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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 91, pp. 1--12.\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-2016/91\hfil Differential inequalities with shifted arguments]
{Blow-up phenomena for second-order differential inequalities with shifted arguments}

\author[M. Jleli, M. Kirane, B. Samet \hfil EJDE-2016/91\hfilneg]
{Mohamed Jleli, Mokhtar Kirane, Bessem Samet}

\address{Mohamed Jleli \newline
Department of Mathematics, College of Science,
 King Saud University, P.O. Box 2455, Riyadh 11451,
Saudi Arabia}
\email{jleli@ksu.edu.sa}

\address{Mokhtar Kirane \newline
 LaSIE, Facult\'e des Sciences et Technologies, Universit\'e de La Rochelle,
Avenue M. Cr\'epeau, 17042 La Rochelle, France}
\email{mokhtar.kirane@univ-lr.fr}

\address{Bessem Samet \newline
Department of Mathematics, College of Science,
King Saud University, P.O. Box 2455, Riyadh 11451,
Saudi Arabia}
\email{bsamet@ksu.edu.sa}

\thanks{Submitted February 22, 2016. Published March 31, 2016.}
\subjclass[2010]{34K38, 35B44}
\keywords{Blow-up; differential inequalities; advanced argument;
\hfill\break\indent delayed argument}

\begin{abstract}
 We provide sufficient conditions for the blow-up of solutions to certain
 second-order differential inequalities and systems with advanced and
 delayed arguments. Our proofs are based on the test function method.
\end{abstract}

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

\section{Introduction}

The study of differential equations and inequalities with shifted
(advanced or delayed) arguments is  an important and significant branch
of mathematical analysis.
Applications of these equations can be found in physics, mechanics,
engineering, biology and so on
(see \cite{Arino, Erneux, MacDonald, Volterra} and the references therein).
Many results concerning existence and uniqueness of solutions,
stability of  solutions,  oscillation and numerical methods have
been published. Results concerning blowing-up solutions in a finite
time are however scarce. It seems that the first result is due to
Kobayashi \cite{Kobayashi} who considered a parabolic equations with delay.
Later, Lightbourne and Rankin \cite{LightbourneRankin} considered also
blowing-up solutions to partial differential equations with delay
(let us remark in passing that their proof is doubtfull).
More recently Ezzinbi and Jazar \cite{EzzinbiJazar} considered blowing-up
solutions to ordinary differential equations with delay,
however their method of proof is different from ours.
Casal, Diaz and Vegas \cite{Casal} also considered blow-up for some
ordinary and parabolic differential equations with time-delay.
Let us mention that many nice results concerning blowing-up solutions
for integral equations have been published mainly by Olmstead and his co-authors,
we just mention the review paper of Roberts \cite{Roberts} and its references.

Recently, Salieva \cite{SAL} obtained sufficient conditions for the blow-up
of solutions to certain classes of first-order differential equations and
inequalities with advanced and delayed arguments using the test function method
\cite{MI}. Motivated by that work, we consider in this paper various classes
 of second-order differential inequalities and systems with shifted arguments,
for which we study the blow-up phenomenon.  As in  \cite{SAL}, using the
test function method, we provide sufficient conditions for the existence
of blow-up. we obtain also an upper bound of the blow-up time.

The paper is organized as follows.
In Section \ref{S2}, we consider a second-order differential inequality with
advanced arguments. Sufficient conditions,  for which any solution of
the considered inequality blows-up
at a finite time, are provided. Moreover, an estimate of the blow-up
time is obtained. In Section \ref{S3}, we consider second-order differential inequalities with delayed arguments in the right-hand side. Finally, in Section \ref{S4}, we study the blow-up of solutions for two systems of differential inequalities with advanced arguments.

\section{Differential inequality with advanced argument}\label{S2}

We consider the differential inequality
\begin{equation}\label{PI}
\begin{gathered}
\frac{d^2y(t)}{dt^2}\ge |y(t+\tau)|^q ,\quad t>0,\\
y(0)=y_0>0,\quad  y'(0)=y_1>0,
\end{gathered}
\end{equation}
where $\tau>0 $ and $q>1$.

If there exists a $T>0$ such that a function $y$ satisfies \eqref{PI}
for every $t\in (0,T)$, then $y$ is called a local solution to \eqref{PI}.
If a function $y$ satisfies \eqref{PI} for every $t>0$, then it is called a
global solution to \eqref{PI}. If $y$ is not a global solution, the largest possible
$T>0$ is called the blow-up time for the function $y$.
We have the following blow-up result.


