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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 46, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/46\hfil Solutions to systems of arbitrary order]
{Solutions to systems of arbitrary-order differential equations 
in complex domains}

\author[R. W. Ibrahim \hfil EJDE-2014/46\hfilneg]
{Rabha W. Ibrahim}  % in alphabetical order

\address{Rabha W. Ibrahim  \newline
Institute of Mathematical Sciences,
University Malaya, 50603,
Kuala Lumpur, Malaysia}
\email{rabhaibrahim@yahoo.com}

\thanks{Submitted November 11, 2013. Published February 12, 2014.}
\subjclass[2000]{34A08, 34A12}
\keywords{Analytic function; fractional calculus; Young inequality;
\hfill\break\indent fractional differential equation;  Cauchy-Schwartz inequality}

\begin{abstract}
 In this article, we study the existence of  solutions for
 a three dimensional fractional system involving
 seven coefficients. We prove that the system has a strong
 global solution which is unique in an appropriate function space.
 We use a method based on analytic technique from the fixed point
 theory, along with the fractional Duhamel principle.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
%\newtheorem{lemma}[theorem]{Lemma}
%\newtheorem{proposition}[theorem]{Proposition}
%\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
%\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


%\newcommand{\sinc}{{\rm sinc} \hspace{0.03in}}
%\newcommand{\sech}{{\rm sech} \hspace{0.03in}}

\section{Introduction}

Fractional calculus (integrals and derivatives)
of any positive order can be considered as a branch of mathematical physics,
associated with differential equations, integral equations and integro-differential
equations, where integrals are of convolution form with weak singular kernels
of power law type. It has gained more and more interest in applications in
several fields of  applied sciences. Fractional differential equations
(real and complex) are viewed as models for nonlinear differential equations;
varieties of them play important roles, not only in mathematics,
but also in physics, dynamical systems, control systems and engineering,
to create the mathematical modeling of many physical  phenomena.
Furthermore, they are employed in social science, such as, food
supplement, climate and economics. Fractional models have been studied by
many researchers, to sufficiently describe the operation of variety of
computational, physical and biological processes and systems. Accordingly,
considerable attention has been paid to the outcomes of fractional differential
equations, integral equations and fractional partial differential equations of
physical phenomena. Most of these fractional differential
equations have analytic solutions, approximation and numerical techniques
\cite{k1,k2,p1,s1,s2}.

In current years, researchers have introduced and studied several types of
nonlinear systems with complex variables.  These systems, which involve
complex variables, are employed to describe the physics of a detuned laser,
rotating fluids, disk dynamos, electronic  circuits, and particle beam dynamics
in high energy accelerators.
As special model, the chaotic complex system  is used to describe
and simulate the physics of detuned lasers and thermal convection
of liquid flows. This model corresponds to the equilibrium state of the atmosphere,
in which surfaces of constant density are not parallel to
the surface of constant gravitational potential \cite{e1,i1,i2}.


Existence and uniqueness of solutions are studied widely in the
field of fractional differential equations \cite{a1,d1,g1,i3,m1,r1,u1,w1}.
In this work, we study fractional system involving seven coefficients
in complex spaces.
We show that the proposed system has a global solution in appropriate
functional spaces. This is strong and unique solution. We  employ a method,
based on analytic methods from the fixed point theory together with the
fractional Duhamel principle.

 \section{Fractional calculus}

The idea of the fractional calculus (that is, calculus of integrals
and derivatives of any arbitrary  real or complex order) was planted over 300
years ago. In 1823, Abel investigated the generalized tautochrone problem, and
he was the pioneer to apply fractional calculus techniques in a
physical problem. Later, Liouville has applied fractional calculus to solve
problems in potential theory.
Since then, the fractional calculus has triggered the attention of many
researchers in all area of sciences.
The following section concerns with some preliminaries
and notation regarding the fractional calculus.

  \begin{definition} \label{def2.1}\rm
The fractional (arbitrary)  order integral of the function $f$ of order
$\alpha >0$  is  defined by
 \[
I^{\alpha}_{a}f(t)=\int^{t}_{a}\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}
f(\tau)d\tau.
\]
When $a=0$, we write $I^{\alpha}_{a}f(t)=f(t)*\phi_{\alpha}(t)$,
where $(*)$ denotes the convolution product (see \cite{p1}),
$\phi_{\alpha}(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}$, $t>0$ and
$\phi_{\alpha}(t)=0$, $t\leq 0$ and $\phi_{\alpha}\to\delta(t)$ as
$\alpha \to 0$ where $\delta(t)$ is the
delta function.
\end{definition}

\begin{definition} \label{def2.2} \rm
The fractional (arbitrary)  order derivative of the function $f$ of order
$0 \leq \alpha <1$  is  defined by
 \[
D^{\alpha}_{a}f(t)=\frac{d}{dt}\int^{t}_{a}\frac{(t-\tau)^{-\alpha}}
 {\Gamma(1-\alpha)}f(\tau)d\tau=\frac{d}{dt}I^{1-\alpha}_{a}f(t).
\]
 In the sequel, we shall use the notation $\partial^{\alpha}_{t}$.
\end{definition}


 \begin{remark} \label{rmk2.1}\rm
From Definitions \ref{def2.1} and  \ref{def2.2}, for $a=0$,   we have
\begin{gather*}
D^{\alpha}t^{\mu}=\frac{\Gamma(\mu+1)}{\Gamma(\mu-\alpha+1)}t^{\mu-\alpha},
\quad \mu> -1; \;0 < \alpha<1, \\
I^{\alpha}t^{\mu}=\frac{\Gamma(\mu+1)}{\Gamma(\mu+\alpha+1)}t^{\mu+\alpha},
\quad \mu> -1; \; \alpha>0.
\end{gather*}
The Leibniz rule is
\[
D^{\alpha}_{a} [f(t)g(t)]= \sum_{k=0}^{\infty}
  \begin{pmatrix}
\alpha \\
k
 \end{pmatrix}
D^{\alpha-k}_{a}f(t) D^{k}_{a} g(t)=  \sum_{k=0}^{\infty}
  \begin{pmatrix}
\alpha \\
k
 \end{pmatrix} D^{\alpha-k}_{a}g(t) D^{k}_{a} f(t),
\]
 where
 \[
  \begin{pmatrix}
\alpha \\
k
 \end{pmatrix}
= \frac{\Gamma(\alpha+1)}{\Gamma(k+1)\Gamma(\alpha+1-k)}.
\]
\end{remark}

