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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 289, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/289\hfil Analysis and application of diffusion equations]
{Analysis and application of diffusion equations involving a new
fractional derivative without singular kernel}

\author[L. Zhang, B. Ahmad, G. Wang \hfil EJDE-2017/289\hfilneg]
{Lihong Zhang, Bashir Ahmad, Guotao Wang}

\address{Lihong Zhang \newline
School of Mathematics and Computer Science,
Shanxi Normal University,
Linfen, Shanxi 041004,  China}
\email{zhanglih149@126.com}

\address{Bashir Ahmad \newline
Department of Mathematics,
Faculty of Science,
 King Abdulaziz University P.O. Box. 80203,
Jeddah 21589, Saudi Arabia}
\email{bashirahmad\_qau@yahoo.com}

\address{Guotao Wang \newline
School of Mathematics and Computer Science,
Shanxi Normal University,
Linfen, Shanxi 041004, China}
\email{wgt2512@163.com}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted September 21, 2017. Published November 21, 2017.}
\subjclass[2010]{26A33, 35B05, 35B50}
\keywords{Caputo-Fabrizio derivative; maximum principle;
\hfill\break\indent multi-term time fractional diffusion equations}

\begin{abstract}
 In this article, a family of nonlinear diffusion equations involving
 multi-term Caputo-Fabrizio time fractional derivative is investigated.
 Some maximum principles are obtained. We also demonstrate the application
 of the obtained results by deriving some  estimation for solution to
 reaction-diffusion equations.
\end{abstract}

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

\section{Introduction}

Luchko \cite{2009} obtained a maximum principle for the generalized time-fractional
diffusion equation on an open bounded domain by applying an extremum principle 
involving Caputo-Dzherbashyan fractional derivative. Later, the maximum principles 
for  generalized  time-fractional diffusion equations (multi-term diffusion 
equation and the diffusion equation of distributed order) with Caputo 
and Riemann-Liouville type derivatives were presented by Luchko \cite{2011}, 
and Al-Refai and Luchko \cite{2015} respectively.  Alsaedi et al.\ \cite{2015a} 
 proved an inequality for fractional derivatives related to the Leibniz rule 
and used it to derive  maximum principles for time and space fractional heat 
equations with nonlinear diffusion. 
For further details on the topic, see \cite{2015B,  2016a,2016, 2017, 2012, 2009S}.

Here, in contrast to the above referenced works, we study  a nonlinear  
diffusion equation with multi-term
Caputo-Fabrizio time fractional derivative (without singular kernel) given by
\begin{equation}\label{101}
\mathcal{P}(^{CF}D_{t}u)(x,t)=-L(u)(x,t)+F(x,t,u(x,t)),
\end{equation}
where $\Omega\subset\mathbb{R}^{N}(N\geq1)$ is a bounded open domain in
$\mathbb{R}^{N}$ with a smooth boundary $\partial\Omega$, 
$L$ is a uniformly elliptic operator given by
\begin{gather*}
L(u)=-\sum_{i,j=1}^{n}a_{ij}(x,t)\frac{\partial^2u}{\partial x_i\partial x_j}
+\sum_{i=1}^{n}b_{i}(x,t)\frac{\partial u}{\partial x_i}, \\
\mathcal{P}(^{CF}D_{t}u)={^{CF}D}_{t}^\alpha u
+\sum_{i=1}^{m}\lambda_i ^{CF}D_t^{\alpha_i}u,\quad 
 0<\alpha_m<\dots<\alpha_1<\alpha<1,\; \lambda_i\ge0,
\end{gather*}
where $i=1,2,\dots,m$, ${}^{CF}D_{t}^\alpha$ denotes Caputo-Fabrizio fractional
 derivative of order $0<\alpha<1$ defined by
$$
{}^{CF}D_{t}^\alpha f(t)
=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}
\int_0^t \exp(-\frac{\alpha}{1-\alpha}(t-s))f'(s)ds,\quad t\ge 0,
$$
 $M(\alpha)$ is a normalization constant depending on $\alpha$.
For details about fractional derivative without singular kernel,
 we refer the reader to \cite{Caputo}.