\begin{theorem}\label{T1}
If $q>1$, then any solution of \eqref{PI} blows-up at a finite time.
\end{theorem}

\begin{proof}
 Suppose that $y$ is a global solution of \eqref{PI}.
Let $\omega$ be a real number such that
\begin{equation}\label{EQ}
\omega>\frac{2q}{q-1}.
\end{equation}
Then
$$
\int_0^\infty \frac{d^2y}{dt^2} \varphi^\omega(t)\,dt \geq
\int_0^\infty |y(t+\tau)|^q \varphi^\omega(t)\,dt,
$$
for every test function $\varphi$, $\varphi(t\geq T)=0$, $\varphi\geq0$.
Integrating by parts,
\begin{equation}\label{w1}
\begin{aligned}
&-y_1[\varphi(0)]^\omega+\omega y_0 [\varphi(0)]^{\omega-1}\varphi'(0)
+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt \\
&\geq  \int_0^\infty |y(t+\tau)|^q \varphi^\omega(t)\,dt,
\end{aligned}
\end{equation}
where
$$
\phi(t)=\omega\Big((\omega-1)\varphi'^2(t)+\varphi(t)\varphi''(t)\Big),\quad t>0.
$$
Observe that from \eqref{PI}, we have $y''(t)\geq 0$, which implies
\begin{gather}\label{ed}
y'(t)\geq y'(0)=y_1>0,\quad t>0,  \\
\label{ey}
y(t)\geq y(0)=y_0>0,\quad t>0.
\end{gather}
Using \eqref{ey} and Taylor-Lagrange formula,  we obtain
\begin{align*}
|y(t+\tau)|^q &=  y(t+\tau)^q\\
&=  y^q(t)+\tau q y^{q-1}(t) y'(t)
 +\frac{q\tau^2}{2} \Big((q-1)y^{q-2}(t+\theta \tau)y'^2(t+\theta\tau)\\
&\quad +y^{q-1}(t+\theta\tau)y''(t+\theta\tau)\Big),
\end{align*}
where $\theta=\theta_t\in (0,1)$. Therefore, by \eqref{PI}, \eqref{ed}
and \eqref{ey}, we obtain