 \section{Fractional system}

In this section, we  propose a one-dimensional setting,
which physically corresponds to the consideration of a Laminar-Couette flow.
This type of flow appropriately models flows in shear rheometers.
Our three unknown fields, the velocity  $u$, the shear stress $\phi$ and the
fluidity $f$ are defined as functions of a space variable
$z \in U:= \{z\in \mathbb{C}, |z| \leq 1\}$. They are also, functions of the
time $t \geq 0. $ The system can be read as
\begin{gather}
 \rho \partial^{\alpha}_{t}u(t,z)
= \eta u_{zz}+ \phi_{z},  \label{e1a} \\
 \lambda \partial^{\alpha}_{t}\phi(t,z) = G u_{z}- f \phi+G \ell, \label{e1b}\\
\partial^{\alpha}_{t}f(t,z) = (-1+\xi|\phi| ) f^2- \nu
f^3, \label{e1c}
\end{gather}
where $\alpha \in (0,1)$,  $\rho$ is the density, $\eta$ is the
viscosity, $\lambda$ is the characteristic relaxation time, $G$ is
the elastic modulus, $\ell$ is a constant scalar in $[0,\infty)$
and $\xi$ and $\nu$ are the evolution of the fluidity $f$.
In the sequel, we assume that $u, \phi$ and $f$ are analytic with
$|f|\leq1$. System \eqref{e1a}--\eqref{e1c} is classified as a fully coupled system of
three equations and seven-dimensionless coefficients. All the
above coefficients are positive and constant in time. The first
one is the equation of conservation of momentum for $u$. The
second equation rules the evolution of the shear stress $\phi$.
The third equation is of the form evolution equations suggested by
many authors. The first two equations are classical in nature,
while the last equation, may differ from one model to another.
Assume that system \eqref{e1a}--\eqref{e1c} supplied with initial conditions
$(u_0,\phi_0,f_0)$ and the homogeneous boundary conditions
$u(t,0)=0$ and $u(t,1)=0$.

 The dimension of a basic physical quantity can
be formulated as a product of the basic physical dimensions:
length, mass,  electric charge, absolute temperature and time
symbolled by sans-serif symbols $ L,M, Q,\Theta$, and $T$,
respectively, each raised to a rational
power. Other physical quantities can be described in phrases of
these fundamental physical dimensions. For example, speed has the
dimension length (or distance) per unit of time, similarly for
velocity, stress  and fluidity. Usually these depending physical
quantities need constant coefficients. In general, these
coefficients are constant with respect to time, therefore they have
positive values.


 \section{Existence and uniqueness}
In this section, we establish  the existence and uniqueness of a solution
for \eqref{e1a}--\eqref{e1c}.


\begin{theorem} \label{thm4.1}
Consider  \eqref{e1a}--\eqref{e1c} with initial condition
$(u_0,\phi_0,f_0) \in H^1(U)^3$ with $\Re(f_0)\geq 0$.
If $\frac{T^{\alpha}(1+\nu)}{\Gamma(\alpha+1)}<1 $ then there exists
a unique global solution $(u,\phi,f)$ for  \eqref{e1a}--\eqref{e1c} subjected
with the boundary condition $u(t,0)=0$ and $u(t,1)= 0$, such that
\begin{equation} \label{e2}
(u,\phi,f) \in \big( C([0,T];H^1)\cap L^2([0,T];H^2)\times
C([0,T];H^1) \times C([0,T];H^1) \big)
\end{equation}
and $\Re(f)\geq 0$ for all $z \in U$ and $t \in [0,T]$.
Moreover,
\begin{equation} \label{e3}
(\partial^{\alpha}_{t}u,\partial^{\alpha}_{t}\phi,\partial^{\alpha}_{t}f)
\in \big( L^2([0,T];L^2)\times C([0,T];L^2) \times
C([0,T];L^2) \big).
\end{equation}
\end{theorem}


The proof consists of eight steps. The first five steps derive the
form of the solution while Step 6 describes the sequence of the
approximate solution. The convergence of this sequence is proven
in Step 7, thereby the existence of a solution is established to
\eqref{e1a}--\eqref{e1c}. Step 8 addresses uniqueness.
\smallskip

 \noindent \textbf{Step 1: Positivity.}
 We prove that $\Re(f) \geq 0$. Define the set
\[
U_0=\{z \in U: \Re(f_0) > 0\}.
\]
For $z \in U \backslash U_0$, we receive $\Re(f_0(z))=0$ and
thus, from \eqref{e1c},
$\Re(f(t,z)) = 0$ for all time $t \in [0,T]$.
Now let $z \in U_0$ we proceed to prove that $\Re(f) > 0$. We
dispute by contradiction and assume, by continuity of $f(., z)$,
that
\[
t_{m}= \inf \{t \in [0,T], f(t,z)=0\}< T.
\]
The Cauchy-Lipschitz theorem employed to \eqref{e1c} with zero as
initial condition at time $t_{m}$ implies that $f(t, z) = 0$ for
$t \in (t_{m} -\varepsilon, t_{m}+\varepsilon) $ and
$\varepsilon> 0$, which contradicts the definition of $t_{m}$.
Hence $\Re(f) \geq 0$.


