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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 20, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/20\hfil Semi-linear pseudo-hyperbolic systems]
{Nonexistence of solutions to Cauchy problems for fractional
time semi-linear pseudo-hyperbolic systems}

\author[S. Abdelmalek, M. Bajneed,  K. Sioud  \hfil EJDE-2016/20\hfilneg]
{Salem Abdelmalek, Maha Bajneed, Khaled Sioud}

\address{Salem Abdelmalek \newline
Department of Mathematics, College of Sciences,
Yanbu, Taibah University,
Saudi Arabia}
\email{sallllm@gmail.com}

\address{Maha Bajneed \newline
Department of Mathematics,
Faculty of Sciences at Yanbu,
Taibah University,
Saudi Arabia}
\email{m\_bajneed@hotmail.com}

\address{Khaled Sioud \newline
Department of Mathematics,
Faculty of Sciences at Yanbu, Taibah University,
Saudi Arabia}
\email{sioudkha@aol.com}

\thanks{Submitted June 25, 2015. Published January 12, 2016.}
\subjclass[2010]{80A23, 65N21, 26A33, 45J05, 34K37, 42A16}
\keywords{Semi-linear pseudo-hyperbolic equation; non-existence of solutions; 
\hfill\break\indent non-linear capacity method}

\begin{abstract}
 We study Cauchy problems time fractional semi-linear
 pseudo-hyperbolic equations and systems. Using the method of
 nonlinear capacity, we show that there are no solutions for certain
 nonlinearities and initial data. Our work complements the work by
 Aliev and col. \cite{Aliev3, Aliev2, Aliev5}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

In this article, we study  Cauchy problems for  time
fractional pseudo-hyperbolic equations and systems. 
 We start by considering the time fractional equation
\begin{equation}
u_{tt}+\eta ( -\Delta ) ^{k}u_{tt}+( -\Delta ) ^{\ell
}u+\xi ( -\Delta ) ^{r}D_{0|t}^{\alpha
}u+\gamma D_{ 0| t}^{\beta }u=f( u) ,
\label{1.1}
\end{equation}
for $x\in\mathbb{R} ^{N}$, $t>0$, supplemented with the initial data
\begin{equation}
u( x,0) =u_{0}( x) ,\quad u_{t}( x,0)=u_1( x) ,\quad x\in
\mathbb{R}^{N},  \label{1.2}
\end{equation}
and with
\begin{equation}
\eta ,\xi ,\gamma \geq 0\quad \text{for }r,k\in\mathbb{N}\cup  \{ 0\} ,\;
\ell \in \mathbb{N}, \; 0<\beta \leq \alpha \leq 1,  \label{1.3}
\end{equation}
$\Delta $ is the Laplacian and $D_{0|t}^{\alpha }$ is the
left-sided Riemann-Liouville fractional derivative of order $\alpha $.

The aim of this paper is to show, using the method of nonlinear capacity
proposed by Pokhozhaev in 1997 \cite{Pokhozhaev} and developed successfully
and jointly with Mitidieri \cite{Mitidieri3, Mitidieri2,Mitidieri}, that
under certain conditions, there are no solutions to \eqref{1.1}-\eqref{1.2}.

For the non fractional case $\alpha =\beta =1$, Lions' monograph \cite{Lions}
considered equation \eqref{1.1} in the case where $\eta =0$ and $f(
u) =-| u| ^{p}u$. A step forward was achieved by
\cite{ Kato, Mitidieri3, Zhang} where they considered the absence of global
solutions for the case where $\eta =\xi =0$ and $f( u)
=| u| ^{p-1}u$ or $f( u) =\pm |
u| ^{p}$. Kato \cite{Kato} showed that for $\ell =1$, $\xi =0$,
and $1<p<1+\frac{2}{N}$, problem \eqref{1.1}-\eqref{1.2} admits no global
solution under a certain condition on the initial data. A further study of
John \cite{John} considered the case $\ell =1$, $\xi =0$, $\eta =0,\gamma =0$
and $f( u) =| u| ^{p}$ for $u$ close to
zero. This was generalized to $\ell \in\mathbb{N}$,
 $\xi =\eta =0$, $\gamma >0$, $\alpha =1$, by Zhang \cite{Zhang} and
Kirane and Qafsaoui \cite{Kirane}. The two studies proved that the critical
exponent for this case is in fact $p=1+\frac{2}{N}$.

The existence of global solutions of problem \eqref{1.1}-\eqref{1.3} for the
non-fractional case $\alpha =\beta =1$, $\eta ,\xi ,\gamma \geq 0$, 
$r,k\in\mathbb{N}\cup \{ 0\} $ and $\ell \in \mathbb{N}$ was achieved by Aliev and Kazymov \cite{Aliev1}.

Recently, by using the method of the test function Aliev and col.
\cite{Aliev3, Aliev2, Aliev5} established sufficient conditions for the
nonexistence of global solutions of problem \eqref{1.1}-\eqref{1.3} for the
non-fractional case $\alpha =\beta =1$:
 Aliev and Lichaei \cite{Aliev2} considered the case
$\alpha =\beta =1$ and $\eta ,\xi ,\gamma >0$ for
$r=k\in\mathbb{N} \cup \{ 0\} $, $\ell \in\mathbb{N}$, and
$f( u) \geq C| u| ^{p}$.
 Aliev and Kazymov \cite{Aliev3} examined the case\newline
$\alpha =\beta =1$, $k=0$, $r=0$, $\ell \in \mathbb{N}$ and
$f( u) =\frac{1}{( 1+| x|^{2}) ^{s}}| u| ^{p}$.
 Aliev and Mamedov \cite{Aliev5} treated the non existence of global
solutions of a semilinear hyperbolic equation with an anisotopic elliptic
part ($\alpha =\beta =1$, $k=r=0$),
\begin{equation*}
u_{tt}+\varepsilon u_{t}+\underset{k=1}{\overset{N}{\sum }}( -1)
^{\ell _{k}}D_{x_{k}}^{2\ell _{k}}u=f( u), \quad
f(u) \geq c| u| ^{p}.
\end{equation*}


Our work will complement the results of \cite{Aliev2} for
$r,k\in\mathbb{N}\cup \{ 0\} $, $\eta ,\xi ,\gamma \geq 0$ and
$\ell \in\mathbb{N}$ and extend it to the time-fractional case
$0<\beta \leq \alpha <1$, using the test function method.

In the second part of this paper, we study the Cauchy problem
for the time-fractional pseudo-hyperbolic system
\begin{equation}
\begin{gathered}
u_{tt}+\eta _1( -\Delta ) ^{k_1}u_{tt}+( -\Delta )
^{\ell _1}u+\xi _1( -\Delta ) ^{r_1}D_{0|t}^{\alpha _1}u+\gamma _1D_{0|t}^{\beta
_1}u=f( v) =| v| ^{p} \\
v_{tt}+\eta _2( -\Delta ) ^{k_2}v_{tt}+( -\Delta )
^{\ell _2}v+\xi _2( -\Delta ) ^{r_2}D_{0|t}^{\alpha _2}v+\gamma _2D_{0|t}^{\beta
_2}v=g( u) =| u| ^{q}
\end{gathered}\label{1.4}
\end{equation}
posed in $Q_{\infty }:=\mathbb{R}^{N}\times ( 0,\infty ) $,
subject to the initial conditions
\begin{equation}
\begin{gathered}
u( x,0) =u_{0}( x) ,u_{t}( x,0) =u_1( x), \quad  x\in\mathbb{R}^{N}, \\
v( x,0) =v_{0}( x) ,v_{t}( x,0)
=v_1( x) x\in\mathbb{R}^{N}
\end{gathered}  \label{1.5}
\end{equation}
with $p,q>1$, $r_i,k_i\in\mathbb{N}\cup \{ 0\}$,
$\ell _i\in\mathbb{N}$, $\eta _i,\xi _i,\gamma _i\geq 0$  and
$0<\beta_i\leq \alpha _i\leq 1$  for $i=1,2$.

