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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 211, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/211\hfil 
 Nonexistence of global solutions of Cauchy problems]
{Nonexistence of global solutions of Cauchy problems for systems of
 semilinear hyperbolic equations with positive initial energy}

\author[A. B. Aliev, G. I. Yusifova \hfil EJDE-2017/211\hfilneg]
{Akbar B. Aliev, Gunay I. Yusifova}

\address[Akbar B. Aliev]
{Azerbaijan Technical University,
 Baku, Azerbaijan. \newline
Institute of Mathematics and Mechanics,
NAS of Azerbaijan, Baku, Azerbaijan}
\email{alievakbar@gmail.com}

\address[Gunay I. Yusifova]
{Ganja State University,
 Ganja, Azerbaijan}
\email{yusifova81@inbox.ru}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted August 17, 2017. Published September 11, 2017.}
\subjclass[2010]{35G25, 35J50, 35Q51}
\keywords{Semilinear hyperbolic equations;
nonexistence of global solutions;
\hfill\break\indent Cauchy problem; blow up}

\begin{abstract}
 In this paper we study the Cauchy problem for a system of semilinear
 hyperbolic equations. We prove a theorem on the nonexistence of global
 solutions with positive initial energy.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

 We study the solution  of some Cauchy  problems for  systems  containing
 nonlinear  wave equations, from  mathematical physics problems in
 \cite{4,8,25,31}.
 We consider the  system of nonlinear Klein-Gordon equations
\begin{equation} \label{e1_1}
u_{ktt} -\Delta u_{k} +u_{k} +\gamma u_{kt} =f_{k} (u_1 ,\dots,u_{m} )\quad
 k=1,2,\dots,m\, ,
\end{equation}
with initial conditions
\begin{equation} \label{e1_2}
u_{k} (0,x)=u_{k0} (x),\quad  u_{kt} (0,x)=u_{k1} (x),\quad
 x\in R^{n} ,\; k=1,{\dots},m ,
\end{equation}
where   $f_{k} (u_1 ,\dots,u_{k} )=|u_1 |^{\rho _{1k} } |u_2 |^{\rho _{2k} }
\dots|u_{m} |^{\rho _{mk} } u_{k}$, $ \rho _{jk} =p_{j} +1$, $ \rho _{kk} =p_{k} -1$,
 $ k$, $j=1,2,\dots,m$,
 $(u_1 ,u_2 ,\dots,u_{m})$  are real functions depending on $t\in R_{+} $
and $x\in\mathbb{R}^n $, $p_1 ,p_2 ,\dots,p_{m} $ are real numbers.
System \eqref{e1_1} describes the model of interaction of various
fields with single masses \cite{8}. The  goal  of this  paper  is to
investigate  nonexistence of global solutions of problem
\eqref{e1_1}, \eqref{e1_2}.

Before going  further, we briefly introduce some results for the wave equation
\begin{equation} \label{e1_3}
u_{tt} -\Delta u=f(u),
\end{equation}
with
\begin{equation} \label{e1_4}
 f(u)\ge (2+\varepsilon )F(u) ,
\end{equation}
 where  $\, F(u)=\int _0^{u}f(s)ds $.  The general nonlinearity $f(u)$
satisfying \eqref{e1_4} was firstly considered  for some abstract
wave equations in \cite{12}, where Levine proved the blow-up result when the
initial energy is negative. But most results concerning the Cauchy problem
of the wave equation were established for the typical form of nonlinearity
as $f(u)=|u|^{p-1} u$  where $1<p<\frac{n+2}{n+1} $ as $n\ge 3$ and
$1<p<\infty $ as $n=1,2$. Here we note that the above power satisfies
the condition \eqref{e1_4}. For the nonlinearity
satisfying \eqref{e1_4}, the wave equations with damping term were
studied by many authors \cite{5,6,7,9,10,11,12,13,14,15,20,22,23,32}.

For  existence and non-existence of global solutions for the Cauchy problem
of equation \eqref{e1_3} with  a damping term, we refer the reader to
\cite{7,14,26,27}.  In particular, recently the wave equation with damping
term was considered in \cite{14}, where Levine and Todorova showed that
for arbitrarily positive initial energy there are choices of initial data
such that the local solution blows up in finite time.
Subsequently, Todorova and Vitillaro \cite{26} established more precise
result regarding the existence of initial values such that the corresponding
solution blows up in finite time for arbitrarily high initial energy.
More recently, Gazzola and Squassina \cite{7} established sufficient
conditions of initial data with arbitrarily positive initial energy such
that the corresponding solution blows up in finite time for the wave
equation with linear damping  and  in the mass free case on a bounded
Lipschitz subset of $R^{n} $. A fairly comprehensive picture of the studies
in this direction can be gained from the monograph \cite{22}.

