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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 172, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/172\hfil Fractional ultra-parabolic equations]
{Cauchy problem for some fractional nonlinear ultra-parabolic equations}

\author[F. Al-Musalhi, S. Kerbal \hfil EJDE-2016/172\hfilneg]
{Fatma Al-Musalhi, Sebti Kerbal}

\address{Fatma Al-Musalhi \newline
Department of Mathematics and Statistics,
Sultan Qaboos University,
P.O. Box 36, Al-Khodh 123, Muscat, Oman}
\email{p070293@student.squ.edu.om}

\address{Sebti  Kerbal \newline
Department of Mathematics and Statistics,
Sultan Qaboos University,
P.O. Box 36, Al-Khodh 123, Muscat, Oman}
\email{skerbal@squ.edu.om}

\thanks{Submitted April 10, 2016. Published July 5, 2016.}
\subjclass[2010]{35A01, 26A33, 35K70}
\keywords{Nonexistence; nonlinear ultra-parabolic equations;
\hfill\break\indent nonlocal space and time operators}

\begin{abstract}
 Blowing-up  solutions to nonlocal nonlinear ultra-parabolic equations
 is presented. The obtained results will contribute in the development of
 ultra-parabolic equations and enrich the existing non-extensive literature
 on fractional nonlinear ultra-parabolic problems. Our method of proof
 relies on a suitable choice of a test function and the weak formulation
 approach of the sought for solutions.
\end{abstract}

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


\section{Introduction} \label{intro}

This article aims to  extend recent results by Kerbal and Kirane \cite{KerKir} 
by considering fractional in time and space nonlinear ultra-parabolic equations 
instead of classical ones.
Indeed, we will present a blow-up result for the nonlocal nonlinear 
ultra-parabolic 2-times equation
\begin{equation}\label{Eq1}
\mathcal{L}u:= u_{t_1}+ D_{0|t_2}^{\alpha}(|u|^{q}-|u_1|^{q}))+
 (-\Delta)^{\beta/2} (|u|^m) = |u|^p
\end{equation}
posed for $(t_1, t_2,  x) \in Q= \mathbb{R}_{+} \times \mathbb{R}_{+}
\times \mathbb{R}^{N}$, $N \in \mathbb{N}$ and supplemented with the initial 
conditions
\begin{equation}\label{C1}
u(t_1, 0; x)=u_1(t_1; x), \quad  u(0, t_2; x)=u_2(t_2; x).
\end{equation}
Here $p> m>1$, $p>q>1$ are real numbers and where  for $0<\alpha<1$ and 
$D^{\alpha}$ is the fractional derivative in the sense of Riemann-Liouville. 
Then, we extend our results to the system of two equations
\begin{gather}
u_{t_1}+ D_{0|t_2}^{\alpha_1}(|u|^s-|u_1|^s)+(-\Delta)^{\beta_1/2}(|u|^m)=|v|^{q}, 
\label{SE1} \\
v_{t_1}+ D_{0|t_2}^{\alpha_2}(|v|^r-|v_1|^r)+(-\Delta)^{\beta_2/2}(|v|^n)=|u|^p,
\label{SE2}
\end{gather}
posed for $(t_1, t_2, x) \in Q= \mathbb{R}^{+} \times \mathbb{R}^{+} 
\times \mathbb{R}^{N}$, $ N \in \mathbb{N}$, and supplemented with the 
initial conditions
\begin{gather}
u(t_1, 0; x)=u_1(t_1; x), \quad u(0, t_2; x)=u_2(t_2; x),\\
v(t_1, 0; x)=v_1(t_1; x), \quad v(0, t_2; x)=v_2(t_2; x).
\end{gather}
Here $p,q, r,s$, are positive real numbers and  
$0<\alpha_1,\alpha_2<1$,  $0<\beta_1,\beta_2\leq 2$.

The nonlocal operator $D_{0|t}^{\alpha}$ is defined, for a an absolutely 
continuous function
$f:\mathbb{R}_{+} \to  \mathbb{R}$, by
\[
  (D_{0|t}^{\alpha})f(t)=\frac{1}{\Gamma(1-\alpha)}
\frac{d}{dt}\int_0^{t}\frac{f(\sigma)}{(t-\sigma)^{\alpha}}d\sigma
\]
and $\Gamma(\alpha)=\int_0^{\infty}r^{\alpha -1}e^{-r}dr$
is the Euler gamma function. The fractional power of
the Laplacian 
$ (-\Delta)^{\beta/2}$ $(0<\beta \leq 2) $ stands for diffusion in media with
impurities and is defined as 
\[
 (-\Delta)^{\beta/2}v(x)= \mathcal{F}^{-1}
\Big(|\xi|^{\beta}\mathcal{F}(v)(\xi)\Big)(x),
\]
where $ \mathcal{F}$ denotes the Fourier transform and 
$  \mathcal{F}^{-1}$ denotes its inverse and the operator $D_{0|t}^{\alpha}$ 
counts for the anomalous diffusion, a recently very much studied topic 
in probability, physics, chemistry, biology, image processing, etc, see for instance
\cite{ Abd,BFW,CC,DMRV,FurKir,Herrmann,Hilfer,Ju,KirLasTat,LP,Magin,SamKil}
and their references. Classical multi-time or ultra-parabolic problems have 
received a special interest and attention by authors due to their 
application in real life problems, see for example  
\cite{KenAbdBer,KerKir,LanPasPol,Ter, Wal}, while the fractional analog 
are in their preliminary steps.