\section{Preliminaries}

In this section,  we present some useful theorems related to our work.

\begin{lemma}\cite{PW,W} \label{L201}
Let  $u\in C^2(\overline{\Omega})$ be a function attaining its maximum at 
a point $x_0$ inside $\Omega\subseteq\mathbb{R}^n$ and 
$A=(a_{ij})_{n\times n},~x\in\Omega$ be a positive definite matrix. Then 
$$
\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^2u}{\partial x_i\partial x_j}
\Big|_{x=x_0}\le0.
$$
\end{lemma}

\begin{theorem} \label{T201}
Let $0<\alpha<1$. Assume that $f\in C^1([0,T])$ attains its
maximum on the interval $[0,T]$ at the point $t_{0}\in(0,T]$. Then
\begin{equation}\nonumber
{^{CF}}D_t^{\alpha}f(t_{0})
\geq\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}
\exp(-\frac{\alpha t_{0}}{1-\alpha})\big[f(t_{0})-f(0)\big]\ge0.
\end{equation}
\end{theorem}

\begin{proof} 
As in \cite{2009S}, we introduce an auxiliary function
$$
y(t)=f(t)-f(t_{0}),\quad t\in[0,T]
.$$
It is easy to deduce that $y\in C^1([0,T])$ and the following properties hold:
\begin{itemize}
\item[(1)] $y(t)\leq0$, for all $t\in[0,T]$;
\item[(2)] $y(t_{0})=y'(t_{0})=0$;
\item[(3)] $y(t)=(t-t_{0})x(t)$ with $x\in C([0,T])$ and $x(t)\le0$ for all 
$t\in[0,t_0]$.
\end{itemize}
In consequence, we obtain
\begin{align*}
{^{CF}}D_t^{\alpha}y(t)
&=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\int_0^t\exp(-\frac{\alpha
}{1-\alpha}(t-s))y'(s)ds\\
&=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\int_0^t\exp(-\frac{\alpha
}{1-\alpha}(t-s))f'(s)ds\\
&={^{CF}}D_t^{\alpha}f(t).
\end{align*}
Note that $y(t)\le0$ for all $t\in[0,t_0] $ and $y(t_0)=0$. 
Then, integrating by parts, we obtain
\begin{align*}
{^{CF}}D_t^{\alpha}y(t_{0})
&=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\int_0^{t_0}\exp(-\frac{\alpha
}{1-\alpha}(t_0-s))y'(s)ds\\
&=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\exp(-\frac{\alpha
}{1-\alpha}(t_0-s))y(s)\Big|_0^{t_0}\\
&\quad -\frac{(2-\alpha)M(\alpha)\alpha}{2(1-\alpha)(1-\alpha)}\int_0^{t_0}\exp(-\frac{\alpha
}{1-\alpha}(t_0-s))y(s)ds\\
&\ge -\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\exp(-\frac{\alpha t_0
}{1-\alpha}[f(0)-f(t_0)])\\
&=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\exp(-\frac{\alpha t_0
}{1-\alpha}[f(t_0)-f(0)])\ge0.
\end{align*}
\end{proof}

\section{Some maximum principles}
In the section, we derive some maximum principles for the following 
parabolic type fractional differential operator without singular kernel
\begin{equation}
\begin{split}\label{Qoper}
Q_\alpha(u)
&=P(^{CF}D_tu)(x,t)+L(u)-h(x,t)u\\
&=P(^{CF}D_tu)(x,t)-\sum_{i,j=1}^{n}a_{ij}(x,t)
\frac{\partial^2u}{\partial x_i\partial x_j} \\
&\quad +\sum_{i=1}^{n}b_{i}(x,t)\frac{\partial u}{\partial x_i}-h(x,t)u,
\end{split}
\end{equation}
where $h(x,t)\le0 ~((x,t)\in\overline{\Omega}\times [0,T])$ 
is a bounded function.
 Let us begin with a weak maximum principle.

\begin{theorem}\label{T301}
If $u\in C(\overline{\Omega}\times[0,T])\cap
C^{2,1}(\Omega\times(0,T))$ satisfies $Q_\alpha(u)\le0$ ($Q$ is defined by 
\eqref{Qoper}), then the following inequality holds:
\[
u(x,t)\le \max\Big\{\max_{x\in \overline{\Omega}}u(x,0),
\max_{(x,t)\in \partial{\Omega}\times(0,T]}u(x,t),\; 0 \Big\}.
\]
\end{theorem}