In \cite{15}, \cite{18} the  authors  obtained  sufficient  conditions
 on initial functions for which the initial boundary value problem for
 second-order quasilinear strongly damped wave equations blow up in a finite time.
The nonexistence of global solutions of a generalized fourth-order Klein-Gordon
 equation with positive initial energy was analyzed in \cite{11}.

A mixed problem  for systems of two semilinear wave equations with viscosity
 and with memory was studied in \cite{10,21,24,29}, where the nonexistence
of global solutions with positive initial energy was proved.

The nonexistence of global solutions of the problem
\begin{equation} \label{e1_5}
\begin{gathered}
 {u_{1tt} -\Delta u_1 +u_1 +\gamma u_{1t} =g_1 (u_1 ,u_2 ),} \\
 {u_{2tt} -\Delta u_2 +u_2 +\gamma u_{2t} =g_2 (u_1 ,u_2 ),}
\end{gathered}
\end{equation}
with
\begin{equation} \label{e1_6}
u_{i} (0,x)=u_{i0} (x),\quad u_{it} (0,x)=u_{i1} (x),\quad   x\in R^{n},\; i=1,2,
\end{equation}
where
\[
g_1 (u_1 ,u_2 )=|u_1 |^{p-1} |u_2 |^{p+1} u_1,\quad
 g_2 (u_1 ,u_2 )=|u_1 |^{p+1} |u_2 |^{p-1} u_2 ,
\]
with negative initial energy was studied in \cite{21}, \cite{27}.  In the case when
\[
g_1 (u_1 ,u_2 )=|u_1 |^{p-1} |u_2 |^{q+1} u_1,\quad
g_2 (u_1 ,u_2 )=|u_1 |^{p+1} |u_2 |^{q-1} u_2.
\]
The absence  of  global solutions  for  problem
\eqref{e1_5}, \eqref{e1_6} was investigated in
 \cite{1,2}. Recently, more  investigations were  carried out in this  field
 \cite{1,7,24,29}.

The absence of global solutions with  positive arbitrary initial energy
 for systems of semilinear hyperbolic equations
\[
u_{itt} -\Delta u_{i} +u_{i} +\gamma u_{it}
=\sum _{\mathop{i,j=1}_{i\ne j} }^{m}|u_{j} |^{p_{j} }
|u_{i} |^{p_{i} } u_{i} \quad i=1,2,\dots,m
\]
 was investigated in [2], where  $n\ge 2$, $p_{j} \ge 0$, $j=1,2,\dots,m$,
 and $\sum _{k=1}^{m}p_{k}  \le \frac{2}{n-2} $  if   $ n\geq3$.

\section{Formulation of the problem and main results}

To state our main results, we briefly mention some facts, notation,
and well known results. We denote the norm on the space $L_2 (R^{n} )$
by $|\cdot|$, the inner product on $L_2 (R^{n} )$  by
$\langle \cdot ,\cdot \rangle $, and the norm on the Sobolev space
 $H^{1} =W_2^{1} (R^{n} )$ by
$\| u\| =[|\nabla u|^2 +|u|^2]^{\frac{1}{2}}$.
The constants $C$ and $c$ used throughout this paper are positive generic
 constants that may be different in various occurrences.

Assume that
\begin{gather} \label{e2_1}
 p_{j} > 0,\quad  j=1,2,\dots,m, \; m=2,3,\dots;\\
\label{e2_2}
\sum _{k=1}^{m}{p_{k}+m-2}  \le \frac{2}{n-2} \quad \text{if } n\geq3.
\end{gather}
Let $E(t)$ be the energy functional
\begin{align*}
E(t)
&=\sum _{j=1}^{m}\frac{p_{j} +1}{2}
\Big[|u'_{jt} (t,\cdot )|^2 +\| u_{j} (t,\cdot )\| ^2
 +2\gamma \int _0^{t}|u'_{jt} (s,\cdot )|^2 ds \Big]\\
&\quad -\int _{R^{n} }\prod _{j=1}^{m}|u_{j} (t,x)|^{p_{j} +1} dx.
\end{align*}
We also set
\begin{equation} \label{e2_3}
I(u _1 ,\dots,u _{m} )=\sum _{j=1}^{m}\frac{p_{j} +1}{\sum _{r=1}^{m}p_r +m}
\| u _{j}(t,\cdot) \| ^2
-\int _{R^{n} }\prod _{j=1}^{m}|u _{j} (t,x)|^{p_{j} +1} dx.
\end{equation}