\begin{align*}
|y(t+\tau)|^q &\geq  y^q(t)+\tau q y_0^{q-1}y_1\\
&\quad +\frac{q\tau^2}{2} \Big((q-1)y_0^{q-2}y_1^2+
y_0^{q-1}y^q(t+(\theta+1)\tau)\Big)\\
&\geq  y^q(t)+\tau q y_0^{q-1}y_1+\frac{q\tau^2}{2} \Big((q-1)y_0^{q-2}y_1^2+
y_0^{2q-1}\Big),
\end{align*}
i.e.,
\begin{equation}\label{esabs}
|y(t+\tau)|^q \geq y^q(t)+\tau q y_0^{q-1}y_1+\frac{q\tau^2}{2}
\Big((q-1)y_0^{q-2}y_1^2+
y_0^{2q-1}\Big),\quad t>0.
\end{equation}
Combining \eqref{w1} with \eqref{esabs}, we obtain
\begin{equation}\label{FW}
\begin{aligned}
\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
&\leq  -y_1[\varphi(0)]^\omega+\omega y_0 [\varphi(0)]^{\omega-1}\varphi'(0)\\
&\quad + \int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
 -\gamma(q,y_0,y_1) \int_0^\infty \varphi^\omega(t)\,dt,
\end{aligned}
\end{equation}
where
$$
\gamma(q,y_0,y_1)=\tau q y_0^{q-1}y_1+\frac{q\tau^2}{2} \Big((q-1)y_0^{q-2}y_1^2+
y_0^{2q-1}\Big)>0.
$$
Using the $\varepsilon$-Young inequality with parameters $q$ and
$q'=\frac{q}{q-1}$, for $\varepsilon>0$, we obtain
\begin{equation}\label{epsi}
\begin{aligned}
&\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt \\
&= \int_0^\infty y(t) \varphi^{\frac{\omega}{q}}(t)
 [\varphi(t)]^{\omega-2-\frac{\omega}{q}}\phi(t)\,dt\\
&\leq  \varepsilon \int_0^\infty y^q(t)\varphi^{\omega}(t)\,dt
 +C_\varepsilon \int_0^\infty [\varphi(t)]^{(\omega-2-\frac{\omega}{q})
 \frac{q}{q-1}}|\phi(t)|^{\frac{q}{q-1}}\,dt\\
&=  \varepsilon \int_0^\infty y^q(t)\varphi^{\omega}(t)\,dt
 +C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
 |\phi(t)|^{\frac{q}{q-1}}\,dt,
\end{aligned}
\end{equation}
where $C_\varepsilon=(\varepsilon q)^{-\frac{q'}{q}}/q'$.
Note that thanks to the choice \eqref{EQ} of the parameter $\omega$, we have
$$
\int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt<\infty.
$$
Inequalities \eqref{FW} and \eqref{epsi} yield
\begin{equation}\label{ouf}
\begin{aligned}
&(1-\varepsilon)\int_0^\infty y^q(t)\varphi^\omega(t)\,dt  \\
&\leq  -y_1\varphi^\omega(0)+\omega y_0\varphi^{\omega-1}(0) \varphi'(0)\\
&\quad +C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt-\gamma(q,y_0,y_1) \int_0^\infty \varphi^\omega(t)
\,dt.
\end{aligned}
\end{equation}
Now, as a test function, we use
\begin{equation}\label{test}
\varphi(t)=\varphi_0(\frac{t}{T}),\quad T>0,
\end{equation}
where $\varphi_0$ is the classical cutoff function, that is,
$\varphi_0$ is a smooth decreasing function such that
\begin{gather*}
0\leq \varphi_0\leq 1, \quad |\varphi_0'(s)|\leq Cs^{-1},\quad  s>0,\\
\varphi_0(s)=\begin{cases}
1 &\text{if } 0\leq s\leq \frac{1}{2},\\
0 &\text{if } s\geq 1.
\end{cases}
\end{gather*}
In this case,
\begin{equation} \label{part}
\varphi^\omega(0)=1\quad\text{and}\quad \varphi'(0)=0.
\end{equation}
Moreover,
\begin{equation} \label{partint}
\int_0^\infty \varphi^\omega(t)\,dt
=T \int_0^{1} \varphi_0^\omega(s)\,ds
\geq T \int_0^{\frac{1}{2}} \varphi_0^\omega(s)\,ds=\frac{T}{2}.
\end{equation}
Combining \eqref{ouf}, \eqref{part} and \eqref{partint}, we obtain
$$
(1-\varepsilon)\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
\leq  -\Big(y_1+\frac{T}{2}\gamma(q,y_0,y_1)\Big)
+C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt.
$$
Taking $\varepsilon=1/q$, we obtain
$$
\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
\leq  -q'\Big(y_1+\frac{T}{2}\gamma(q,y_0,y_1)\Big)
+q'C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt.
$$
On the other hand, we have $q'C_\varepsilon=1$.
Therefore,
\begin{equation}\label{aya}
\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
\leq  -q'\Big(y_1+\frac{T}{2}\gamma(q,y_0,y_1)\Big)
+\int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}|\phi(t)|^{\frac{q}{q-1}}\,dt.
\end{equation}
Using the change of variable $t=Ts$, we obtain
\begin{equation} \label{YAALLAH}
\begin{aligned}
 &\int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}|\phi(t)|^{\frac{q}{q-1}}(t)\,dt\\
 &\leq\frac{1}{T^\frac{q+1}{q-1}} \omega^{\frac{2q}{q-1}}
 \int_0^\infty [\varphi_0(s)]^{\omega-\frac{2q}{q-1}}
\Big((\omega-1)\varphi_0'^2(s)+\varphi_0(s)|\varphi_0''(s)|\Big)^{\frac{q}{q-1}}\,ds\\
&=\frac{C'}{T^\frac{q+1}{q-1}},
\end{aligned}
\end{equation}
where
\begin{equation}\label{Sobhanak}
C'=\omega^{\frac{2q}{q-1}} \int_0^\infty [\varphi_0(s)]^{\omega-\frac{2q}{q-1}}
\Big((\omega-1)\varphi_0'^2(s)+\varphi_0(s)|\varphi_0''(s)|
\Big)^{\frac{q}{q-1}}\,ds>0.
\end{equation}
Then, by \eqref{aya}, we have
\begin{equation}\label{lim}
\int_0^\infty y^q(t)\varphi_0^\omega\big(\frac{t}{T}\big)\,dt
\leq  -q'\Big(y_1+\frac{T}{2}\gamma(q,y_0,y_1)\Big)
+\frac{C'}{T^\frac{q+1}{q-1}}.
\end{equation}
Passing to the limit as $T\to \infty$ and using the monotone convergence theorem,
we obtain
$$
\int_0^\infty y^q(t)\,dt\leq -\infty,
$$
which is a contradiction. Therefore, for any $q>1$, we have nonexistence
of a global solution for \eqref{PI}.
\end{proof}

\begin{remark} \label{rmk2} \rm
Observe that if $T^*$ is  the zero of the right-hand side of \eqref{lim},
then  the blow-up time for \eqref{PI} does not exceed $T^*$.
Moreover, we have the estimate
$$
T^*< \mathbb{T}=\Big(\frac{C'}{q'y_1}\Big)^{\frac{q-1}{q+1}},
$$
since the right-hand side of \eqref{lim} tends to $+\infty$ as
$T\to 0^+$, and it is negative for $t=\mathbb{T}$.
\end{remark}


\section{Differential inequalities with delayed arguments}\label{S3}


We start with  the second-order differential inequality
\begin{equation}\label{PII}
\begin{gathered}
\frac{d^2y(t)}{dt^2}\ge |y(t-\tau)|^q, \quad t>0,\\
y(t)=\psi(t),\quad t\in [-\tau,0],
\end{gathered}
\end{equation}
where $\tau> 0$, $q>1$ and $\psi$ is a given continuously differentiable
real-valued function on the initial interval $[-\tau, 0]$.
We have the following result.

