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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 09, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/09\hfil Nonexistence of global solutions]
{Nonexistence of global solutions to the system of semilinear
parabolic equations with biharmonic operator and singular potential}

\author[S. Bagirov \hfil EJDE-2018/09\hfilneg]
{Shirmayil Bagirov}

\address{Shirmayil Bagirov \newline
Institute of Mathematics and Mechanics of NAS of Azerbaijan,
Baku, Azerbaijan}
\email{sh\_bagirov@yahoo.com}

\dedicatory{Communicated by Ludmila S. Pulkina}

\thanks{Submitted November 12, 2017. Published January 6, 2018.}
\subjclass[2010]{35A01, 35B33, 35K52, 35K91}
\keywords{System of semilinear parabolic equation; biharmonic operator;
\hfill\break\indent global solution; critical exponent; method of test functions}

\begin{abstract}
 In the domain  $Q_{R}'= \{ x:| x| >R\}\times( 0,+\infty)$
 we consider the  problem
 \begin{gather*}
 \frac{\partial u_1}{\partial t}+\Delta^2 u_1-\frac{C_1}{|x| ^4}u_1
 =| x| ^{\sigma _1}| u_2| ^{q_1}, \quad u_1| _{t=0}=u_{10}( x)\geq0, \\
 \frac{\partial u_2}{\partial t}+\Delta^2 u_2-\frac{C_2}{|
 x| ^4}u_2=| x| ^{\sigma _2}| u_1| ^{q_2},\quad  u_2|
 _{t=0}=u_{20}( x)\geq0, \\
 \int_0^\infty \int_{\partial B_{R}} u_i\,ds\,dt\geq 0, \quad
 \int_0^\infty \int_{\partial B_{R}}\Delta u_i\,ds\,dt\leq 0,
 \end{gather*}
 where $\sigma_i\in \mathbb{R} $, $ q_i>1 $,
 $ 0\leq C_i<( \frac{n( n-4) }{4}) ^2$, $ i=1,2 $.
 Sufficient condition for the nonexistence of global solutions
 is obtained.The proof is based on the method of test functions.
\end{abstract}

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

\section{Introduction}

Let us introduce the following notation:
$x=(x_1,\dots ,x_{n})\in \mathbb{R}^{n} $, $n>4$,
$r=|x|=\sqrt{x_1^2 +\dots +x_{n}^2 } $, $B_{R} =\{x;|x|<R\}$,
 $B_{R}' =\{x;|x|>R\}$, $B_{R_1 ,R_2 } =\{x;R_1 <|x|<R_2 \}$,
$Q_{R}=B_{R}\times(0;+\infty)$, $Q_{R}'=B_{R}'\times(0;+\infty)$,
$\partial B_{R} =\{x;|x|=R\}$,
$\nabla{u}=( \frac{\partial u}{\partial x_1},\dots ,
\frac{\partial u}{\partial x_{n}})$,
$C_{x,t}^{4,1}(Q_{R}') $ is the set of functions that are four times
continuously differentiable with respect to $x$ and continuously
differentiable with respect to $t$ in $Q_{R}'$.

In the domain $Q_{R}'$ we consider the system of equations
\begin{equation} \label{e1.1}
\begin{gathered}
\frac{\partial u_1}{\partial t}+\Delta ^2u_1-\frac{C_1}{
| x| ^4}u_1=| x| ^{\sigma
_1}| u_2| ^{q_1} \\
\frac{\partial u_2}{\partial t}+\Delta ^2u_2-\frac{C_2}{
| x| ^4}u_2=| x| ^{\sigma
_2}| u_1| ^{q_2},
\end{gathered}
\end{equation}
with the initial condition
\begin{equation} \label{e1.2}
 u_i| _{t=0}=u_{i0}( x) \geq 0,
\end{equation}
and the conditions
\begin{equation} \label{e1.3}
\int_0^\infty \int_{\partial B_{R}}u_i\,dx\,dt\geq 0,\quad
\int_0^\infty \int_{\partial B_{R}}\Delta u_i\,dx\,dt\leq 0,
\end{equation}
where $n>4$, $q_i>1$,  $\sigma _i\in \mathbb{R}$, $0\leq C_i<( \frac{n( n-4) }{4}) ^2$,
$u_{i0}(x) \in C( B_{R}') $,
$\Delta ^2u=\Delta( \Delta u) $,
 $\Delta u=\underset{k=1}{\overset{n}{\sum }}\frac{
\partial ^2u}{\partial x_{k}^2}$, $i=1,2$.

We will study the nonexistence of a global solution of problem
\eqref{e1.1}-\eqref{e1.3}.
By a global solution of problem \eqref{e1.1}-\eqref{e1.3} we
understand a pair of functions $(u_1,u_2)$ such that
$u_1(x,t),u_2(x,t)\in C^{4,1}_{x,t}(Q_R')\cap C^{3,0}_{x,t}
(\overline{B'_R}\times(0;+\infty))\cap C(B'_R \times [0;+\infty)) $ and
satisfy the system \eqref{e1.1} at every point of $Q_{R}'$,
 the initial condition \eqref{e1.2} and conditions \eqref{e1.3}.

The problems of nonexistence of global solutions for
differential equations and inequalities play a key role in theory
and applications. Therefore, they have a constant attention of mathematicians,
and a great number of works were devoted to them
 \cite{1,2,3,4,9,12,13,16,21,22}.
A survey of such results can be found in the monograph \cite{17}.

In the classical paper \cite{7}, Fujita considered the following initial
value problem
\begin{equation} \label{e1.4}
\begin{gathered}
\frac{{\partial u}}{{\partial t}} = \Delta u + u^q,  \quad
 (x,t) \in \mathbb{R}^n  \times (0,+\infty),\\
 u |_{t = 0}  = u_0 (x),\quad x\in \mathbb{R}^{n},\\
\end{gathered}
\end{equation}
and proved that positive global solutions of problem \eqref{e1.4} do not
exist for   $ 1 <q <q ^{*} = 1+ \frac {2}{n} $.
If $ q> q^{*} $, then there are positive global solutions for small $ u_0 (x) $.
The case $ q = q^* $ was investigated in \cite{10,11} and it was proved that
in this case there are no positive global solutions. Pinsky \cite{19} showed the
existence and nonexistence of global solutions in $\mathbb{R}^n  \times (0,+\infty)$
to the equation $u_{t}- \Delta u =a(x) u^{q}$,
where $q > 1$ and $a(x)$ behaves like $ | x| ^{\sigma}$  with
$\sigma >-2$ for large $| x|$. The results of Fujita's work \cite{7} aroused
great interest in the problem of the nonexistence of global solutions,
and they were expanded in several directions. For example,
various bounded and unbounded domains were considered instead of
$ R ^ n $, as well as more general operators than the Laplace operator
including different type nonlinear operators  were considered
(for more comprehensive treatment of such problems, see
\cite{14,17,20} and references there in).