The main result of this article is stated in the following theorem.

\begin{theorem} \label{thm2.1}
Let conditions \eqref{e2_1} and \eqref{e2_2} be satisfied.
Assume  $u_{k0} \in H^{1} $ and   $u_{k1} \in L_2 (R^{n} )$, $k=1,2,\dots,m$,
and
\begin{gather} \label{e2_4}
E(0)>0, \\
\label{e2_5}
I(u_{10} ,u_{20} ,\dots,u_{m0} )<0, \\
\label{e2_6}
\sum _{k=1}^{m}\langle u_{k0} ,u_{k1} \rangle  \ge 0,  \\
\label{e2_7}
\sum _{j=1}^{m}\frac{p_{j} +1}{2} |u_{j0} |^2
>\frac{\sum _{j=1}^{m}p_{j} +m }{\sum _{j=1}^{m}p_{j}  } E(0).
\end{gather}
Then the solution of the Cauchy problem \eqref{e1_1},
\eqref{e1_2} blows up in finite time.
\end{theorem}

Note that, in the case of $m=2$, this result was obtained in
 \cite{1}, and in  the  case   $m=2$, $p_1 =p_2 \ge 1$, it was obtained
 in \cite{29}.

\section{Auxiliary assertions}


In the  Hilbert  space
$\tilde{H}=L_2 (R^{n} )\times L_2 (R^{n} )\times \dots\times L_2 (R^{n} )$
we write  problem \eqref{e1_1}, \eqref{e1_2} as the
Cauchy problem
\begin{gather} \label{e3_1}
w''+Bw'+Aw=F(w), \\
\label{e3_2}
w(0)=w_0 ,\quad  w' (0)=w_1 ,
\end{gather}
where
\[
w=\begin{bmatrix} {u_1 } \\ {u_2 }
 \\ {\dots} \\ {u_{m} } \end{bmatrix},\quad
w_0 =\begin{bmatrix} {u_{10} (x)} \\ {u_{20} (x)}  \\ {\dots} \\ {u_{m0} (x)}
 \end{bmatrix},\quad
w_1 =\begin{bmatrix} {u_{11} (x)} \\ {u_{21} (x)}  \\ {\dots} \\ {u_{m1} (x)}
 \end{bmatrix}
\]
Here $A$  and   $B$  are linear operators in  $\tilde{H}$ defined by
\begin{gather*}
A=\begin{pmatrix} {-\Delta +1} & {0} & {\dots} & {0} \\
{0} & {-\Delta +1} & {\dots} & {0} \\ {\dots} & {\dots} & {\dots} & {\dots} \\
{0} & {0} & {\dots} & {-\Delta +1} \end{pmatrix},\\
 D(A)=\tilde{H }_2 =H^2 \times H^2 \times \dots\times H^2,\\
 B=\begin{pmatrix} {\gamma } & {0} & {\dots} & {0} \\
{0} & {\gamma } & {\dots} & {0} \\
{\dots} & {\dots} & {\dots} & {\dots} \\
{0} & {0} & {\dots} & {\gamma } \end{pmatrix},\\
D(B)=L_2 (R_{n} )\times L_2 (R_{n} )\times \dots\times L_2 (R_{n} ),\\
F(w)=\begin{pmatrix}
f_1 (u_1 ,u_2 ,\dots,u_{m} ) \\
f_2 (u_1 ,u_2 ,\dots,u_{m} ) \\ {\dots} \\
f_{m} (u_1 ,u_2 ,\dots,u_{m} )
 \end{pmatrix}.
\end{gather*}

Note that $A$ is a self-adjoint  positive  definite operator,
$B$ is  a linear  bounded  operator acting in  $\tilde{H}$
and  conditions  \eqref{e2_1}, \eqref{e2_2}  imply
that $F(w)$ is a   nonlinear operator acting from
$\tilde{H }_1 =H^{1} \times H^{1} \times \dots\times H^{1} $ to
 $\tilde{H }$.