\begin{theorem}\label{T2}
Let $q>1$. If
\begin{equation}\label{ass}
\psi'(0)+\int_{-\tau}^0 |\psi(t)|^q\,dt >0,
\end{equation}
then any solution of \eqref{PII} has a blow-up.
\end{theorem}

\begin{proof}
Suppose that $y$ is a global solution of \eqref{PII}.
 Let $\omega$ be a real number such that
$$
\omega>\frac{2q}{q-1}.
$$
As a test function, we take the function $\varphi^\omega$,
 where $\varphi$ is defined by \eqref{test}. By integration by parts, we obtain
\begin{equation}\label{ww1}
-\psi'(0)+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
\geq  \int_0^\infty |y(t-\tau)|^q \varphi^\omega(t)\,dt,
\end{equation}
where
$$
\phi(t)=\omega\Big((\omega-1)\varphi'^2(t)+\varphi(t)\varphi''(t)\Big),\quad t>0.
$$
On the other hand, for $T>2\tau$, we have
\begin{equation}\label{ESB}
\begin{aligned}
 \int_0^\infty |y(t-\tau)|^q \varphi^\omega(t)\,dt
&=  \int_{-\tau}^\infty |y(t)|^q \varphi^\omega(t+\tau)\,dt\\
&\geq  \int_{-\tau}^0|\psi(t)|^q\,dt
+\int_{0}^\infty |y(t)|^q\varphi^\omega(t)\,dt.
\end{aligned}
\end{equation}
Using \eqref{ww1} and \eqref{ESB}, we obtain
\begin{equation}\label{besmellah}
\int_0^\infty |y(t)|^q\varphi^\omega(t)\,dt
\leq -\Big(\psi'(0)+\int_{-\tau}^0|\psi(t)|^q\,dt\Big)
+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt.
\end{equation}
As in the proof of Theorem \ref{T1}, using the $\varepsilon$-Young inequality
 with parameters $q$ and $q'=\frac{q}{q-1}$, we obtain
\begin{equation}\label{hamdellah}
\int_0^\infty y(t)\varphi^{\omega-2}(t)\phi(t)\,dt
\leq \varepsilon \int_0^\infty |y(t)|^q\varphi^{\omega}(t)\,dt
+C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt,
\end{equation}
where $C_\varepsilon>0$. Combining \eqref{besmellah} and \eqref{hamdellah}, we obtain
\begin{align*}
&(1-\varepsilon)\int_0^\infty |y(t)|^q\varphi^\omega(t)\,dt \\
&\leq -\Big(\psi'(0)+\int_{-\tau}^0|\psi(t)|^q\,dt\Big)
+ C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt.
\end{align*}
Taking $0<\varepsilon<1$ and using \eqref{ass}, we obtain
\begin{equation}\label{YaRabb}
\psi'(0)+\int_{-\tau}^0|\psi(t)|^q\,dt
\leq C_\varepsilon\int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt.
\end{equation}
From \eqref{YAALLAH}, we have
$$
\int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}|\phi(t)|^{\frac{q}{q-1}}\,dt
\leq \frac{C'}{T^\frac{q+1}{q-1}},
$$
where $C'$ is given by \eqref{Sobhanak}. Therefore, by \eqref{YaRabb}, we obtain
$$
\psi'(0)+\int_{-\tau}^0|\psi(t)|^q\,dt
\leq \frac{C_\varepsilon C'}{T^\frac{q+1}{q-1}}.
$$
Passing to the limit as $T\to \infty$, we obtain
$$
\psi'(0)+\int_{-\tau}^0|\psi(t)|^q\,dt\leq  0
$$
which is a contradiction with \eqref{ass}.
\end{proof}

Now, we consider the differential inequality
\begin{equation}\label{PIII}
\begin{gathered}
\frac{d}{dt}\Big(\frac{dy(t)}{dt}+y(t)\Big)\ge |y(t-\tau)|^q, \quad t>0,\\
y(t)=\psi(t),\quad t\in [-\tau,0],
\end{gathered}
\end{equation}
where $\tau> 0$, $q>1$, and $\psi$ is a given continuously differentiable
real-valued function on the initial interval $[-\tau, 0]$.
We have the following result.

\begin{theorem}\label{T3}
Let $q>1$. If
\begin{equation}\label{asss}
\psi'(0)+\psi(0)+\int_{-\tau}^0 |\psi(t)|^q\,dt >0,
\end{equation}
then any solution of \eqref{PIII} has a blow-up.
\end{theorem}