The non-existence of global solutions in the case of a non fractional system
of two (or more) equations with $\alpha _i=0$ or $1$ and $\beta _i=0$ or
$1$, is investigated in numerous studies of Aliev and colleagues:
 Aliev, Mammadzada, and Lichaei \cite{Aliev4} considered the case
$\beta _i=1$, $\gamma _i=\eta _i=1$, $\xi _i=0$, $\ell _1=1$, $\ell _2=2$,
$p=\frac{7}{2}$ and $q=\frac{5}{2}$;
Aliev and Kazymov \cite{Aliev6} examined the case
$\beta _i=1$, $\gamma _i=\eta _i=1$, $\xi _i=0 $,
$\ell _i\in\mathbb{N}$, and $f_i( u,v) \geq C_{i,1}| u|^{p_i}+C_{i,2}| v| ^{q_i}$;
Aliev and Kazymov \cite{Aliev7} considered the case
$\beta _i=1$, $\gamma _i=\eta _i=1$, $\xi _i=0$, $\ell _i\in\mathbb{N}$, and 
$f( v) \geq C| v| ^{p}$ and $g(u) \geq C| u| ^{q}$;
 Aliev and Kazymov \cite{Aliev8} dealt with a system of three equations
that is similar to the case presented in \cite{Aliev6}.

Our work will complement these papers for the system of two equations in the
cases $\gamma _i,\eta _i,\xi _i >0$, 
$r_i,k_i\in\mathbb{N}\cup \{ 0\}$, $\ell _i\in\mathbb{N}$ and extend it to 
the time-fractional case, using again the test function
method.

\section{Preliminaries}

For the convenience of the reader, we start by recalling some basic
definitions and properties which will be useful throughout this paper.

\begin{definition} \label{def1}\rm
The left- and right-sided Riemann-Liouville integrals of order $0<\alpha <1$
for an integrable function\ are defined as
\begin{gather}
\big( I_{0| t }^{\alpha }f\big) ( t) 
:=\frac{1 }{\Gamma ( \alpha ) }\int_{0}^{t}( t-s) ^{\alpha -1}f( s) ds,  \label{2.1} \\
\big( I_{t| T}^{\alpha }f\big) ( t) 
:=\frac{1 }{\Gamma (\alpha ) }\int_{t}^{T}( s-t) ^{\alpha -1}f( s) ds,  \label{2.2}
\end{gather}
where $\Gamma $ is the Euler gamma function.
\end{definition}

\begin{definition} \label{def2}\rm
Let $AC[0,T]$ be the space of functions $f$ which are
absolutely continuous on $[0,T]$. The left and right-handed
Riemann-Liouville fractional derivatives of order$\ n-1<\gamma <n$ for a
function
\begin{equation*}
f\in AC^{n}[ 0,T] :=\{ f:[ 0,T] \to\mathbb{R}, D^{n-1}f\in AC[ 0,T] \} ,\quad
n\in \mathbb{N}
\end{equation*}
is defined as (see \cite{Kilbas})
\begin{gather}
D_{0| t}^{\gamma }f( t) :=D^{n}( I_{0|t}^{n-\gamma }f) ( t) ,\quad t>0,
\label{2.3} \\
D_{t|T}^{\gamma }f( t) :=( -1)^{n}D^{n}( I_{t|T}^{n-\gamma }f) (t) ,  \label{2.4}
\end{gather}
where $D$ is the usual time derivative.
\end{definition}

Furthermore, for every $f,g\in C( [ 0,T] ) $ such that
$D_{0|t}^{\alpha }f( t) ,D_{0|t}^{\alpha }g( t) $ exist and are continuous 
for all $t\in [ 0,T] $, $0<\alpha <1$, the formula of integration by parts
can be given according to Love and Young \cite{Love} by
\begin{equation}
\int_{0}^{T}g( t) ( D_{0|t}^{\alpha}f) ( t) dt
=\int_{0}^{T}f( t) ( D_{t|T}^{\alpha }g) ( t) dt.  \label{2.7}
\end{equation}
In addition, \cite[Lemma 2.2]{Samko} provides us with the formula
\begin{equation}
D_{t|T}^{\alpha }f( t) :=\frac{1}{\Gamma
( 1-\alpha ) }[ \frac{f( T) }{( T-t)
^{\alpha }}-\int_{t}^{T}( t-s) ^{-\alpha }f'(s) ds]  \label{2.9}
\end{equation}
or
\begin{equation}
D_{t|T}^{\alpha }f( t) :=\frac{1}{\Gamma
( 1-\alpha ) }\frac{d}{dt}\int_{t}^{T}( t-s) ^{-\alpha}f( s) ds.  \label{2.10}
\end{equation}

\section{Non-existence of global solutions of one equation} \label{Sec3}

In this section, we study the  non-existence of global solutions for the
time-fractional semi-linear pseudo-hyperbolic equation \eqref{1.1} for
certain initial data with $f( u) =| u| ^{p}$. Before we state our result. 
let us define the weak solution of problem \eqref{1.1}-\eqref{1.3}.

In this article, $Q_T$ denotes the set $Q_T:=\mathbb{R}^{N}\times ( 0,T) $,
$0<T\leq +\infty $. We set 
\begin{gather*}
\int_{Q_T}f:=\int_{\mathbb{R}^{N}}\int_{0}^{T}f( x,t) \,dx\,dt,\quad
\int_{Q_{\infty }}f:=\int_{\mathbb{R}^{N}}\int_{0}^{\infty }f( x,t) \,dx\,dt,\\
\int_{\mathbb{R}^{N}}f:=\int_{\mathbb{R}^{N}}f( x,0) dx.
\end{gather*}

\begin{definition} \label{DefWeak} \rm
The function $u\in L_{\rm loc}^{p}( Q_{\infty }) $ is
a weak solution of problem \eqref{1.1}-\eqref{1.3} on $Q_T$ with initial
data $u_{0}( x) ,u_1( x) \in L_{\rm loc}^{1}(\mathbb{R}^{N}) $ if it satisfies
\begin{equation}
\begin{aligned}
& \int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) \varphi ( x,0)
+\eta \int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0) \\
& =\int_{\mathbb{R}^{N}}u_{0}( x) \varphi _{t}( x,0) +\eta \int_{\mathbb{R}
^{N}}u_{0}( x) ( -\Delta ) ^{k}\varphi _{t}(x,0)
+\int_{Q_T}u\varphi _{tt}\\
&\quad +\eta \int_{Q_T}u( -\Delta) ^{k}\varphi _{tt}
 -\xi \int_{Q_T}u( -\Delta ) ^{r}D_{ t|T}^{\alpha }\varphi
+\gamma \int_{Q_T}uD_{t| T}^{\beta }\varphi \\
&\quad +\int_{Q_T}u( -\Delta ) ^{\ell }\varphi ,
\end{aligned} \label{3.1}
\end{equation}
for any test-function
$\varphi \in C_{x\text{ \ }t}^{2d\text{ \ }2}(Q_T) $
with $d=\max \{ \ell ,k,r\} $ such that $\varphi$ is positive,
$\varphi \equiv 0$ outside a compact
 $K\subset\mathbb{R}^{n}$, $\varphi ( x,T) =\varphi _{t}( x,T) =0$ and
$D_{t| T}^{\beta }\varphi ,D_{t|T}^{\alpha }\varphi \in C( Q_T) $.
\end{definition}

