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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 110, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/110\hfil 
 Nonexistence of solutions in the Heisenberg group]
{Nonexistence results for a pseudo-hyperbolic equation
in the Heisenberg group}

\author[M. Kirane, L. Ragoub \hfil EJDE-2015/110\hfilneg]
{Mokhtar Kirane, Lakhdar Ragoub}

\address{Mokhtar Kirane \newline
 Laboratoire de Math\'ematiques, Image et Applications\\
P\^ole Sciences et Technologies, Universit\'e de La Rochelle \\
Avenue M. Cr\'epeau, 17042 La Rochelle, France.\newline
NAAM Research Group, Department of Mathematics,
Faculty of Science, King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{mkirane@univ-lr.fr}

\address{Lakhdar Ragoub \newline
 Al Yamamah University,
 College of Computers and Information Systems\newline
P.O. Box 45180, Riyadh 11512, Saudi Arabia}
\email{l\_ragoub@yu.edu.sa}

\thanks{Submitted January 5, 2015. Published April 22, 2015.}
\subjclass[2010]{47J35, 34A34, 35R03}
\keywords{Nonexistence; nonlinear pseudo-hyperbolic equation;
\hfill\break\indent systems of pseudo-hyperbolic equations; Heisenberg group}

\begin{abstract}
 Sufficient conditions are obtained for the nonexistence of solutions
  to the nonlinear pseudo-hyperbolic equation
 $$
 u_{tt} -\Delta_{\mathbb H} u_{tt}-\Delta_{\mathbb H} u=|u|^p, \quad
  (\eta, t) \in   \mathbb{H} \times (0,\infty), \;  p>1,
 $$
 where $\Delta_\mathbb{H}$ is  the Kohn-Laplace operator on the
 $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$.
 Then, this result is extended to the case of a $2 \times 2$-system of the
 same type. Our technique  of proof is based on judicious choices of
 the test functions in the weak formulation of the sought solutions.
\end{abstract}

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

\section{Introduction}

In this article, we are concerned with the nonexistence of weak solutions
to the nonlinear pseudo-hyperbolic equation
\begin{equation}\label{P1}
u_{tt} -\Delta_{\mathbb H} u_{tt}-\Delta_{\mathbb H} u=|u|^p, \quad
(\eta, t) \in   \mathbb{H} \times (0,\infty), \;  p>1,
\end{equation}
under  the initial conditions
\begin{equation}\label{P2}
u(\eta, 0)=u_0(\eta),\quad  u_t(\eta, 0)=u_1(\eta), \quad \eta\in \mathbb{H},
\end{equation}
where $\Delta_\mathbb{H}$ is  the Kohn-Laplace operator on the
$(2N+1)$-dimensional Heisenberg group $\mathbb{H}$.
In the Euclidean case, pseudo-hyperbolic
equations served as  models for the unidirectional propagation of
nonlinear dispersive long waves
\cite{BBM}, creep buckling \cite{Hoff} for example. For further applications,
one is referred to the valuable book \cite{Korpusov}
where a sizeable number of pseudo-hyperbolic equations are studied.
Our proofs rely on the test function method \cite{MitidieriPohozaev, Zhang}.
For the reader convenience, some background facts used in the sequel are recalled.

The $(2N+1)$-dimensional Heisenberg group $\mathbb{H}$ is the
space $\mathbb{R}^{2N+1}$ equipped with the group operation
$$
\eta\circ \eta'=(x+x',y+y',\tau+\tau'+2(x\cdot y'-x'\cdot y)),
$$
for all $\eta=(x,y,\tau),\eta'=(x',y',\tau')\in \mathbb{R}^N
\times \mathbb{R}^N\times \mathbb{R}$, where $\cdot$ denotes the
 standard scalar product in $\mathbb{R}^N$.
This group operation endows $\mathbb{H}$ with the structure of a Lie group.

On $\mathbb{H}$ it is natural to define a distance from
$\eta=(x,y,\tau)=:(z, \tau)$ to the origin by
$$
|\eta|_\mathbb{H}=\Big(\tau^2+\Big(\sum_{i=1}^N (x_i^2+y_i^2) \Big)^2\Big)^{1/4}
=\big(\tau^2+ |z|^{4}\big)^{1/4},
$$
where $x=(x_1,\cdots,x_N)$ and $y=(y_1,\cdots,y_N)$.