\begin{lemma} \label{lem3.1}
Let $n=1,2$, $p_j \geq 1$,  $j=1,2,\dots,n$, $ m=2,3,\dots$ or $n=3$, $m=2$,
$p_1=p_2=1$. Then the nonlinear operator
$w\to F(w):\tilde{H }_1 \to \tilde{H }$ satisfies
the local  Lipchitz  condition, that is  for  any
$w^{1}, \,w^2 \in \tilde{H }_1 $  we have
\begin{equation}
\| F(w^{1} )-F(w^2 )\| _{\tilde{H }}
\le c(r)\| w^{1} -w^2 \| _{\tilde{H }_1 } ,
\end{equation}\label{e3_3}
where  $c(\cdot )\in C(R_{+} )$, $c(r)\ge 0$,
$r=\sum _{i=1}^2\| w^{i} \|  _{\tilde{H }_1 } $.
\end{lemma}

\begin{proof}
Let us take $w^j=(u^j_1,u^j_2,\dots,u^j_m)\in \tilde{H}^1$, $j=1,2$.
Then, by the mean value theorem we have
\begin{equation}\label{e3_4}
\begin{aligned}
\| F( w^{1}) -F( w^2) \| _{\tilde{H}}^2
&\leq c\sum_{k=1}^{m}\sum_{j=1}^{m}\underset{R_{n}}{\int }
( | u_{j}^{1}| ^{2( \rho _{jk}-1)}+| u_{j}^2| ^{2( \rho _{jk}-1) })\\
&\quad\times  \prod _{i=1,i\neq j}^{m}( | u_{j}^{1}|
^{2\rho _{jk}}+| u_{j}^2| ^{2\rho _{jk}})
| u_{j}^{1}-u_{j}^2| ^2dx.
\end{aligned}
\end{equation}
Let $n\geq 2$. By Holder  inequality with exponents,
\begin{gather*}
\alpha _{k}^{i}=\frac{\sum_{r=1}^m p_r+m}{\rho _{ki}}
\quad\text{if } i\neq j,\; i=1,\dots,m,\\
\alpha _{k}^{j}= \frac{\sum_{r=1}^m p_r+m}{\rho _{kj}-1},\quad
\alpha _{k}^{0}= \sum_{r=1}^m p_r+m
\end{gather*}
and using interpolation inequalities of Gagliardo and Nirenberg
in the case  $n=2$ or Sobolev inequality in case $n = 3$ we have
\begin{equation}\label{e3_5}
\| F( w^{1}) -F( w^2) \| _{\widetilde{H}}^2
\leq c\Big( \| w^{1}\| _{\widetilde{H}_1} ^{\sum_{r=1}^m p_r+m-1}
+\| w^2\| _{\widetilde{H}_1}^{\sum_{r=1}^m p_r+m-1}\Big)
\| w^{1}-w^2\| _{\widetilde{H}_1}.
\end{equation}		
In case $n=1$, from \eqref{e3_4} using embedding theorem
 we again obtain \eqref{e3_3}.
\end{proof}

By the theorem of solvability of the Cauchy problem for the evolution
equation \cite{3}, we have the following local solvability theorem
for problem \eqref{e2_3}, \eqref{e2_4}.

\begin{theorem} \label{thm3.1}
Let $n=1,2$, $p_j \geq 1$,  $j=1,2,\dots,m$, $ m=2,3,\dots$ or
 $n=3$, $m=2$, $p_1=p_2=1$. Then for arbitrary
 $w_0 \in \tilde{H }_1 $, $w_1 \in {\rm \tilde H} $,
there exists $T'>0$ such that problem \eqref{e3_1}, \eqref{e3_2}
has a unique solution $w(\cdot )\in C([0,T^{*} ]; \tilde{H }_1 )
\cap C^{1} ([0,T^{*} ];\tilde{H })$.
If $T_{\rm max} =\sup T^{*} $, i.e., $T_{\rm max} $
is the length of the maximal existence interval of the solution
 $w(\cdot )\in C([0,T_{\rm max} );\tilde{H }_1 )
\cap C^{1} ([0,T_{\rm max} );\tilde{H })$, then either
\begin{itemize}
\item[(i)] $T_{\rm max} =+\infty$, or

\item[(ii)] $\limsup_{t\to T_{\rm max} -0} [\| w(\cdot )\| _{\tilde{H }_1 }
 +\| \dot{w}(\cdot )\| _{\tilde{H }} ]= + \infty $.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm3.2} Let conditions \eqref{e2_1} and \eqref{e2_2}
 be satisfied.
Then for arbitrary $w_0\in \tilde{H}_1$ and
$w_1\in \tilde{H}$ there exists $T'>0$ such  that problem \eqref{e3_1},
\eqref{e3_2} has a solution $w(\cdot)\in C( [ 0,T'] ;\widetilde{H}
_1) \cap C^{1}( [ 0,T'] ;\widetilde{H}) $ and
 $w( t) $ is either global or blow-up in a finite time.
\end{theorem}