As for the result on the non-existence of a global solution, the constants 
$\eta ,\xi $ and $\gamma $ will not play a role, and thus will be taken equal
to one.

\begin{theorem} \label{TheoCase2}
Assume that
\begin{itemize}
\item[(1)] $r,k\in\mathbb{N}\cup \{ 0\}$, $\ell \in\mathbb{N}$ and 
$0<\beta \leq \alpha <1$;

\item[(2)] $u_{0},u_1,\in L^{1}(\mathbb{R}^{N}) $ such that 
$\int_{\mathbb{R}^{N}}u_{0}( x) dx>0$, 
$\int_{\mathbb{R}^{N}}u_1( x) dx>0$ 

\item[(3)] $1<p\leq 1+\dfrac{2\ell }{N+2\ell ( \frac{1}{\beta }-1) }
=:p_{c}$.
\end{itemize}
Then problem \eqref{1.1}--\eqref{1.3} does not admit any global in time
 nontrivial solution.
\end{theorem}

\begin{proof}
The proof is by contraction. Let $u$ be a global weak solution of problem 
\eqref{1.1}-\eqref{1.3} and $\varphi $ be a non-negative function (satisfying
the conditions of Definition \ref{DefWeak}) that will be specified later.

Using $\epsilon$-Young's inequality
\begin{equation*}
ab\leq \epsilon a^{p}+c( \epsilon ) b^{\widetilde{p}},\quad
p>1,\;a\geq 0,\; b\geq 0,\; p+\widetilde{p}=p\widetilde{p},
\;\epsilon >0,
\end{equation*}
we can write
\begin{equation}
\begin{gathered}
\int_{Q_T}u\varphi _{tt}
\leq \epsilon \int_{Q_T}| u| ^{p}\varphi +c( \epsilon ) \int_{Q_T}| \varphi
_{tt}| ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}, \\
\int_{Q_T}u( -\Delta ) ^{k}\varphi _{tt}\leq \epsilon
\int_{Q_T}| u| ^{p}\varphi +c( \epsilon )
\int_{Q_T}| ( -\Delta ) ^{k_1}\varphi _{tt}|
 ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}, \\
\int_{Q_T}u( -\Delta ) ^{r}D_{t| T}^{\alpha
}\varphi \leq \epsilon \int_{Q_T}| u| ^{p}\varphi
+c( \epsilon ) \int_{Q_T}| ( -\Delta )
^{r}D_{t| T}^{\alpha }\varphi | ^{\widetilde{p}
}\varphi ^{-\widetilde{p}/p}, \\
\int_{Q_T}uD_{t| T}^{\beta }\varphi \leq \epsilon
\int_{Q_T}| u| ^{p}\varphi +c( \epsilon )
\int_{Q_T}| D_{t| T}^{\beta }\varphi | ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}, \\
\int_{Q_T}u( -\Delta ) ^{\ell }\varphi \leq \epsilon
\int_{Q_T}| u| ^{p}\varphi +c( \epsilon )
\int_{Q_T}| ( -\Delta ) ^{\ell }\varphi |
^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}.
\end{gathered}\label{3.3}
\end{equation}
Using inequalities \eqref{3.3} in \eqref{3.1}, we obtain the inequality
\begin{equation}
\begin{aligned}
& \int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) \varphi ( x,0) 
+\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0)  \\
& -\int_{\mathbb{R}^{N}}u_{0}( x) \varphi _{t}( x,0) 
-\int_{\mathbb{R}^{N}}u_{0}( x) ( -\Delta ) ^{k}\varphi (x,0)  \\ 
&\leq  \epsilon _1\int_{Q_T}| u| ^{p}\varphi
+C_1\Big\{ \int_{Q_T}| \varphi _{tt}| ^{\widetilde{p}}
\varphi ^{-\widetilde{p}/p}+\int_{Q_T}|
( -\Delta ) ^{k}\varphi _{tt}| ^{\widetilde{p}}\varphi
^{-\widetilde{p}/p}  \\
&  +\int_{Q_T}| ( -\Delta ) ^{r}D_{
t| T}^{\alpha }\varphi | ^{\widetilde{p}}\varphi ^{
\frac{-\widetilde{p}}{p}}+\int_{Q_T}| D_{t|
T}^{\beta }\varphi | ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}
+\int_{Q_T}| ( -\Delta ) ^{\ell }\varphi
| ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}\Big\}.
\end{aligned}\label{3.4}
\end{equation}
Setting
\begin{gather*}
A_1=\int_{Q_T}| \varphi _{tt}| ^{\widetilde{p} }\varphi ^{-\widetilde{p}/p}, \quad
A_2=\int_{Q_T}| ( -\Delta ) ^{k}\varphi_{tt}| ^{\widetilde{p}}
\varphi ^{-\widetilde{p}/p}, \\
A_3=\int_{Q_T}| ( -\Delta ) ^{r}D_{t| T}^{\alpha }\varphi
| ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}, \quad
A_4=\int_{Q_T}| D_{t| T}^{\beta }\varphi
 | ^{\widetilde{p}}\varphi ^{-\widetilde{p}/p}, \\
A_5=\int_{Q_T}| ( -\Delta ) ^{\ell }\varphi | ^{\widetilde{p}}
 \varphi ^{-\widetilde{p}/p},
\end{gather*}
and taking $\epsilon =1/2$, inequality \eqref{3.4} becomes
\begin{equation}
\begin{aligned}
& \int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) \varphi ( x,0)
 +\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0)  \\
& -\int_{\mathbb{R}^{N}}u_{0}( x) \varphi _{t}( x,0) 
-\int_{\mathbb{R}^{N}}u_{0}( x) ( -\Delta ) ^{k}\varphi _{t}(x,0)  \\
&\leq  C\{ A_1+A_2+A_3+A_4+A_5\} .
\end{aligned} \label{3.5}
\end{equation}
At this stage, we set
\begin{equation}
\varphi ( x,t) =\Psi ^{\nu }\Big( \frac{t^{2}+|
x| ^{4\rho }}{R^4}\Big) ,\quad  R>0,\; \nu \gg 1,\; \rho >0,  \label{3.6}
\end{equation}
where $\Psi \in C_{c}^{\infty }(\mathbb{R}^{+}) $ is a decreasing function
 defined as
\begin{equation*}
\Psi ( r) =\begin{cases}
1 & \text{if }r\leq 1 \\
0 & \text{if }r\geq 2,
\end{cases}
\end{equation*}
with $0\leq \Psi \leq 1$ and $r| \Psi '( r) | <C$.