The Laplacian $\Delta_{\mathbb{H}}$ over $\mathbb{H}$ can be defined from
the vectors fields
$$
X_i=\partial_{x_i}+2y_i\partial_{\tau}\quad\text{and}\quad
Y_i=\partial_{y_i}-2x_i\partial_{\tau},
$$
for $i=1,\cdots,N$, as follows
$$
\Delta_{\mathbb{H}}=\sum_{i=1}^N (X_i^2+Y_i^2).
$$
A simple computation gives the expression
$$
\Delta_{\mathbb{H}}u=\sum_{i=1}^N
\big(\partial^2_{x_ix_i}u+\partial_{y_iy_i}^2u+4y_i\partial^2_{x_i\tau}u
-4x_i\partial^2_{y_i\tau}u+4(x_i^2+y_i)^2
\partial^2_{\tau\tau}u\big).
$$

The operator $\Delta_{\mathbb{H}}$ satisfies the following properties:
\begin{itemize}
\item  It is invariant with respect to the left multiplication in the group,
i.e., for all $\eta,\eta'\in \mathbb{H}$, we have
$$
\Delta_{\mathbb{H}}(u(\eta\circ\eta'))=\Delta_{\mathbb{H}}u(\eta\circ\eta');
$$
\item  It is homogeneous with respect to a dilatation. More precisely,
for $\lambda\in \mathbb{R}$ and $(x,y,\tau)\in \mathbb{H}$, we have
    $$
    \Delta_{\mathbb{H}}(u(\lambda x,\lambda y,\lambda^2\tau))=\lambda^2 (\Delta_\mathbb{H}u)(\lambda x,\lambda y,\lambda^2\tau);
    $$
\item  If $u(\eta)=v(|\eta|_\mathbb{H})$, then
$$
\Delta_{\mathbb{H}}v(\rho)
=a(\eta)\Big(\frac{d^2v}{d\rho^2}+\frac{Q-1}{\rho}\frac{dv}{d\rho}\Big),
$$
where $\rho=|\eta|_\mathbb{H}$, $a(\eta)=\rho^{-2}\sum_{i=1}^N (x_i^2+y_i^2)$
and $Q=2N+2$ is the homogeneous dimension of $\mathbb{H}$.
\end{itemize}
For more details on Heisenberg groups, we refer to \cite{Folland,Lanconelli}.

In this work, we first provide a sufficient condition for the nonexistence
of weak solutions to the nonlinear problem
\eqref{P1}-\eqref{P2}, then we extend the result to the case of the
$2\times 2$ system
\begin{equation}\label{S1}
\begin{gathered}
u_{tt} -\Delta_{\mathbb H}u_{tt} -\Delta_{\mathbb H} u=|v|^q,\quad
 (\eta, t)\in \mathbb{H}\times (0,\infty),\\
v_{tt} -\Delta_{\mathbb H} v_{tt} -\Delta_{\mathbb H} v=|u|^p,\quad
  (\eta, t)\in  \mathbb{H}\times (0,\infty),\\
u(\eta, 0)=u_0(\eta),\,\, u_1(\eta, 0)=u_1(\eta),  \quad \eta \in \mathbb{H}\\
v(\eta, 0)=v_0(\eta), \,\, v(\eta, 0)=v_1(\eta),\,\, \quad \eta \in \mathbb{H},
\end{gathered}
\end{equation}
where $p,q>1$ are real numbers, for which we provide a sufficient
 condition for the nonexistence of weak solutions.