 \noindent \textbf{Step 2: Boundedness.} From the evolution equation \eqref{e1a}
 on $u$, we obtain
\begin{equation} \label{e4}
\frac{1}{\alpha+1} \rho
\partial^{\alpha}_{t}\|u(t,.)\|^2_{L^2}+
\eta\|u_{z}(t,.)\|^2_{L^2}=\int_{U}(\phi_{z}u)(t,.).
\end{equation}
Similarly,  the evolution equation \eqref{e1b} implies
\begin{equation} \label{e5}
\frac{1}{\alpha+1}\lambda
\partial^{\alpha}_{t}\|\phi(t,.)\|^2_{L^2}+
\|\sqrt{f}\phi(t,.)\|^2_{L^2}
=G\int_{U}(\phi u_{z}\phi)(t,.)+G\ell\widehat{\phi},
\end{equation}
where
\[
\widehat{g}(t)=\int_{U}g(t,z)dz.
\]
Combining estimates \eqref{e4} and \eqref{e5} and using the fact that $u$
vanishes on the boundary, yields
\begin{equation} \label{e6}
\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u(t,.)\|^2_{L^2}+ \lambda
\|\phi(t,.)\|^2_{L^2}]+\|\sqrt{f}\phi(t,.)\|^2_{L^2}+
G\eta\|u_{z}(t,.)\|^2_{L^2}
 =G\ell \widehat{\phi}(t).
\end{equation}
Now integrating \eqref{e1c} over $U$ implies
\begin{equation} \label{e7}
\partial^{\alpha}_{t} \|f(t,.)\|_{L^1}+
\|f(t,.)\|^2_{L^2}+\nu \|f(t,.)\|^3_{L^3}= \xi \int_{U}
(|\phi|f^2)(t,.).
\end{equation}
By  the  Young inequality, we have
\[
\xi |\phi|f^2 = \sqrt{\nu} |f|^{3/2}\frac{\xi}{\sqrt{\nu}}|\phi||f|^{1/2}
\leq \frac{\nu}{2}|f|^3+\frac{\xi^2}{2 \nu} |f|\phi^2 \] and
hence we have
\begin{equation} \label{e8}
\partial^{\alpha}_{t} \|f(t,.)\|_{L^1}+
\|f(t,.)\|^2_{L^2}+\frac{\nu}{2} \|f(t,.)\|^3_{L^3}
\leq \frac{\xi^2}{2\nu}\|\sqrt{f}\phi(t,.)\|_{L^2}^2.
\end{equation}
Summing \eqref{e6} and \eqref{e8}, we obtain
\begin{equation} \label{e9}
\begin{aligned}
&\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u(t,.)\|^2_{L^2}+ \lambda \|\phi(t,.)\|^2_{L^2}+
\frac{2\nu}{\xi^2} \|f(t,.)\|_{L^1}]\\
&+\frac{1}{2}\|\sqrt{f}\phi(t,.)\|^2_{L^2}+
G\eta\|u_{z}(t,.)\|^2_{L^2} \\
&\leq K \ell \|\phi(t,.)\|_{L^2},
\end{aligned}
\end{equation}
where $K$ is a positive constant depending on the coefficients
$G,\eta, \lambda,\nu,\xi$ and $\rho$. By applying the fact that
\[
\|\phi(t,.)\|_{L^2} \leq \frac{\|\phi(t,.)\|_{L^2}^2+1}{2}
\]
and using the generalized Gronwall lemma to \eqref{e9}, we attain
\begin{equation} \label{e10}
\begin{aligned}
&\sup_{t \in [0,T]} [  \|u(t,.)\|^2_{L^2}+
\|\phi(t,.)\|^2_{L^2}+\|f(t,.)\|_{L^1}]\\
&+\frac{T^{\alpha-1}}{\Gamma(\alpha)}\int^{T}_0
\Big(\|\sqrt{f}\phi(t,.)\|^2_{L^2}+
\|u_{z}(t,.)\|^2_{L^2}\Big)dt 
\leq \widetilde{K},
\end{aligned}
\end{equation}
where $\widetilde{K}$ is a positive constant depending on the
seven coefficients $G$, $T$,  $u_0$, $\phi_0$, $f_0$, $\alpha$, $\eta$,
$\lambda$, $\nu$, $\xi$, $\rho$ and $\ell$. Note that when $\ell =0$, in
\eqref{e9}, then  $\widetilde{K} $ does not depend on $T$ and hence, we
have uniform bounds in time.
\smallskip

\noindent\textbf{Step 3: Auxiliary functions.}
Define a function $q$ as follows
\[
q(t,z)= \int^{z}_0 \big(\phi(t,\zeta)-\widehat{\phi}(t)\big)d\zeta
\]
satisfying the Dirichlet boundary conditions which solves
\[
\frac{\partial^2q}{\partial z^2}= \frac{\partial \phi}{\partial z}.
\]
Applying \eqref{e1a} and \eqref{e1b}, which respectively impose
\[
\rho \partial^{\alpha}_{t} u
= \eta \frac{\partial^2}{\partial z^2}(u+ \frac{1}{\eta}q)
\]
and
\[
\lambda \partial^{\alpha}_{t}q
=- \int^{z}_0 \big(f\phi(t,\zeta)-\widehat{f\phi}(t)\big)d\zeta+Gu.
\]
Define the function
\begin{equation} \label{e11}
\Lambda =u+\frac{1}{\eta} \int^{z}_0(\phi-\widehat{\phi})
= u+ \frac{1}{\eta} q.
\end{equation}
A fractional derivative yields
 \begin{equation} \label{e12}
\begin{split}
\partial^{\alpha}_{t} \Lambda
&=\partial^{\alpha}_{t}u+\frac{1}{\eta}\partial^{\alpha}_{t}q\\
&=\frac{\eta}{\rho} \frac{\partial^2}{\partial z^2}(u+
\frac{1}{\eta}q)+\frac{1}{\lambda\eta}[- \int^{z}_0
\big(f\phi(t,\zeta)-\widehat{f\phi}(t)\big)d\zeta+Gu]\\
&=\frac{\eta}{\rho} \frac{\partial^2}{\partial
z^2}\Lambda-\frac{1}{\lambda\eta} \int^{z}_0
\big(f\phi(t,\zeta)-\widehat{f\phi}(t)\big)d\zeta+\frac{G}{\eta \lambda}u.
\end{split}
\end{equation}
Multiplying by �$\frac{\partial^2}{\partial
z^2}\Lambda$ and integrating over $U$ yields
\begin{align*}
&\frac{1}{\alpha+1} \partial^{\alpha}_{t} \|
\frac{\partial \Lambda}{\partial z}(t,.)\|^2_{L^2}
+\frac{\eta}{2\rho}  \|\frac{\partial ^2\Lambda}{\partial
z^2}(t,.)\|^2_{L^2}\\
&\leq C\Big(\|(f\phi)(t,.)\|_{L^1}
 \int_{U} |\frac{\partial ^2\Lambda}{\partial z^2}|(t,.)
 + \int_{U} |u \frac{\partial ^2\Lambda}{\partial z^2}| (t,.)\Big).
\end{align*}
The Young and the Cauchy-Schwartz inequalities imply that
\begin{equation} \label{e13}
\partial^{\alpha}_{t} \|\frac{\partial
\Lambda}{\partial z}(t,.)\|^2_{L^2} + \|\frac{\partial
^2\Lambda}{\partial z^2}(t,.)\|^2_{L^2}
\leq C_{\alpha,\eta,\rho}\Big(\|(f (t,.)\|_{L^1}
\|\sqrt{f}\phi\|^2_{L^2}+ \|u (t,.)\|^2_{L^2}\Big).
\end{equation}
Since
\[
\frac{\partial \Lambda}{\partial z}|_{t=0}= \frac{\partial
u_0}{\partial z}+\frac{1}{\eta} (\phi_0-\widehat{\phi}_0)\in
L^2(U);
\]
hence, we deduce from \eqref{e13} that
\begin{equation*}
\Lambda \in L^{\infty} ([0,T], H^1)\cap L^2([0,T], H^2),
\end{equation*}
and consequently
\[
u \in L^{\infty} ([0,T], H^1)\cap L^2([0,T], H^2).
\]