Note that with this choice of $\varphi $, we have
\begin{equation*}
\varphi _{t}( x,t) =2\nu tR^{-4}\Psi ^{\nu -1}( (
t^{2}+| x| ^{4\rho }) /R^4) \Psi '( ( t^{2}+| x| ^{4\rho })/R^4) ,
\end{equation*}
leading to
\begin{equation}
\varphi _{t}( x,0) =0.  \label{3.7}
\end{equation}
We also assume  that $\varphi $ satisfies
\begin{equation*}
\int_{Q_T}\varphi ^{-p/\widetilde{p}}
( | \varphi _{tt}| ^{\widetilde{p}}
+| ( -\Delta )^{k}\varphi _{tt}| ^{\widetilde{p}}+| ( -\Delta
) ^{r}D_{t| T}^{\alpha }\varphi | ^{
\widetilde{p}}+| D_{t| T}^{\beta }\varphi | ^{\widetilde{p}}
+| ( -\Delta ) ^{\ell }\varphi| ^{\widetilde{p}}) <\infty ,
\end{equation*}
for that, we will choose $\nu \gg 1$. Therefore, the inequality \eqref{3.5}
becomes
\begin{equation}
\begin{aligned}
& \int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) \varphi ( x,0) 
+\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0) \\
&\leq  C\{ A_1+A_2+A_3+A_4+A_5\} .
\end{aligned} \label{3.9}
\end{equation}

Let us pass to the scaled variables $y=R^{-1/\rho}x$, 
$\tau=R^{-2}t $ and the function $\widetilde{\varphi }$ given by 
$\varphi (x,t) =\widetilde{\varphi }( y,\tau ) $. In doing so, it
follows that
\begin{equation*}
\varphi _{t}=R^{-2}\widetilde{\varphi }_{\tau },\quad
\varphi _{tt}=R^{-4} \widetilde{\varphi }_{\tau \tau },\quad
D_{t| T}^{\beta }\varphi =R^{-2\beta }D_{\tau | T^{\ast }}^{\beta }
\widetilde{\varphi },\quad
( -\Delta ) ^{m}\varphi =R^{\frac{-2m}{\rho }}\Delta \widetilde{\varphi },
\end{equation*}
where $T=R^{2}T^{\ast }$ and $T^{\ast }$is a positive constant.

Now, let us set
\begin{equation*}
Q=\{ ( y,\tau ) ,\text{ }0\leq \tau ^{2}+y^{4\rho }\leq 2\} .
\end{equation*}
Using these definitions, $A_1,\dots ,A_5$ can be rewritten as
\begin{equation}
\begin{gathered}
A_1\leq R^{-4\widetilde{p}+\frac{N}{\rho }+2}\int_{Q}|
\widetilde{\varphi }_{\tau \tau }| ^{\widetilde{p}}\widetilde{
\varphi }^{-\widetilde{p}/p}, \\
A_2\leq R^{-( \frac{2k}{\rho }+4) \widetilde{p}+\frac{N}{\rho }
+2}\int_{Q}| ( -\Delta ) ^{k}\widetilde{\varphi }
_{\tau \tau }| ^{\widetilde{p}}\widetilde{\varphi }^{\frac{-
\widetilde{p}}{p}}, \\
A_3\leq R^{-( \frac{2r}{\rho }+2\alpha ) \widetilde{p}+\frac{N}{
\rho }+2}\int_{Q}| ( -\Delta ) ^{r}D_{\tau
| T^{\ast }}^{\alpha }\widetilde{\varphi }| ^{
\widetilde{p}}\varphi ^{-\widetilde{p}/p}, \\
A_4\leq R^{-2\beta \widetilde{p}+\frac{N}{\rho }+2}\int_{Q}|
D_{\tau | T^{\ast }}^{\beta }\widetilde{\varphi }| ^{\widetilde{p}}\widetilde{\varphi }^{\frac{-\widetilde{p}}{p}
}, \\
A_5\leq R^{\frac{-2\ell }{\rho }\widetilde{p}+\frac{N}{\rho }
+2}\int_{Q}| ( -\Delta ) ^{\ell }\widetilde{\varphi }
| ^{\widetilde{p}}\widetilde{\varphi }^{-\widetilde{p}/p}.
\end{gathered} \label{3.10}
\end{equation}
In a short form, we can write
\begin{equation*}
A_i=C_iR^{\theta _i}\quad \text{for } i=1,2,\dots ,5,
\end{equation*}
where
\begin{equation}
\begin{gathered}
\theta _1=-4\widetilde{p}+\frac{N}{\rho }+2, \quad
\theta _2=-\big( \frac{2k}{\rho }+4\big) \widetilde{p}+\frac{N}{\rho }+2, \\
\theta _3=-\big( \frac{2r}{\rho }+2\alpha \big) \widetilde{p}
+\frac{N}{\rho }+2, \quad
\theta _4=-2\beta \widetilde{p}+\frac{N}{\rho }+2, \\
\theta _5=\frac{-2\ell }{\rho }\widetilde{p}+\frac{N}{\rho }+2.
\end{gathered}\label{3.11}
\end{equation}
As $\beta \leq \alpha <1$, we observe that
\begin{equation*}
\theta _2\leq \theta _1\leq \theta _4\quad\text{and}\quad \theta _3
\leq \theta _4.
\end{equation*}

For $R\geq 1$, we have $R^{\theta _i}\leq R^{\theta _4}+R^{\theta _5}$
for $i=1,2,\dots ,5$ and then inequality \eqref{3.9} becomes
\begin{equation}
\int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) \varphi ( x,0) 
+\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (
x,0) \leq K( R^{\theta _4}+R^{\theta _5}) ,
\label{3.12}
\end{equation}
where $K$ is a positive constant.
We have
\begin{align*}
\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0) 
&=\int_{\overline{Q_{R}}}u_1( x) ( -\Delta) ^{k}\varphi ( x,0) \\
&=R^{\frac{N}{\rho }-\frac{2k}{\rho }}
 \int_{\overline{Q}}u_1( R^{1/\rho}y) ( -\Delta
^{k}) \widetilde{\varphi }( y,0) ,
\end{align*}
where
\begin{equation*}
\overline{Q_{R}}=\{ ( x,0);R^{1/\rho}
\leq | x| \leq 2^{1/(4\rho)}R^{1/\rho}\} , \quad 
\overline{Q}=\{ ( y,0) ;1\leq | y| \leq 2\} .
\end{equation*}

We assume that $\varphi $ satisfies
\begin{equation*}
\left \Vert ( -\Delta ^{k}) \widetilde{\varphi }( \cdot
,0) \right \Vert _{\infty }<\infty ,
\end{equation*}
for that, we will choose $\nu \gg 1$.
It follows that
\[
\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0) 
\leq CR^{\frac{N}{\rho }-\frac{2k}{\rho }}\int_{\overline{Q}}| u_1( R^{1/\rho}y) | 
\leq CR^{-\frac{2k}{\rho }}\int_{\overline{Q_{R}}}| u_1(x) | ,
\]
then, passing to the limit as $R\to +\infty $, we have
\begin{equation}
\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k}\varphi (x,0) \to 0.  \label{3.8}
\end{equation}
Now, if $\theta _4<0$ and $\theta _5<0$, (i.e. $\max ( \theta _4,\theta _5) <0$), 
that means
\begin{gather*}
p<p_1( \rho ) =1+\frac{2\beta \rho }{N+2\rho ( 1-\beta) }, \\
p<p_2( \rho ) =1+\frac{2\ell }{N+2( \rho -\ell ) }
\end{gather*}
which is equivalent to
\begin{equation*}
p<\overline{p}( \rho ) =\min ( p_1( \rho ) ,p_2( \rho ) ) .
\end{equation*}

As $p_1$ is a decreasing function and $p_2$ is an increasing function,
the maximum value of the function $\min ( p_1,p_2) $ will be
at the point $\rho =\frac{\ell }{\beta }$, when the two functions $p_1$
and $p_2$ are equal
\begin{equation*}
p_1( \frac{\ell }{\beta }) =p_2( \frac{\ell }{\beta }
) =1+\frac{2\ell }{N+2\ell ( \frac{1}{\beta }-1) }=:p_{c},
\quad (\text{i.e. } \theta _4=\theta _5).
\end{equation*}
Therefore, for
\begin{equation}
1<p<p_{c},  \label{siod}
\end{equation}
we have $R^{\theta _4}+R^{\theta _5}$ tends to zero when
 $R\to \infty $ and the inequalities \eqref{3.12} and \eqref{3.8} yield
\begin{equation*}
\int_{Q_{\infty }}| u| ^{p}+\int_{\mathbb{R}^{N}}u_1( x) \leq 0.
\end{equation*}
As
\begin{equation*}
\int_{\mathbb{R}^{N}}u_1( x) >0,
\end{equation*}
we get a contradiction. This proves the theorem in the case \eqref{siod}.