Another may of extending of Fujita's result is to investigate a system
of Fujita-type reaction-diffusion equations, and this is what we do here.
For example, many authors have investigated the existence and nonexistence
of global and local solutions to the  initial value problem
\begin{equation} \label{e1.5}
\begin{gathered}
\frac{\partial u}{\partial t}=\alpha _1\Delta u+t^{k_1}| x| ^{\sigma _1}v^{q_1},\quad
  u|_{t=0}=u_0( x)\geq0 \\
\frac{\partial v}{\partial t}=\alpha _2\Delta v+t^{k_2}| x| ^{\sigma _2}u^{q_2},
\quad
  v|_{t=0}=v_0( x)\geq0\,.
\end{gathered}
\end{equation}
Escobedo and Herrero \cite{5} considered  problem \eqref{e1.5} on
$ \mathbb{R}^n  \times (0,+\infty)$ with $\alpha_ i=1$,$k_i=0, \sigma_i=0$,
$q_i>0,q_1q_2>1, i=1,2 $ and proved that if
 $\max(\frac{q_1+1}{q_1q_2-1},\frac{q_2+1}{q_1q_2-1})\geq \frac{n}{2}$,
then for any nontrivial initial functions there are no nonnegative global solutions.
Fila, Levine and Uda \cite{6} considered  problem\eqref{e1.5} on
$ \mathbb{R}^n  \times (0,+\infty)$  with $0 \leq \alpha_ {1}\leq 1$,
$\alpha_ 2 = 1$, $k_i=0$, $\sigma_i=0$, $q_i\geq 0$, $q_1q_2>1$, $i=1,2 $
and studied the  existence of nonnegative global and non-global  solutions.
In the case $\alpha_ i = 1$, $k_i=0$, $i=1,2 $,  Mochizuki  and Huang \cite{18}
proved the existence and nonexistence theorems for global solutions and
studied asymptotic behavior of the global solution of problem \eqref{e1.5}
on $ \mathbb{R}^n  \times (0,+\infty)$.  Caristi \cite{8} considered
 problem \eqref{e1.5} for $k_i,\sigma_i\in \mathbb{R} $, $q_1,q_2>1 $ on
$ \mathbb{R}^n\times (0,+\infty)$, and nonexistence of global solution is studied.
Levine \cite{15}  studied nonnegative solutions of the initial boundary value
 problem   for the system of equations in \eqref{e1.5} for $\alpha_ i = 1$, $k_i=0$,
$\sigma_i=0$, $i=1,2$ in domain $D\times(0,+\infty)$, where $D$ is a cone or
 the exterior of a bounded domain.
In the present paper we consider a system of semilinear parabolic equations
with  biharmonic operator and singular potential in the exterior domain
$ Q'_{R} $. Using the technique of test functions worked out by Mitidieri
and Pohozaev in \cite{16},\cite{17}, we find a sufficient condition
for nonexistence of global nontrivial solution.


\section{Main result and its proof}

The avoid complications, we introduce the following denotation:
\begin{gather*}
D_i=\sqrt{( n-2)^2+C_i},\quad
 \lambda_i^{\pm }=\sqrt{\big( \frac{n-2}{2}\big) ^2+1\pm D_i}, \\
\mu _i=\frac{1}{2}\Big( 1+\frac{D_i-\lambda_i^{+}}{\lambda _i^{-}}
\Big) ,\quad \overline{\mu }_i=\frac{1}{2}\Big( 1-\frac{D_i-\lambda
_i^{+}}{\lambda_i^{-}}\Big) , \\
\alpha _1= \frac{\lambda _1^{-}+\sigma _1+\frac{n+4}{2}}{\lambda _2^{-}+\frac{
n+4}{2}},\quad
 \alpha _2= \frac{\lambda _2^{-}+\sigma _2+\frac{n+4}{2}}{\lambda _1^{-}
 +\frac{n+4}{2}}, \\
\beta _1= \frac{\lambda _1^{-}+\sigma _1+4+\frac{n+4}{2}}{\lambda _2^{-}+\frac{
n-4}{2}},\quad \beta _2= \frac{\lambda _2^{-}+\sigma _2+4+\frac{n+4}{
2}}{\lambda _1^{-}+\frac{n-4}{2}}, \\
\theta _1=\frac{\sigma _1+4+q_1( \sigma _2+4) }{q_1q_2-1}-\lambda_1^{-}
-\frac{n+4}{2}, \\
\theta _2=\frac{\sigma _2+4+q_2( \sigma _1+4) }{
q_1q_2-1}-\lambda_2^{-}-\frac{n+4}{2},\quad i=1,2.
\end{gather*}

Let us consider the functions
\begin{equation*}
\xi _i( x) =\mu _i| x| ^{-\frac{n-4}{2}
+\lambda_i^{-}}+\overline{\mu }_i| x| ^{-\frac{n-4}{2}
-\lambda_i^{-}}-| x| ^{-\frac{n-4}{2}-\lambda_i^{+}},\quad  i=1,2.
\end{equation*}

It is easy to verify that $\xi_i(x)$ are the solution of the equation
\begin{equation} \label{e2.1}
\Delta ^2u-\frac{C_i}{| x| ^4}u=0
\end{equation}
in $R^{n}\backslash \{ 0\} $ and for  $|x| =1$,
\begin{equation} \label{e2.2}
\xi _i=0,\quad \frac{\partial \xi _i}{\partial r}=D_i\geq 0,\quad
\Delta \xi _i=0,\quad  \frac{\partial ( \Delta \xi _i) }{\partial r}\leq 0.
\end{equation}
The main result of this paper reads as follows.

\begin{theorem} \label{thm2.1}
Assume that $n>4$, $\beta_i>1$, $0\leq C_i< ( \frac{n( n-4) }{4}) ^2$
and $1<q_i\leq \beta_i$, $\max(\theta_1,\theta_2)\geq 0 $,
$(q_1,q_2)\neq (\alpha_1, \beta_2)$ in case $\alpha_1>1$,
$(q_1,q_2)\neq (\beta_1, \alpha_2)$ in case $\alpha_2>1$, $ i=1,2$.
Then there is no nontrivial global solution of \eqref{e1.1}-\eqref{e1.3}.
\end{theorem}

\begin{proof}
For simplicity  we take $R=1$.
 Assume that $(u_1(x,t),u_2(x,t))$ is a nontrivial solution of
 \eqref{e1.1}-\eqref{e1.3}. Let us consider the following two functions:
\begin{gather*}
\varphi ( x) =\begin{cases}
1, &\text{for }1\leq | x| \leq \rho , \\
( 2-\frac{| x| }{\rho }) ^{\kappa },& \text{for }\rho \leq | x| \leq 2\rho  \\
0, & \text{for }| x| \geq 2\rho,
\end{cases} \\
T_{\rho }( t) =\begin{cases}
1, &\text{for }0\leq t\leq \rho ^4, \\
(2-\rho ^{-4}t) ^{\gamma }, &\text{for } \rho ^4\leq t\leq 2\rho ^4 \\
0, &\text{for }t\geq 2\rho ^4,
\end{cases}
\end{gather*}
where $\kappa,  \gamma$ are large positive,  and  $\kappa$ is such number that
 for $|x|=2\rho$,
\begin{equation} \label{e2.3}
\varphi =\frac{\partial\varphi}{\partial r} =\frac{\partial^2 \varphi}{\partial r^2}
=\frac{\partial^3 \varphi}{\partial r^3}=0.
\end{equation}