 \noindent \textbf{Step 4: $L^{\infty}-$bounds.}
By using the definition of $\Lambda$ and $\widehat{\phi}$, we
rewrite \eqref{e1b} as
\[
\lambda \partial^{\alpha}_{t} \phi
=G \frac{\partial \Lambda}{\partial z}- \big(f+\frac{G}{\eta}\big)\phi
+\frac{G}{\eta} \widehat{\phi}+ G \ell.
\]
Multiplying the last assertion by $\phi$, we conclude that
\[
\frac{\lambda}{2} \partial^{\alpha}_{t} |\phi|^2
+\big(|f|+\frac{G}{\eta}\big)|\phi|^2
\leq C \Big(  |\phi| |\frac{\partial \Lambda}{\partial z}|
+|\phi|\|\phi\|_{L^2} +\ell |\phi|\Big),
\]
consequently, the Young inequality yields
\begin{equation} \label{e15}
\frac{\lambda}{2} \partial^{\alpha}_{t} |\phi|^2
+\big(|f|+\frac{G}{\eta}\big)|\phi|^2
\leq C \Big(  |\frac{\partial \Lambda}{\partial z}|^2
+\|\phi\|_{L^2} +\ell \Big).
\end{equation}
Since $\phi_0 \in H^1(U)$ (Step 2) and
 $\frac{\partial \Lambda}{\partial z} \in L^2([0,T], L^{\infty})$ (Step 3),
then the generalized Gronwall lemma to \eqref{e15} shows that
\begin{equation} \label{e16}
\|\phi(t,.)\|_{L^{\infty}} \leq \widetilde{K},
\end{equation}
where $\widetilde{K}$ is defined in \eqref{e10}, that is
$\phi \in L^{\infty}([0,T],L^{\infty})$.

  We proceed to prove that $f \in L^{\infty}([0,T],L^{\infty})$.
For this purpose, we apply the fractional Duhamel principle which can be
found in \cite{u1}. Then \eqref{e1c} reduces to
\[
\partial^{\alpha}_{t}f(t,z) = (-f-\nu f^2 ) f + \xi|\phi|f^2.
\]
Assume that $F$ is a solution for the problem
\begin{equation} \label{e17}
\partial^{\alpha}_{t}F + (F+\nu F^2 ) F=0
\end{equation}
subjected to the initial condition
\[
I^{1-\alpha} F|_{t=0}=h(0),\quad \textrm{where } h:= \xi|\phi|f^2,
\;  |F|\leq1.
\]
Then
\[
f(t)= \int^{t}_0F(s)ds
\]
is a solution of the problem
\[
\partial^{\alpha}_{t}f(t,z) + (f+\nu f^2 ) f = \xi|\phi|f^2.
\]
It suffices to prove that $F \in L^{\infty}([0,T],L^{\infty})$;
from \eqref{e17}, we obtain
\[
\Big(1-\frac{T^{\alpha}}{\Gamma(\alpha+1)}(\|F\|+\nu \|F\|^2)\Big)\|F\|
\leq \xi f_0^2 \|\phi_0\|.
\]
Since $|F|\leq1$, the above inequality becomes
\begin{equation} \label{e18}
\|F\|\leq \frac{\xi f_0^2
\|\phi_0\|}{\big(1-\frac{T^{\alpha}(1+\nu)}{\Gamma(\alpha+1)}\big)}.
\end{equation}
Hence we obtain that $F \in L^{\infty}([0,T],L^{\infty})$ (because
$\phi \in L^{\infty}([0,T],L^{\infty})$) and consequently
$f \in L^{\infty}([0,T],L^{\infty})$.
\smallskip

 \noindent \textbf{Step 5: Second estimate bounds of $u$.}
Differentiate with respect to $z$ the evolution equation \eqref{e1b}, we
have
 \begin{equation} \label{e19}
\begin{split}
\lambda \partial^{\alpha}_{t}(\frac{\partial \phi}{\partial z})
&= G \frac{\partial^2 u}{\partial z^2}
 - \frac{\partial \phi}{\partial z}f- \frac{\partial f}{\partial z}\phi\\
&=G \frac{\partial^2 \Lambda}{\partial z^2}- \frac{G}{\eta}
\frac{\partial \phi}{\partial z}- \frac{\partial
\phi}{\partial z}f- \frac{\partial f}{\partial z}\phi.\\
\end{split}
\end{equation}
Moreover, we differentiate with respect to $z$ the evolution
equation \eqref{e1c} to obtain
 \begin{equation} \label{e20}
 \partial^{\alpha}_{t}(\frac{\partial f}{\partial z})
= \xi\frac{\partial|\phi|}{\partial z}  f^2
 +2 (\xi |\phi|-1)f \frac{\partial f}{\partial z}-3 \nu
f^2\frac{\partial f}{\partial z}.
  \end{equation}
Multiplying  \eqref{e19} and \eqref{e20} by
 $\frac{\partial \phi}{\partial z}$ and �$\frac{\partial f}{\partial z}$
respectively, integrating over the domain $U$, summing up and using that both
$f$ and $\phi$ are in $L^{\infty}([0,T],L^{\infty})$, we have
\begin{align*}
&\partial^{\alpha}_{t} \Big(  \lambda \|
\frac{\partial \phi}{\partial z}(t,.)\|^2_{L^2}
+\|\frac{\partial f}{\partial z}(t,.)\|^2_{L^2}\Big) \\
&\leq K_{\alpha} \int_{U} \Big(\frac{\partial^2 \Lambda}{\partial
z^2}\frac{\partial \phi}{\partial z}+ \big(\frac{\partial
\phi}{\partial z}\big)^2 + \frac{\partial f}{\partial
z}\frac{\partial \phi}{\partial z}+ \frac{\partial |
\phi|}{\partial z}\frac{\partial f}{\partial z}+
\big(\frac{\partial f}{\partial z}\big)^2 \Big)(t,.).
\end{align*}
Since $\phi \in L^2([0,T],H^1)$ then in view of the Young
inequality, we have
\begin{equation} \label{e21}
\begin{aligned}
&\partial^{\alpha}_{t}  \Big(  \lambda \|\frac{\partial
\phi}{\partial z}(t,.)\|^2_{L^2} +\|\frac{\partial f}{\partial
z}(t,.)\|^2_{L^2}\Big)\\
&\leq K_{\alpha} \Big(   \|\frac{\partial
\phi}{\partial z}(t,.)\|^2_{L^2} +\|\frac{\partial f}{\partial
z}(t,.)\|^2_{L^2}+ \|\frac{\partial^2 \Lambda}{\partial
z^2}(t,.)\|^2_{L^2}\Big).
\end{aligned}
\end{equation}
Since $\phi, f \in L^{\infty}([0,T],H^1)$ and
$\phi_0, f_0 \in H^1(U)$,  the generalized Gronwall inequality together
with  \eqref{e11} imply that
$u \in L^{\infty} ([0,T], H^1)\cap L^2([0,T], H^2)$.
\smallskip