\section{Preliminaries}

Here, we  need the right-hand fractional derivative in the sense
 of Riemann-Liouville
\[
  (D_{t|T}^{\alpha})f(t)=-\frac{1}{\Gamma(1-\alpha)}\frac{d}{dt}
\int_{t}^{T}\frac{f(\sigma)}{(\sigma-t)^{\alpha}}d\sigma,
\]
for an absolutely continuous function $f:\mathbb{R}_{+}  \to  \mathbb{R}$.
Note that for a differentiable function $f$, we have the so-called 
Caputo's fractional derivative
\[
  D_{0|t}^{\alpha}(f-f(0))(t)=\frac{1}{\Gamma(1-\alpha)}
\int_0^{t}\frac{f' (\sigma)}{(\sigma-t)^{\alpha}}d\sigma.
\]
It is shown in \cite[Corollary 2, p.46]{SamKil} that for $f,g$ possessing 
appropriate regularity,
the formula of integration by parts holds true
\[
\int_0^{T}f(t) D^{\alpha}_{0|t}g(t)dt=\int_0^{T}g(t)D^{\alpha}_{t|T}f(t)dt.
\]

We also need some preparatory lemmas  based on the function $\phi$ defined by
\begin{equation}\label{Eq2}
 \phi(t)=\begin{cases}
\big(1-\frac{t}{T}\big)^{\lambda}, & 0\leq t\leq T,\\
0,& t> T,
 \end{cases}
\end{equation}
where  $\lambda \geq 2$. 

\begin{lemma} \label{lem2.1}
 Let $\phi$ be as in \eqref{Eq2}. We have
\begin{equation}
 \int_0^{T}D^{\alpha}_{t,T}\phi(t)dt=C_{\alpha,\lambda}T^{1-\alpha},
\end{equation}
where
\[
{ C_{\alpha,\lambda}=\frac{\lambda\Gamma(\lambda-\alpha)}
{(\lambda-\alpha+1)\Gamma(\lambda-2\alpha+1)}}.
\]
\end{lemma}

For a proof of the above lemma, see \cite{KirLasTat,FurKir}.

\begin{lemma}  \label{lem2.2}
Let $\phi$ be as in \eqref{Eq2} and $p>1$. Then for $p<\lambda +1$,
\[
 \int_0^{T}\phi^{1-p}(t)|\phi^{'}(t)|^p=C_{p}T^{1-p},
\]
where 
\[
 C_{p}=\frac{\lambda^p}{1+\lambda-p}.
\]
For $\lambda>\alpha p-1$,
\begin{equation*}
 \int_0^{T}\phi(t)^{1-p}| D^{\alpha}_{t,T}\phi(t)|^pdt
=C_{p,\alpha}T^{1-\alpha p},
\end{equation*}
where  
\[
 C_{p,\alpha}=\frac{\lambda^p}{(\lambda+1-p\alpha)}
\big\{\frac{\Gamma(\lambda-\alpha)}{\Gamma(\lambda-2\alpha+1)}\big\}^p.
\]
\end{lemma}

For a proof of the above lemma, see \cite{KirLasTat,FurKir}.
We define the regular function $\psi$:
\begin{equation}\label{Eq7}
\psi(\xi)=\begin{cases}
1,&\text{if } 0 \leq \xi \leq 1,\\
\text{decreasing},&\text{if } 1 \leq \xi \leq 2,\\
0,&\text{if } \xi \geq 2,
\end{cases}
\end{equation}
which will be used hereafter.

\section{Results}

Solutions to  \eqref{Eq1} subject to conditions \eqref{C1} are meant in the
following weak sense.

\begin{definition} \rm
A function $u\in L^m(Q)\bigcap
L^p(Q)$ is called a weak solution to \eqref{Eq1} if
\begin{equation}\label{WFEQ1}
\begin{aligned}
&\int_{Q} |u|^p \, \varphi \, dP+
\int_{S} u(0,t_2;x) \varphi (0, t_2 ;x) \, dP_2+\int_{Q}|u(t_1,0;x)|^q 
 D_{t_2|T}^{\alpha}\varphi \,dP   \\
&= - \int_{Q} u \varphi_{t_1}\, dP +\int_{Q} |u|^q \,D_{t_2|T}^{\alpha}\varphi\, dP
 + \int_{Q} |u|^m \, (-\Delta)^{\beta/2} \varphi \, dP
\end{aligned}
\end{equation}
for any test function $\varphi \in C^{\infty}_0(Q)$; 
$ S= \mathbb{R}_{+} \times \mathbb{R}^{N}$,
$P=(t_1, t_2, x)$ and $ P_2=(t_2, x)$, such that 
$\varphi(T,t_2;x)=\varphi(t_1,T;x)=0$.
\end{definition}

Note that every weak solution is a classical solution near the points 
$ (t_1, t_2, x)$ where $u(t_1, t_2, x)$ is positive.

Our main result dealing with  equation \eqref{Eq1}  subject to 
\eqref{C1} is given by the following theorem.

\begin{theorem} \label{thm3.2}  Assume that
\[
{{ \int_{S} u(0,t_2;x) \varphi (0, t_2 ;x)dP_2>0,\quad
\int_{Q}|u(t_1,0;x)|^q D_{t_2|T}^{\alpha}\varphi \,dP>0}} .
\]
If $ 1< p\leq \min\big(1+\frac{1}{N+1}, q(1+\frac{\alpha}{N+2-\alpha}) ,
m(1+\frac{\beta}{N+2-\beta})  \big)$,
 then Problem \eqref{Eq1}-\eqref{C1} does not admit  global weak solutions.
\end{theorem}

For the proof, we need to recall the following proposition from 
\cite[proposition 3.3]{Ju}.

\begin{proposition}[\cite{Ju}] \label{tabsP1} 
Suppose that $\delta \in [0,2]$, $\beta+1\geq 0$, and
${ \theta \in {\mathcal{C}}^{\infty}_0({R}^{N})} $.
Then, the following point-wise inequality holds:
$$
|\theta(x)|^{\beta}\theta(x)(-\Delta)^{\delta/2} \theta(x)\geq 
\frac{1}  {\beta+2}(-\Delta)^{\delta/2} |\theta(x)|^{\beta+2}.
$$
\end{proposition}

\begin{proof}[Proof of Theorem \ref{thm3.2}]
Our strategy of proof is to use the weak formulation  of the solution 
with a suitable choice  of the test function (see for example \cite{MitPok}). 
We assume that the solution is nontrivial and global.
We choose the test function $\varphi(t_1,t_2,x)$ in the form
\begin{equation}\label{testft}
\varphi(t_1,t_2;x)= \varphi_1(t_1)\varphi_2(t_2)\varphi_3(x)
\end{equation}
where  $\varphi_1(t_1)=\psi(t_1/T)$,
 $\varphi_2(t_2)=(1-t_2/T)^{\lambda}$   and
 $\varphi_3(x)=\psi(| x|^{2}/T^{2})$.