\begin{proof}
Suppose that $y$ is a global solution of \eqref{PIII}. Let $\omega$ be a real
 number such that
$$
\omega>\frac{2q}{q-1}.
$$
As a test function, we take the function $\varphi^\omega$,
where $\varphi$ is defined by \eqref{test}. Integrating by parts, we obtain
\begin{equation}\label{Astarfirallah}
\begin{aligned}
&-\psi'(0)+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
-\psi(0)-\omega\int_0^\infty y(t)[\varphi(t)]^{\omega-1}\varphi'(t)\,dt \\
&\geq  \int_0^\infty |y(t-\tau)|^q \varphi^\omega(t)\,dt,
\end{aligned}
\end{equation}
where
$$
\phi(t)=\omega\Big((\omega-1)\varphi'^2(t)+\varphi(t)\varphi''(t)\Big),\quad t>0.
$$
From the proof of Theorem \ref{T2}, for $T>2\tau$, we have
$$
\int_0^\infty |y(t-\tau)|^q \varphi^\omega(t)\,dt
\geq  \int_{-\tau}^0|\psi(t)|^q\,dt+\int_{0}^\infty |y(t)|^q\varphi^\omega(t)\,dt.
$$
Therefore, from \eqref{Astarfirallah}, we obtain
\begin{equation} \label{Astarfirallah2}
\begin{aligned}
 &-\Big(\psi'(0)+\psi(0)+\int_{-\tau}^0|\psi(t)|^q\,dt\Big)
+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt\\
& -\omega\int_0^\infty y(t)[\varphi(t)]^{\omega-1}\varphi'(t)\,dt \\
& \geq  \int_0^\infty |y(t)|^q \varphi^\omega(t)\,dt.
\end{aligned}
\end{equation}
As in the proof of Theorem \ref{T2}, using the $\varepsilon$-Young
inequality with parameters $q$ and $q'=\frac{q}{q-1}$, we obtain
\begin{equation}\label{hamdellah2}
\begin{aligned}
&\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt\\
&\leq \varepsilon \int_0^\infty |y(t)|^q\varphi^{\omega}(t)\,dt
+C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt,
\end{aligned}
\end{equation}
where $C_\varepsilon>0$. Similarly, we obtain
\begin{equation}\label{hamdellah3}
\begin{aligned}
&-\omega\int_0^\infty y(t)[\varphi(t)]^{\omega-1}\varphi'(t)\,dt \\
&\leq \omega \varepsilon \int_0^\infty |y(t)|^q\varphi^{\omega}(t)\,dt
 +\omega C_\varepsilon \int_0^\infty [\varphi(t)]^{\omega-\frac{q}{q-1}}
 |\varphi'(t)|^{\frac{q}{q-1}}\,dt.
\end{aligned}
\end{equation}
Combining \eqref{Astarfirallah2}, \eqref{hamdellah2} and \eqref{hamdellah3},
 we obtain the estimate
\begin{equation}\label{AST}
\begin{aligned}
&\big(1-(\omega+1)\varepsilon\big) \int_0^\infty |y(t)|^q\varphi^{\omega}(t)\,dt\\
&\leq -\Big(\psi'(0)+\psi(0)+\int_{-\tau}^0|\psi(t)|^q\,dt\Big)
+C\Big(L_1(\varphi)+L_2(\varphi)\Big),
\end{aligned}
\end{equation}
where
\begin{gather*}
C =  \omega C_\varepsilon,\\
L_1(\varphi)=  \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
 |\phi(t)|^{\frac{q}{q-1}}\,dt,\\
L_2(\varphi)=  \int_0^\infty [\varphi(t)]^{\omega-\frac{q}{q-1}}
 |\varphi'(t)|^{\frac{q}{q-1}}\,dt.
\end{gather*}
Taking $0<\varepsilon<\frac{1}{\omega+1}$, we deduce from \eqref{AST} that
\begin{equation}\label{BISMILLAHARAHMAN}
\psi'(0)+\psi(0)+\int_{-\tau}^0|\psi(t)|^q\,dt \leq
C\Big(L_1(\varphi)+L_2(\varphi)\Big).
\end{equation}

\noindent $\bullet$ Estimate of $L_1(\varphi)$. From \eqref{YAALLAH}, we have
\begin{equation}\label{L1ph}
L_1(\varphi)\leq \frac{C'}{T^\frac{q+1}{q-1}},
\end{equation}
where $C'$ is given by \eqref{Sobhanak}.
\smallskip

\noindent $\bullet$ Estimate of $L_2(\varphi)$. Using the change of
variable $t=Ts$, we obtain
\begin{equation}\label{L2ph}
L_2(\varphi)=\frac{C''}{T^{q'-1}},
\end{equation}
where
$$
C''=\int_0^\infty [\varphi_0(s)]^{\omega-\frac{q}{q-1}}
|\varphi_0'(s)|^{\frac{q}{q-1}}\,ds.
$$

From \eqref{BISMILLAHARAHMAN}, the estimates \eqref{L1ph} and \eqref{L2ph},
we deduce that
$$
\psi'(0)+\psi(0)+\int_{-\tau}^0|\psi(t)|^q\,dt \leq
C\Big(\frac{C'}{T^\frac{q+1}{q-1}}+\frac{C''}{T^{q'-1}}\Big).
$$
Passing to the limit as $T\to \infty$, we obtain
$$
\psi'(0)+\psi(0)+\int_{-\tau}^0|\psi(t)|^q\,dt \leq
0,
$$
which is a contradiction with \eqref{asss}.
\end{proof}