\noindent\textbf{Step 6. Approximate solution.}
In this step, we construct a sequence of approximating solutions
to system \eqref{e1a}--\eqref{e1c}. Consider the system
\begin{gather}
\rho \partial^{\alpha}_{t}u_n(t,z)
= \eta \frac{\partial^2 u_n}{\partial z^2}+
 \frac{\partial\phi_n}{\partial z},  \label{e22a}\\
 \lambda \partial^{\alpha}_{t}\phi_n(t,z)
= G  \frac{\partial u_n}{\partial z}- f_{n-1} \phi_n+G \ell, \label{e22b}\\
\partial^{\alpha}_{t}f_n(t,z) = (-1+\xi|\phi_n| )
f_{n-1}f_n- \nu f_{n-1}f_n^2, \label{e22c}
\end{gather}
subjected to the boundary conditions
\[
u_n(t,0)=0, \quad u_n(t,1)=0, \quad \forall \, t \in [0,T]
\]
and the initial condition
$(u_{n0},\phi_{n0},f_{n0})=(u_0,\phi_0,f_0)$. Our aim is to
show that  \eqref{e22a}--\eqref{e22c} has a unique solution
$(u_n,\phi_n,f_n)$ in the space
\[
\big(C([0,T];H^1)\cap L^2([0,T];H^2)\times C([0,T];H^1) \times
C([0,T];H^1) \big).
\]
 For this purpose, we split the system
\eqref{e22a}--\eqref{e22c}
into two subsystems \eqref{e22a}-\eqref{e22b} on the one hand and \eqref{e22c}
 on the other hand.

 First we prove the existence of unique
solution. Let $(u_{n1}, \phi_{n1})$ and $(u_{n2}, \phi_{n2})$ be
two solutions in the aforementioned class; the functions
$u_n=u_{n1}-u_{n2} $ and $\phi_n= \phi_{n1}- \phi_{n2}$
satisfy the system
\begin{gather} \label{e23a}
\rho \partial^{\alpha}_{t}u_n(t,z) = \eta \frac{\partial^2
u_n}{\partial z^2}+
 \frac{\partial\phi_n}{\partial z}, \\
\lambda \partial^{\alpha}_{t}\phi_n(t,z) = G  \frac{\partial
u_n}{\partial z}- f_{n-1} \phi_n.  \label{e23b}
\end{gather}
Multiplying  \eqref{e23a} by $u_n$ and integrating over the domain
$U$, we obtain
\begin{equation} \label{e24}
\frac{1}{\alpha+1} \rho
\partial^{\alpha}_{t}\|u_n(t,.)\|^2_{L^2}+
\eta\|\frac{\partial u_n}{\partial z}(t,.)\|^2_{L^2}
=\int_{U}(\frac{\partial \phi_n}{\partial z}u_n)(t,.).
\end{equation}
Similarly,  Multiplying  \eqref{e23b} by $\phi_n$ and and integrating
over the domain $U$, implies
\begin{equation} \label{e25}
\frac{1}{\alpha+1}\lambda
\partial^{\alpha}_{t}\|\phi_n(t,.)\|^2_{L^2}+
\|\sqrt{f_{n-1}}\phi_n(t,.)\|^2_{L^2}
=G\int_{U}(\phi_n \frac{\partial u_n}{\partial z})(t,.),
\end{equation}
Adding estimates \eqref{e24} and \eqref{e25} and using and using integration by
parts together with the fact that $u_n$ vanishes on the
boundary, yields
\begin{equation} \label{e26}
\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u_n(t,.)\|^2_{L^2}+ \lambda
\|\phi_n(t,.)\|^2_{L^2}]+\|\sqrt{f_{n-1}}\phi_n(t,.)\|^2_{L^2}+
G\eta\|\frac{\partial u_n}{\partial z}(t,.)\|^2_{L^2}=0,
\end{equation}
which gives  $u_n=0$ and $\phi_n=0$. Hence system \eqref{e23a}--\eqref{e23b} has a
unique bound uniform solution in the space
 $(C([0, T ];H^1) �\cap L^2([0, T ];H^2)) �\times C([0, T ];H^1)$.

 Second we show that $f_n$ exists in $C([0, T ];H^1)$ and $\Re(f_n)\geq 0$.
Equation \eqref{e22c} can be reduced to
\begin{equation} \label{e27}
\begin{gathered}
\partial^{\alpha}_{t}f_n(t,z) = \Theta(t,f_n,z),
f_n|_{t=0}= f_0,
\end{gathered}
\end{equation}
where $\Theta: [0,T] \times \mathbb{C} \times U \to \mathbb{C}$.
 The function $\Theta$ is continuous in its first two variables and locally
Lipschitz in its second variable. The Cauchy-Lipschitz theorem
imposes there exists a unique local solution with $f_0(z)$ as
initial condition. Let $[0, T^{*})$ be the interval of existence
of the maximal solution for positive time. For all $t \in  [0,
T^{*})$, we have $\Re (f_n )\geq  0$, using Step 1. Furthermore,
\begin{equation} \label{e28}
|\partial^{\alpha}_{t}f_n(t,z) | \leq \xi |\phi_n||f_{n-1}||f_n|
 \leq \xi
\|\phi_n\|_{L^{\infty}}\|f_{n-1}\|_{L^{\infty}}|f_n|;
\end{equation}
using that both $\phi_n$ and $f_{n-1}$ belong to $C([0, T];H^1)$.
The Gronwall lemma then shows that $f_n$ remains
bounded on $[0, T^{*}]$ and thus we have established existence and
uniqueness on $[0, T^{*}]$.

 Now we  prove the boundedness of the solution. From \eqref{e22a} and \eqref{e22b},
we may have
\begin{equation} \label{e29}
\begin{aligned}
&\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u_n(t,.)\|^2_{L^2}+ \lambda
\|\phi_n(t,.)\|^2_{L^2}]\\
&+\|\sqrt{f_{n-1}}\phi_n(t,.)\|^2_{L^2}
 + G\eta\|\frac{\partial u_n}{\partial z}(t,.)\|^2_{L^2} 
=G\ell \widehat{\phi_n}(t).
\end{aligned}
\end{equation}
and from \eqref{e22c}, we obtain
\begin{equation} \label{e30}
\partial^{\alpha}_{t} \|f_n(t,.)\|_{L^1}+
 \int_{U} (|f_{n-1}||f_n)|(t,.) +
\frac{\nu}{2} \int_{U} (|f_{n-1}||f|^2_n)(t,.)
\leq \frac{\xi^2}{2 \nu}\|\sqrt{f_{n-1}}\phi_n(t,.)\|_{L^2}^2.
\end{equation}
Collecting \eqref{e29} and \eqref{e30},
we obtain
\begin{equation} \label{e31}
\begin{split}
&\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u_n(t,.)\|^2_{L^2}+ \lambda
\|\phi_n(t,.)\|^2_{L^2}]+ \frac{\nu}{\xi^2}
\|f_n(t,.)\|_{L^1}\\
& +\frac{1}{2}\|\sqrt{f_{n-1}}\phi_n(t,.)\|^2_{L^2}+
G\eta\|\frac{\partial u_n}{\partial z}(t,.)\|^2_{L^2}\\
& \leq K \ell \|\phi_n(t,.)\|_{L^2}.
\end{split}
\end{equation}
Hence the solution $(u_n,\phi_n,f_n)$ is bounded.