Now, replacing $\varphi$ by $\varphi^{\mu}$ in \eqref{WFEQ1}, we estimate 
$ \int_{Q_T} u  \varphi_{t_1}^{\mu} dP$ using
the $\varepsilon$-Young inequality as follows
\begin{equation}\label{Est1}
\int_{Q}| u |\, |\varphi_{t_1}^{\mu}| \, dP \leq \varepsilon \int_{Q} |u|^p 
\varphi^{\mu} \, dP
 + C_{\varepsilon} \int_{Q} \varphi^{\mu-\frac{p}{p-1}}
 |\varphi_{t_1}|^{\frac{p}{p-1}} \, dP.
\end{equation}
Similarly, we have
\begin{equation}\label{Est2}
\int_{Q} |u|^{q}  D_{t_2|T}^{\alpha}\varphi^{\mu} \,dP
\leq \varepsilon \int_{Q} |u|^p  \varphi^{\mu} \, dP 
+ C_{\varepsilon} \int_{Q}|D_{t_2|T}^{\alpha} \varphi^{\mu}|^{\frac{p}{p-q}}
\varphi^{-\frac{\mu q}{p-q}} \,dP,
\end{equation}
where $p>q$.
Observe that
\begin{equation}\label{Est3}
\begin{aligned}
&\int_{Q} |u(t_1,0;x)|^q D_{t_2|T}^{\alpha}\varphi^{\mu} \,dP\\
&=\Big(\int_0^{T}D_{t_2|T}^{\alpha}\varphi_2^{\mu}(t_2) dt_2\Big) 
\int_{S}|u(t_1,0;x)|^q \varphi_3^{\mu}(x) \varphi_1^{\mu}(t_1) \,dP_1
\end{aligned}
 \end{equation}
 with the help of Lemma \ref{lem2.1} one can rewrite the equation \eqref{Est3} as
\begin{equation}\label{Est4}
 \int_{Q} |u(t_1,0;x)|^q D_{t_2|T}^{\alpha}\varphi^{\mu} \,dP\\
 =C_{\alpha,\lambda\mu}\; T^{1-\alpha} \int_{S}|u(t_1,0;x)|^q 
\varphi_3^{\mu}(x)\varphi_1^{\mu}(t_1) \,dP_1,
 \end{equation}
where $P_1=(t_1, x)$.
 Using the convexity inequality in proposition \ref{tabsP1}  and the 
$\varepsilon$-Young inequality, the last term in the right hand side of 
equation \eqref{WFEQ1} can be estimated by
 \begin{equation}\label{Est5}
 \begin{aligned}
&\int_{Q} |u|^m \, (-\Delta)^{\beta/2} \varphi^{\mu} \, dP\\
&\leq\int_{Q}\mu \varphi^{\mu-1}|u|^m  (-\Delta)^{\beta/2} \varphi \, dP\\
&\leq \varepsilon\int_{Q}\varphi^{\mu} |u|^p \,dP
+C(\varepsilon)\int_{Q} |(-\Delta)^{\beta/2} \varphi|^{\frac{p}{p-m}}
 \varphi^{(\mu-1-\frac{m\mu}{p})\frac{p}{p-m}} \, dP.
\end{aligned}
\end{equation}
Now, using \eqref{Est1}, \eqref{Est2}, \eqref{Est3}, and \eqref{Est5}, we obtain
\begin{equation}\label{EstF1}
\begin{aligned}
&\int_{Q} |u|^p  \varphi^{\mu} \, dP 
+\int_{S } u(0, t_2;x) \varphi^{\mu}(0,t_2;x) \, dP_2 \\
&\ + C_{\alpha,\lambda\mu} T^{1-\alpha} 
 \int_{S}|u(t_1,0;x)|^q \varphi_3^{\mu}(x)\varphi_1^{\mu}(t_1) \,dP_1 \\
&\leq 3 \varepsilon \int_{Q} |u|^p \, \varphi^{\mu} \, dP  
+ C_{\varepsilon} \Big(\int_{Q_T} \varphi^{\mu-\frac{p}{p-1}}
 |\varphi_{t_1}|^{\frac{p}{p-1}} \, dP \\
&\quad +\int_{Q}|D_{t_2|T}^{\alpha}\varphi^{\mu}|^{\frac{p}{p-q}}
\varphi^{-\frac{\mu q}{p-q}} \,dP\\
 &\quad +\int_{Q} |(-\Delta)^{\beta/2} \varphi|^{\frac{p}{p-m}} 
\varphi^{(p(\mu-1)-m\mu)\frac{1}{p-m}} \, dP\Big).
   \end{aligned}
\end{equation}
If we choose $ \varepsilon=1/6 $ (for example), then we obtain the estimate
\begin{equation}\label{EstF2}
\begin{aligned}
&\int_{Q} |u|^p \, \varphi^{\mu} \, dP +
 2\int_{S } u(0, t_2;x) \varphi^{\mu}(0,t_2;x) \, dP_2\\
&+ C_{\alpha,\lambda\mu} T^{1-\alpha} \int_{S}|u(t_1,0;x)|^q
  {\varphi_3^{\mu}(x)}\varphi_1^{\mu}(t_1) \,dP_1 \\
&\leq  C \Big(\int_{Q} \varphi^{\mu-\frac{p}{p-1}}
 |\varphi_{t_1}|^{\frac{p}{p-1}} \, dP 
+\int_{Q}|D_{t_2|T}^{\alpha}\varphi^{\mu}|^{\frac{p}{p-q}}
\varphi^{-\frac{\mu q}{p-q}} \,dP\\
 &\quad +\int_{Q} |(-\Delta)^{\beta/2} \varphi|^{\frac{p}{p-m}} 
\varphi^{(p(\mu-1)-m\mu)\frac{1}{p-m}} \, dP\Big)
\end{aligned}
\end{equation}