The  argument used for the proof of Theorem \ref{T3} yields the
following blow-up result for the more general differential inequality
\begin{equation}\label{PIV}
\begin{gathered}
\frac{d}{dt}\Big(\frac{dy(t)}{dt}+y(t)\Big)\ge |t-\tau|^p |y(t-\tau)|^q,
 \quad t>0,\\
y(t)=\psi(t),\quad t\in [-\tau,0],
\end{gathered}
\end{equation}
where $\tau> 0$, $q>1$, $p\geq 0$, and $\psi$ is a given continuously
differentiable real-valued function on the initial interval $[-\tau, 0]$.

\begin{theorem}\label{T4}
Let $q>1$ and $p\geq 0$. If
$$
\psi'(0)+\psi(0)+\int_{-\tau}^0 |t|^p|\psi(t)|^q\,dt >0,
$$
then any solution of \eqref{PIV} has a blow-up.
\end{theorem}

\section{Systems of differential inequalities with advanced arguments}  \label{S4}

We consider the system of differential inequalities
\begin{equation}\label{SI}
\begin{gathered}
\frac{d^2y(t)}{dt^2} \ge  |z(t+\tau_1)|^p, \quad t>0,\\
\frac{d^2z(t)}{dt^2} \ge  |y(t+\tau_2)|^q, \quad t>0
\end{gathered}
\end{equation}
with the initial conditions
\begin{equation}\label{IN1}
y(0)=y_0>0, \quad y'(0)=y_1>0,\quad z(0)=z_0>0,\quad z'(0)=z_1>0,
\end{equation}
where $\tau_i>0$, $i=1,2$, $p>1$, and $q>1$.
We have the following result.

\begin{theorem}\label{TS1}
Let $p>1$ and $q>1$. Then a blow-up situation takes place for
problem \eqref{SI}-\eqref{IN1}.
\end{theorem}

\begin{proof}
Suppose that $(y,z)$ is a global solution of \eqref{SI}-\eqref{IN1}.
Let $\omega$ be a real number such that
$$
\omega>\max\big\{\frac{2p}{p-1}, \frac{2q}{q-1}\big\}.
$$
Then
\begin{gather*}
-y_1+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
\geq  \int_0^\infty |z(t+\tau_1)|^p \varphi^\omega(t)\,dt, \\
-z_1+\int_0^\infty z(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
\geq  \int_0^\infty |y(t+\tau_2)|^q \varphi^\omega(t)\,dt,
\end{gather*}
where $\varphi$ is the test function defined by \eqref{test} and
$$
\phi(t)=\omega\Big((\omega-1)\varphi'^2(t)+\varphi(t)\varphi''(t)\Big),\quad t>0.
$$
Applying the Taylor-Lagrange formula to the functions $|z(t+\tau_1)|^p$
and $|y(t+\tau_2)|^q$,  we obtain
\begin{align*}
|z(t+\tau_1)|^p
&=  z(t+\tau_1)^p\\
&=  z^p(t)+\tau_1 p z^{p-1}(t) z'(t)
 +\frac{p\tau_1^2}{2} \Big((p-1)z^{p-2}(t+\theta_1 \tau_1)z'^2(t+\theta_1\tau_1)\\
&\quad +z^{p-1}(t+\theta_1\tau_1)z''(t+\theta_1\tau_1)\Big)
\end{align*}
and
\begin{align*}
|y(t+\tau_2)|^q
&=  y(t+\tau_2)^q\\
&=  y^q(t)+\tau_2 q y^{q-1}(t) y'(t)
 +\frac{q\tau_2^2}{2} \Big((q-1)y^{q-2}(t+\theta_2 \tau_2)y'^2(t+\theta_2\tau_2)\\
&\quad +y^{q-1}(t+\theta_2\tau_2)y''(t+\theta_2\tau_2)\Big),
\end{align*}
where $\theta_i=\theta_i(t)\in (0,1)$, $i=1,2$. Therefore, by \eqref{SI}, we obtain
\begin{align*}
|z(t+\tau_1)|^p
&\geq  z^p(t)+\tau_1 p z_0^{p-1}z_1\\