  The arguments given in Step 3 to derive \eqref{e13} and in Step 4 for the
$L^{\infty}$ estimates can  simulate for the approximate system in
$(u_n, \phi_n, f_n)$ instead of  $(u, \phi, f)$, and the
corresponding auxiliary functions $q_n$ and $\Lambda_n$. In
this place, we have the estimate for the solution
$(u_n,\phi_n,f_n)$
\begin{equation} \label{e32}
\sup _n \sup _{t\in[0,T ]} \Big(\|u_n(t,.)\|_{L^2} +
\|\phi_n(t,�.)\|_{L^2} + \|f_n(t,.)\|_{L^1}\Big)
\leq \widetilde{K}
\end{equation}
and
\begin{equation} \label{e33}
\sup _n \sup _{t\in[0,T ]} \Big(\|\Lambda_n(t,.)\|_{H^1} +
\|\phi_n(t,�.)\|_{L^{\infty}} +
\|f_n(t,.)\|_{L^{\infty}}\Big)+ \|\Lambda_n\|_{L^2} \leq
\widetilde{K},
\end{equation}
where we recall that $\widetilde{K}$ denotes various constants
which depends on the coefficients in system \eqref{e1a}--\eqref{e1c},
the initial data $u_0, \phi_0, f_0$ and the time $T$.

 Similar arguments as the ones in Step 5, show that
\begin{equation} \label{e34}
\begin{split}
&\partial^{\alpha}_{t}  \Big(  \lambda \|\frac{\partial
\phi_n}{\partial z}(t,.)\|^2_{L^2} +\|\frac{\partial
f_n}{\partial z}(t,.)\|^2_{L^2}\Big) \\
&\leq \widetilde{C} \Big(\|\frac{\partial \phi_n}{\partial z}(t,.)\|^2_{L^2}
+\|\frac{\partial f_n}{\partial z}(t,.)\|^2_{L^2}
 +\|\frac{\partial f_{n-1}}{\partial z}(t,.)\|^2_{L^2}
 + \|\frac{\partial^2 \Lambda_n}{\partial z^2}\|^2_{L^2}\Big).
\end{split}
\end{equation}
Let
\[
Z_n(t):= \|\frac{\partial
\phi_n}{\partial z}(t,.)\|^2_{L^2} +\|\frac{\partial
f_n}{\partial z}(t,.)\|^2_{L^2}.
\]
Applying the operator $I^{\alpha}$, yields
\begin{equation} \label{e35}
\begin{split}
Z_n(t)
&\leq Z_0+\widetilde{C}\int^{t}_0\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}
Z_n(\tau)d
\tau+\widetilde{C}\int^{t}_0\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}
Z_{n-1}(\tau)d \tau+\widetilde{C} \| \Lambda_n\|^2_{L^2}\\
& \leq \widetilde{C}_{\alpha,0}
+\widetilde{C}_{\alpha,1}\int^{t}_0\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}
Z_{n-1}(\tau)d \tau\\
& \leq \widetilde{M}
+\widetilde{M}\int^{t}_0\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}
Z_{n-1}(\tau)d \tau,
\end{split}
\end{equation}
where
$\widetilde{M}:=\max(\widetilde{C}_{\alpha,1},\widetilde{C}_{\alpha,0})$.
By induction, we may find that for all $t \in [0,T] $ and all $n$,
\begin{equation} \label{e36}
Z_n(t) \leq \widetilde{M}
\sum^{n-1}_{j=0}\frac{(\widetilde{M}t)^{j}}{\Gamma(\alpha j+1)} +
\frac{(\widetilde{M}t)^{n}}{\Gamma(\alpha n+1)} ,
\end{equation}
consequently,
\begin{equation} \label{e37}
Z_n(t) \leq \widetilde{M} E_{\alpha,1}(\widetilde{M} t),
\end{equation}
where, $E_{\alpha,1}(.)$ is a Mittag-Leffler function. It follows
that
\begin{equation} \label{e38}
\sup_n \sup _{t \in [0,T]}Z_n(t) \leq \widetilde{M}
E_{\alpha,1}(\widetilde{M} T).
\end{equation}
Thus inequalities \eqref{e33} and \eqref{e38} imply the inequalities
\begin{equation} \label{e39}
\sup_n \sup _{t \in [0,T]}\Big(\|u_n(t, .)\|_{H^1} +
\|\phi_n(t, .)\|_{H^1} + \|f_n(t, .)\|_{H^1}\Big) +
\|u_n\|_{L^2}\leq M
\end{equation}
and
\begin{equation} \label{e40}
\sup_n \Big(\|\partial^{\alpha}_{t} u_n(t, .)\|_{L^2} +
\|\partial^{\alpha}_{t}\phi_n(t, .)\|_{L^2} +
\|\partial^{\alpha}_{t} f_n(t, .)\|_{L^2}\Big)\leq
M_{T,\alpha}.
\end{equation}
\smallskip