For the border case where $p=p_{c}$ which corresponds to $\theta _4=\theta
_5=0$ and $\rho =\frac{\ell }{\beta }$, let
\begin{equation*}
Q_{T,R}=\{ ( x,t) ,R^4\leq t^{2}+| x| ^{4\rho }\leq 2R^4\} ,
\end{equation*}
if we use the H\"{o}lder inequality in the estimate of 
$\int_{Q_T}u \varphi _{tt}$ instead of the $\epsilon$-Young inequality, 
we obtain
\begin{align*}
\int_{Q_T}u\varphi _{tt}
&=\int_{Q_{T,R}}u\varphi _{tt}
\leq \Big(\int_{Q_{T,R}}| u| ^{p}\varphi \Big) ^{1/p}
\Big( \int_{Q_{T,R}}\varphi ^{-\widetilde{p}/p}|
\varphi _{tt}| ^{\widetilde{p}}\Big) ^{1/\widetilde{p}}\\
&\leq ( A_1) ^{1/\widetilde{p}}(
\int_{Q_{T,R}}| u| ^{p}\varphi ) ^{1/p},
\end{align*}
and similarly
\begin{gather*}
\int_{Q_T}u( -\Delta ) ^{k}\varphi _{tt}
=\int_{Q_{T,R}}u( -\Delta ) ^{k}\varphi _{tt}\leq ( A_2) ^{1/\widetilde{p}}
\int_{Q_{T,R}}| u| ^{p}\varphi ,
\\
\int_{Q_T}u( -\Delta ) ^{r}D_{t| T}^{\alpha
}\varphi =\int_{Q_{T,R}}u( -\Delta ) ^{r}D_{t|
T}^{\alpha }\varphi 
\leq ( A_3) ^{1/\widetilde{p}
}\Big( \int_{Q_{T,R}}| u| ^{p}\varphi \Big) ^{1/p},
\\
\int_{Q_T}uD_{t| T}^{\beta }\varphi
=\int_{Q_{T,R}}uD_{t| T}^{\beta }\varphi \leq (
A_4) ^{1/\widetilde{p}}\Big( \int_{Q_{T,R}}|
u| ^{p}\varphi \Big) ^{1/p},
\\
\int_{Q_T}u( -\Delta ) ^{\ell }\varphi 
=\int_{Q_{T,R}}u( -\Delta ) ^{\ell }\varphi \leq ( A_5) ^{1/\widetilde{p}}
\Big( \int_{Q_{T,R}}| u| ^{p}\varphi \Big) ^{1/p}.
\end{gather*}
Thus, we obtain
\begin{align*}
\int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) 
& \leq \big( A_1^{1/\widetilde{p}
}+A_2^{1/\widetilde{p}}+A_3^{1/\widetilde{p}}+A_4^{
\frac{1}{\widetilde{p}}}+A_5^{1/\widetilde{p}}\big) 
\Big(\int_{Q_{T,R}}| u| ^{p}\varphi \Big) ^{1/p}
\\
& \leq C\Big( \int_{Q_{T,R}}| u| ^{p}\varphi \Big)^{1/p}.
\end{align*}
As $\int_{Q_T}| u| ^{q}<+\infty $, we have
\begin{equation*}
\lim_{R\to +\infty} \int_{Q_{T,R}}| u| ^{q}\varphi 
\leq \lim_{R\to +\infty} \int_{Q_{T,R}}| u| ^{q}=0.
\end{equation*}
Passing to the limit as $R\to +\infty $, we find that 
$\int_{Q_{\infty }}| u| ^{q}+\int_{\mathbb{R}^{N}}u_1( x) =0$, which 
contradicts  $\int_{\mathbb{R}^{N}}u_1>0$. This prove the theorem 
in the case $p=p_{c}$.
\end{proof}

\section{A pseudo-hyperbolic system}

This section is concerned with the fractional time pseudo-hyperbolic system 
\eqref{1.4})-\eqref{1.5}.

\begin{definition} \label{DefWeakSys} \rm
The couple of functions $( u,v) $, $u\in L_{\rm loc}^{q}( Q_{\infty }) $ and 
$v\in L_{\rm loc}^{p}( Q_{\infty }) $ is a weak solution of \eqref{1.4}-\eqref{1.5} 
on $Q_T$ with initial data $u_{0}( x) ,u_1( x) ,v_{0}( x) $ and 
$v_1( x) \in L_{\rm loc}^{1}(\mathbb{R}^{N}) $, if it satisfies
\begin{equation}
\begin{aligned}
& \int_{Q_T}| v| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1( x) \varphi ( x,0)
+\eta _1\int_{\mathbb{R}^{N}}u_1( x) ( -\Delta ) ^{k_1}\varphi (x,0)   \\
& =\int_{\mathbb{R}^{N}}u_{0}( x) \varphi _{t}( x,0)
+\eta _1\int_{\mathbb{R}^{N}}u_{0}( x) ( -\Delta ) ^{k_1}\varphi _{t}(x,0)
+\int_{Q_T}u\varphi _{tt} \\
&\quad +\int_{Q_T}u( -\Delta )^{\ell _1}\varphi
 +\eta _1\int_{Q_T}u( -\Delta ) ^{k_1}\varphi _{tt}
 -\xi_1\int_{Q_T}u( -\Delta ) ^{r_1}D_{t|T}^{\alpha _1}\varphi  \\
&\quad +\gamma _1\int_{Q_T}uD_{t|T}^{\beta _1}\varphi ,  \label{4.1}
\end{aligned}
\end{equation}
and
\begin{equation}
\begin{aligned}
& \int_{Q_T}| u| ^{q}\varphi +\int_{\mathbb{R}^{N}}v_1( x) \varphi ( x,0)
+\eta _2\int_{\mathbb{R}^{N}}v_1( x) ( -\Delta ) ^{k_2}\varphi (x,0)   \\
& =\int_{\mathbb{R}^{N}}v_{0}( x) \varphi _{t}( x,0)
+\eta _2\int_{\mathbb{R}^{N}}v_{0}( x) ( -\Delta ) ^{k_2}\varphi _{t}(x,0)
+\int_{Q_T}v\varphi _{tt} \\
&\quad +\int_{Q_T}v( -\Delta ) ^{\ell _2}\varphi
 +\eta _2\int_{Q_T}v( -\Delta ) ^{k_2}\varphi _{tt}
 -\xi_2\int_{Q_T}v( -\Delta ) ^{r_1}D_{t| T}^{\alpha _2}\varphi \\
&\quad +\gamma _2\int_{Q_T}vD_{t|T}^{\beta _2}\varphi ,
\end{aligned}  \label{4.2}
\end{equation}
for any test-function $\varphi \in C_{x\text{ \ }t}^{2\ell 2}( Q_T) $,
$\ell =\max \{ \ell _1,\ell _2\} $
being positive, $\varphi \equiv 0$ outside a compact
$K\subset\mathbb{R}^{n}$, $\varphi ( x,T) =\varphi _{t}( x,T) =0$ and
$D_{t| T}^{\alpha _1}\varphi ,D_{t|T}^{\beta _1}\varphi ,
D_{t| T}^{\alpha _2}\varphi,D_{t| T}^{\beta _2}\varphi \in C( Q_T)$.
\end{definition}

\begin{theorem} \label{TheoCaseSys}
Assume that 
\begin{itemize}
\item[(1)] $r_i,k_i\in \mathbb{N}\cup \{ 0\}$, 
$\ell _i\in\mathbb{N}$ and $0<\beta _i\leq \alpha _i<1$, $i=1,2$;