&\quad +\frac{p\tau_1^2}{2} \Big((p-1)z_0^{p-2}z_1^2+
z_0^{p-1}y^q(t+\theta_1\tau_1+\tau_2)\Big)\\
&\geq  z^p(t)+\tau_1 p z_0^{p-1}z_1+\frac{p\tau_1^2}{2} \Big((p-1)z_0^{p-2}z_1^2+
z_0^{p-1}y_0^{q}\Big),
\end{align*}
i.e.,
$$
|z(t+\tau_1)|^p\geq z^p(t)+\mathcal{A},
$$
where
$$
\mathcal{A}=\tau_1 p z_0^{p-1}z_1+\frac{p\tau_1^2}{2} \Big((p-1)z_0^{p-2}z_1^2+
z_0^{p-1}y_0^{q}\Big)>0.
$$
Similarly,
$$
|y(t+\tau_2)|^q\geq y^q(t)+\mathcal{B},
$$
where
$$
\mathcal{B}=\tau_2 q y_0^{q-1}y_1+\frac{q\tau_2^2}{2} \Big((q-1)y_0^{q-2}y_1^2+
y_0^{q-1}z_0^{p}\Big)>0.
$$
Therefore, we have the system of integral inequalities
\begin{gather*}
  \int_0^\infty   z^p(t)\varphi^\omega(t)\,dt
\leq -y_1+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
  -\mathcal{A} \int_0^\infty
  \varphi^\omega(t)\,dt,\\
\int_0^\infty   y^q(t)\varphi^\omega(t)\,dt
\leq  -z_1+\int_0^\infty z(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
  -\mathcal{B} \int_0^\infty \varphi^\omega(t)\,dt.
 \end{gather*}
Using the $\varepsilon$-Young inequality with parameters $\varepsilon_1>0$,
  $q$, $\frac{q}{q-1}$, and the $\varepsilon$-Young inequality with parameters
$\varepsilon_2>0$,   $p$, $\frac{p}{p-1}$, we obtain
\begin{gather*}
\int_0^\infty   z^p(t)\varphi^\omega(t)\,dt
\leq -y_1+\varepsilon_1\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
  +C_{\varepsilon_1}L(q,\varphi)-\mathcal{A}\int_0^\infty
  \varphi^\omega(t)\,dt,\\
\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
\leq  -z_1+\varepsilon_2 \int_0^\infty z^p(t)\varphi^\omega(t)\,dt
  +C_{\varepsilon_2}L(p,\varphi)-\mathcal{B}\int_0^\infty
  \varphi^\omega(t)\,dt,
\end{gather*}
where
$$
L(q,\varphi)=\int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt.
$$
Combining the two  integral inequalities above, we obtain
\begin{align*}
&y_1+\varepsilon_1z_1+(1-\varepsilon_1\varepsilon_2)
 \int_0^\infty z^p(t)\varphi^\omega(t)\,dt\\
&\leq C_{\varepsilon_1}L(q,\varphi)
 +\varepsilon_1 C_{\varepsilon_2}L(p,\varphi)-
(\mathcal{B}\varepsilon_1+\mathcal{A})\int_0^\infty
  \varphi^\omega(t)\,dt.
\end{align*}
Following the proof of Theorem \ref{T1}, we have
\begin{gather*}
L(q,\varphi) \leq  \frac{C_1}{T^{\frac{q+1}{q-1}}}\,,\quad
L(p,\varphi) \leq  \frac{C_2}{T^{\frac{p+1}{p-1}}}\,,\\
\int_0^\infty \varphi^\omega(t)\,dt \geq  \frac{T}{2}\,,
\end{gather*}
where $C_i>0$, $i=1,2$, are constants. Therefore, taking
$0<\varepsilon_1\varepsilon_2<1$, we obtain the estimate
$$
y_1+\varepsilon_1z_1 \leq \frac{C_{\varepsilon_1}C_1}{T^{\frac{q+1}{q-1}}}+
\frac{\varepsilon_1C_{\varepsilon_2}C_2}{T^{\frac{p+1}{p-1}}}-
(\mathcal{B}\varepsilon_1+\mathcal{A})\frac{T}{2}\,.
$$
Passing to the limit as $T\to \infty$, we obtain
$y_1+\varepsilon_1z_1 \leq -\infty$
which is a contradiction.
\end{proof}