\noindent \textbf{Step 7: Convergence of the approximate solutions.}
The bounds introduced in the previous steps, namely \eqref{e39} and
\eqref{e40} imply  that, at least up to extraction of a subsequence, we have
the weak convergence
\[
(u_n,\phi_n,f_n)\to (u,\phi,f), \quad \text{weakly  in }
 L^{\infty}([0,T],H^1)^3.
\]
In this step, we establish a strong convergence in
$L^{\infty}([0,T],L^2(U))^3$. Denoted by
$h_n^{*}=h_n-h_{n-1}$ and derive the evolution equations for
$(u^{*}_n,\phi^{*}_n,f^{*}_n)$,
\begin{gather}
\rho \partial^{\alpha}_{t}u^{*}_n(t,z) = \eta
\frac{\partial^2 u^{*}_n}{\partial z^2}+
 \frac{\partial\phi^{*}_n}{\partial z},  \label{e41a}\\
 \lambda \partial^{\alpha}_{t}\phi^{*}_n(t,z) = G
\frac{\partial u_n^{*}}{\partial z}- f_{n-1} \phi^{*}_n-
f^{*}_{n-1} \phi_{n-1}, \label{e41b} \\
\begin{aligned}
\partial^{\alpha}_{t}f^{*}_n(t,z)
&= (-1+\xi|\phi_{n-1}| )
(f^{*}_{n-1}f_n+f_{n-1}f^{*}_n)  \\
&\quad- \nu f_{n-1}f^{*}_n(f_n+f_{n-1})-\nu
f_{n-1}^2f^{*}_{n-1}+\xi |\phi^{*}_n|f_nf_{n-1},
\end{aligned}\label{e41c}
\end{gather}
Since $(u_n,\phi_n,f_n)$ satisfies the assertions \eqref{e2} and  \eqref{e3},
then the same holds for $(u^{*}_n,\phi^{*}_n,f^{*}_n)$.
We multiply  \eqref{e41a}, \eqref{e41b}, \eqref{e41c}, respectively by
$u^{*}_n,\phi^{*}_n,f^{*}_n$, integrate over $U$, and then sum them,
\[
\partial^{\alpha}_{t}\Big(G \rho\| u^{*}_n(t, .)\|_{L^2}^2 +
\lambda\|\phi^{*}_n(t, .)\|_{L^2}^2 + \| f^{*}_n(t,.)\|_{L^2}^2\Big)
\leq \int_{U} \Phi\big(|\phi_{n-1}|,|\phi_n|,|f_{n-1}|,|f_n|\big),
\]
where $\Phi$ is a positive valued function. Let
\[\Psi_n(t):= \| u^{*}_n(t, .)\|_{L^2}^2 +  \|\phi^{*}_n(t,.)
\|_{L^2}^2 + \| f^{*}_n(t, .)\|_{L^2}^2.
\]
Now by using the $L^{\infty}-$ bound in \eqref{e32} on
$|\phi_{n-1}|,|\phi_n|,|f_{n-1}|,|f_n|$, Young inequality
yields
\begin{equation} \label{e42}
\partial^{\alpha}_{t}\Psi_n(t)\leq \widetilde{K}
\Big(\Psi_n(t) + \Psi_{n-1}(t)\Big).
\end{equation}
Applying the Gronwall lemma to \eqref{e42}, we may find
\begin{equation} \label{e43}
\Psi_n(t)\leq  \widetilde{L}_{\alpha} \int^{t}_0\Psi_{n-1}(s)
ds,
\end{equation}
where $ \widetilde{L}_{\alpha}$ is a constant depending on all the
coefficients of the System \eqref{e1a}--\eqref{e1c} and its initial condition.
Thus we have
\[
\Psi_n(t)\leq \frac{(\widetilde{L}_{\alpha} t)^{n-1}}{(n-1)!}\sup_{s \in [0, T]}
\Psi_1(s);
\]
therefore, the sequence $(u_n,\phi_n,f_n)$ is a  Cauchy
sequence in $L^{\infty}([0,T],L^2(U))^3$ which
implies that $f_{n-1} \to f$ strongly. This completes the
existence proof.
\smallskip

\noindent \textbf{Step 8: Uniqueness.}
Consider $(u_1, \phi_1, f_1)$ and $(u_2, \phi_2, f_2)$
satisfying \eqref{e2} and solutions to system \eqref{e1a}--\eqref{e1c}
supplied with the same
initial condition $(u_0, \phi_0, f_0) \in H^1(U)$. Assume
that $u=u_1-u_2$, $\phi=\phi_1-\phi_2$ and $f=f_1-f_2$ satisfying the system
\begin{gather*}
 \rho \partial^{\alpha}_{t}u(t,z) = \eta \frac{\partial^2 u}{\partial z^2}
+ \frac{\partial \phi}{\partial z}, \\
 \lambda \partial^{\alpha}_{t}\phi(t,z) = G \frac{\partial u}{\partial z}-
 f \phi, \\
\partial^{\alpha}_{t}f(t,z) = (-1+\xi|\phi| ) f^2- \nu f^3.
\end{gather*}
Multiply these three equations by
$u,\phi,f$, respective, then integrate over $U$, and then summing  up,
we have
\[
\partial^{\alpha}_{t}\Big(G \rho\| u(t, .)\|_{L^2}^2 +
\lambda\|\phi(t, .)\|_{L^2}^2 + \| f(t, .)\|_{L^2}^2\Big)
\leq \widetilde{\ell}_{\alpha}
(\|\phi\|_{L^2}^2+\|f\|_{L^2}^2).
\]
The Gronwall lemma then implies uniqueness.
This completes the proof of Theorem \ref{thm4.1}.



 \section{Convergence to the steady point}

In this section, we  discuss the convergence of solution
$(u,\phi,f)$ to the steady point
$(0,0,0)$ in the space $H^1(U) \times L^{\infty} (U)  \times
L^{\infty} (U)$.

\begin{theorem} \label{thm5.1}
Consider Systen \eqref{e1a}--\eqref{e1c}. If
$\ell =0$ and $\Re (f_0)\neq 0$ then
\begin{equation} \label{e45}
\|u(t,.)\|_{H^1}+ \|\phi(t,.)\|_{L^{\infty}}+
\|f(t,.)\|_{L^{\infty}}\to 0.
\end{equation}
\end{theorem}

The proof will be done in three steps. The first step derive the
lower bound of the fluidity $f$, while Step 2 proves the
convergence in $L^2(U)$ and consequently Step 3, provides the
convergence in $L^{\infty}(U)$.
\smallskip

 \noindent \textbf{Step 1: Lower bound of the fluidity $f$.}
Since $ \Re(f_0)\neq 0$, there exists, by analytically of
$f_0$ (assumed in $H^1)$, a non-empty closed interval $U_0$
in $U$ where $f_0$ does not vanish. We rewrite the evolution
equation \eqref{e1c} on $f$ as follows:
\begin{equation} \label{e46}
\partial^{\alpha}_{t}f^{-1}(t,z) = (1-\xi|\phi| ) f^2+
\nu f^3;
\end{equation}
but the $L^{\infty}-$ bounds of $f$ and $\phi$ are uniform in time (see Step 4),
thus for all $z \in U_0$, we obtain that
\begin{equation*}
\partial^{\alpha}_{t}|f^{-1}(t,z)| \leq \kappa
\end{equation*}
and therefore,
\[ % \label{e47}
|f(t,z)| \geq
\frac{\Gamma(\alpha+1)}{\frac{\Gamma(\alpha+1)}{|f_0|}+ \kappa
t^{\alpha}}, \quad t \in [0,T]
\]
and this implies the lower bound.
\smallskip