\begin{proof}
Assume that the function $u(x,t)$ attains its positive maximum $u(x_0,t_0)$ 
at a point $(x_0,t_0)\in \Omega\times(0,T]$.
By Lemma \ref{L201}, we obtain
\begin{equation}\label{303}
Lu(x,t)|_{(x_0,t_0)}
=-\sum_{i,j=1}^{n}a_{ij}(x,t)\frac{\partial^2u}{\partial x_i\partial x_j}|_{(x_0,t_0)}
+\sum_{i=1}^{n}b_{i}(x,t)\frac{\partial u}{\partial x_i}|_{(x_0,t_0)}\ge0.
\end{equation}
By  Theorem \ref{T201}, we have
\begin{equation}\label{304}
\begin{split}
&\mathcal{P}(^{CF}D_{t}u)(x_0,t_0) \\
&={}^{CF}D_{t}^\alpha u(x_0,t_0)+\sum_{i=1}^{m}\lambda_i {}^{CF}D_t^{\alpha_i}
u(x_0,t_0)\\
&=\frac{(2-\alpha)M(\alpha)}{2(1-\alpha)}\exp(-\frac{\alpha
t_{0}}{1-\alpha})\big[u(x_0,t_{0})-u(x_0,0)\big]\\
&\quad +\sum_{i=1}^{m}\lambda_i\frac{(2-\alpha_i)M(\alpha_i)}{2(1-\alpha_i)}\exp(-\frac{\alpha_i
t_{0}}{1-\alpha_i})\big[u(x_0,t_{0})-u(x_0,0)\big]
>0.
\end{split}
\end{equation}
Applying the condition $h(x,t)\le 0$, one can easily deduce from 
inequalities \eqref{303} and \eqref{304} that
\begin{equation*}
(Q_\alpha u)(x_0,t_{0})>P(^{CF}D_t)u(x_0,t_0)-h(x_0,t_0)u(x_0,t_0)>0,
\end{equation*}
which contradicts the condition $(Q_\alpha u)(x,t)\le 0$ for all
$(x,t)\in\Omega\times(0,T]$. 
\end{proof}

 Analogously, we can prove the following minimum principle.

\begin{theorem}\label{T302}
If $u\in C(\overline{\Omega}\times[0,T])\cap
C^{2,1}(\Omega\times(0,T))$ satisfies $Q_\alpha(u)\ge0$, then the 
following inequality holds:
\[
u(x,t)\ge \min\big\{\min_{x\in \overline{\Omega}}u(x,0),\min_{(x,t)
\in \partial{\Omega}\times(0,T]}u(x,t),\;0 \big\}.
\]
\end{theorem}

\section{Applications of maximum and minimum principles}

Here we present some new results for multi-dimensional time-fractional
 diffusion equation by using the maximum and minimum principles 
derived in the previous section.
Consider the  linear problem
\begin{equation}\label{401}
\begin{gathered}
P(^{CF}D_tu)(x,t)+L(u)-h(x,t)u=f(x,t),\quad (x,t)\in \Omega\times(0,T],\\
u(x,0)=g(x),\quad x\in \Omega,\\
u(x,t)=\mu(x,t),\quad (x,t)\in \partial\Omega\times(0,T).
\end{gathered}
\end{equation}
 A direct application of Theorems \ref{T301} and \ref{T302} leads 
to the following two comparison results for the problem \eqref{401}.

\begin{theorem}\label{T401}
Assume that $f(x,t)\le0$, $g(x)\le0$ and $\mu(x,t)\le0$.
 If $u(x,t)\in C^{2,1}(\Omega\times(0,T))$ is a solution of the problem 
\eqref{401}, then $u(x,t)\le0$, $(x,t)\in \overline{\Omega}\times[0,T]$.
\end{theorem}

\begin{theorem}\label{T402}
 Assume that $f(x,t)\ge0,~g(x)\ge0$ and $\mu(x,t)\ge0$. If 
$u(x,t)\in C^{2,1}(\Omega\times(0,T))$ is a solution of the  problem \eqref{401}, 
then $u(x,t)\ge0$, $(x,t)\in \overline{\Omega}\times[0,T]$.
\end{theorem}

\begin{theorem}\label{T405}
There exists at most one solution for the problem \eqref{401}.
\end{theorem}

\begin{proof}
Let us suppose that \eqref{401} has two solutions $u_1$ and $u_2$. Letting 
$u=u_1-u_2$, we obtain
\begin{gather*}
P(^{CF}D_tu)(x,t)+L(u)-h(x,t)u=0,\quad (x,t)\in\Omega\times(0,T),\\
u(x,0)=0,\quad x\in \Omega,\\
u(x,t)=0\quad (x,t)\in \partial\Omega\times(0,T).
\end{gather*}
By Theorems \ref{T401} and \ref{T402}, it follows that $u=0$, that is,
 $u_1=u_2$. 
\end{proof}


