 2015, \textbf{26}: 306-314.

\bibitem{RT-JY}  R. Torrej\'on, J. Young;
 On a quasilinear wave equation with memory. \emph{Nonlinear Anal.},
1991, \textbf{16}: 61-78.

\bibitem{YW}  Y. Wang;
 A global nonexistence theorem for viscoelastic equations with arbitrarily 
positive initial energy. \emph{Appl. Math. Lett.}, 2009, \textbf{22}: 1394-1400.

\bibitem{STW-LYT} S-T. Wu, L-Y. Tsai;
 Blow-up of solutions for some nonlinear wave equations of Kirchhoff type with
some dissipation. \emph{Nonlinear Anal.}, 2006, \textbf{65}: 243-246.

\bibitem{STW-LYT2}  S-T. Wu, L-Y. Tsai;
 Blow-up of positive-initial-energy solutions for an integro-differential 
equation with nonlinear damping. \emph{Taiwan. J. Math.}, 2010, \textbf{14}: 
2043-2058.

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