\item[(2)] $u_{0},u_1,v_{0},v_1\in L^{1}(\mathbb{R}^{N}) $ such that  
$\int_{\mathbb{R}^{N}}u_{0}( x) >0,\int_{\mathbb{R}^{N}}u_1( x) >0$, 
$\int_{\mathbb{R}^{N}}v_{0}( x) >0$ and 
$\int_{\mathbb{R}^{N}}v_1( x) >0$;
 
\item[(3)] $p>1$, $q>1$,
\[
pq\leq \min \Big(1+\frac{2( p\beta _2+\beta _1)
\overline{\rho }}{N+2( 1-\beta _1) \overline{\rho }},1+\frac{
2( q\beta _1+\beta _2) \overline{\rho }}{N+2( 1-\beta
_2) \overline{\rho }}\Big) 
\]
 where $\overline{\rho }=\min (\frac{\ell _1}{\beta _1},\frac{\ell _2}{\beta _2}) $.
\end{itemize}
Then problem \eqref{1.4}-\eqref{1.5} does not admit any global non trivial
solution.
\end{theorem}

\begin{proof}
The proof is by contraction. Let $( u,v) $ be a global weak
solution of \eqref{1.4}-\eqref{1.5} and $\varphi $ be a non-negative
function (satisfying the conditions of Definition \ref{DefWeakSys}).