We multiply the first equation by  $\psi_1(x,t)=T_\rho(t)\xi_1(x)\varphi(x) $,
the second by  $\psi_2(x,t)=T_\rho(t)\xi_2(x)\varphi(x) $ and integrate over
$Q'_1$. After integration by parts, we obtain the following relations

\begin{equation} \label{e2.4}
\begin{aligned}
&\iint_{Q_1'} | x| ^{\sigma_i}| u_j| ^{q_i}T_{\rho }( t) \xi_i( x)
 \varphi ( x) \,dx\,dt\\
&=-\iint_{Q_1'} u_i\xi _i\varphi \frac{dT_{\rho} }{dt}\,dx\,dt
+\iint_{Q_1'} u_iT_{\rho }\Delta ^2( \xi _i\varphi ) \,dx\,dt \\
&\quad -\iint_{Q_1'} \frac{C_i}{| x| ^4} u_iT_{\rho }\xi _i\varphi \,dx\,dt
 -\int_{B_1'} u_{i0}( x) \xi _i( x) \varphi ( x) dx \\
&\quad + \int _0^{\infty}T_{\rho }( t) dt
\Big[ \int_{\partial B_{1,2}\rho }  \frac{\partial ( \Delta u_i) }{
\partial \nu}\xi _i\varphi ds
-\int_{\partial B_{1,2}\rho } \Delta u_i
 \frac{\partial ( \xi _i\varphi ) }{\partial \nu}ds\\
&\quad +\int_{\partial B_{1,2}\rho }
 \frac{\partial {u _i}}{\partial \nu}\Delta ( \xi _i\varphi ) ds
 -\int_{\partial B_{1,2}\rho} u_i\frac{\partial }{\partial \nu}
 \Delta(\xi _i\varphi) ds\Big],
\end{aligned}
\end{equation}
where $\nu$  is a unit vector of external normal to  $\partial B_{1,2}\rho$,
$i,j=1,2$, $i\neq j$.

In order not to be repeated, in what follows, we will take into account that
 $i,j=1,2$, $i\neq j$ and in all expressions will write the same constant $C$,
 but in fact, in each expression  $C$ indicates different constants.

Using \eqref{e2.2}, \eqref{e2.3}, we estimate the integrals in square brackets
 in \eqref{e2.4}.
\begin{gather*}
\int_{\partial B_{1,2}\rho} \frac{\partial ( \Delta u_i) }
{\partial \nu}\xi _i\varphi ds=0, \\
\begin{aligned}
-\int_{\partial B_{1,2}\rho} \Delta u_i\frac{\partial (
\xi _i\varphi ) }{\partial \nu}ds
&=-\int_{| x| =1} \Delta u_i\frac{\partial ( \xi _i\varphi ) }{\partial \nu}ds \\
&=\underset{| x| =1}{\int }\Delta u_i( \frac{\partial \xi _i }{\partial r}\varphi
 +\xi _i\frac{\partial \varphi }{\partial r}) ds \\
&=\int_{| x| =1} \Delta u_i\frac{\partial \xi _i}{\partial r}ds\leq 0,
\end{aligned}\\
\begin{aligned}
\int_{\partial B_{1,2}\rho} \frac{\partial u_i}{\partial \nu}
\Delta ( \xi _i\varphi ) ds
&=\underset{\partial B_{1,2}\rho }{\int }\frac{\partial u_i}
{\partial \nu}( \Delta \xi _i\varphi+2( \nabla \xi _i,\nabla \varphi )
+\xi _i\Delta \varphi ) ds \\
&= -\int_{| x| =1} \frac{\partial u_i}{\partial r}\Delta \xi _ids=0,
\end{aligned} \\
\begin{aligned}
 -\int_{\partial B_{1,2}\rho} u_i\frac{\partial }{\partial \nu}
( \Delta ( \xi _i\varphi ) ) ds 
&=-\int_{| x| =1} u_i\frac{\partial }{\partial \nu}(\Delta \xi _i\varphi ) ds \\
&=\int_{| x| =1} u_i\frac{\partial (\Delta \xi _i) }{\partial r}ds\leq 0.
\end{aligned}
\end{gather*}
Since
\[
\int_{B_1'} u_{i0}( x) \xi _i(x) \varphi ( x) dx\geq 0,\quad\text{and}\quad
\int_0^\infty T_{\rho }( t) dt\geq 0,
\]
 taking into account that $\xi_i$ is the solution of n \eqref{e2.1} and using
the above  estimates, from \eqref{e2.4} we obtain
\begin{equation} \label{e2.5}
\begin{aligned}
&\iint_{Q_1'} | x| ^{\sigma_i}| u_j| ^{q_i}T_{\rho }( t) \xi
_i( x) \varphi ( x) \,dx\,dt \\
&\leq
-\iint_{Q_1'} u_i\xi _i\varphi \frac{dT\rho }{dt}\,dx\,dt
+\iint_{Q_1'} u_iT_{\rho }\Delta ^2(\xi _i\varphi ) \,dx\,dt
-\iint_{Q_1'} \frac{C_i}{| x| ^4}u_iT_{\rho }\xi _i\varphi \,dx\,dt \\
&=-\iint_{Q_1'} u_i\xi _i\varphi \frac{dT\rho }{dt}\,dx\,dt
 +\iint_{Q_1'} u_iT_{\rho }\varphi (\Delta ^2\xi _i
 -\frac{C_i}{| x| ^4}\xi_i) \,dx\,dt \\
&\quad +\iint_{Q_1'} u_iT_{\rho }\Big[ 4(
\nabla ( \Delta \xi _i) ,\nabla \varphi ) +4( \nabla
\xi _i,\nabla ( \Delta \varphi ) )  +2\Delta \xi
_i\Delta \varphi  \\
&\quad  +4\sum_{k,m=1}^{n}\frac{\partial ^2\xi _i}{\partial x_{k}\partial
x_{m}}\frac{\partial ^2\varphi }{\partial x_{k}\partial x_{m}}\Big]
\,dx\,dt \\
&\leq -\int_{\rho ^4}^{2\rho ^4} \int_{B_1'}
u_i\xi _i\varphi \frac{dT_{\rho }}{dt}\,dx\,dt
+\int_0 ^{2\rho ^4} \int_{B_{\rho ,2\rho }}
u_i T_{\rho }H_i( \xi _i,\varphi ) \,dx\,dt,
\end{aligned}
\end{equation}
where $H_i( \xi _i,\varphi )$ denotes the expression in the square brackets, i.e.
\begin{equation} \label{e2.6}
\begin{aligned}
H_i( \xi _i,\varphi )
&=4( \nabla ( \Delta \xi _i),\nabla \varphi ) +4( \nabla \xi _i,\nabla ( \Delta
\varphi ) )  +2\Delta \xi _i\Delta \varphi  \\
&\quad +4\sum_{k,m=1}^{n}\frac{\partial ^2\xi _i}{\partial x_{k}\partial x_{m}}
\frac{\partial ^2\varphi }{\partial x_{k}\partial x_{m}}.
\end{aligned}
\end{equation}
Using the Holder's inequality, we estimate the right-hand side of \eqref{e2.5}.
We can write:
\begin{align*}
&\iint_{Q_1'} | x| ^{\sigma_i}| u_j| ^{q_i}T_{\rho }\xi _i\varphi \,dx\,dt \\
&\leq
\Big( \int_{\rho ^4}^{2\rho ^4} \int_{B_1'} | x| ^{\sigma _j}|
u_i| ^{q_j}T_{\rho }\xi _j\varphi \,dx\,dt\Big) ^{1/q_j} \\
&\quad \times \Big( \int_{\rho ^4}^{2\rho ^4} \int_{B_1'}
\frac{| \frac{dT_{\rho }}{dt}|^{q_j'}\xi _i^{q_j'}\varphi }{T_{\rho
}^{q_j'-1}| x| ^{\sigma _j(
q_j'-1) }\xi _j^{q_j'-1}}\,dx\,dt\Big) ^{1/q_j'} \\
&\quad +\Big( \int_0^{2\rho^4 } \int_{B_{\rho,2\rho }}
 | x| ^{\sigma _j}| u_i| ^{q_j}T_{\rho }\xi _j\varphi \,dx\,dt\Big) ^{1/q_j}\\
&\quad\times \Big( \int_0^{2\rho ^4} \int_{B_{\rho,2\rho }}
\frac{| H_i( \xi _i,\varphi )| ^{q_j'}T_{\rho }}{| x| ^{\sigma_j( q_j'-1) }
\xi _j^{q_j'-1}\varphi^{q_j'-1}}\,dx\,dt\Big) ^{1/q_j'},
\end{align*}
where $\frac{1}{q_j}+\frac{1}{q_j'}=1$.