\section{Results and proofs}

Let $\mathcal {H_{T}} =\mathbb{H}\times (0, T)$,
$\mathcal {H} =\mathbb{H}\times (0,\infty)$. For $R>0$, let
$$
\mathcal{U}_R=\{(x,y,\tau, t)\in \mathcal{H}:
0\leq t^{4}+|x|^{4}+|y|^{4}+\tau^{2} \leq 2R^4\}.
$$

\subsection{Case of a single equation}

The definition of solutions we adopt for \eqref{P1}-\eqref{P2} is:

We say that $u$ is a local weak solution to \eqref{P1}-\eqref{P2}
on ${\mathcal H}$ with initial data $u(0,\cdot)=u_0\in L^1_{\rm loc}(\mathbb{H})$, if
$u\in L^{p}_{\rm loc}({\mathcal H})$ and satisfies
\begin{align*}
&\int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt +\int_{\mathbb{H}}u_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta+\int_{\mathbb{H}}u_1(\vartheta)\Delta_{\mathbb H}\varphi(\vartheta, 0) \,d\vartheta\\
&=\int_{\mathcal H} u\varphi_{tt}\,d\vartheta\,dt +\int_{\mathcal H} u \Delta_{\mathbb H}\varphi_{tt} \,dt\,d\vartheta-\int_{\mathcal H}u \Delta_{\mathbb{H}}\varphi\,dt\,d\vartheta,
\end{align*}
for any test function $\varphi$, $\varphi(\cdot,t)=0$,
$\varphi_t(\cdot,t)=0, t \geq T$.
The solution $u$ is said global if it exists on $(0, \infty)$.

Our first main result is given by the following theorem.

\begin{theorem}\label{T1}
Let $u_1\in L^1(\mathbb{H})$.  Suppose that
\begin{equation}\label{IH}
\int_{\mathbb{H}}u_0\,d\vartheta > 0.
\end{equation}
If
$$
1<p\leq 1+\frac{2}{Q-1},
$$
then any weak solution to \eqref{P1}-\eqref{P2}  blows-up in a finite time.
\end{theorem}


\begin{proof}
 Suppose that $u$ is a  weak solution to \eqref{P1}-\eqref{P2}.
Then for any regular test function $\varphi$, we have
\begin{equation} \label{I1}
\begin{aligned}
&\int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt
 +\int_{\mathbb{H}}u_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta\\
&\leq \int_{\mathcal H} |u| |\varphi_{tt}|\,d\vartheta\,dt
 +\int_{\mathcal H} |u|  |\Delta_{\mathbb H}\varphi_{tt}|\,dt\,d\vartheta\\
&\quad +\int_{\mathcal H}|u| |\Delta_{\mathbb{H}}\varphi|\,dt\,d\vartheta
 +\int_{\mathbb{H}}|u_1(\vartheta)| |\Delta_{\mathbb H}\varphi(\vartheta, 0)|
 \,d\vartheta.
\end{aligned}
\end{equation}
Using the $\varepsilon$-Young inequality
\[
ab \leq \varepsilon a^{p}+C_{\varepsilon}b^{p'}, \quad
 a, b, \varepsilon, C_{\varepsilon} >0, \; 1<p,p',\; p+p'=pp',
\]
with parameters $p$ and $p'=p/(p-1)$, we obtain
\begin{equation}\label{es1}
\begin{aligned}
\int_{\mathcal H} |u| |\varphi_{tt}|\,d\vartheta\,dt
& = \int_{\mathcal H} |u|\varphi^{1/p}
 \varphi^{-\frac{1}{p}} |\varphi_{tt}|\,d\vartheta\,dt \\
&\leq \varepsilon \int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt +
c_\varepsilon \int_{\mathcal H}
\varphi^{- \frac{1}{p-1}}|\varphi_{tt}|^{\frac{p}{p-1}}\,d\vartheta\,dt,
\end{aligned}
\end{equation}
for some positive constant $c_\varepsilon$.


