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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 44, pp. 1--12.\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-2017/44\hfil Moving-boundary problems]
{Moving-boundary problems for the time-fractional
 diffusion equation}

\author[S. D. Roscani \hfil EJDE-2017/44\hfilneg]
{Sabrina D. Roscani}

\address{Sabrina D. Roscani \newline
 CONICET - Departamento de Matem\'atica FCE,
Universidad Austral, Paraguay 1950,
S2000FZF Rosario, Argentina}
\email{sabrinaroscani@gmail.com}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted August 26, 2016. Published February 14, 2017.}
\subjclass[2010]{26A33, 35R37, 33E12, 33E20}
\keywords{Fractional diffusion equation; Caputo derivative;
\hfill\break\indent moving-boundary problem;
 maximum principle; asymptotic behaivor}

\begin{abstract}
 We consider a one-dimensional moving-boundary problem
 for the time-fractional diffusion equation. The time-fractional derivative
 of order $\alpha\in (0,1)$ is taken in the sense of Caputo.
 We study the asymptotic behaivor, as $t$ tends to infinity, of a
 general solution by using a fractional weak maximum principle.
 Also, we give some particular exact solutions in terms of Wright functions.
\end{abstract}

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

\section{Introduction}

The beginning of fractional calculus dates to the 19th century.
 Abel, Liouville, Riemann and Letnikov proposed several definitions of
fractional derivatives motivated by the idea of providing a novel operator
that includes the classical concept of derivative. However, these
definitions were not used until a century later. It was in the 1950s when
the study of fractional differential equations gained relevance.
From that moment, many authors pointed out that derivatives and integrals of
 non-integer order were useful for describing properties of various real-world
materials such as polymers and some types of non-homogeneous solids
\cite{AST-libro, Diethelm, MK:2000, Podlubny}.

Among the fractional partial differential equations, we have the fractional
diffusion equation (FDE), obtained from the standard diffusion
equation by replacing the first order time-derivative by a fractional
derivative. This equation was studied in
\cite{EidelmanKochubei:2004, Fujita:1989, FM-TheFundamentalSolution,
 GoReRoSa:2015,Pskhu:2009, Wyss:1986}.
Several applications have been considered in the past two decades:
 Mainardi studied the FDE in the context of the theory of
linear viscoelasticity in \cite{FM-Libro}.
Voller et al.\ \cite{Voller:2013} stated that the FDE can be derived if
 we consider a new kind of heat flux involving the memory of the material,
instead of the classical local flux and we replace it in the balance
heat equation (see \cite{GuPi:1968} where a general theory of heat conduction
for materials with memory is presented).
FDEs ha been solved numerically by several authors; see a
 general discussion  in \cite{DiFoFrLu:2005}.
Some numerical results for a fractional Stefan problem based on a
finite difference method adapted to the FDE are presented in
\cite{BlKl:2014}.
A finite difference scheme for an initial-boundary value problem is presented
in \cite{Lin:2007}.

In this article, we consider a moving-boundary problem for the FDE in which a
time-fractional derivative of order $\alpha \in (0,1) $ in the sense of Caputo
\cite{Ca:1967}. More precisely, we consider the problem
\begin{equation}\label{(P)}\begin{gathered}
 _0D^{\alpha}_t u(x,t)=\lambda^2\, u_{xx}(x,t), \quad
 s_1(t)<x<s_2(t), \; 0<t\leq T, \; 0<\alpha<1, \\
 u(s_1(t),t)=g(t),\quad 0<t\leq T,\\
 u(s_2(t),t)=h(t), \quad 0<t\leq T,\\
 u(x,0)=f(x), \quad a\leq x\leq b, \quad s_1(0)=a, \quad s_2(0)=b,
 \end{gathered}
\end{equation}
where
\begin{equation}\label{FDE}
{}_0 D^{\alpha}_t u(x,t)
= \frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-\tau)^{-\alpha} u_t(x,\tau)d\tau,
\end{equation}
and the given functions $s_1$, $s_2$, $g$, $h$ and $f$ are continuous functions
is their respective domains.

 Problem \eqref{(P)} has not yet been deeply studied
(some works related to it are \cite{Atkinson, Li-Xu-Jiang, Luchko1, Ro:2016,VMD:2012,
Voller:2014}) and our purpose is to present some explicit solutions
and then analyze the asymptotic behavior of a general solution
as $t$ tends to infinity.
In partial differential equations of parabolic type, the asymptotic behavior
of a solution is closely linked to the maximum principles (weak and strong)
valid for this kind of problems.
These statements are not yet valid for fractional parabolic operators,
but we gather some results known at the moment for the FDE
(which is a particular case of of a fractional differential operator of
parabolic type) related to maximum principles
\cite{AlRefai-Luchko, Luchko2, Luchko3}. Using these results in conjunction
with the benefits of the Mittag-Leffler functions, the asymptotic behaviour
is obtained.