Let us denote the second integral in the first addend above by $I_i$,  and
the second integral in the second addend by $J_i$.
If we write separately, then from \eqref{e2.6} we obtain the following:
\begin{gather}
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma
_1}| u_2| ^{q_1}T_{\rho }\xi _1\varphi \,dx\,dt \\
&\leq \Big( \int_{Q_1'}\int | x|
^{\sigma _2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi
\,dx\,dt\Big) ^{1/q_2}
\big[ I_1^{1/q_2'}+J_1^{1/q_2'}\big] ,
\end{aligned} \label{e2.7}\\
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma_2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi \,dx\,dt\\
&\leq
\Big( \int_{Q_1'}\int | x|^{\sigma _1}| u_2| ^{q_1}Ò_{\rho }\xi _1\varphi
\,dx\,dt\Big) ^{1/q_1}
\big[ I_2^{1/q_1'}+J_2^{\frac{1}{
q_1'}}\big] . \label{e2.8}
\end{aligned}
\end{gather}
Using \eqref{e2.6}, from these inequalities we obtain
\begin{equation} \label{e2.9}
\begin{aligned}
&\iint_{Q_1'} | x| ^{\sigma_1}| u_2| ^{q_1}T_{\rho }\xi _1\varphi \,dx\,dt \\
&\leq \Big[ \Big( \int_{\rho ^4} ^{2\rho ^4} \int_{B_1'}
| x| ^{\sigma _1}| u_2| ^{q_1}T_{\rho }\xi _1\varphi  \,dx\,dt\Big)
^{1/q_1} I_2^{1/q_1'} \\
&\quad+ \Big( \int_0^{2\rho^4} \int_{B_{\rho ,2\rho }} | x| ^{\sigma _1}
|u_2| ^{q_1}T_{\rho }\xi _1\varphi \,dx\,dt\Big) ^{1/q_1}
 J_2^{1/q_1'}\Big] ^{1/q_2}
\big[I_1^{1/q_2'}+J_1^{1/q_2'}\big],
\end{aligned}
\end{equation}
\begin{equation} \label{e2.10}
\begin{aligned}
&\iint_{Q_1'} | x| ^{\sigma_2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi  \,dx\,dt \\
&\leq \Big[ \Big( \int_{\rho ^4}^{2\rho ^4}\int_{B_1'} | x| ^{\sigma _2}|
u_1| ^{q_2}T_{\rho }\xi _2\varphi  \,dx\,dt\Big) ^{1/q_2}
I_1^{1/q_2'} \\
&\quad +\Big(\int_0^{2\rho^4 }
\int_{B_{\rho ,2\rho }} | x| ^{\sigma _2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi
 \,dx\,dt) ^{1/q_2} J_1^{1q_2'}\Big] ^{1/q_1}
\big[ I_2^{1/q_1'}+J_2^{1/q_1'}\big].
\end{aligned}
\end{equation}
Substituting \eqref{e2.8} in \eqref{e2.7} and \eqref{e2.7}
in \eqref{e2.8}, we obtain
\begin{gather*}
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma
_1}| u_2| ^{q_1}T_{\rho }\xi _1\varphi \,dx\,dt \\
&\leq\Big( \int_{Q_1'}\int | x|
^{\sigma _1}| u_2| ^{q_1}T_{\rho }\xi _1\varphi
\,dx\,dt\Big) ^{\frac{1}{q_1q_2}}
 \big[ I_1^{1/q_2'}+J_1^{1/q_2'}\big]
\big[ I_2^{1/q_1'}+J_2^{1/q_1'}\big] ^{1/q_2},
\end{aligned} \\
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma_2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi \,dx\,dt\\
&\leq\Big( \int_{Q_1'}\int | x|
^{\sigma _2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi\,dx\,dt\Big) ^{\frac{1}{q_1q_2}}
 \big[ I_2^{1/q_1'}+J_2^{1/q_1'}\big]
\big[ I_1^{1/q_2'}+J_1^{1/q_2'}\big] ^{1/q_1}.
\end{aligned}
\end{gather*}
Hence
\begin{gather} \label{e2.11}
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma_1}| u_2| ^{q_1}T_{\rho }\xi _1\varphi \,dx\,dt \\
&\leq \big[ I_1^{1/q_2'}+J_1^{1/q_2'}
\big] ^{\frac{q_1q_2}{q_1q_2-1}}
\big[ I_2^{\frac{1}{q_1'}}+J_2^{1/q_1'}
\big] ^{\frac{q_1}{q_1q_2-1}},
\end{aligned} \\
\label{e2.12}
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma_2}| u_1| ^{q_2}T_{\rho }\xi _2\varphi \,dx\,dt\\
&\leq \big[ I_2^{1/q_1'}+J_2^{1/q_1'}\big] ^{\frac{q_1q_2}{q_1q_2-1}}
\big[ I_1^{1/q_2'}+J_1^{1/q_2'}
\big] ^{\frac{q_2}{q_1q_2-1}}.
\end{aligned}
\end{gather}
Making the substitutions
\begin{equation} \label{e2.13}
\begin{gathered}
t=\rho ^4\tau ,\quad r=\rho s,\quad x=\rho y,\quad
\widetilde{T}( \tau) =T_{\rho }( \rho ^4\tau ) ,\\
\widetilde{\xi }_i( y) =\xi _i( \rho y) ,\quad
\widetilde{\varphi }( y) =\varphi ( \rho y),
\end{gathered}
\end{equation}
we estimate the right-hand sides of \eqref{e2.11} and \eqref{e2.12}.