Similarly, we have
\begin{gather}\label{es2}
\int_{\mathcal H} |u|  |\Delta_{\mathbb H}\varphi_{tt}|\,dt\,d\vartheta
\leq \varepsilon \int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt +
c_\varepsilon \int_{\mathcal H}\varphi^{-\frac{1}{p-1}} |
\Delta_{\mathbb H}\varphi_{tt}|^{\frac{p}{p-1}}\,dt\,d\vartheta,
\\
\label{es3}
\int_{\mathcal H} |u|  |\Delta_{\mathbb H}\varphi|\,dt\,d\vartheta
\leq \varepsilon \int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt
+ c_\varepsilon \int_{\mathcal H}\varphi^{-\frac{1}{p-1}}
|\Delta_{\mathbb H}\varphi|^{\frac{P}{P-1}}\,dt\,d\vartheta.
\end{gather}
Using \eqref{I1}, \eqref{es1}, \eqref{es2} and \eqref{es3}, for
$\varepsilon>0$ small enough, we obtain
\begin{equation} \label{EQ}
\begin{aligned}
&\int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt
 +\int_{\mathbb{H}}u_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta\\
&\leq C \Big(A_p(\varphi)+B_p(\varphi)+C_p(\varphi)+
\int_{\mathbb{H}}|u_1(\vartheta)| |\Delta_{\mathbb H}\varphi(\vartheta, 0)|
\,d\vartheta\Big),
\end{aligned}
\end{equation}
where
\begin{gather}
\label{1} A_p(\varphi)
=\int_{\mathcal H} \varphi^{-\frac{1}{p-1}}|\varphi_{tt}|^{\frac{p}{p-1}}
 \,d\vartheta\,dt,\\
\label{2} B_p(\varphi)
=\int_{\mathcal H} \varphi^{-\frac{1}{p-1}}|\Delta_{\mathbb H}
 \varphi_{tt}|^{\frac{p}{p-1}}\,d\vartheta\,dt,\\
\label{3} C_p(\varphi)
=\int_{\mathcal H} \varphi^{-\frac{1}{p-1}}|\Delta_{\mathbb H}
\varphi|^{\frac{p}{p-1}}\,d\vartheta\,dt.
\end{gather}

Now, let us consider the test function
\begin{equation}\label{test}
\varphi_R(t,\vartheta)=\phi^\omega
\Big(\frac{t^4+|x|^4+|y|^4+\tau^2}{R^4}\Big),\quad  R>0, \; \omega\gg 1,
\end{equation}
where $\phi\in C_0^\infty(\mathbb{R}^+)$ is a decreasing function satisfying
$$
\phi(r)=\begin{cases}
1 &\text{if } 0\leq r\leq 1,\\
0 &\text{if } r\geq 2.
\end{cases}
$$
Observe that $\operatorname{supp}(\varphi_R)$ is a subset of $\mathcal{U}_R$,
while $\operatorname{supp}({\varphi_R}_{tt})$,
 $\operatorname{supp}(\Delta_\mathbb{H}\varphi_R)$ and
$\operatorname{supp}(\Delta_{\mathbb H}(\varphi_R)_{tt})$  are subsets of
$$
\Theta_R=\{(t,x,y,\tau)\in \mathcal H:\, R^4
\leq t^4+|x|^4+|y|^4+\tau^2\leq 2R^4\}.
$$
Let
$$
\rho=\frac{t^4+|x|^4+|y|^4+\tau^2}{R^4}.
$$
Then we have
\begin{align*}
&\Delta_{\mathbb H} \varphi_R(t,\vartheta)\\
&= \frac{4\omega(N+4)}{R^4}
 \Big(|x|^{2}+|y|^{2}\Big)\phi'(\rho)\phi^{\omega-1}(\rho)\\
&\quad +\frac{16\omega(\omega-1)}{R^8}
 \Big((|x|^6+|y|^6)+2\tau(|x|^2-|y|^{2})x\cdot y+\tau^{2}(|x|^2+|y|^2)\Big)
 \phi'^2(\rho)\phi^{\omega-2}(\rho)\\
&\quad +\frac{16\omega}{R^{8}}
 \Big((|x|^{6}+|y|^{6})+2\tau(|x|^{2}-|y|^{2})x\cdot y+\tau^{2}(|x|^2+|y|^2)\Big)
  \phi''(\rho)\phi^{\omega-1}(\rho)
\end{align*}
for example.