for some positive constant $C$. The right hand side of \eqref{EstF2}
is now free of the unknown function $u$.
Let us now pass to the new variables
\begin{equation}\label{scaledvariables}
\tau_1=T^{-1}t_1 , \quad \tau_2=T^{-1}t_2, \quad y=T^{-1}x.
\end{equation}
We have
\begin{equation}\label{Eq10}
\begin{aligned}
\int_{Q} \varphi^{\mu-\frac{p}{p-1}}\, |\varphi_{t_1}|^{\frac{p}{p-1}} \, dP
&= \Big(\int_{S} \varphi_2^{\mu} \varphi_3^{\mu} dP_2\Big)
\Big(\int_0^{T}\varphi_1^{\mu-\frac{p}{p-1}}\,| \varphi_{1,t_1}|^{\frac{p}{p-1}}
dt_1\Big) \\
&= C_1 T^{2+N -\frac{p}{p-1}}
\end{aligned}
\end{equation}
where   
\[
{ C_1=\Big(\int_{\Omega_2}  \varphi_2^{\mu} \varphi_3^{\mu}
 dP_{\tau_2}\Big) \Big(\int_0^{1}\psi^{\mu-\frac{p}{p-1}}
| \psi_{\tau_1}|^{{\frac{p}{p-1}}}d\tau_1\Big)<\infty   }
\]
 with $\mu>\frac{p}{p-1}$ and $P_{\tau_2}=(\tau_2,y)$,
$\Omega_2=\{ 1\leq \tau_2+| y|\leq 2 \}$.
Similarly, we obtain
\begin{equation} \label{Eq11}
\begin{aligned}
&\int_{Q}|D_{t_2|T}^{\alpha}\varphi^{\mu}|^{\frac{p}{p-q}}\varphi^{\frac{\mu q}{q-p}}
\,dP \\
& = \Big(\int_{S}\varphi_1^{\mu}\varphi_3^{\mu} dP_1\Big)
\Big(\int_0^{T} \varphi_2^{-{\frac{\mu q}{p-q}}}  |D_{t_2|T}^{\alpha} 
\varphi_2^{\mu}|^{\frac{p}{p-q}}dt_2\Big) \\
& = C_2 T^{2+N -\frac{\alpha p}{p-q}}
\end{aligned}
\end{equation}
where 
\[
 C_2=\Big(\int_{\Omega_1}\varphi_1^{\mu}\varphi_3^{\mu} dP_{\tau_1}\Big)
\Big(\int_0^{1} \varphi_2^{-{\frac{\,\mu q}{p-q}}}  |D_{\tau_2}^{\alpha} 
\varphi_2^{\mu}|^{\frac{p}{p-q}}d\tau_2\Big)
<\infty
\]
 and $P_{\tau_1}=(\tau_1,y)$,
$\Omega_1=\{ 1\leq \tau_1+| y|\leq 2 \}$, 
and
\begin{equation}\label{Eq12}
\begin{aligned}
&\int_{Q} |(-\Delta)^{\beta/2} \varphi|^{\frac{p}{p-m}} \varphi^{(p(\mu-1)-m\mu)
\frac{1}{p-m}} \, dP \\
&= \Big(\int_{\mathbb{R}^{N}} |(-\Delta)^{\beta/2} \varphi_3|^{\frac{p}{p-m}} 
\varphi_3^{(p(\mu-1)-m\mu)\frac{1}{p-m}} dx\Big)
\Big(\int_{Q_T}\varphi_1^{\mu} \varphi_2^{\mu}\, dt_1\,dt_2 \Big) \\
&= C_3 T^{2+N -\frac{\beta p}{p-m}}
\end{aligned}
\end{equation}
where 
\[
 C_3= \int_{\operatorname{support}\psi)} |(-\Delta_{y})^{\beta/2} 
\psi|^{\frac{p}{p-m}} 
\psi^{(p(\mu-1)-m\mu)\frac{1}{p-m}} dy
\int_{Q_{T}}\varphi_1^{\mu} \varphi_2^{\mu}\, d\tau_1\,d\tau_2 <\infty  
\]
with $\mu>\frac{p}{p-m}$ and ${Q_{T}}=[0,T]\times [0,T]$.
By  \eqref{Eq10}-\eqref{Eq12}, we obtain for \eqref{EstF2} the following estimate
\begin{equation}\label{Eq13}
\begin{aligned}
&\int_{Q} |u|^p \, \varphi^{\mu} \, dP 
 + 2\int_{S } u(0, t_2;x) \varphi^{\mu}(0,t_2;x) \, dP_2\\
&+ C_{\alpha,\lambda\mu} T^{1-\alpha} \int_{S}|u(t_1,0;x)|^q 
{\varphi_3^{\mu}(x)}\varphi_1^{\mu}(t_1) \,dP_1 \\
&\leq C_1 T^{2+N -\frac{p}{p-1}}+C_2 T^{2+N -\frac{\alpha p}{p-q}}
+C_3 T^{2+N -\frac{\beta p}{p-m}},
 \end{aligned}
\end{equation}
 then
\begin{equation}\label{Eq14}
\begin{aligned}
&\int_{Q} |u|^p  \varphi^{\mu} \, dP +2\int_{S } u(0, t_2;x) 
\varphi^{\mu}(0,t_2;x) \, dP_2 \\
& +C_{\alpha,\lambda\mu} T^{1-\alpha}\int_{S}|u(t_1,0;x)|^q 
 {\varphi_3^{\mu}(x)}\varphi_1^{\mu}(t_1) \,dP_1\\
&\leq {\tilde{C}}\Big( T^{2+N -\frac{p}{p-1}}
 +T^{2+N -\frac{\alpha p}{p-q}} +T^{2+N -\frac{\beta p}{p-m}}\Big)
 \end{aligned}
\end{equation}
where ${\tilde{C}}=\max\{ C_1, C_2,C_3 \}$.
Now, for the first case, we require:
\begin{itemize}
\item[(a)] $ 2+N -\frac {p}{p-1}<0$  or
 $ 1<p\leq 1+\frac{1}{N+1}$, for $p>q$ and $m>1$.