 First, we estimate the integrals $I_i$, $i=1,2$.
\begin{equation} \label{e2.14}
\begin{aligned}
I_i
&=\int_{\rho ^4}^{2\rho ^4}\int_{B_1'} \frac{| \frac{dT\rho }{dt}|
^{q_j'}\xi _i^{q_j'}\varphi }{T_{\rho }^{q_j'-1}| x| ^{\sigma _j( q_j'-1)
}\xi _j^{q_j'-1}}\,dx\,dt \\
& \leq \int_{\rho ^4}^{2\rho ^4}\frac{| \frac{
dT\rho }{dt}| ^{q_j'}}{T_{\rho }^{q_j'-1}}dt
\int_{B_1'}\frac{\xi _i^{q_j'}}{
| x| ^{\sigma _j( q_j'-1) }\xi_j^{q_j'-1}}dx \\
&\leq C \rho ^{-4( q_j'-1) }
\int_1^2 \frac{| \frac{d\widetilde{T}}{d\tau }|
^{q_j'}}{\widetilde{T}^{q_j'-1}}d\tau
\int_{B_1'} \frac{\xi _i^{q_j'}}{|x| ^{\sigma _j( q_j'-1) }\xi _j^{q_j'-1}}dx\\
&\leq C \rho^{-4q_j'/q_j} \widetilde{I}_j ( \widetilde{T})\int_{B_1'}
\frac{ \xi_i^{q_j'}}{| x| ^{\sigma _j(q_j'-1) }\xi _j^{q_j'-1}}dx,
\end{aligned}
\end{equation}
where
\[
\widetilde{I}_j( \widetilde{T}) =\int_1^2
\frac{| \frac{d\widetilde{T}}{d\tau }|^{q_j'}}{\widetilde{T}^{q_j'-1}}d\tau .
\]
Since for $|x|=1$ in the last integral \eqref{e2.14} there is a singularity,
then we estimate it separately.
\begin{equation} \label{e2.15}
\begin{aligned}
&\int_{B_1'}\frac{\xi _i^{q_j'}}{| x| ^{\sigma _j( q_j'-1) }\xi_j^{q_j'-1}}dx \\
&=\int_1^{2\rho }\frac{( \mu _ir^{-\frac{n-4}{2}
+\lambda_i^{-}}+\overline{\mu }_ir^{-\frac{n-4}{2}-\lambda_i^{-}}-r^{-
\frac{n-4}{2}-\lambda_i^{+}}) ^{q_j'}r^{n-1}}{r^{\sigma
_j( q_j'-1) }( \mu _jr^{-\frac{n-4}{2}+\lambda
_j^{-}}+\overline{\mu }_jr^{-\frac{n-4}{2}-\lambda _j^{-}}-r^{-\frac{n-4
}{2}-\lambda _j^{+}}) ^{q_j'-1}}dr\\
&=\int_1^{2\rho }r^{ \lambda_i^{-}q_j-\lambda_j^{-}(q_j'-1)-\sigma _j(q_j'-1)
-\frac{n-4}{2}+n-1} \\
&\quad\times \frac{( \mu _i+\overline{\mu }_ir^{-2\lambda
_i^{-}}-r^{-\lambda_i^{+}-\lambda_i^{-}}) ^{q_j'}}{
( \mu _j+\overline{\mu }_jr^{-2\lambda _j^{-}}-r^{-\lambda
_j^{+}-\lambda _j^{-}}) ^{q_j'-1}}dr.
\end{aligned}
\end{equation}
Using the L'Hopital's rule, we obtain
\begin{align*}
&\lim_{r\to 1}\frac{\mu _i+\overline{\mu }
_ir^{-2\lambda_i^{-}}-r^{-\lambda_i^{+}-\lambda_i^{-}}}{\mu _j+
\overline{\mu }_jr^{-2\lambda _j^{-}}-r^{-\lambda _j^{+}-\lambda _j^{-}}} \\
& =\lim_{r\to 1}\frac{-2\lambda_i^{-}\overline{\mu }
_ir^{-2\lambda_i^{-}-1}+( \lambda_i^{+}+\lambda_i^{-})
r^{-\lambda_i^{+}-\lambda_i^{-}-1}}{-2\lambda _j^{-}\overline{\mu }
_jr^{-2\lambda _j^{-}-1}+( \lambda _j^{+}+\lambda _j^{-})
r^{-\lambda_j^{+}-\lambda _j^{-}-1}} \\
&=\frac{-\lambda_i^{-}+D_i-\lambda_i^{+}+\lambda_i^{+}+\lambda_i^{-}
}{-\lambda _j^{-}+D_j-\lambda _j^{+}+\lambda _j^{+}+\lambda _j^{-}}=
\frac{D_i}{D_j}.
\end{align*}
Then there exists $r_0>1$ such that for $r<r_0$,
\begin{equation*}
\frac{D_i}{D_j}-1
<\frac{\mu _i+\mu _i^{-}r^{-2\lambda _i^{-}}-r^{-\lambda_i^{+}
 -\lambda_i^{-}}}{\mu _j+\mu_j^{-}r^{-2\lambda _j^{-}}-r^{-\lambda_j^{+}
-\lambda _j^{-}}}<\frac{D_i}{D_j}+1.
\end{equation*}
So, for $r<r_0$,
\[
\mu _i+\mu _i^{-}r^{-2\lambda_i^{-}}-r^{-\lambda_i^{+}-\lambda
_i^{-}}<\big( \frac{D_i}{D_j}+1\big) \big( \mu _j+\mu
_j^{-}r^{-2\lambda _j^{-}}-r^{-\lambda _i^{+}-\lambda _j^{-}}\big).
\]

On the other hand, for $r\geq r_0$,
\[
\frac{\mu _j+\overline{\mu }_jr^{-\lambda _j^{-}}-r^{-\lambda
_j^{+}-\lambda _j^{-}}}{\mu _j+\overline{\mu }_jr^{-\lambda
_j^{-}}-r^{-\lambda_j^{+}-\lambda _j^{-}}}\leq C( r_0).
\]
Taking into account the above two relations, from \eqref{e2.15} we obtain
\begin{equation} \label{e2.16}
\begin{aligned}
\int_{B_1'}\frac{\xi _i^{q_j'}}{|x| ^{\sigma _j( q_j'-1) }\xi_j^{q_j'-1}}dx
&\leq C\int_1^{2\rho }r^{\lambda _i^{-}q_j'-\lambda _j^{-}( q_j'-1)
-\sigma _j( q_j'-1) +\frac{n+4}{2}-1}dr \\
&=C\int_1^{2\rho }r^{\frac{q_j'}{q_j}
( \lambda _i^{-}q_j-\lambda _j^{-}-\sigma _j+\frac{n+4}{2}
( q_j-1) ) -1}dr \\
&\leq
 C\begin{cases}
\rho ^{\frac{q_j'}{q_j}\eta _i},&\text{for }\eta _i>0 \\
\ln (2\rho) ,& \text{for }\eta _i=0 \\
1, & \text{for}\eta _i<0,
\end{cases}
\end{aligned}
\end{equation}
where
\[
\eta _i=\lambda _i^{-}q_j-\lambda _j^{-}-\sigma _j+\frac{n+4}{2}(q_j-1).
\]