Observe first that $(\varphi_R)_t(\vartheta, 0) =0$ as required in the definition. 
It follows that there is a positive constant $C>0$, independent of $R$, 
such that for all $(t,\vartheta)\in \mathcal{\Theta}_R$, we have
\begin{equation}\label{HL}
|\Delta_{\mathbb H} \varphi_R(t,\vartheta)|
\leq C R^{-2}\phi^{\omega-2}(\rho)\chi(\rho),
\end{equation}
where
$$
\chi(\rho)=|\phi'(\rho)|\phi(\rho)+\phi'^2(\rho)+|\phi''(\rho)|\phi(\rho),
$$
and
\begin{gather}\label{HL2}
|(\Delta_{\mathbb H} \varphi_R)_t(t,\vartheta)|\leq C R^{-3}, \\
\label{HL3}
|(\varphi_R)_{tt}(t,\vartheta)|\leq C R^{-4}.
\end{gather}
Using \eqref{HL} and \eqref{HL2}, we obtain
\begin{gather}
\label{A} A_p(\varphi_R)\leq  C R^{Q+1-\frac{2p}{p-1}}, \\
\label{B} B_p(\varphi_R)\leq  C R^{Q+1-\frac{4p}{p-1}},\\
\label{C} C_p(\varphi_R)\leq  C R^{Q+1-\frac{2p}{p-1}}.
\end{gather}

Let us consider now the change of variables
\begin{equation}\label{V}
(t,x,y,\tau)=(t,\vartheta)\mapsto 
(\widetilde{t},\widetilde{v})=(\widetilde{t},\widetilde{x},
\widetilde{y},\widetilde{\tau}),
\end{equation}
where
$$
\widetilde{t}=R^{-1}t,\quad \widetilde{\tau}=R^{-2}\tau,
\widetilde{x}=R^{-1}x,\quad \widetilde{y}=R^{-1}y.
$$
Let
\begin{gather*}
\widetilde{\rho}= \widetilde{t}^4+|\widetilde{x}|^{4}
+|\widetilde{y}|^4+\widetilde{\tau}^2,\\
\widetilde{\mathcal{C}_R}= \{(\widetilde{t},\widetilde{x},\widetilde{y},
\widetilde{\tau})\in \mathcal{H}: 1\leq \widetilde{\rho}\leq 2\},\\
\mathcal{C}_R =  \{(x,y,\tau)\in \mathbb{H}: R^4\leq |x|^4+|y|^4+\tau^2\leq 2R^4\}.
\end{gather*}
Using \eqref{EQ}, \eqref{B} and \eqref{C}, we obtain
\begin{equation} \label{LIMIT2}
\begin{aligned}
&\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
 +\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta\\
&\leq C \Big(R^{\vartheta_1}+R^{\vartheta_2}
 +\int_{\mathcal{C}_R}|u_1(\vartheta)| |\Delta_{\mathbb H}
 \varphi_R(\vartheta, 0)|\,d\vartheta\Big),
\end{aligned}
\end{equation}
where
$$
\vartheta_1=Q+1-\frac{2p}{p-1}\quad\text{and}\quad
\vartheta_2=Q+1-\frac{4p}{p-1}.
$$
On the other hand, we have
\begin{align*}
&\liminf_{R\to \infty} \int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
+\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta \\
&\geq \liminf_{R\to \infty}\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
+\liminf_{R\to \infty}\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)
\,d\vartheta.
\end{align*}
Using the monotone convergence theorem, we obtain
$$
\liminf_{R\to \infty}\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
=\int_{\mathcal H} |u|^p\,d\vartheta\,dt.
$$
Since $u_1\in L^1(\mathbb{H})$, by the dominated convergence theorem, we have
$$
\liminf_{R\to \infty}\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)
\,d\vartheta
= \int_{\mathbb{H}}u_1(\vartheta)\,d\vartheta.
$$
Now, we have
\[
\liminf_{R\to \infty} \Big(\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt+\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta
\Big)
\geq \int_{\mathcal H} |u|^p\,d\vartheta\,dt+\ell,
\]
where from \eqref{IH},
$$
\ell=\int_{\mathbb{H}}u_1(\vartheta)\,d\vartheta>0.
$$
By the definition of the limit inferior, for every $\varepsilon>0$,
there exists $R_0>0$ such that
\begin{align*}
&\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt+\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta\\
&> \liminf_{R\to \infty} \Big(\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
+\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta \Big)
-\varepsilon\\
&\geq \int_{\mathcal H} |u|^p\,d\vartheta\,dt+\ell-\varepsilon,
\end{align*}
for every $R\geq R_0$. Taking $\varepsilon=\ell/2$, we obtain
\[
\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
+\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta
\geq \int_{\mathcal H} |u|^p\,d\vartheta\,dt+\frac{\ell}{2},
\]
for every $R\geq R_0$.  Then from \eqref{LIMIT2}, we have
\begin{equation}\label{LIMIT}
\int_{\mathcal H} |u|^p\,d\vartheta\,dt+\frac{\ell}{2}
\leq C \Big(R^{\vartheta_1}+R^{\vartheta_2}+\int_{\mathcal{C}_R}|u_0(\vartheta)|
|\Delta_{\mathbb H}\varphi_R(\vartheta, 0)|\,d\vartheta\Big),
\end{equation}
for $R$ large enough.