\begin{proof}
We carry out the proof by Galerkin's method, using some
considerations from the work [18]. Let $\{w_1,w_2,\dots,w_r\dots\} $
be the basis of the space $\widetilde{H}_1$ and
$w_r( t,\cdot) =\sum_{j=1}^r g_{rj}( t) w_{j}$,
 $r=1,2,\dots$ be defined as a solution of the
system
\begin{equation}\label{e3_6}
( w_r''( t) ,\omega _{j}) _{\widetilde{H}}+( Bw_r'( t) ,w_{j}) _{
\widetilde{H}}+( w_r( t) ,\omega _{j}) _{\widetilde{H
}_1}=( F( w_r) ,\omega _{j}) _{\widetilde{H}}
\end{equation}
with initial data
\begin{equation}\label{e3_7}
w_r( 0,\cdot ) =w_{0r},\quad
 w_r'( 0,\cdot) =w_{1r},
\end{equation}
where $w_{0r}$ and $w_{1r}$ belongs to the subspace
$[ \omega_1,\omega _2,\dots,\omega _r] $\ generated by the $r$ first
vectors of the basis $\{ \omega _{j}\} $, and
\begin{equation}\label{e3_8}
w_{0r}\to w_0\text{ in }\widetilde{H}_1\quad \text{and}\quad
w_{1r}\to w_1\text{ in  }\widetilde{H}\text{ if  }
r\to \infty .
\end{equation}

By multiplying the equation \eqref{e3_6} by $g_{rj}'( t) $ and
summing by $k$, where $k$ takes the values from $1$ to $r$, we get that
\begin{equation}\label{e3_9}
\begin{aligned}
&\frac{1}{2}\frac{d}{dt}[ \| w_{rt}( t,\cdot )
\|_{\widetilde{H}}^2  + \| w_r( t,\cdot ) \|_{\widetilde H_1}^2]
+( Bw_{rt}( t,\cdot ) ,w_{rt}( t,\cdot ) )
_{\widetilde{H}} \\
&=( F( w( t,\cdot ) ) ,w'( t,\cdot ) ) _{\widetilde{H}}.
\end{aligned}
\end{equation}
Then using Holder's inequalities and \eqref{e3_5}, for
\begin{equation}\label{e3_10}
y_r( t) =\| w_{rt}( t,\cdot ) \|_{\widetilde{H}}
^2+\| w_r( t,\cdot ) \| _{\widetilde{H}
_1}^2
\end{equation}
from \eqref{e3_9} we get $y_r'( t) \leq c( y_r(t) ) ^{\sum_{r=1}^m p_r+m}$.

Integrating this inequality and taking into account the inequality \eqref{e3_4},
we find that there exists $T'>0$  and $r_0$ such that
\begin{equation}\label{e3_11}
y_r( t) \leq c,\quad t\in [ 0,T'] ,\;  r\geq r_0.
\end{equation}

From \eqref{e3_10}, \eqref{e3_11} it follows that there exists a subsequence still
denoted by the same symbols, such that
\begin{gather*}
w_r\to w \quad \text{weak star in } L_{\infty }( 0,T';\widetilde{H}_1),\\
w_r'\to w'\quad \text{weak star in }L_{\infty}( 0,T';\widetilde{H}),
F( w_r) \to \chi  \quad \text{weak star in }L_{\infty}( 0,T';\widetilde{H}) .
\end{gather*}
Further, using the method given in \cite{16}, we obtain that $\chi =F( w) $.

Passing to the limit is carried out by the standard method
(for example, see [\cite{16,18}). Thus,  problem \eqref{e3_1}, \eqref{e3_2}
 has the solution $w \in L_{\infty }(0,T';\widetilde{H}_1) $, such that
$w' \in L_{\infty }( 0,T';\widetilde{H}) $ and
$F( w) \in L_{\infty }( 0,T';\widetilde{H}) $.

Further applying the linear theory of the hyperbolic equations, considering
equation \eqref{e3_1} as a linear equation with a given right-hand side of
 $\chi( t) =F( w) \in L_{\infty }( 0,T';\widetilde{H}) $, we find that
$w\in C( [ 0,T'] \widetilde{H}_1) \cap C^{1}( [ 0,T']
\widetilde{H}) $  (see \cite{17}).
\end{proof}

\begin{remark}label{rmk3.1} \rm
If $w_0 \in \tilde{H }_2 $ and $w_1 \in \tilde{H }_1$,
then $w(\cdot )\in C([0,T_{\rm max} );\tilde{H }_2 )\cap C^{1} ([0,T_{\rm max} );
\tilde{H }_1 )$.
\end{remark}