Now, we consider the system of differential inequalities
\begin{equation}\label{SII}
\begin{gathered}
\frac{d^2y(t)}{dt^2} \ge  |z(t+\tau_1)|^p, \quad t>0,\\
\frac{dz(t)}{dt} \ge  |y(t+\tau_2)|^q, \quad t>0
\end{gathered}
\end{equation}
with the initial conditions
\begin{equation}\label{IN2}
y(0)=y_0>0, \quad y'(0)=y_1>0,\quad z(0)=z_0>0,
\end{equation}
where $\tau_i>0$, $i=1,2$, $p>1$, and $q>1$.
We have the following result.

\begin{theorem}
Let $p>1$ and $q>1$. Then a blow-up situation takes place for
problem \eqref{SII}-\eqref{IN2}.
\end{theorem}

\begin{proof}
Suppose that $(y,z)$ is a global solution of \eqref{SII}-\eqref{IN2}.
Let $\omega$ be a real number such that
$$
\omega>\max\big\{\frac{p}{p-1}, \frac{2q}{q-1}\big\}.
$$
Then
\begin{align*}
-y_1+\int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
\geq  \int_0^\infty |z(t+\tau_1)|^p \varphi^\omega(t)\,dt, \\
-z_0+\omega\int_0^\infty z(t)[\varphi(t)]^{\omega-1}|\varphi'(t)|\,dt
\geq \int_0^\infty |y(t+\tau_2)|^q \varphi^\omega(t)\,dt,
\end{align*}
where $\varphi$ is the test function defined by \eqref{test} and
$$
\phi(t)=\omega\Big((\omega-1)\varphi'^2(t)+\varphi(t)\varphi''(t)\Big),\quad t>0.
$$
Applying the Taylor-Lagrange formula to the functions $|z(t+\tau_1)|^p$
and $|y(t+\tau_2)|^q$,  we obtain
\[
|z(t+\tau_1)|^p =  z(t+\tau_1)^p
=  z^p(t)+\tau_1 p z^{p-1}(t+\theta_1 \tau_1) z'(t+\theta_1 \tau_1)
\]
and
\begin{align*}
|y(t+\tau_2)|^q
&=  y(t+\tau_2)^q\\
&=  y^q(t)+\tau_2 q y^{q-1}(t) y'(t)
 +\frac{q\tau_2^2}{2} \Big((q-1)y^{q-2}(t+\theta_2 \tau_2)y'^2(t+\theta_2\tau_2)\\
&\quad +y^{q-1}(t+\theta_2\tau_2)y''(t+\theta_2\tau_2)\Big),
\end{align*}
where $\theta_i=\theta_i(t)\in (0,1)$, $i=1,2$.  Therefore, by \eqref{SII}, we obtain
$$
|z(t+\tau_1)|^p\geq z^p(t)+\mathcal{A},
$$
where
$$
\mathcal{A}=\tau_1 p z_0^{p-1}y_0^q >0.
$$
Similarly,
$$
|y(t+\tau_2)|^q\geq y^q(t)+\mathcal{B},
$$
where
$$
\mathcal{B}=\tau_2 q y_0^{q-1}y_1+\frac{q\tau_2^2}{2} \Big((q-1)y_0^{q-2}y_1^2+
y_0^{q-1}z_0^{p}\Big)>0.
$$
Therefore, we obtain the system of integral inequalities
\begin{gather*}
 \int_0^\infty z^p(t) \varphi^\omega(t)\,dt
\leq -y_1+  \int_0^\infty y(t)[\varphi(t)]^{\omega-2}\phi(t)\,dt
 -\mathcal{A} \int_0^\infty  \varphi^\omega(t)\,dt,\\
 \int_0^\infty   y^q(t)\varphi^\omega(t)\,dt
\leq  -z_0+\omega\int_0^\infty z(t)[\varphi(t)]^{\omega-1}|\varphi'(t)|\,dt
  -\mathcal{B} \int_0^\infty \varphi^\omega(t)\,dt.
 \end{gather*}
Using the $\varepsilon$-Young inequality with parameters $\varepsilon_1>0$,
  $q$, $\frac{q}{q-1}$, and the $\varepsilon$-Young inequality with parameters
$\varepsilon_2>0$,   $p$, $\frac{p}{p-1}$, we obtain
\begin{gather*}
  \int_0^\infty   z^p(t)\varphi^\omega(t)\,dt
\leq -y_1+\varepsilon_1\int_0^\infty y^q(t)\varphi^\omega(t)\,dt
  +C_{\varepsilon_1}L(q,\varphi)-\mathcal{A}\int_0^\infty
  \varphi^\omega(t)\,dt,\\
 \int_0^\infty   y^q(t)\varphi^\omega(t)\,dt
\leq  -z_0+\omega\varepsilon_2 \int_0^\infty z^p(t)\varphi^\omega(t)\,dt
  +\omega C_{\varepsilon_2}L_p(\varphi)-\mathcal{B}\int_0^\infty
  \varphi^\omega(t)\,dt,
\end{gather*}
where
\begin{gather*}
L(q,\varphi)= \int_0^\infty [\varphi(t)]^{\omega-\frac{2q}{q-1}}
|\phi(t)|^{\frac{q}{q-1}}\,dt,\\
L_p(\varphi) =  \int_0^\infty [\varphi(t)]^{\omega-\frac{p}{p-1}}
|\varphi'(t)|^{\frac{p}{p-1}}\,dt.
\end{gather*}
 Combining the above inequalities and taking
 $0<\varepsilon_1\varepsilon_2<\omega^{-1}$, we obtain
 $$
 y_1+\varepsilon_1 z_0\leq \omega \varepsilon_1C_{\varepsilon_2}L_p(\varphi)-
 (\varepsilon_1\mathcal{B}+\mathcal{A})\int_0^\infty \varphi^\omega(t)\,dt
 +C_{\varepsilon_1}L(q,\varphi),
 $$
which yields
$$
y_1+\varepsilon_1 z_0
\leq \frac{\omega \varepsilon_1C_{\varepsilon_2}C_1}{T^{\frac{1}{p-1}}}
-\frac{ (\varepsilon_1\mathcal{B}+\mathcal{A})T}{2}+
\frac{C_{\varepsilon_1}C_2}{T^{\frac{q+1}{q-1}}},
$$
for some constants $C_i>0$, $i=1,2$. Passing to the limit as $T\to \infty$, we
obtain
$y_1+\varepsilon_1 z_0\leq -\infty$
which is a contradiction.
\end{proof}

\subsection*{Acknowledgements}: 
The third authors extends his appreciation to Distinguished Scientist 
Fellowship Program (DSFP) at King Saud University (Saudi Arabia).


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\end{document}