\item[(b)] $ 2+N -\frac{\alpha p}{p-q}<0$  or 
 $ 1<p\leq q(1+\frac{\alpha}{N+2-\alpha})$, for $p>m>1$.

\item[(c)] $ 2+N -\frac{\beta p}{p-m}<0$  or
 $ 1<p\leq m\big(1+ \frac{\beta}{N+2-\beta}\big)$.
\end{itemize}
 Letting $T$ aproach infinity in \eqref{Eq14}, we obtain a contradiction 
as the left hand side is positive while the right hand side goes to zero. 

 For the second case, we assume the  exponents of $T$ in \eqref{Eq14} are zeros.
Applying H\"older's inequality to the right hand side of inequality \eqref{EstF2},
 we obtain
\begin{equation}\label{Eq15}
\begin{aligned}
&\int_{Q} |u|^p \, \varphi^{\mu} \, dP +
 2\int_{S } u(0, t_2;x) \varphi^{\mu}(0,t_2;x) \, dP_2\\
&+ C_{\alpha,\lambda\mu} T^{1-\alpha} \int_{S}|u(t_1,0;x)|^q 
 {\varphi_3^{\mu}(x)}\varphi_1^{\mu}(t_1) \,dP_1\\
&\leq \Big( \int_{C_T} |u|^p  \varphi^{\mu} \, dP \Big)^{1/p}
 C(\varphi)
\end{aligned}
\end{equation}
where 
\begin{align*}
C(\varphi)
&= C \Big(\int_{Q} \varphi^{-\frac{\mu}{p-1}}
 |\varphi_{t_1}^{\mu}|^{\frac{p}{p-1}} \, dP 
+\int_{Q}|D_{t_2|T}^{\alpha}\varphi^{\mu}|^{\frac{p}{p-q}}
\varphi^{-\frac{\mu q}{p-q}} \,dP\\
&\quad+\int_{Q} |(-\Delta)^{\beta/2} \varphi|^{\frac{p}{p-m}} 
\varphi^{(p(\mu-1)-m\mu)\frac{1}{p-m}} \, dP\Big).
\end{align*}
Whereupon, using Lebesgue's dominated convergence theorem we have
\begin{equation*}
\int_{Q}|u|^p\varphi\, dP\leq {\tilde{C}}
\;\Longrightarrow\;\lim_{T\to\infty} {\int_{C_T}|u|^p dP}=0,
\end{equation*}
where $C_T=\{ (t_1,t_2,x)|\; T \leq t_1+t_2+|x|\leq 2 T\}$.

Then, letting $T$ appraoch infinity in \eqref{Eq15}, the right-hand side approaches
 zero, which is again contradiction.
\end{proof}

\section{A $2\times 2$ system with a 2-dimensional fractional time}

We consider
\begin{gather}\label{se1}
 u_{t_1}+ D_{0|t_2}^{\alpha_1}(|u|^s-|u_1|^s)+(-\Delta)^{\beta_1/2}(|u|^m)=|v|^{q},\\
\label{se2}
v_{t_1}+ D_{0|t_2}^{\alpha_2}(|v|^r-|v_1|^r)+(-\Delta)^{\beta_2/2}(|v|^n)=|u|^p,
\end{gather}
posed for $(t_1, t_2, x) \in Q= \mathbb{R}^{+} \times \mathbb{R}^{+}
 \times \mathbb{R}^{N}$, $N \in \mathbb{N}$, and
supplemented with the initial conditions
\begin{gather}
u(t_1, 0; x)=u_1(t_1; x), \quad u(0, t_2; x)=u_2(t_2; x),\\
v(t_1, 0; x)=v_1(t_1; x), \quad v(0, t_2; x)=v_2(t_2; x).
\end{gather}
Here $p,q, r,s$, are positive real numbers and  $0<\alpha_1,\alpha_2<1$,
$0<\beta_1,\beta_2\leq 2$.
Let us set
\begin{gather*}
I_0=\int_{S} u_2 (0,t_2,x)\varphi(0,t_2,x) dP_2
+\int_{Q}|u_1|^s D_{t_2|T}^{\alpha_1}\varphi\,dP\\
J_0=\int_{S} v_2 (0,t_2,x)\varphi(0,t_2,x) dP_2
+\int_{Q}|v_1|^r D_{t_2|T}^{\alpha_2}\varphi\,dP
\end{gather*}

\begin{definition} \label{def4.1} \rm
We say that $(u,v)\in (L^p\cap L^m) \times (L^{q}\cap L^{n})$ is a weak 
formulation to system \eqref{se1}-\eqref{se2} if
\begin{equation} \label{wfs}
\begin{gathered}
\int_{Q}|v|^{q}\varphi\,dP+I_0
= -\int_{Q}u\, \varphi_{t_1} \,dP
 +\int_{Q}|u|^s\,D_{t_2|T}^{\alpha_1}\varphi\,dP
 +\int_{Q}|u|^m\,(-\Delta)^{\beta_1/2}\varphi\,dP
\\
\int_{Q}|u|^p\varphi\,dP+J_0
=-\int_{Q}v\, \varphi_{t_1} \,dP+\int_{Q}|v|^r\,D_{t_2|T}^{\alpha_2}\varphi\,dP
+\int_{Q}|v|^{n}\,(-\Delta)^{\beta_2/2}\varphi\,dP
\end{gathered}
\end{equation}
for any test function $\varphi\in C_0^{\infty}$.
Now, set
\begin{gather*}
\sigma_1=-\frac{ q[  1-p(N+1)] +N+2 }{pq-1},\\
\sigma_2=-\frac{ q[ \alpha_1-p(N+1)] + r(N+2) }{pq-r},\\
\sigma_3=-\frac{ q[ \beta_1-p(N+1)] + n(N+2) }{pq-n},\\
\sigma_4=-\frac{ q[ s-p(N+2-\alpha_1)  ] + s(N+2) }{pq-s},\\
\sigma_5=-\frac{ q[ s \alpha_2-p(N+2-\alpha_1)  ] + s r(N+2) }{pq-sr},\\
\sigma_6=-\frac{ q[ s \beta_2-p(N+2-\alpha_1)  ] + s n(N+2) }{pq-s n},\\
\sigma_7=-\frac{ q[m-p(N+2-\beta_1)  ] + m(N+2) }{pq-m},\\
\sigma_8=-\frac{ q[m \alpha_2 -p(N+2-\beta_1)  ] + r m(N+2) }{pq-rm}\\
\sigma_9=-\frac{ q[m \beta_2-p(N+2-\beta_1)  ] + nm(N+2) }{pq-nm}.
\end{gather*}
\end{definition}