Now, we require that $\vartheta_1=\max\{\vartheta_1,\vartheta_2\}\leq 0$, 
which is equivalent to $1<p\leq 1+\frac{2}{Q-1}$.
We distinguish two cases.
\smallskip

\noindent\textbf{Case 1:} $1<p< 1+\frac{2}{Q-1}$.
In this case, letting $R\to \infty$ in \eqref{LIMIT} and using the 
dominated convergence theorem,  we obtain
$$
\int_{\mathcal H} |u|^p\,d\vartheta\,dt+\frac{\ell}{2}\leq 0,
$$
which is a contradiction as $\ell>0$.
\smallskip

\noindent\textbf{Case 2:} $p=1+\frac{2}{Q-1}$.
From \eqref{LIMIT}, we obtain
\begin{equation}\label{F}
\int_{\mathcal H} |u|^p\,d\vartheta\,dt \leq C<\infty \; \; \Rightarrow \; \; 
\lim_{R\to \infty} \int_{{\mathcal C}_R} |u|^p\varphi_R\,d\vartheta\,dt=0.
\end{equation}
Using the H\"{o}lder inequality with parameters $p$ and $p/(p-1)$, 
from \eqref{I1}, we obtain
$$
\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt
+\frac{\ell}{2}
\leq  C \Big(\int_{\Theta_R} |u|^p\varphi_R\,d\vartheta\,dt\Big)^{1/p}.
$$
Letting $R\to \infty$ in the above inequality and using \eqref{F}, we obtain
$$
\int_{\mathcal H} |u|^p\,d\vartheta\,dt+\frac{\ell}{2}=0.
$$
A contradiction; the proof of the theorem is complete.
\end{proof}


\subsubsection{The case of system \eqref{S1}}

The definition of solutions we adopt for \eqref{S1} is:

We say that the pair $(u,v)$ is a local  weak solution to \eqref{S1} 
on $\mathcal{H}$ with initial data 
$(u(0,\cdot),v(0,\cdot))=(u_0,v_0)\in L^1_{\rm loc}(\mathbb{H})
\times L^1_{\rm loc}(\mathbb{H})$, if
$(u,v)\in L^{p}_{\rm loc}(\mathcal H)\times L^{q}_{\rm loc}(\mathcal H)$ 
and satisfies
\begin{align*}
&\int_{\mathcal H} |v|^q\varphi\,d\vartheta\,dt 
 +\int_{\mathbb{H}}u_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta\\
&=\int_{\mathcal H} u\varphi_{tt}\,d\vartheta\,dt 
 +\int_{\mathcal H} u (\Delta_{\mathbb H}\varphi)_{tt}\,dt\,d\vartheta
 -\int_{\mathcal H}u \Delta_{\mathbb{H}}\varphi\,dt\,d\vartheta 
 +\int_{\mathbb{H}}u_1(\vartheta)\Delta_{\mathbb H}\varphi(\vartheta, 0)\,d\vartheta
\end{align*}
and
\begin{align*}
&\int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt 
 +\int_{\mathbb{H}}v_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta\\
&=\int_{\mathcal H} v\varphi_{tt}\,d\vartheta\,dt 
 +\int_{\mathcal H} v (\Delta_{\mathbb H}\varphi)_{tt}\,dt\,d\vartheta
 -\int_{\mathcal H}v \Delta_{\mathbb{H}}\varphi\,dt\,d\vartheta 
 +\int_{\mathbb{H}}v_1(\vartheta)\Delta_{\mathbb H}\varphi(\vartheta, 0)\,d\vartheta,
\end{align*}
for any test function $\varphi$, $\varphi(\cdot, t)=0$, 
$\varphi_t(\cdot, t)=0$, $t \geq T$.
The solution is said global if it exists for $T=+\infty$.