\begin{lemma} \label{lem3.2}
Let conditions \eqref{e2_1}, \eqref{e2_2} and \eqref{e2_4}-\eqref{e2_7} be satisfied.
 Then
\[
I(u_1 (t,.),u_2 (t,.),\dots,u_{m} (t,.))<0,\quad  t\in [0,T_{\rm max} ).
\]
\end{lemma}

\begin{proof}
By  \eqref{e2_5} there exists  $T_1 >0$, such that
\begin{equation} \label{e3_12}
I(u_1 (t,\cdot ),u_2 (t,\cdot ),\dots,u_{m} (t,\cdot ))<0,\quad  t\in [0,T_1 ).
\end{equation}
We shall prove that $T_1 =T_{\rm max} $. Assume that $T_1 <T_{\rm max}$.
Then by the continuity of $I(u_1 (t ,\cdot ),u_2 (t ,\cdot ),\dots,u_{m} (t,\cdot ))$
we have
\begin{equation} \label{e3_13}
I(u_1 (T_1 ,\cdot ),u_2 (T_1 ,\cdot ),\dots,u_{m} (T_1 ,\cdot ))=0.
\end{equation}
We introduce the functional $F(t)=\sum _{j=1}^{m}(p_{j} +1)|u_{j} (t,\cdot )|^2$.
Taking into account Remark 2.1 and using \eqref{e1_1}, \eqref{e1_2} we obtain:
\[
F'(t)=2\sum _{j=1}^{m}(p_{j} +1)\langle u_{j} (t,\cdot ),\dot{u}_{j}
(t,\cdot )\rangle,
\]
and
\begin{align*}
F''(t)
&=2\sum _{j=1}^{m}(p_{j} +1)|u'_{j} (t,\cdot )|^2
 - 2\sum _{j=1}^{m}(p_{j} +1)
\big[ \| u_{j} (t,\cdot )\| ^2 +\gamma \langle u_{j} (t,\cdot ),\dot{u}_{j}
 (t,\cdot )\rangle \big] \\
&\quad +2\Big(\sum _{k=1}^{m}p_{k} +m \Big)\int _{R^{n} }
\prod _{j=1}^{m}|u_{j} (t,x)|^{p_{j} +1} dx.
\end{align*}
Therefore,
\begin{equation} \label{e3_14}
F''(t)+\gamma \, F'(t)=\varphi (t), \quad  t\in [0,T_1 ),
\end{equation}
where
\begin{align*}
\varphi (t)
&=2\sum _{j=1}^{m}(p_{j} +1)|u'_{j} (t,\cdot )|^2
 -2\Big(\sum _{k=1}^{m}p_{k} +m \Big)
 I(u_1 (t,\cdot ),u_2 (t,\cdot ),\dots,u_{m} (t,\cdot )).
\end{align*}
Taking into account  inequality  \eqref{e3_12}, we obtain
\begin{equation} \label{e3_15}
\varphi (t)>0, \quad  t\in [0,T_1 ).
\end{equation}

It follows from condition \eqref{e2_6} and relations \eqref{e3_14} and \eqref{e3_15}
 that
\[
F'(t)>0, \quad  t\in [0,T_1 ).
\]
Therefore, the function $F(t)$ is monotone increasing on $[0,T_1 )$.
 Consequently,
\begin{equation} \label{e3_16}
F(t)>F(0)=\sum _{j=1}^{m}(p_{j} +1)|u_{j0} |^2.
\end{equation}

By taking into account the continuity of the function $F(t)$,
from condition \eqref{e2_7} and inequalities \eqref{e3_16}, we obtain
\begin{equation} \label{e3_17}
F(T_1)>\frac{2\big[\sum_{j=1}^{m}p_{j} +m \big]}{\sum _{j=1}^{m}p_{{j}_{3}} } E(0).
\end{equation}
On the other hand it follows from \eqref{e1_1} and \eqref{e1_2}  that
\begin{equation} \label{e3_18}
E(t)= E(0)
\end{equation}
for every $t\in [0,T_{\rm max} )$. From   \eqref{e3_13} and \eqref{e3_18}
 we obtain the inequality
\[
\Big(1-\frac{2}{\sum _{j=1}^{m}p_{j} +m } \Big)\sum _{j=1}^{m}\frac{p_{j} +1}{2}
 \| u_{j} (T_1,\cdot ) \| ^2  \le E(0).
\]
It follows that
\begin{equation} \label{e3_19}
F(T_1)\le \frac{2\big[\sum _{j=1}^{m}p_{j} +m \big]}
{\sum _{j=1}^{m}p_{j} +m -2} E(0).
\end{equation}
The resulting contradiction \eqref{e3_17} with \eqref{e3_19}
shows that our assumption  fails. Therefore $T_1 =T_{\rm max} $.