\begin{theorem} \label{thm4.2}
Let $p>1$, $q>1$, $p> m$, $p>s$, $q>n$, $q>r$ and assume that
\begin{gather*}
\int_{S} u_2 (0,t_2,x)\varphi^{\mu}(0,t_2,x) dP_2>0,\quad
\int_{Q}|u_1|^s D_{t_2|T}^{\alpha_1}\varphi^{\mu}\,dP>0,\\
\int_{S} v_2 (0,t_2,x)\varphi^{\mu}(0,t_2,x) dP_2>0,\quad 
\int_{Q}|v_1|^r D_{t_2|T}^{\alpha_2}\varphi^{\mu}\,dP>0,
\end{gather*}
then solutions to system \eqref{se1}-\eqref{se2}  blow-up  whenever
$$
\max\{ \sigma_1,\dots,\sigma_9;\,\delta_1,\dots,\delta_9\} \leq 0.
 $$
\end{theorem}

\begin{proof}[Proof of theorem \ref{thm4.2}]
Assume that the solution is nontrivial and global.  
Next, replacing $\varphi$ by $\varphi^{\mu}$ in \eqref{wfs} and then using 
H\"older's inequality to estimate the RHS, we obtain the following estimates:

\noindent $\bullet$ For $p>1$,
\begin{equation}\label{es1}
-\int_{Q}u \varphi_{t_1}^{\mu} \,dP\leq \mu \Big( \int_{Q}|u|^p\varphi^{\mu}\, dP\Big)^{1/p}
\Big(\int_{Q} \varphi^{\mu-\frac{p}{p-1}}\, |\varphi_{t_1}|^{\frac{p}{p-1}} \,dP
\Big)^{\frac{p-1}{p}}.
\end{equation}

\noindent$\bullet$ For $p>s$,
\begin{equation}\label{es2}
\int_{Q}|u|^s D_{t_2|T}^{\alpha_1}\varphi^{\mu}\,dP
\leq\Big( \int_{Q} |u|^p\varphi^{\mu} dP\Big)^{s/p}
\Big(\int_{Q} \varphi^{-\frac{s\mu}{p-s}} \,|D_{t_2|T}^{\alpha_1}
\varphi^{\mu}|^{\frac{p}{p-s}}dP\Big) ^{\frac{p-s}{p}}.
\end{equation}

\noindent$\bullet$ For $p>m$,
\begin{equation}\label{es3}
\int_{Q}|u|^m(-\Delta)^{\frac{\beta_1}{2}}\varphi^{\mu}
\leq\mu\Big( \int_{Q}|u|^p\varphi^{\mu}\Big)^{\frac{m}{p}} 
\Big( \int_{Q}\varphi^{\mu-\frac{p}{p-m}}|(-\Delta)^{\frac{\beta_1}{2}}
\varphi|^{\frac{p}{p-m}}\Big) ^{\frac{p-m}{p}}.
\end{equation}

Similarly, we have

\noindent$\bullet$ For $q>1$,
\begin{equation}\label{es4}
 -\int_{Q}v \varphi_{t_1}^{\mu} \,dP\leq \mu
\Big(  \int_{Q}|v|^{q}\varphi^{\mu}\, dP \Big)^{\frac{1}{q}} 
\Big( \int_{Q} \varphi^{\mu-\frac{q}{q-1}}\, |\varphi_{t_1}|^{\frac{q}{q-1}} \,dP
\Big)^{\frac{q-1}{q}} .
\end{equation}

\noindent$\bullet$ For $q>r$,
\begin{equation}\label{es5}
\int_{Q}|v|^r\, \,D_{t_2|T}^{\alpha_2}\varphi^{\mu}\,dP
\leq\Big(  \int_{Q} |v|^q\varphi^{\mu} dP\Big)^{\frac{r}{q}}
\Big(\int_{Q} \varphi^{-\frac{r\mu}{q-r}} \,|D_{t_2|T}^{\alpha_2}
\varphi^{\mu}|^{\frac{q}{q-r}}dP\Big)^{\frac{q-r}{q}}.
\end{equation}

\noindent$\bullet$ For $q>n$
\begin{equation}\label{es6}
\int_{Q}|v|^{n}(-\Delta)^{\frac{\beta_2}{2}}\varphi^{\mu}
\leq\mu\Big(  \int_{Q}|v|^{q}\varphi^{\mu}\Big)^{\frac{n}{q}} 
\Big(\int_{Q}\varphi^{\mu-\frac{q}{q-n}}|
(-\Delta)^{\frac{\beta_2}{2}}\varphi|^{\frac{q}{q-n}}\Big)^{\frac{q-n}{q}}.
\end{equation}