Using \eqref{e2.16}, from \eqref{e2.14} we obtain
\begin{equation} \label{e2.17}
I_i\leq C\begin{cases}
\widetilde{I}_j( \widetilde{T}) \rho ^{\frac{q_j'}{q_j}(\eta _i-4 ) },
&\text{for }\eta _i>0 \\
\ln (2\rho)\rho ^{-4q_j'/q_j}, & \text{for }\eta _i=0 \\
\rho ^{-4q_j'/q_j}, & \text{for }\eta _i<0.
\end{cases}
\end{equation}

To estimate $J_i$, $i=1,2, $ we estimate each addend  of
$H_i(\xi_i, \varphi)$ separately.
\begin{gather*}
\begin{aligned}
| ( \nabla ( \Delta \xi _i) ,\nabla \varphi) |
&\leq \big| \frac{\partial ^{3}\xi _i}{\partial r^{3}}+\frac{n-1}{r}
\frac{\partial ^2\xi _i}{\partial r^2}-\frac{n-1}{r^2}\frac{\partial
\xi _i}{\partial r}\big| \big| \frac{\partial \varphi }{
\partial r}\big| \\
&\leq Cr^{-\frac{n-4}{2}+\lambda_i^{-}-3}\big|
\frac{\partial \varphi }{\partial r}\big| ,
\end{aligned}
\\
\begin{aligned}
| \Delta \xi_i\Delta \varphi |
&\leq \big| \frac{\partial ^2\xi _i}{\partial r^2}+\frac{n-1}{r}
\frac{\partial \xi _i}{\partial r}\big| 
\big| \frac{\partial ^2\varphi }{\partial r^2
}+\frac{n-1}{r}\frac{\partial \varphi }{\partial r}\big| \\
&\leq Cr^{-\frac{n-4}{2}+\lambda_i^{-}-2}\big| \frac{\partial
^2\varphi }{\partial r^2}+\frac{n-1}{r}\frac{\partial \varphi }{\partial
r}\big| ,
\end{aligned} \\
| ( \nabla \xi_i \nabla ( \Delta \varphi ) )
| \leq Cr^{-\frac{n-4}{2}+\lambda_i^{-}-1} 
\big| \frac{\partial^{3}\varphi}{\partial r^{3}}+\frac{n-1}{r}
\frac{\partial ^2\varphi }{\partial r^2}+\frac{n-1}{r^2}
\frac{\partial \varphi }{\partial r}\big| , \\
\begin{aligned}
&\big| \sum_{i,j=1}^n \frac{\partial ^2\xi _i}{
\partial x_i\partial x_j}\frac{\partial ^2\varphi }{\partial x_i\partial x_j}\big|\\
&\leq \big| \sum_{i,j=1}^n \frac{\partial }{\partial x_j}( \frac{\partial \xi _i}{
\partial r}\frac{x_i}{r}) \frac{\partial }{\partial x_j}(
\frac{\partial \varphi }{\partial r}\frac{x_i}{r}) \big| \\
&\leq \sum_{i,j=1}^n | \frac{\partial ^2\xi
_i}{\partial r^2}\frac{x_ix_j}{r^2}+\frac{\partial \xi _i}{
\partial r}( \frac{\delta _{ij}}{r}-\frac{x_ix_j}{r^{3}})|\,
| \frac{\partial ^2\varphi }{\partial r^2}\frac{
x_ix_j}{r^2}+\frac{\partial \varphi }{\partial r}\big( \frac{\delta
_{ij}}{r}-\frac{x_ix_j}{r^{3}}\big) | \\
& \leq C\Big( | \frac{\partial ^2\xi _i}{\partial r^2}
| +\frac{1}{r}| \frac{\partial \xi _i}{\partial r}
| \Big) \Big( | \frac{\partial ^2\varphi }{\partial
r^2}| +\frac{1}{r}| \frac{\partial \varphi }{\partial r}
| \Big) \\
&\leq Cr^{-\frac{n-4}{2}+\lambda_i^{-}-2}\Big( | \frac{\partial
^2\varphi }{\partial r^2}| +\frac{1}{r}| \frac{
\partial \varphi }{\partial r}| \Big) .
\end{aligned}
\end{gather*}

Now, taking into account these relations and   \eqref{e2.13}, we estimate
$J_i$, $ i=1,2$:
\begin{equation} \label{e2.18}
\begin{aligned}
J_i&=\int_0^{2\rho ^4} \int_{B_{\rho ,2\rho }} \frac{| H_i( \xi _i,\varphi ) |
^{q_j'}T_{\rho}}{| x| ^{\sigma _j(
q_j'-1) }\xi _j^{q_j'-1}\varphi
^{q_j'-1}}\,dx\,dt\\
&\leq \int_0^{2\rho ^4} T_{\rho }dt\int_{B_{\rho,2\rho }}
 \frac{| H_i( \xi _i,\varphi ) | ^{q_j'}}{| x| ^{\sigma _j(
q_j'-1) }\xi _j^{q_j'-1}\varphi
^{q_j'-1}}dx \\
&\leq C\rho ^{( -\frac{n-4}{2}+\lambda _i^{-}-4) q_j'-\sigma _j( q_j'-1)
-( -\frac{n-4}{2}+\lambda_j^{-}) ( q_j'-1) +n+4}\\
&\quad \times \int_1^2 \frac{( | \frac{d^{3}\widetilde{\varphi }}{ds^{3}}|
+| \frac{d^2\widetilde{\varphi }}{ds^2}| +| \frac{d\widetilde{\varphi }}{ds}
| ) ^{q_j'}}{s^{\sigma_j(q_j-1)}\widetilde{\varphi }^{q_j'-1}}ds\\
&\leq C\rho ^{-4( q_j'-1) +\lambda_i^{-}q_j'-\lambda _j^{-}( q_j'-1)
-\sigma _j( q_j'-1) +\frac{n+4}{2}}\widetilde{J}
_j( \widetilde{\varphi }) \\
&=C\rho ^{\frac{q_j'}{q_j}(\eta _i-4 ) }\widetilde{
J}_j( \widetilde{\varphi }) ,
\end{aligned}
\end{equation}
where  $\widetilde{J}_j( \widetilde{\varphi })$ denotes the last integral.


Using the  estimates \eqref{e2.17},\eqref{e2.18}, we estimate
the right-hand sides of \eqref{e2.11}, \eqref{e2.12}.
 It is known that for large $\kappa$ and $\gamma$, the integrals
$\widetilde {I}_j(\widetilde{T})$, $ \widetilde{J}_j( \widetilde{\varphi })$
are bounded \cite{17}.

Depending on the sign of $\eta_i$, $i=1,2 $, we consider various variants.
\smallskip

\noindent I.  $\alpha_1>1$, $ \alpha_2>1$.
This is equivalent to
\begin{equation} \label{e2.19}
\lambda _1^{-}-\lambda _2^{-} +\sigma _1>0\quad\text{and}\quad
\lambda _2^{-}-\lambda _1^{-} +\sigma_2>0.
\end{equation}
Subject to relation \eqref{e2.19}, we consider the following cases.