Our second main result is given by the following theorem.

\begin{theorem}\label{T2}
Let $(u_1,v_1)\in L^1(\mathbb{H})\times L^1(\mathbb{H})$.
Suppose that
$$
\int_{\mathbb{H}}u_1\,d\vartheta >0\quad\text{and}\quad 
\int_{\mathbb{H}}v_1\,d\vartheta >0.
$$
If $1<pq\leq (pq)^*$, where
$$
(pq)^*=1+\frac{2}{Q-1}\max\{p+1,q+1\},
$$
then  there exists no nontrivial  weak solution to \eqref{S1}.
\end{theorem}


\begin{proof} Suppose that $(u,v)$ is a nontrivial weak solution to \eqref{S1}. 
Then for any regular test function $\varphi$, we have
\begin{align*}
&\int_{\mathcal H} |v|^q\varphi\,d\vartheta\,dt 
+\int_{\mathbb{H}}u_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta\\
&\leq \int_{\mathcal H} |u||\varphi_{tt}|\,d\vartheta\,dt
 +\int_{\mathcal H} |u| |(\Delta_{\mathbb H}\varphi)_{tt}|\,dt\,d\vartheta\\
&\quad +\int_{\mathcal H}|u| |\Delta_{\mathbb{H}}\varphi|\,dt\,d\vartheta 
+\int_{\mathbb{H}}|u_1(\vartheta)||\Delta_{\mathbb H}\varphi(\vartheta, 0)|\,d\vartheta
\end{align*}
and
\begin{align*}
&\int_{\mathcal H} |u|^p\varphi\,d\vartheta\,dt 
+\int_{\mathbb{H}}v_1(\vartheta)\varphi(\vartheta, 0)\,d\vartheta\\
&\leq\int_{\mathcal H} |v||\varphi_{tt}|\,d\vartheta\,dt
 +\int_{\mathcal H} |v| |(\Delta_{\mathbb H}\varphi)_{tt}|\,dt\,d\vartheta\\
&\quad +\int_{\mathcal H}|v| |\Delta_{\mathbb{H}}\varphi|\,dt\,d\vartheta 
+\int_{\mathbb{H}}|v_1(\vartheta)||\Delta_{\mathbb H}\varphi(\vartheta, 0)|
\,d\vartheta.
\end{align*}