If we set
\begin{gather*}
I_{u}:=\int_{Q} |u|^p\varphi^{\mu} dP,\quad
I_{v}:=\int_{Q}|v|^{q}\varphi^{\mu}\,dP,\\
A(p)=\mu\Big( \int_{Q} \varphi^{\mu-\frac{p}{p-1}} |\varphi_{t_1}|^{\frac{p}{p-1}} 
\,dP\Big)^{\frac{p-1}{p}},\\
 A(q)=\mu\Big( \int_{Q} \varphi^{\mu-\frac{q}{q-1}} |\varphi_{t_1}|^{\frac{q}{q-1}} 
\,dP\Big) ^{\frac{q-1}{q}},\\
B(p,s)=\Big(\int_{Q} \varphi^{-\frac{s\mu}{p-s}} 
|D_{t_2|T}^{\alpha_1}\varphi^{\mu}|^{\frac{p}{p-s}}dP\Big) ^{\frac{p-s}{p}},\\ 
B(q,r)=\Big( \int_{Q} \varphi^{-\frac{r\mu}{q-r}} 
 |D_{t_2|T}^{\alpha_2}\varphi^{\mu}|^{\frac{q}{q-r}}dP\Big)^{\frac{q-r}{q}}, \\
C(p,m)=\mu\Big(\int_{Q}\varphi^{\mu-\frac{p}{p-m}}|(-\Delta)^{\frac{\beta_1}{2}}
\varphi|^{\frac{p}{p-m}}dP\Big) ^{\frac{p-m}{p}},\\
C(q,n)=\mu\Big(\int_{Q}\varphi^{\mu-\frac{q}{q-n}}
|(-\Delta)^{\frac{\beta_2}{2}}\varphi|^{\frac{q}{q-n}}\,dP\Big)^{\frac{q-n}{q}}, \\
I_0^{\mu}=\int_{S} u_2 (0,t_2,x)\varphi^{\mu}(0,t_2,x) dP_2
+\int_{Q}|u_1|^s D_{t_2|T}^{\alpha_1}\varphi^{\mu}\,dP,\\
J_0^{\mu}=\int_{S} v_2 (0,t_2,x)\varphi^{\mu}(0,t_2,x) dP_2
+\int_{Q}|v_1|^r D_{t_2|T}^{\alpha_2}\varphi^{\mu}\,dP,
\end{gather*}
then, using estimates \eqref{es1}-\eqref{es6}, we can write
\eqref{wfs} as 
\begin{gather*}
I_{v}+I_0^{\mu}\leq I_{u}^{1/p}A(p)+I_{u}^{s/p}B(p,s)
+ I_{u}^{\frac{m}{p}}C(p,m), \\
I_{u}+J_0^{\mu}\leq  I_{v}^{\frac{1}{q}}A(q)+I_{v}^{\frac{r}{q}}B(q,r)
+ I_{v}^{\frac{n}{q}}C(q,n).
\end{gather*}
Since $I_0^{\mu},J_0^{\mu}>0$,  we have
\begin{gather}\label{eta}
I_{v}\leq I_{u}^{1/p}A(p)+I_{u}^{s/p}B(p,s)+ I_{u}^{\frac{m}{p}}C(p,m),
\\ \label{xi}
I_{u}\leq  I_{v}^{\frac{1}{q}}A(q)+I_{v}^{\frac{r}{q}}B(q,r)
+ I_{v}^{\frac{n}{q}}C(q,n).
\end{gather}
Now, from \eqref{eta} and \eqref{xi}, we have
\begin{align*}
I_{v} +I_0^{\mu}
&\leq \Big( I_{v}^{\frac{1}{pq}}A^{1/p}(q)+I_{v}^{\frac{r}{pq}}B^{1/p}(q,r)
+ I_{v}^{\frac{n}{pq}}C^{1/p}(q,n) \Big) A(p)\\
&\quad+ ( I_{v}^{\frac{s}{pq}}A^{s/p}(q)+I_{v}^{\frac{rs}{pq}}
B^{s/p}(q,r)+ I_{v}^{\frac{ns}{pq}}C^{s/p}(q,n) )B(p,s)\\
&\quad + \Big( I_{v}^{\frac{m}{pq}}A^{\frac{m}{p}}(q)+I_{v}^{\frac{rm}{pq}}
B^{\frac{m}{p}}(q,r)+ I_{v}^{\frac{nm}{pq}}C^{\frac{m}{p}}(q,n) \Big)C(p,m).
\end{align*}
Then  Young's inequality implies
\begin{align*}
I_{v}+I_0^{\mu} 
&\leq K \Big\{ \Big(A^{1/p}(q)A(p) \Big)^{\frac{pq}{pq-1}}
+\Big( B^{1/p}(q,r)A(p)\Big)^{\frac{pq}{pq-r}}  \\
&\quad +\Big(C^{1/p}(q,n)A(p) \Big)^{\frac{pq}{pq-n}}
+ \big(  A^{s/p}(q)B(p,s) \Big)^{\frac{pq}{pq-s}}\\
&\quad +\Big( B^{s/p}(q,r)B(p,s)\Big)^{\frac{pq}{pq-rs}} 
+\Big(C^{s/p}(q,n)B(p,s) \Big)^{\frac{pq}{pq-ns}}  \\
&\quad +\Big(A^{\frac{m}{p}}(q)C(p,m) \Big)^{\frac{pq}{pq-m}}
 +\Big( B^{\frac{m}{p}}(q,r)C(p,m)\Big)^{\frac{pq}{pq-rm}}\\
&\quad +\Big(C^{\frac{m}{p}}(q,n)C(p,m) \Big)^{\frac{pq}{pq-nm}}
\Big\}
\end{align*}
for some positive constant $K$.
Using the scaled variables  \eqref{testft} we obtain
\begin{gather*}
A(p)=C T^{-1+(N+2)(1-1/p)}, \quad
A(q)=C T^{-1+(N+2)(1-1/q)},\\
B(p,s)=C T^{-\alpha_1+(N+2)(1-s/p)}, \quad
B(q,r)=C T^{-\alpha_2+(N+2)(1-r/q)},\\
C(p,m)=C T^{-\beta_1+(N+2)(1-m/p)}, \quad 
C(q,n)=C T^{-\beta_2+(N+2)(1-n/q)},
\end{gather*}
for some positive constant $C$.
Hence, we obtain
\begin{equation}
I_{v}+I_0^{\mu} \leq K \{ T^{\sigma_1}+T^{\sigma_2}+\dots+T^{\sigma_9} \}.
\end{equation}
Similarly, we obtain for $I_{u}$ the estimate
\begin{equation}
I_{u}+J_0^{\mu}\leq K \{ T^{\delta_1}+T^{\delta_2}+\dots+T^{\delta_9} \}.
\end{equation}
Finally, passing to the limit as $T\to\infty$,  we observe that:

 Either $\max\{ \sigma_1,\dots,\sigma_9;\delta_1,\dots,\delta_9\}<0$ 
and in this case, the right hand side tends to zero while the left hand side 
is strictly positive. Hence, we obtain a contradiction.