Applying H\"{o}lder inequality to $\int_{Q_T}u\varphi _{tt}$, we obtain
\begin{equation*}
\int_{Q_T}u\varphi _{tt}
\leq \Big( \int_{Q_T}| u| ^{q}\varphi \Big) ^{1/q}
\Big( \int_{Q_T}\varphi ^{\frac{-
\widetilde{q}}{q}}| \varphi _{tt}| ^{\widetilde{q}
}\Big) ^{1/\widetilde{q}}
\leq ( A_1) ^{1/\widetilde{q}}\Big( \int_{Q_T}| u| ^{q}\varphi
\Big) ^{1/q}
\end{equation*}
and similarly
\begin{equation}
\begin{gathered}
\int_{Q_T}u( -\Delta ) ^{k_1}\varphi _{tt}\leq (
A_2) ^{1/\widetilde{q}}\Big( \int_{Q_T}|u| ^{q}\varphi \Big) ^{1/q}, \\
\int_{Q_T}u( -\Delta ) ^{r_1}D_{t|
T}^{\alpha _1}\varphi \leq ( A_3) ^{\frac{1}{\widetilde{q}}
}\Big( \int_{Q_T}| u| ^{q}\varphi \Big) ^{1/q}, \\
\int_{Q_T}uD_{t| T}^{\beta _1}\varphi \leq (
A_4) ^{1/\widetilde{q}}\Big( \int_{Q_T}| u| ^{q}\varphi \Big) ^{1/q}, \\
\int_{Q_T}u( -\Delta ) ^{\ell _1}\varphi \leq (
A_5) ^{1/\widetilde{q}}\Big( \int_{Q_T}|u| ^{q}\varphi \Big) ^{1/q},
\end{gathered}\label{4.3}
\end{equation}
where
\begin{gather*}
A_1 =\int_{Q_T}| \varphi _{tt}| ^{\widetilde{q}
}\varphi ^{\frac{-\widetilde{q}}{q}}\,dx\,dt, \\
A_2 =\int_{Q_T}| ( -\Delta ) ^{k_1}\varphi
_{tt}| ^{\widetilde{q}}\varphi ^{-\widetilde{q}/q}\,dx\,dt, \\
A_3 =\int_{Q_T}| ( -\Delta ) ^{r_1}D_{
t| T}^{\alpha _1}\varphi | ^{\widetilde{q}}\varphi ^{-\widetilde{q}/q}\,dx\,dt, \\
A_4 =\int_{Q_T}| D_{t| T}^{\beta
_1}\varphi | ^{\widetilde{q}}\varphi ^{-\widetilde{q}/q}\,dx\,dt, \\
A_5 =\int_{Q_T}| ( -\Delta ) ^{\ell _1}\varphi
| ^{\widetilde{q}}\varphi ^{-\widetilde{q}/q}\,dx\,dt.
\end{gather*}
Now, let $\varphi $ be the test function defined by the expression 
\eqref{3.6}. Using the previous estimates \eqref{4.3} and the 
properties \eqref{3.6} and \eqref{3.7} of the function $\varphi $
 in equation \eqref{4.1}, we obtain the inequality
\begin{equation}
\begin{aligned}
& \int_{Q_T}| v| ^{p}\varphi +\int_{\mathbb{R}^{N}}u_1\varphi ( x,0)  \\
&\leq  \Big( \int_{\mathbb{R}^{N}}| u| ^{q}\varphi \Big) ^{1/q}
\Big[A_1^{1/\widetilde{q}}+A_2^{1/\widetilde{q}}
+A_3^{1/\widetilde{q}}+A_4^{1/\widetilde{q}}+A_5^{1/\widetilde{q}}\Big] .
\end{aligned} \label{4.4}
\end{equation}
Similarly, for the equation \eqref{4.2}, we have
\begin{equation}
\begin{aligned}
& \int_{Q_T}| u| ^{p}\varphi +\int_{\mathbb{R}^{N}}v_1\varphi ( x,0)  \\
&\leq  \Big( \int_{\mathbb{R}^{N}}| v| ^{q}\varphi \Big) ^{1/q}
\big[B_1^{1/\widetilde{q}}+B_2^{1/\widetilde{q}}+B_3^{\frac{
1}{\widetilde{q}}}+B_4^{1/\widetilde{q}}+B_5^{\frac{1}{
\widetilde{q}}}\big] ,
\end{aligned}\label{4.5}
\end{equation}
where
\begin{gather*}
B_1 =\int_{Q_T}| \varphi _{tt}| ^{\widetilde{q}
}\varphi ^{\frac{-\widetilde{q}}{q}}\,dx\,dt, \\
B_2 =\int_{Q_T}| u( -\Delta ) ^{k_2}\varphi
_{tt}| ^{\widetilde{q}}\varphi ^{\frac{-\widetilde{q}}{q}
}\,dx\,dt, \\
B_3 = \int_{Q_T}| ( -\Delta ) ^{r_2}D_{
t| T}^{\alpha _2}\varphi | ^{\widetilde{q}}\varphi ^{
\frac{-\widetilde{q}}{q}}\,dx\,dt, \\
B_4 =\int_{Q_T}| D_{t| T}^{\beta
_2}\varphi | ^{\widetilde{q}}\varphi ^{\frac{-\widetilde{q}}{q}
}\,dx\,dt, \\
B_5 =\int_{Q_T}| ( -\Delta ) ^{\ell _2}\varphi
| ^{\widetilde{q}}\varphi ^{\frac{-\widetilde{q}}{q}}\,dx\,dt.
\end{gather*}
Now, we estimate $A_1,\dots ,A_5$ and $B_1,\dots ,B_5$ in the same way as
in Section \ref{Sec3}, we obtain inequalities similar to those given in 
\eqref{3.9} and \eqref{3.10}
\begin{equation}
A_i=C_iR^{\theta _i}, \quad B_i=D_iR^{\delta _i}\quad \text{for }
i=1,2,\dots ,5,  \label{4.6}
\end{equation}
where
\begin{equation}
\begin{gathered}
\theta _1=-4\widetilde{q}+\frac{N}{\rho }+2, \quad
\delta _1=-4 \widetilde{p}+\frac{N}{\rho }+2, \\
\theta _2=-\big( \frac{2k_1}{\rho }+4\big) \widetilde{q}+\frac{N}{
\rho }+2, \quad
 \delta _2=-\big( \frac{2k_2}{\rho }+4\big)
\widetilde{p}+\frac{N}{\rho }+2, \\
\theta _3=-\big( \frac{2r_1}{\rho }+2\alpha _1\big) \widetilde{q}+
\frac{N}{\rho }+2, \quad
 \delta _3=-\big( \frac{2r_2}{\rho }
+2\alpha _2\big) \widetilde{p}+\frac{N}{\rho }+2, \\
\theta _4=-2\beta _1\widetilde{q}+\frac{N}{\rho }+2, \quad
 \delta_4=-2\beta _2\widetilde{p}+\frac{N}{\rho }+2, \\
\theta _5=\frac{-2\ell _1}{\rho }\widetilde{q}+\frac{N}{\rho }+2, \quad
 \delta _5=\frac{-2\ell _2}{\rho }\widetilde{p}+\frac{N}{\rho }+2.
\end{gathered}\label{4.7}
\end{equation}
If we set
\begin{equation*}
I=\Big( \int_{Q_T}| v| ^{p}\varphi \Big) ^{1/p}\quad  \text{and}\quad
J=( \int_{Q_T}| u|^{q}\varphi ) ^{1/q},
\end{equation*}
inequalities \eqref{4.4} and \eqref{4.5} become
\begin{gather}
I^{P}+\int_{\mathbb{R}^{N}}u_1\varphi ( x,0) 
\leq J( C_1R^{\theta _1/\widetilde{q}}
+C_2R^{\theta _2/\widetilde{q}}
+C_3R^{\theta _3/\widetilde{q}}
+C_4R^{\theta _4/\widetilde{q}}
+C_5R^{\theta _5/\widetilde{q}}) , \label{4.8}
\\
J^{q}+\int_{\mathbb{R}^{N}}v_1\varphi ( x,0) 
\leq I( D_1R^{\frac{\delta _1}{\widetilde{q}}}
+D_2R^{\delta _2/\widetilde{q}}
+D_3R^{\delta _3/\widetilde{q}}
+D_4R^{\delta _4/\widetilde{q}}
+D_5R^{\delta _5/\widetilde{q}}) .  \label{4.9}
\end{gather}
We can observe that under the conditions on $\alpha _i$ and $\beta _i$,
we have
\begin{equation*}
\theta _2\leq \theta _1\leq \theta _4,\quad
\theta _3\leq \theta _4,\quad 
\delta _2\leq \delta _1\leq \delta _4, \quad 
\delta _3\leq \delta _4.
\end{equation*}
Hence, for $R\geq 1$, we have 
$R^{\theta i}\leq R^{\theta _4}+R^{\theta_5}$ and 
$R^{\delta i}\leq R^{\delta _4}+R^{\delta _5}$, and
consequently, inequalities \eqref{4.8} and \eqref{4.9} can be rewritten as
\begin{gather}
I^{p}+\int_{\mathbb{R}^{N}}u_1\varphi ( x,0) 
\leq CJ\big( R^{\frac{\theta _4}{
\widetilde{q}}}+R^{\theta _5/\widetilde{q}}\big) ,  \label{4.10}
\\
J^{P}+\int_{\mathbb{R}^{N}}v_1\varphi ( x,0) \leq 
DI( R^{\frac{\delta _4}{ \widetilde{q}}}+R^{\frac{\delta _5}{\widetilde{q}}}) ,
  \label{4.11}
\end{gather}
where
\begin{equation*}
C=\sum_{i=1}^5 C_i,\quad \text{and}\quad
D=\sum_{i=1}^5 D_i.
\end{equation*}
Since $\int_{\mathbb{R}^{N}}u_1\varphi ( x,0) \geq 0$ and 
$\int_{\mathbb{R}^{N}}v_1\varphi ( x,0) \geq 0$, inequalities \eqref{4.10} 
and \eqref{4.11} yield
\begin{gather}
I^{p}\leq CJ\big( R^{\theta _4/\widetilde{q}}
+R^{\theta _5//\widetilde{q}} \big) ,  \label{4.12}
\\
J^{P}\leq DI\big( R^{\delta _4/\widetilde{q}}
+R^{\delta _5/\widetilde{q}}\big) .  \label{4.13}
\end{gather}
The constants $C$ and $D$ will be updated at each step of the calculation
and will not play a role. This implies that
\begin{equation*}
I^{pq}\leq CI( R^{\delta _4/\widetilde{p}}
 +R^{\delta _5/\widetilde{p}}) ( R^{\theta _4/\widetilde{q}}+R^{
\frac{\theta _5}{\widetilde{q}}}) ^{q},\ J^{pq}\leq CJ( R^{
\frac{\theta _4}{\widetilde{q}}}+R^{\frac{\theta _5}{\widetilde{q}}
}) ( R^{\delta _4/\widetilde{p}}+R^{\delta _5/\widetilde{p}}) ^{p},
\end{equation*}
leading to
\begin{equation}
I^{pq-1}\leq C( R^{\delta _4/\widetilde{p}}+R^{\delta _5/\widetilde{p}}) 
( R^{\theta _4/\widetilde{q}}+R^{
\frac{\theta _5}{\widetilde{q}}}) ^{q},\ J^{pq-1}\leq C( R^{
\frac{\theta _4}{\widetilde{q}}}+R^{\frac{\theta _5}{\widetilde{q}}
}) ( R^{\delta _4/\widetilde{p}}+R^{\delta _5/\widetilde{p}}) ^{p}.  \label{4.14}
\end{equation}
Now, let
\begin{equation*}
S_1=\frac{1}{\widetilde{p}}\max ( \delta _4,\delta _5) +
\frac{q}{\widetilde{q}}\max ( \theta _4,\theta _5) ,\quad
S_2= \frac{1}{\widetilde{q}}\max ( \theta _4,\theta _5) +\frac{p}{
\widetilde{p}}\max ( \delta _4,\delta _5) .
\end{equation*}

If
\begin{equation}
S_1<0,\quad\text{and}\quad S_2<0,  \label{4.15}
\end{equation}
we have $( R^{\delta _4/\widetilde{p}}
+R^{\delta _5/\widetilde{p}}) 
( R^{\theta _4/\widetilde{q}}
+R^{\theta _5/\widetilde{q}}) ^{q}\to 0$ and 
$( R^{\frac{ \theta _4}{\widetilde{q}}}
+R^{\theta _5/\widetilde{q}}) ( R^{\delta _4/\widetilde{p}}
+R^{\delta _5/\widetilde{p}}) ^{p}\to 0$ as $R\to \infty $. Hence,
by \eqref{4.14}, both $I$ and $J$ vanish as $R\to \infty $. This
implies that $J^{q}=\int_{Q_T}| u| ^{q}\varphi $
converges to $\int_{Q_{\infty }}| u| ^{q}\varphi =0$
and $I^{p}=\int_{Q_T}| v| ^{p}\varphi $ converges to
$\int_{Q_{\infty }}| v| ^{p}\varphi =0$. Consequently,
$u\equiv 0$ and $v\equiv 0$.

As $S_1<0$, then we have $\max ( \theta _4,\theta _5) <0$
or $\max ( \delta _4,\delta _5) <0$. Suppose that
 $\max( \theta _4,\theta _5) <0$ and let again $R\to \infty$. 
By \eqref{4.10}, we obtain $\int_{\mathbb{R}^{N}}u_1=0$, which 
contradicts $\int_{\mathbb{R}^{N}}u_1>0$.