\begin{theorem}\label{T406}
The solution $u$ of \eqref{401}  depends continuously on the given initial 
value $g(x)$ and the boundary value $\mu(x,t)$.
\end{theorem}

\begin{proof}
Let $u_i$ denote the solution of \eqref{401} with the corresponding data 
$g_i(x)$ and $\mu_i(x,t)$, $i=1,2$.
Fix $\eta=\frac{\varepsilon}{2}, \, \varepsilon>0$ and suppose that
$\max_{x\in \Omega}|g_1(x)-g_2(x)|\le \eta$ and 
$\max_{(x,t)\in \partial\Omega\times (0,T)}|\mu_1(x,t)-\mu_2(x,t)|\le \eta$. 
Taking $u=u_1-u_2$, we obtain
\begin{gather*}
P(^{CF}D_tu)(x,t)+L(u)=0,\quad (x,t)\in\Omega\times(0,T),\\
u(x,0)=g_1(x)-g_2(x),\quad x\in \Omega,\\
u(x,t)=\mu_1(x,t)-\mu_2(x,t)\quad (x,t)\in \partial\Omega\times(0,T).
\end{gather*}
Since $h(x,t)\le0$,  it follows by Theorems \ref{T301} and \ref{T302} that
\begin{gather*}
u(x,t)\le \max\big\{\max_{x\in \overline{\Omega}}(g_1-g_2),
\max_{(x,t)\in \partial{\Omega}\times(0,T]}(\mu_1-\mu_2),\; 0 \big\}<\varepsilon, \\
u(x,t)\ge \min \big\{\min_{x\in \overline{\Omega}}(g_1-g_2),
 \min_{(x,t)\in \partial{\Omega}\times(0,T]}(\mu_1-\mu_2),\; 0 \big\}>-\varepsilon.
\end{gather*}
\end{proof}

Next, we consider the nonlinear problem  
\begin{equation}\label{404}
\begin{gathered}
P(^{CF}D_tu)(x,t)+L(u)=F(x,t,u),\quad (x,t)\in \Omega\times(0,T],\\
u(x,0)=g(x),\quad x\in \Omega,\\
u(x,t)=\mu(x,t),\quad (x,t)\in \partial\Omega\times(0,T).
\end{gathered}
\end{equation}


\begin{theorem}\label{T403}
Let $F(x,t,u)$ be a smooth function. If $\frac{\partial F}{\partial u}\le0$, 
then  \eqref{404} has at most one solution.
\end{theorem}

\begin{proof}
Suppose that nonlinear problem \eqref{404} has two solutions $u_1$ and $u_2$. 
Letting $u=u_1-u_2$, we obtain
\begin{gather*}
P(^{CF}D_tu)(x,t)+L(u)=F(x,t,u_1)-F(x,t,u_2),\quad
(x,t)\in\Omega\times(0,T),\\
u(x,0)=0,\quad x\in \Omega,\\
u(x,t)=0,\quad (x,t)\in \partial\Omega\times(0,T).
\end{gather*}
Since $F$ is a smooth function, there exists 
$\xi=(1-\lambda)u_1+\lambda u_2 (\lambda\in(0,1))$ such that
$$
F(x,t,u_1)-F(x,t,u_2)=\frac{\partial F}{\partial u}(\xi)u.
$$
Using the condition $\frac{\partial F}{\partial u}<0$ together with Theorems
\ref{T401} and \ref{T402}, we obtain  $u=0$. This implies that  $u_1=u_2$. 
\end{proof}

\begin{theorem}\label{T404}
Let $F(x,t,u)$ be a smooth function such that $\frac{\partial F}{\partial u}\le0$. 
Then the solution $u$ of  \eqref{404}  depends continuously on the given 
initial and boundary data $g(x)$ and $\mu(x,t)$ respectively.
\end{theorem}