 Or $\max\{ \sigma_1,\dots,\sigma_9;\delta_1,\dots,\delta_9\}=0 $ and in this 
case, following the analysis similar as in one equation, we prove a contradiction.
\end{proof}


\subsection*{Acknowledgements}
The authors acknowledge financial support from The Research Council (TRC),
 Oman. This work is funded by TRC under the research agreement no. 
ORG/SQU/CBS/13/030.

\begin{thebibliography}{99}

\bibitem{Abd} A. Berrabah;
 \emph{\'{E}tude d'un probl\'{e}me d'\'{e}volution non locale et 
 r\'{e}gularisation d'un probl\'{e}me elliptique mal pos\'{e} },
 Doctoral thesis, Universit\'{e} Badji-Mokhtar, Annaba, Algeria, 2011.

\bibitem{BFW} P. Biler, T. Funaki, WA. Woyczynski;
 \emph{ Fractal Burgers equations}, Journal of Differential Equations,  
148(1)(1998), 9-46.

\bibitem{CC} A. C\'{o}rdoba, D. C\'{o}rdoba;
\emph{ A maximum principle applied to quasi-geostrophic equations}, 
Communications in Mathatimatical Physics, 249 (2004), 511-528.

\bibitem{DMRV} J. L. A. Dubbeldam, A. Milchev, V. G. Rostiashvili, T. A. Vilgis;
\emph{ Polymer translocation through a nanopore: a showcase of anomalous 
diffusion}, Physical Review, E 76 (2007), 010801(R).

\bibitem{FurKir}  K. M. Furati, M. Kirane;
\emph{ Necessary Conditions for the Existence of Global Solutions to Systems 
of Fractional Differential Equations}, Fractional Calculus and Applied Analysis, 
11(3),(2008),  281-298.

\bibitem{Herrmann} R. Herrmann;
\emph{ Fractional Calculus: An Introduction to Physicists}, 
World Scientific: Singapore, 2011.

\bibitem{Hilfer} R. Hilfer (editor);
\emph{Applications of Fractional Calculus in Physics}, World Scientific, 
Singapore, 2000.

\bibitem{Ju} N. Ju;
 \emph{The Maximum Principle and the Global Attractor for the Dissipative 
2D Quasi-Geostrophic Equations}, Communications in Pure and 
Applied Analysis, (2005), 161-181.

\bibitem{KenAbdBer} K. K. Kenzhebaev, G. A. Abdikalikova, 
 A. B. Berzhanov;
\emph{Multiperiodic solution of  a boundary-value problem for one 
class of prabolic equations with multidimensional  time},
Ukrainian Mathematical Journal, Vol. 66, No. 5(2014), 645-655.

\bibitem{KerKir} S. Kerbal, M. Kirane;
\emph{Nonexistence Results for the Cauchy Problem for Nonlinear Ultraparabolic
 Equations}, Abstract and Applied Analysis, 2011 (2011).

\bibitem{KirLasTat} M. Kirane, Y. Laskri,  N.-E. Tatar;
\emph{ Critical exponents of Fujita type for certain evolution equations 
and systems with Spatio-Temporal Fractional derivatives}, 
J. Math. Anal. Appl., 312(2005),  488-501.

\bibitem{LanPasPol} E. Lanconelli, A. Pascucci,  S. Polidoro;
\emph{Linear and nonlinear ultraparabolic equations of Kolmogorov type arising 
in diffusion theory and in finance}, in Nonlinear Problems in Mathematical
 Physics and Related Topics, II, vol. 2 of Int. Math. Ser. (N. Y.),
 Kluwer/Plenum, New York, NY, USA, (2002), 243-265.

\bibitem{LP} Y. Luchko, A. Punzi;
\emph{Modeling anomalous heat transport in geothermal reservoirs via fractional 
diffusion equations}, International Journal on Geomathematics, 1(2), (2011),
 257-276.

\bibitem{Magin} R. L. Magin;
\emph{Fractional calculus in bioengineering: Part 1, Part 2 and Part 3.}, 
Critical Reviews in Biomedical Engineering, 32(1), (2004), 104, 105-193, 195-377.

\bibitem{MitPok} E. Mitidieri, S. N. Pokhozhaev;
\emph{A priori estimates and blow-up of solutions to nonlinear partial differential 
equations and inequalities}, Proc. Steklov Inst. Math., 234(3),(2001), 1-362.

\bibitem{SamKil} S. G. Samko, A. A. Kilbas, O. I. Marichev;
\emph{Fractional integrals and derivatives, Theory and Applications}, 
Gordon and Breach Science Publishers, Yverdon, 1993.

\bibitem{Ter} A. S. Tersenov;
\emph{Ultraparabolic equations and unsteady heat transfer}, 
Journal of Evolution Equations, 5(2), (2005), 277-289.

\bibitem{VIKH} L. Vlahos, H. Isliker, Y. Kominis, K. Hizonidis;
 \emph{Normal and anomalous diffusion: a tutorial. In press at order and chaos}, 
Bountis T (ed.). Patras University Press, Patras, Vol.10, (2008).

\bibitem{Wal} W. Walter;
\emph{Parabolic differential equations and inequalities with several 
time variables}, Mathematische Zeitschrift, 191(2), (1986), 319-323.

\end{thebibliography}


\end{document}