 \noindent \textbf{Step 2: Convergence in  $L^2(U)$.}
From \eqref{e6} and \eqref{e9}, we have
\begin{equation} \label{e48}
\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u(t,.)\|^2_{L^2}+ \lambda
\|\phi(t,.)\|^2_{L^2}]+\|\sqrt{f}\phi(t,.)\|^2_{L^2}+
G\eta\|u_{z}(t,.)\|^2_{L^2} =0
\end{equation}
and
\begin{equation} \label{e49}
\begin{aligned}
&\frac{1}{\alpha+1} \partial^{\alpha}_{t}[ G \rho
\|u(t,.)\|^2_{L^2}+ \lambda \|\phi(t,.)\|^2_{L^2}]+
\frac{\nu}{\xi^2}\|f(t,.)\|_{L^1}\\
&+\frac{1}{2}\|\sqrt{f}\phi(t,.)\|^2_{L^2}+
G\eta\|u_{z}(t,.)\|^2_{L^2} =0.
\end{aligned}
\end{equation}
Combining \eqref{e48} and \eqref{e49}
and applying the Gronwall lemma, we obtain
\begin{equation} \label{e50}
\lim _{t \to \infty} \Big(  \|u(t,.)\|^2_{L^2}+
 \|\phi(t,.)\|^2_{L^2}+ \|f(t,.)\|_{L^1} \Big) \to 0.
\end{equation}

 \noindent \textbf{Step 3: Convergence in  $L^{\infty}(U)$.}
Combining \eqref{e13} and \eqref{e15} and using the $L^{\infty}-$bound of $f$
yield
\begin{align*}
&\partial^{\alpha}_{t} \Big( \|\frac{\partial \Lambda}{\partial z}(t,.)\|^2_{L^2}+
\lambda |\phi(t,z)|^2 \Big)+ \Big( \|\frac{\partial^2
\Lambda}{\partial z^2}(t,.)\|^2_{L^2}+ |\phi(t,z)|^2 \Big)\\
&\leq \sigma \Big( \|u(t,.)\|^2_{L^2}+
 \|\phi(t,.)\|^2_{L^2}+  \|\frac{\partial \Lambda}{\partial z}(t,.)\|^2_{L^2}\Big),
\end{align*}
where $\sigma$ is a positive constant depending on all the
coefficients of \eqref{e1a}--\eqref{e1c}.
Employing the Gronwall lemma and using the last convergence \eqref{e50},
we obtain
 \begin{equation} \label{e51}
\lim _{t \to \infty} \|\phi(t,.)\|^2_{L^{\infty}}= 0.
\end{equation}
Using this convergence in  \eqref{e1c}, we have
\begin{equation} \label{e52}
\lim _{t \to \infty} \|f(t,.)\|^2_{L^{\infty}}= 0.
\end{equation}
Finally, definition \eqref{e11} and convergence \eqref{e50} supply the
convergence of $u_{z}$ to zero in $L^2(U)$; hence we have
\[
(u,\phi,f) \in H^1 (U)\times L^{\infty}(U) \times L^{\infty}(U) .
\]
This completes the proof.


\subsection*{Conclusion}
In this article, we had illustrated an analytic method for establishing
the existence and uniqueness of solutions to fractional differential system
in a complex domain.
The proposed method  depends on fractional Duhamel principle, which can be
applied in various kinds of fractional systems.
 Throughout the article, we had used the homogeneous boundary value problem.
For future work, one may try the non-homogeneous case.


\subsection*{Acknowledgements}
The author thankful to the referees for  helpful suggestions for the
improvement of this article. This research has been funded by
university of Malaya, under Grant RG208-11AFR.

\begin{thebibliography}{00}

\bibitem{a1} S. Abbas, M. Benchohray;
\emph{Darboux problem for implicit  impulsive partial hyperbolic
fractional order differential equations},
 Electronic Journal of Differential Equations,
  2011 (2011), No. 150, pp. 1-14.

\bibitem{d1} B. K. Dutta, L. K. Aroray;
\emph{On the existence and uniqueness of solution of a class of
initial value problems of fractional order},
 Mathematical Sciences 7(17) (2013) 1-21.

\bibitem{e1} A. M. A. El-Sayed, E. Ahmed, H. A. A. El-Sakay;
\emph{Dynamic properties of the fractional-order
logistic equation of complex variables}, Abstract and Applied
Analysis, vol 2012, Article ID 251715, 12 pages,
doi:10.1155/2012/251715.

\bibitem{g1} Z. Guo, M. Liuy;
\emph{Existence and uniqueness of solutions for a system of higher-order
nonlinear fractional differential equations},
Lobachevskii Journal of Mathematics,  34 (1) (2013) 68-75.

\bibitem{i1} R. W. Ibrahim;
\emph{Complex transforms for systems of fractional
differential equations}, Abstract and Applied Analysis, vol 2012,
Article ID 814759, 15 pages, doi:10.1155/2012/814759

\bibitem{i2} R. W. Ibrahim;
\emph{Stability and stabilizing of fractional complex Lorenz
systems}, Abstract and Applied Analysis, vol 2013, Article ID
127103, 13 pages, doi.org/10.1155/2013/127103.

\bibitem{i3} R. W. Ibrahim;
\emph{Existence and uniqueness of holomorphic solutions for fractional Cauchy
problem}, Journal of Mathematical Analysis and Applications,
380(1)( 2011) 232-240.

\bibitem{k1}  A. A. Kilbas, H. M. Srivastava, J. J. Trujilloy;
\emph{Theory and applications of fractional differential
 equations}. North-Holland, Mathematics Studies, Elsevier 2006.

\bibitem{k2} J. Klafter, S. C. Lim, R. Metzler (Eds.)y;
\emph{Fractional Dynamics: Recent Advances}, World Scientific Publishing
Company, Singapore, 2011.

\bibitem{m1} J. Mu, Y. Liy;
\emph{Monotone iterative technique for impulsive
fractional evolution equations}, Journal of Inequalities and
Applications 2011, 2011:125.

\bibitem{p1} I. Podlubny;
\emph{Fractional Differential Equations}, Acad. Press, London, 1999.

\bibitem{r1} A. Razminia, A. F. Dizaji, V. J. Majdy;
\emph{Solution existence for non-autonomous variable-order fractional
differential equations},
Mathematical and Computer Modelling, 55(2012) 1106-1117.

\bibitem{s1} J. Sabatier, O. P. Agrawal, J. A. Machadoy;
\emph{Advance in Fractional Calculus: Theoretical Developments
and Applications in Physics and Engineering}, Springer, 2007.

\bibitem{s2} H. M. Srivastava,  S. Oway;
\emph{Univalent Functions, Fractional Calculus,
and Their Applications}, Halsted Press, John Wiley and Sons, New
York, Chichester, Brisban, and Toronto, 1989.

\bibitem{u1} S. Umarovy;
\emph{On fractional Duhamel's principle and its applications},
Journal of Differential Equations 252 (2012), 5217-5234.

\bibitem{w1} G. Wang, R. P. Agarwal, A. Cabaday;
\emph{Existence results and the monotone iterative technique for systems
of nonlinear fractional differential equations}, Applied
Mathematics Letters, 25\eqref{e6} (2012) 1019-1024.

\end{thebibliography}

\end{document}