Taking $\varphi=\varphi_R$, the test function given by \eqref{test}, 
and using the H\"{o}lder inequality with parameters $p$ and $p/(p-1)$, we obtain
\begin{align*}
&\int_{\mathcal H} |v|^q\varphi_R\,d\vartheta\,dt 
+\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta
 -\int_{\mathbb{H}}|u_1(\vartheta)||\Delta_{\mathbb H}\varphi_R(\vartheta, 0)
|\,d\vartheta\\
&\leq \left(A_p(\varphi_R)^{\frac{p-1}{p}}+B_p(\varphi_R)^{\frac{p-1}{p}}
+C_p(\varphi_R)^{\frac{p-1}{p}}\right) 
\Big(\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt\Big)^{1/p},
\end{align*}
where $A_p(\varphi)$, $B_p(\varphi)$ and $C_p(\varphi)$ are given 
respectively by \eqref{1}, \eqref{2} and \eqref{3}.
Similarly, by the H\"{o}lder inequality with parameters $q$ and $q/(q-1)$, 
we get
\begin{align*}
&\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt 
+\int_{\mathbb{H}}v_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta
-\int_{\mathbb{H}}|v_1(\vartheta)||\Delta_{\mathbb H}\varphi_R(\vartheta, 0)|
 \,d\vartheta\\
&\leq \left(A_q(\varphi_R)^{\frac{q-1}{q}}+B_q(\varphi_R)^{\frac{q-1}{q}}
+C_q(\varphi_R)^{\frac{q-1}{q}}\right) 
\Big(\int_{\mathcal H} |v|^q\varphi_R\,d\vartheta\,dt\Big)^{1/q}.
\end{align*}
Without restriction of the generality, we may assume that for $R$ 
large enough, we have
\begin{gather*}
\int_{\mathbb{H}}u_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta
-\int_{\mathbb{H}}|u_1(\vartheta)||\Delta_{\mathbb H}\varphi_R(\vartheta, 0)|
\,d\vartheta\geq 0, \\
\int_{\mathbb{H}}v_1(\vartheta)\varphi_R(\vartheta, 0)\,d\vartheta
-\int_{\mathbb{H}}|v_1(\vartheta)||\Delta_{\mathbb H}\varphi_R(\vartheta, 0)|
\,d\vartheta\geq 0.
\end{gather*}
Slight modifications yield the proof in the general case 
(see the proof of Theorem \ref{T1}). Then for $R$ large enough, we have
\begin{equation}\label{salam}
\begin{aligned}
&\int_{\mathcal H} |v|^q\varphi_R\,d\vartheta\,dt \\
&\leq \left(A_p(\varphi_R)^{\frac{p-1}{p}}+B_p(\varphi_R)^{\frac{p-1}{p}}
+C_p(\varphi_R)^{\frac{p-1}{p}}\right) 
\Big(\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt\Big)^{1/p}
\end{aligned}
\end{equation}
and
\begin{equation}\label{salam2}
\begin{aligned}
&\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt\\
&\leq \left(A_q(\varphi_R)^{\frac{q-1}{q}}+B_q(\varphi_R)^{\frac{q-1}{q}}
+C_q(\varphi_R)^{\frac{q-1}{q}}\right) 
\Big(\int_{\mathcal H} |v|^q\varphi_R\,d\vartheta\,dt\Big)^{1/q}.
\end{aligned}
\end{equation}
Using the change of variables \eqref{V}, from \eqref{salam} and \eqref{salam2},
 we obtain
\begin{gather}\label{salamm}
\int_{\mathcal H} |v|^q\varphi_R\,d\vartheta\,dt 
\leq CR^{\frac{Q(p-1)-2}{p}} \Big(\int_{\mathcal H} 
|u|^p\varphi_R\,d\vartheta\,dt\Big)^{1/p},
\\
\label{salamm2}
\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt 
\leq CR^{\frac{Q(q-1)-2}{q}}\Big(\int_{\mathcal H} 
|v|^q\varphi_R\,d\vartheta\,dt\Big)^{1/q}.
\end{gather}
Combining \eqref{salamm} with \eqref{salamm2}, we obtain
\begin{gather}\label{MIR}
\Big(\int_{\mathcal H} |u|^p\varphi_R\,d\vartheta\,dt\Big)^{1-\frac{1}{pq}} 
\leq CR^{\upsilon_1}, \\
\label{KIR2}
\Big(\int_{\mathcal H} |v|^q\varphi_R\,d\vartheta\,dt\Big)^{1-\frac{1}{pq}} 
\leq CR^{\upsilon_2},
\end{gather}
where
$$
\upsilon_1=\frac{Q(pq-1)-2(p+1)}{pq-1}\quad\text{and}\quad 
\upsilon_2=\frac{Q(pq-1)-2(q+1)}{pq-1}.
$$

We require that $\upsilon_1\leq 0$ or $\upsilon_2\leq 0$ which is 
equivalent to $1<pq\leq 1+\frac{2}{Q}\max\{p+1,q+1\}$. We distinguish two cases.
\smallskip

\noindent\textbf{Case 1:}  $1<pq< 1+\frac{2}{Q}\max\{p+1,q+1\}$.
Without loss of the generality, we may suppose that  $0<q\leq p$. 
In this case, letting $R\to \infty$ in \eqref{MIR}, we obtain
$$
\int_{\mathcal H} |u|^p\,d\vartheta\,dt=0,
$$
which is a contradiction.
\smallskip

\noindent\textbf{Case 2:} $pq=1+\frac{2}{Q}\max\{p+1,q+1\}$.
This case can be treated in the same way as in the proof of Theorem \ref{T1}.
\end{proof}

\begin{remark} \rm
If $p=q$ and $u=v$ in Theorem \ref{T2}, we obtain the result for
 a single equation given by Theorem \ref{T1}.
\end{remark}

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\end{document}