Now, we return to the condition \eqref{4.15} that lead to the contradiction.
Inequalities $S_1<0$ and $S_2<0$ are equivalent to
\begin{equation}
\begin{gathered}
S_1=-2\Big( q\min ( \beta _1,\frac{\ell _1}{\rho }) +\min
( \beta _2,\frac{\ell _2}{\rho }) \Big) 
+( \frac{N}{ \rho }+2) \frac{pq-1}{p}<0 \\
S_2=-2\Big( p\min ( \beta _2,\frac{\ell _2}{\rho }) +\min
( \beta _1,\frac{\ell _1}{\rho }) \Big) 
+( \frac{N}{ \rho }+2) \frac{pq-1}{q}<0.
\end{gathered}\label{4.15.5}
\end{equation}
Let us take $\rho =\overline{\rho }=\min ( \frac{\ell _1}{\beta _1},
\frac{\ell _2}{\beta _2}) $. We have
\begin{equation*}
\min \big( \beta _1,\frac{\ell _1}{\rho }\big) =\beta _1\quad\text{and}\quad
\min \big( \beta _2,\frac{\ell _2}{\rho }\big) =\beta _2.
\end{equation*}
The inequalities in \eqref{4.15.5} can now be written as
\begin{equation}
\begin{gathered}
S_1=-2( q\beta _1+\beta _2) \overline{\rho }+( N+2
\overline{\rho }) \frac{pq-1}{p}<0 \\
S_2=-2( p\beta _2+\beta _1) \overline{\rho }+( N+2
\overline{\rho }) \frac{pq-1}{p}<0,
\end{gathered} \label{4.16}
\end{equation}
which are equivalent to
\begin{equation*}
1<pq<\min \Big( 1+\frac{2( p\beta _2+\beta _1) \overline{
\rho }}{N+2( 1-\beta _1) \overline{\rho }},1+\frac{2(
q\beta _1+\beta _2) \overline{\rho }}{N+2( 1-\beta
_2) \overline{\rho }}\Big) .
\end{equation*}

 Let us now consider the border case where
\begin{equation*}
pq=\min \Big( 1+\frac{2( p\beta _2+\beta _1) \overline{\rho }
}{N+2( 1-\beta _1) \overline{\rho }},1+\frac{2( q\beta
_1+\beta _2) \overline{\rho }}{N+2( 1-\beta _2)
\overline{\rho }}\Big) ,
\end{equation*}
which corresponds to
\begin{equation}
S_1=0,\; S_2\leq 0\quad \text{or}\quad S_1\leq 0,\; S_2=0.  \label{4.16.5}
\end{equation}
Let us take the case $S_1=0$, $S_2\leq 0$ (the second case: 
$S_1\leq 0$, $S_2=0$ is similar). We have
\begin{gather}
\widetilde{p}\widetilde{q}S_1=\widetilde{q}\max ( \delta _4,\delta
_5) +q\widetilde{p}\max ( \theta _4,\theta _5) =0,\label{4.18}
\\
\widetilde{p}\widetilde{q}S_2=\widetilde{p}\max ( \theta _4,\theta
_5) +p\widetilde{q}\max ( \delta _4,\delta _5) \leq 0.
\label{4.19}
\end{gather}
From \eqref{4.18} and \eqref{4.19}, we have
\begin{equation*}
\widetilde{p}\max ( \theta _4,\theta _5) =-\frac{\widetilde{q}
\max ( \delta _4,\delta _5) }{q}\quad\text{and}\quad
\frac{\widetilde{q}}{q}( pq-1) \max ( \delta _4,\delta_5) \leq 0,
\end{equation*}
which implies
\begin{equation*}
\max ( \delta _4,\delta _5) =\max ( \delta _1,\delta _2,\dots ,\delta _5) \leq 0.
\end{equation*}
Moreover, using Young's inequality $( a+b) ^{r}\leq 2^{r-1}(
a^{r}+b^{r}) $ for $r\geq 1$, the inequalities in \eqref{4.14} lead to
\begin{equation*}
I^{pq-1}\leq 2R^{1/\widetilde{p}\max ( \delta _4,\delta
_5) } 2^{q-1} R^{\frac{q}{\widetilde{q}}\max ( \theta
_4,\theta _5) }=2^{q}R^{S_1}=2^{q},
\end{equation*}
and similarly
\begin{equation*}
J^{pq-1}\leq 2^{p}R^{S_2}=2^{p},
\end{equation*}
for every $T\in ( 0,+\infty ) $. Hence, $I<+\infty $ and 
$J<+\infty $ for every $T\in ( 0,+\infty ) $, and thus 
$\int_{Q_{\infty }}| u| ^{q}<+\infty $ and 
$\int_{Q_{\infty }}| v| ^{q}<+\infty $.

Now, let $Q_{T,R}=\big\{ ( x,t) ,\text{ }R^4\leq
t^{2}+| x| ^{4\overline{\rho }}\leq 2R^4\big\} $.
For the first inequality of \eqref{4.3}, we obtain
\begin{align*}
\int_{Q_T}u\varphi _{tt}
&=\int_{Q_{T,R}}u\varphi _{tt}
\leq \Big(\int_{Q_{T,R}}| u| ^{q}\varphi \Big) ^{1/q}
\Big( \int_{Q_{T,R}}\varphi ^{\frac{-\widetilde{q}}{q}}|
\varphi _{tt}| ^{\widetilde{q}}\Big) ^{1/\widetilde{q}}
\\
&\leq  ( A_1) ^{1/\widetilde{q}}\Big(
\int_{Q_{T,R}}| u| ^q \varphi \Big) ^{1/q}
\leq C_1R^{\theta _1/\widetilde{q}}.
\end{align*}
Doing the same for the remaining inequalities of \eqref{4.3}, we obtain a
new estimate of \eqref{4.1},
\begin{align*}
0&<\int_{\mathbb{R}^{N}}u_1( x) \\
& \leq  \int_{Q_{T,R}}| v|^{p}\varphi 
 +\int_{\mathbb{R}^{N}}\{ u_1( x) +u_{0}( x) \} \varphi( x,0) \\
& \leq  \Big( \int_{Q_{T,R}}| u| ^{q}\varphi \Big)^{1/q}
\big[ C_1R^{\theta _1/\widetilde{q}}
+C_2R^{\theta _2/\widetilde{q}}
+C_3R^{\theta _3/\widetilde{q}}
+C_4R^{\theta _4/\widetilde{q}}
+C_5R^{\theta _5/\widetilde{q}}\big] \\
& \leq  C\Big( \int_{Q_{T,R}}| u| ^{q}\varphi
\Big) ^{1/q}
\Big( R^{\theta _4/\widetilde{q}}+R^{\theta _5/\widetilde{q}}\Big) \\
& \leq  C\Big( \int_{Q_{T,R}}| u| ^{q}\varphi\Big) ^{1/q}.
\end{align*} %\label{4.17}
Let $R\to \infty $. Since 
$\int_{Q_{\infty }}|u| ^{q}<+\infty $, the right-hand side of the above inequality 
approaches zero when $R\to \infty $, while the left-hand side 
$\int_{\mathbb{R}^{N}}u_1( x) $ is assumed to be positive, this is a contradiction.

Similarly, the second case $S_1\leq 0$, $S_2=0$ leads to a contradiction.
\end{proof}

\subsection*{Acknowledgements} 
The authors would like to thank Taibah University
for grant No 4113 which made this work possible. 
Special thanks also go to Prof. M. Kirane for his continuous support 
and guidance, which have helped improve the quality of this paper.

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\end{document}