Let  $T_2 >0$, $T_{3} >0$ and  $k>0$ be some numbers. Consider the functional
\begin{equation} \label{e3_20}
\begin{aligned}
R(t)&=\sum _{j=1}^{m}\frac{p_{j} +1}{2}  \Big[|u_{j} (t,\cdot )|^2
+\gamma \int _0^{t}|u_{j} (s,\cdot )|^2 ds +\gamma |u_{j0} |^2 (T_2 -t)\Big]\\
&\quad +k(T_{3} +t)^2 .
\end{aligned}
\end{equation}
\end{proof}

\begin{lemma} \label{lem3.3}
Let  \eqref{e2_4}--\eqref{e2_7} be satisfied.
Then $\ddot{R}(t)>0$ for $t\in [0,T_{\rm max} )$.
\end{lemma}

\begin{proof} A simple computation gives us
\begin{equation} \label{e3_21}
\begin{aligned}
R'(t)=\sum _{j=1}^{m}\frac{p_{j} +1}{2}
\big[2\langle u_{j} (t,\cdot ),u'_{j} (t,\cdot )\rangle
 +\gamma |u_{j} (t,\cdot )|^2 -\gamma |u_{j0} |^2 \big]
 +2k(t+T_{3} ).
\end{aligned}
\end{equation}

Next, from \eqref{e3_18}, \eqref{e3_21}  by using relations \eqref{e1_1}
and \eqref{e1_2}, we obtain
\begin{equation} \label{e3_22}
\begin{aligned}
R''(t) &=\sum _{j=1}^{m}(p_{j} +1) [|u'_{j} (t,\cdot )|^2
-\| u_{j} (t,\cdot )\| ^2 ]\\
&\quad +\Big[\sum _{j=1}^{m}p_{j} +m \Big]
\int _{R^{n} }\prod _{j=1}^{m}|u_{j} (t,x)|^{p_{j} +1} dx  +2k.
\end{aligned}
\end{equation}
It follows from \eqref{e2_3} and \eqref{e3_22} that
\[
R''(t)\ge -\Big[\sum _{j=1}^{m}p_{j} +m \Big]
 I(u_1 (t,.),\dots,\, u_{3} (t,.)) +2k,\quad   t\in [0,T_{\rm max} ).
\]
By  Lemma \ref{lem3.2},  for sufficiently small  $k$ it holds
\begin{equation} \label{e3_23}
R''(t)>0, \quad t \in [0,T_{\rm max} ).
\end{equation}
\end{proof}

\section{Proof of  main result}

We first assume that $u_{i0} \in H^2$, $u_{i1} \in H^{1}$,
$ i=1,2,\dots,m$. We shall prove that under conditions
\eqref{e2_1}, \eqref{e2_2} and \eqref{e2_4}-\eqref{e2_7},
$T_{\rm max} <+\infty $. Suppose the contrary: $T_{\rm max} =+\infty $.
 It follows  from   \eqref{e1_1} and \eqref{e1_2}  that
\begin{align*}
&\int _{R^{n} }\prod _{j=1}^{m}|u_{j} (t,x)|^{p_{j} +1} dx \\
& =-E(0) +\sum _{j=1}^{m}\frac{p_{j} +1}{2}  [|u'_{j} (t,\cdot )|^2
+\| u_{j} (t,\cdot )\| ^2 +2\gamma \int _0^{t}|u'_{j} (s,\cdot )|^2 ds ].
\end{align*}
Taking into account this relation in \eqref{e3_22}, we obtain
\begin{equation} \label{e4_1}
\begin{aligned}
R''(t)&=\frac{\sum _{j=1}^{m}p_{j} +m +2}{2} \sum _{j=1}^{m}(p_{j} +1)|u'_{j}
 (t,\cdot )|^2 \\
&\quad +\frac{\sum _{j=1}^{m}p_{j} +m -2}{2} \sum _{j=1}^{m}(p_{j} +1)
\| u_{j} (t,\cdot )\| ^2
+\gamma \Big[\sum _{j=1}^{m}p_{j} +m \Big]\\
&\quad\times \sum _{j=1}^{3}(p_{j} +1) \int _0^{t}|u'_{j} (s,\cdot )|^2 ds
-\Big[\sum _{j=1}^{m}p_{j} +m \Big]E(0)+2k.
\end{aligned}
\end{equation}