\begin{proof}
Let $u_i$ be the solutions of \eqref{404} with the corresponding data $g_i(x)$ 
and $\mu_i(x,t)$, $i=1,2$.
Fixing $\eta=\frac{\varepsilon}{2}$,  $\varepsilon>0$, assume that 
$\max_{x\in \Omega}|g_1(x)-g_2(x)|\le \eta$ and 
$\max_{(x,t)\in \partial\Omega\times (0,T)}|\mu_1(x,t)-\mu_2(x,t)|\le \eta$.
Define $u=u_1-u_2$ so that $u$ satisfies
\begin{gather*}
P(^{CF}D_tu)(x,t)+L(u)=F(x,t,u_1)-F(x,t,u_2),\quad (x,t)\in\Omega\times(0,T),\\
u(x,0)=g_1(x)-g_2(x),\quad x\in \Omega,\\
u(x,t)=\mu_1(x,t)-\mu_2(x,t),\quad (x,t)\in \partial\Omega\times(0,T).
\end{gather*}
As in the proof of Theorem \ref{T403}, we have
$$
F(x,t,u_1)-F(x,t,u_2)=\frac{\partial F}{\partial u}(\xi)u,
$$
where $\xi$ is between $u_1$ and $u_2$. Since $\frac{\partial F}{\partial u}\le0$, 
 by Theorems \ref{T301} and \ref{T302} it follows that
\begin{gather*}
u(x,t)\le \max \big\{\max_{x\in \overline{\Omega}}(g_1-g_2),
\max_{(x,t)\in \partial{\Omega}\times(0,T]}(\mu_1-\mu_2),\; 0 \big\}<\varepsilon,\\
u(x,t)\ge \min \big\{\min_{x\in \overline{\Omega}}(g_1-g_2),
\min_{(x,t)\in \partial{\Omega}\times(0,T]}(\mu_1-\mu_2),\; 0 \big\}
>-\varepsilon.
\end{gather*}
\end{proof}

\subsection*{Acknowledgments} 
The authors thank the editor and the reviewers for their constructive remarks
 that led to the improvement of the original manuscript.
This work is partially supported by National Natural Science Foundation of
 China (No. 11501342) and  the Natural Science Foundation for Young Scientists
 of Shanxi Province, China (No. 201701D221007).

\begin{thebibliography}{10}

\bibitem{2015} M. Al-Refai, Y. Luchko;
 Maximum principle for the multi-term time-fractional diffusion equations with
the Riemann-Liouville fractional derivatives,
 \emph{Appl. Math. Comput.}, \textbf{257} (2015), 40--51.

\bibitem{2015a} A. Alsaedi, B. Ahmad, M. Kirane;
Maximum principle for certain generalized time and space fractional
diffusion equations, \emph{Quart. Appl. Math.}, \textbf{73}  (2015),  163-175.

\bibitem{2015B} H. Brunner,  H. Han,  D. Yin;
The maximum principle for time-fractional diffusion equations and its application,
\emph{Numer. Funct. Anal. Optim.} , \textbf{36}  (2015),  1307--1321.

\bibitem{Caputo} M. Caputo, M. Fabrizio;
 A new definition of fractional derivative without singular kernel,
\emph{Progr. Fract. Differ. Appl.}, \textbf{1} (2015), 73--85.

\bibitem{2016a} M. H. Duong;
Comparison and maximum principles for a class of flux-limited diffusions
with external force fields, \emph{Adv. Nonlinear Anal.}  \textbf{5} (2016), 167--176.

\bibitem{2016} J. Jia, K. Li;
Maximum principles for a time-space fractional diffusion equation,
\emph{Appl. Math. Lett.}, \textbf{62} (2016), 23--28.

\bibitem{2017} Y. Liu;
Strong maximum principle for multi-term time-fractional diffusion equations
and its application to an inverse source problem, \emph{Comput. Math. Appl.},
 \textbf{73} (2017),  96--108.

\bibitem{2009} Y. Luchko;
 Maximum principle for the generalized time-fractional diffusion equation, 
\emph{J. Math. Anal. Appl.}, \textbf{351} (2009), 218-223.

\bibitem{2011} Y. Luchko;
  Maximum principle and its application for the time-fractional diffusion equations,
 \emph{Fract. Calc. Appl. Anal.},  \textbf{14}  (2011),  110--124.

\bibitem{2012} Y. Luchko;
Initial-boundary-value problems for the one-dimensional time-fractional diffusion
equation, \emph{Fract. Calc. Appl. Anal.}, \textbf{15} (2012),  141--160,

\bibitem{PW} M. H. Protter, H. F. Weinberger;
\emph{Maximum Principles in Differential Equations}, Springer, Berlin, 1999.

\bibitem{2009S} A. Shi, S. Zhang;
Upper and lower solutions method and a fractional differential equation
 boundary value problem, \emph{Electron. J. Qual. Theory Differ. Equ.} (2009),
 No. 30, 13 pp.

\bibitem{W} W. Walter;
 On the strong maximum principle for parabolic differential equations, 
\emph{Proc. Edinb. Math. Soc.}, \textbf{29} (1986), 93--96.

\end{thebibliography}

\end{document}