(a) $\eta _1\leq 0$, $\eta _2\leq 0$ or $q_1\leq \alpha_1$, $q_2\leq \alpha_2$.
Then, taking into account \eqref{e2.17}, \eqref{e2.18}, from \eqref{e2.11},
\eqref{e2.12} we obtain
\begin{align*}
&\int_{Q_1'}\int | x| ^{\sigma_i}| u_j| ^{q_i}T_{\rho }\xi _i\varphi \,dx\,dt \\
&\leq C\rho ^{-\frac{4}{q_1q_2-1}( q_i+1) }\big[ f_i^{
\frac{1}{q_j'}}\widetilde{I}_j^{1/q_j'}+\widetilde{J}_j^{1/q_j'}
 \big] ^{\frac{q_1q_2}{q_1q_2-1}}\big[ f_j^{\frac{1}{
q_i'}}\widetilde{I}_i^{\frac{1}{q_i'}}+
\widetilde{J}_i^{\frac{1}{q_i'}}\big] ^{\frac{q_i}{q_1q_2-1}},
\end{align*}
where
\[
f_i( \rho ) =
\begin{cases}
1,& \text{if }\eta _i<0 \\
\ln ( 2\rho ) ,& \text{if }\eta _i=0.
\end{cases}
\]
When we pass to limit as $\rho\to +\infty$, we obtain
\begin{equation*}
\int_{Q_1'}\int | x| ^{\sigma_i}| u_j| ^{q_i}T_{\rho }\xi _i\varphi \,dx\,dt\leq
0.
\end{equation*}
Hence  $u_1\equiv 0$, $u_2\equiv 0$.



(b) Now let $\eta_1>0$, $\eta_2>0$ or $q_1>\alpha_1,q_2>\alpha_2$.
Again using \eqref{e2.17}, \eqref{e2.18}, from \eqref{e2.11},
\eqref{e2.12} we obtain
\begin{equation} \label{e2.20}
\begin{aligned}
&\int_{Q_1'}\int | x| ^{\sigma_i}| u_j| ^{q_i}T_{\rho }\xi _i\varphi  \,dx\,dt\\
&\leq C\rho ^{\frac{1}{q_1q_2-1}( q_i(\eta_i-4)+ \eta_j-4)}
\big[ I_j^{1/q_j'}( \widetilde{T}) +\widetilde{J}_j^{1/q_j'}
( \widetilde{T}) \big] ^{\frac{q_1q_2}{q_1q_2-1}}\\
&\quad\times \big[ \widetilde{I}_i^{\frac{1}{q_i'}}( \widetilde{T}
) +\widetilde{J}_i^{\frac{1}{q_i'}}(
\widetilde{T}) \big] ^{\frac{q_i}{q_1q_2-1}}.
\end{aligned}
\end{equation}
Assume that
\begin{equation} \label{e2.21}
\min \{ q_1( \eta _1-4) +\eta _2-4,\quad
 q_2(\eta _2-4) +\eta _1-4\} <0.
\end{equation}
Since
\begin{align*}
&q_i(\eta_i-4)+\eta_j-4\\
&=\lambda_i^{-}q_iq_j-\lambda_j^{-}q_i-\sigma_jq_i
+\frac{n+4}{2}(q_iq_j-q_i)+\lambda_j^{-}q_i-\lambda_i^{-}-\sigma_i\\
&\quad +\frac{n+4}{2}(q_i-1)-4-4q_i\\
&= -(q_iq_j-1)\theta_i,
\end{align*}
then we can write \eqref{e2.21} as $\max(\theta_1,\theta_2)>0$.

 For definiteness, we assume  $q_1( \eta _1-4) +\eta _2-4<0$.
Then for $i=1$, from \eqref{e2.20} we obtain
\begin{align*}
&\int_{Q_1'}\int | x| ^{\sigma _1}| u_2|
^{q_1}T_{\rho }\xi _1\varphi  \,dx\,dt\\
&\leq C\rho ^{\frac{1}{q_1q_2-1}( q_1( \eta _1-4)
+\eta _2-4) }\big[ \widetilde{I}_2^{\frac{1}{q_2'}
}+\widetilde{J}_2^{1/q_2'}\big] ^{\frac{q_1q_2}{q_1q_2-1}}
 \big[ \widetilde{I}_1^{1/q_1'}+
\widetilde{J}_1^{1/q_1'}\big] ^{\frac{q_1}{q_1q_2-1}}.
\end{align*}
Passing  to the limit as  $\rho \to +\infty$, we obtain
\begin{equation*}
\int_{Q_1'}\int | x| ^{\sigma _1}| u_2|
^{q_1}\xi _1\,dx\,dt\leq 0.
\end{equation*}

Hence $u_2\equiv 0$. Then from the second equation of the system it follows
that  $u_1\equiv 0$. Similarly, for $q_2(\eta_2-4)+\eta_1-4<0$, we obtain
$u_1\equiv 0$, $u_2\equiv 0$. Now let
$\min\{ q_1( \eta _1-4) +\eta _2-4,q_2( \eta_2-4) +\eta _1-4\} =0$ or the same
$\max(\theta_1,\theta_2)=0$. For example, take
$q_1( \eta_1-4) +\eta _2-4=0$.  Then from \eqref{e2.20} it follows
\begin{gather*}
\int_{Q_1'}\int | x| ^{\sigma_1}| u_2| ^{q_1}T_{\rho }\xi _1\,dx\,dt\leq C.
\end{gather*}
From the properties of the integral, it follows that
\begin{gather} \label{e2.22}
\overset{\infty}{\underset{0}{\int }}\underset{B_{\rho ,2\rho }}{\int }|
x| ^{\sigma _1}| u_2| ^{q_1}\xi _1\,dx\,dt\to 0, \\
\label{e2.23}
\int_{\rho ^4}^{2\rho ^4}\underset{B_1'
}{\int }| x| ^{\sigma _1}| u_2|
^{q_1}\xi _1\,dx\,dt\to 0.
\end{gather}
Then from \eqref{e2.9}, by \eqref{e2.22} and \eqref{e2.23} we obtain
\begin{align*}
&\int_{Q_1'}\int | x| ^{\sigma _1}| u_2|^{q_1}T_{\rho }\xi _1\varphi \,dx\,dt \\
&\leq \Big[ \Big( \int_{\rho ^4}^{2\rho ^4}\underset
{B_1'}{\int }| x| ^{\sigma _1}|
u_2| ^{q_1}\xi _1\,dx\,dt\Big) ^{1/q_1}I_2^{1/q_1'} \\
&\quad  +\Big( \int_0^\infty \underset{
B_1'}{\int }| x| ^{\sigma _1}|
u_2| ^{q_1}\xi _1\,dx\,dt\Big) ^{1/q_1}J_2^{1/q_1'}\Big] ^{1/q_2}
\big[ I_1^{1/q_2'}+J_1^{1/q_2'}\big] \\
&\leq C\rho ^{-\frac{1}{q_1q_2}( q_1(\eta_1-4)+ \eta_2-4)}
\Big[\Big( \int_{\rho ^4}^{2\rho ^4}\underset{
B_1'}{\int }| x| ^{\sigma _1}|
u_2| ^{q_1}\xi _1\,dx\,dt) ^{\frac{1}{q_1'}
} \widetilde{I}_1^{1/q_1'}\\
&\quad  +\Big( \int_0^\infty \underset{
B_{\rho ,2\rho }}{\int }| x| ^{\sigma _1}|
u_2| ^{q_1}\xi _1\,dx\,dt\Big) ^{1/q_1'} \widetilde{J}_2^{1/q_1'}
 \Big] ^{1/q_2}\big[ \widetilde{I}_2^{1/q_2'}
 +\widetilde{J}_2^{1/q_2'}\big] \to 0.
\end{align*}
So, again
\begin{equation*}
\iint_{Q_1'} | x| ^{\sigma_1}| u_2| ^{q_1}\xi _1\,dx\,dt\leq 0.
\end{equation*}
Hence $u_2\equiv0$ and respectively $u_1\equiv0$. If
$q_2( \eta _2-4) +\eta _1-4=0$, then in the same way, we obtain
$u_1\equiv0$, $u_2\equiv0$.