By \eqref{e3_9} we have
\begin{equation} \label{e4_2}
\begin{aligned}
R'^2 (t)
&\le \Big[\sum _{j=1}^{m}(p_{j} +1) (|u_{j} (t,\cdot )|^2
 +\gamma \int _0^{t}|u_{j} (s,\cdot )|^2 ds )+k(t+T_{3} )^2 \Big]\\
&\quad \times \Big[\sum _{j=1}^{m}(p_{j} +1) (|u'_{j} (t,\cdot )|^2
+\gamma \int _0^{t}|u'_{j} (s,\cdot )|^2 ds )+k\Big]  .
\end{aligned}
\end{equation}
By choosing a sufficiently large   $T_{3} $, from Lemma \ref{lem3.2} and relations
\eqref{e3_19}, \eqref{e4_1}, and \eqref{e4_2}, we obtain
\begin{align}
&R(t) R''(t)-\frac{\sum _{j=1}^{m}p_{j} +m +2}{4} (R' (t))^2 \nonumber \\
&\ge R(t)\cdot R''(t)-\frac{\sum _{j=1}^{m}p_{j} +m +2}{4}
\Big[2R(t)-(T_1 -t)\sum _{j=1}^{m}(p_{j} +1) \cdot |u_{j0} |^2 \Big]\nonumber \\
&\quad \times \Big[\sum _{j=1}^{m}(p_{j} +1) (|u'_{j} (t,\cdot )|^2
+\gamma \int _0^{t}|u'_{j} (s,\cdot )|^2 ds )+k] \nonumber \\
&\ge R(t)\Big\{\frac{\sum _{j=1}^{m}p_{j} +m }{2}
\sum _{j=1}^{m}(p_{j} +1) |u'_{j} (t,\cdot )|^2 \nonumber \\
&\quad +\frac{\sum _{j=1}^{m}p_{j} +m -2}{2}
 \sum _{j=1}^{m}(p_{j} +1) \| u_{j} (t,\cdot )\| ^2 \nonumber \\
&\quad +[\sum _{j=1}^{m}p_{j} +m ] \sum _{j=1}^{m}(p_{j} +1)
  \int _0^{t}|u'_{j} (s,\cdot )|^2 ds
 -\Big[\sum _{j=1}^{m}p_{j} +m \Big]E(0)+2k \nonumber \\
&\quad  - \frac{\sum _{j=1}^{m}p_{j} +m +2}{2}
\sum _{j=1}^{m}(p_{j} +1) \Big(|u'_{j} (t,\cdot )|^2
+\int _0^{t}|u'_{j} (s,\cdot )|^2 ds \Big)+k\Big\} \nonumber \\
&=R(t)y(t)+\frac{\sum _{j=1}^{m}p_{j} +m -2}{2}
\sum _{j=1}^{m}\int _0^{t}|u'_{j} (s,\cdot)|^2 ds  , \label{e4_3}
\end{align}
where
\begin{align*}
y(t)&=\frac{\sum _{j=1}^{m}p_{j} +m -2}{2}
 \sum _{j=1}^{m}(p_{j} +1) \| u_{j} (t,\cdot )\| ^2 \\
&\quad -\Big[\sum _{j=1}^{m}p_{j} +m \big]E(0)-\frac{p_1 +p_2 +p_{3} +1}{2} k.
\end{align*}

Having in mind Lemma \ref{lem3.2}, and choosing a sufficiently small $k>0$,
we obtain that $y(t)\ge 0$. Thus, for sufficiently large $T_2 >0$,
$T_{3} >0$, and for sufficiently small $k>0$ we ahve
\begin{equation} \label{e4_4}
R(t)\cdot R''(t)-\frac{\sum _{j=1}^{m}p_{j} +m +2}{4} R'^2 (t)\ge 0\,.
\end{equation}
On the other hand,
\[
R'(0)=\sum _{j=1}^{m}(p_{j} +1) \langle u_{j0} ,u_{j1} \rangle +2kT_2 \, .
\]
Therefore, $R'(0)>0 $.
Using this inequality and \eqref{e4_4} bya standard procedure, we obtain
 that there exists $0<T^{*} <+\infty $  such    that
 $\lim_{t\to T^{*} -0} R(t)=+\infty$. We obtain a contradiction, which shows that
$T_{\rm max} <+\infty$.

If  $u_{i0} \in H^{1} $ and $ u_{i1} \in L_2 (R^{n} )$, $i=1,2,\dots,m$, then
the justification can be carried out in a standard way,
 by approximation of the initial data by functions  from  $H^2$ and
$H^{1}$, respectively.

\subsection*{Acknowledgements}
The authors want to thank the anonymous referees for the careful reading
of the paper and his comments for improvements.

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\end{document}