(c) Let us consider the case when $\eta_i\leq0$, $\eta_j\geq0$.
At first, let $\eta_1\leq0$, $\eta_2\geq0$.
As in the previous cases, from \eqref{e2.11}, \eqref{e2.12} we obtain
\begin{equation} \label{e2.24}
\begin{aligned}
&\iint_{Q_1'}| x| ^{\sigma _1}| u_2| ^{q_1}\xi _1\,dx\,dt \\
&\leq C\rho ^{\frac{1}{q_1q_2-1}( -4( q_1+1) +\eta_2) }
\big[ f_1^{1/q_2'}
\widetilde{I}_2^{1/q_2'}+\widetilde{J}_2^{1/q_2'}
\big] ^{\frac{q_1q_2}{q_1q_2-1}}\big[
\widetilde{I}_1^{1/q_1'}+\widetilde{J}_1^{1/q_1'}\big] ^{\frac{q_1}{q_1q_2-1}}.
\end{aligned}
\end{equation}

If  $\eta_2<4(q_1+1)$, then passing to limit as $\rho\to +\infty$,
from \eqref{e2.24} we have
\begin{equation*}
\iint_{Q_1'}| x| ^{\sigma_1}| u_2| ^{q_1}\xi _1\,dx\,dt\leq 0.
\end{equation*}
Hence $u_2\equiv0$ and from the second equation of the system it follows
$u_1\equiv0$. Note that if $\eta_1<0$, then for $\eta_2=4(q_1+1)$, f
rom \eqref{e2.24} we obtain
\begin{equation*}
\iint_{Q_1'}| x| ^{\sigma_1}| u_2| ^{q_1}\xi _1\,dx\,dt<C.
\end{equation*}

As in the previous case, we can show again that
$u_1\equiv0$, $u_2\equiv0$.
Note that the condition $\eta_1 < 0$, $0\leq \eta_2 \leq 4(q_1+1)$
is equivalent to the condition
\begin{equation*}
1<q_2 < \alpha_2, \alpha_1\leq q_1 \leq \beta_1,
\end{equation*}
and the condition $\eta_1 = 0$, $0\leq \eta_2 \leq 4(q_1+1)$ to the condition
\begin{equation*}
q_2=\alpha_2, \quad  \alpha_1\leq q_1< \beta_1.
\end{equation*}

Now let $\eta_1\geq0$, $\eta_2\leq0$.
Then similar to the previous case we obtain that for  $\eta_2<0$,
$0\leq\eta_1\leq4(q_2+1)$ and for $\eta_2=0$, $\eta_1<4(q_2+1)$,
 $u_1\equiv0$, $u_2\equiv0$.

The same condition  $\eta_2<0$, $0\leq\eta_1\leq4(q_2+1)$ is equivalent
to the condition
\begin{equation*}
q_1< \alpha_1, \alpha_2\leq q_2\leq \beta_2,
\end{equation*}
and the condition $\eta_2=0$, $0\leq\eta_1<4(q_2+1)$ to the condition
 \begin{equation*}
q_1=\alpha_1,  \alpha_2\leq q_2< \beta_2.
\end{equation*}
\smallskip


\noindent II. $\alpha_1\leq1$, $\alpha_2>1$.
Herewith, the cases  $\eta_1\leq0$, $\eta_2>0$ and $\eta_1>0$, $\eta_2>0$
should be considered.
For $\eta_1\leq0$, $\eta_2>0$ as in the previous cases, we obtain  $u_1\equiv0$,
$u_2\equiv0$ if $\eta_1<0$,  $\eta_2\leq4(q_1+1)$  and $\eta_1=0$,
$\eta_2<4(q_1+1)$.

From the inequality  $\eta_2\leq4(q_1+1)$ it follows that $1<q_1\leq\beta_1$.
Since
\[
\beta_1=\frac{\lambda_1^{-}+\sigma_1+4+\frac{n+4}{2}}{\lambda_2^{-}
+\frac{n-4}{2}},
\]
 this case has meaning for
 $\lambda_1^{-}+\sigma_1+8>\lambda_2^{-}$.

Now let  $\eta_1>0$, $\eta_2>0$. Then similar to case (b), we obtain that
$u_1\equiv0$, $u_2\equiv0$ if
\[
q_1>\alpha_1, q_2>\alpha_2,\quad \max\{\theta_1,\theta_2\}\geq0.
\]
\smallskip

\noindent III. $\alpha_1>0$, $\alpha_2\leq1$.
Herewith, it is necessary to consider the case when $\eta_1>0$,
$\eta_2\leq0$ and $\eta_1>0$, $\eta_2>0$.
 For $\eta_1>0$, $\eta_2\leq0$, \ \ $u_1,u_2\equiv0$ if  $q_1<\alpha_1$,
$1<q_2\leq\beta_2$ and $q=\alpha_1$, $1<q_2<\beta_2$, and in the case
$\eta_1>0$, $\eta_2>0$, for $q_1>\alpha_1$, $1<q_2<\beta_2$,
$\max\{\theta_1,\theta_2\}\geq0$.
Obviously, this case has meaning for $\beta_2>1$ or for
$\lambda_2^{-}+\sigma_2+8>\lambda_1^{-}$.
\smallskip

\noindent IV. $\alpha_1\leq1$, $\alpha_2\leq1$.
Here it is necessary to consider the only case when  $\eta_1>0$,
$\eta_2>0$. Then  $u_1\equiv0$, $u_2\equiv0$, if $1<q_1<\beta_1$,
$1<q_2<\beta_2$ and $\max\{ \theta_1, \theta_2\}\geq0$. Obviously,
this set is not empty if  $\lambda_1^{-}+\sigma_1+8>\lambda_2^{-}$,
$\lambda_2^{-}+\sigma_2+8 > \lambda_1^{-}$.
This completely proves the theorem.
\end{proof}

Note that remains open the cases  $q_1=\alpha_1, q_2=\beta_2$ and
$q_1=\beta_1, q_2=\alpha_2$.

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\end{document}