\section{Some exact solutions}

\begin{definition} \label{defi frac} \rm
Let $[a,b]\subset \mathbb{R}$, $\alpha \in \mathbb{R}^+$ and $n \in \mathbb{N}$
be such that $n-1<\alpha< n$, and let
$ f \in W^n(a,b)=\{ f  \in  \mathcal{C}^n(a,b]: f^{(n)}\in L^1[a,b] \}$ be.
The \textsl{fractional Caputo derivative of order $\alpha$} is defined by
$$
{}_{a} D^{\alpha}f(x)
=\begin{cases} \frac{1}{\Gamma(n-\alpha)}\int^{x}_{a}(x-\tau)^{n-\alpha-1}
f^{(n)}(\tau)d\tau, & n-1<\alpha<n\\[4pt]
f^{(n)}(x), & \alpha=n.
\end{cases}
$$
\end{definition}

\begin{proposition}\label{PropiedadesDervCaputo}
 Let $\alpha \in \mathbb{R}^+$. Then the fractional Caputo derivative of
 order $\alpha$ is a linear operator such that:
\begin{itemize}
	\item[(a)] ${}_a D^{\alpha} (C)=0$ for every $C \in \mathbb{R}$.
	\item[(b)] ${}_a D^{\alpha} ( (t-a)^\beta )
=\frac{\Gamma(1+\beta)}{\Gamma(1+\beta-\alpha)}(t-a)^{\beta-\alpha}$.
	\end{itemize}
\end{proposition}


\begin{definition}  \label{def2} \rm
For every $x\in \mathbb{R}$, $\rho>0$ the \textit{Mittag-Leffler} function is
defined by
\begin{equation}\label{E}
E_\rho(x)=\sum_{k=0}^{\infty}\frac{x^k}{\Gamma(\rho k +1)}, \quad x\in \mathbb{R}, \;
 \rho >0.
\end{equation}
\end{definition}

\begin{definition} \label{def3} \rm
 For every $x\in \mathbb{R}$, $\rho>-1$ and $\beta\in \mathbb{R}$ the \textit{Wright}
function \cite{Wr1:1934} is defined by
\begin{equation}\label{W}
W(x;\rho;\beta)=\sum^{\infty}_{k=0}\frac{x^{k}}{k!\Gamma(\rho k+\beta)}.
\end{equation}
The Mainardi function \cite{FM:1999} is a special case of the Wright
function defined by
\begin{equation}\label{M}
M_\rho (x)= W(-x,-\rho,1-\rho)=\sum^{\infty}_{n=0}
\frac{(-x)^n}{n! \Gamma( -\rho n+ 1-\rho )}, \quad x\in \mathbb{C}, \;
 0<\rho<1.
\end{equation}
\end{definition}

\begin{proposition}\label{PropiedadesML}
The Mittag-Leffler function defined in \eqref{E} satisfies the following properties:
\begin{itemize}
	\item[(a)] $E_\rho$ is an entire function if $\rho>0$.
	\item[(b)] $\lim_{t\to \infty}E_\rho(-t)=0$ for every $\rho>0$.
\item[(c)] $ _0D^\rho (E_\rho(\mu x^\rho))=\mu E_\rho(\mu x^\rho)$ for every
$x, \mu \in \mathbb{R}$, $\rho>0$.
	\item[(d)] $E_\rho$ is completely monotonic on the negative real axis.
In particular $E_\rho(-t)$ is a decreasing function in $\mathbb{R}^+$.
	\end{itemize}
\end{proposition}

See \cite{Kilbas} for items (a), (b) and (c), and \cite{FM-TheFundamentalSolution}
for item (d).

\begin{proposition}\label{WrightProp}
The Wright function satisfies the following properties:
\begin{itemize}
	\item[(a)] The Wright function \eqref{W} is an entire function if $\rho>-1$.

	\item[(b)] The derivative of the Wright function can be computed as
 $ \frac{\partial}{\partial x} W(x,\rho,\beta) = W(x,\rho,\rho+\beta)$.

\item[(c)] For all $\alpha,c \in \mathbb{R}^+$, $\rho \in (0,1)$, $\beta \in \mathbb{R}$ we have
$$
{}_0D^\alpha(x^{\beta-1}W(-cx^{-\rho},-\rho,\beta) )=x^{\beta-\alpha-1}W(-cx^{-\rho},
-\rho,\beta-\alpha).
$$

\item[(d)] The following limits involving the parameter $\alpha$ hold:
		$$
\lim_{\alpha\nearrow 1}M_{\alpha/2}(x)=M_{1/2}(x)
=\frac{e^{-\frac{x^2}{4}}}{\sqrt{\pi}},\quad 
\lim_{\alpha\nearrow 1}[1-W(-x,-\frac{\alpha}{2},1)]= \operatorname{erf}(\frac{x}{2})
$$
where $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-\xi^2}d\xi$.
 This result allows us to call function $1-W(-\cdot,-\frac{\alpha}{2},1)$ as the
``fractional erf function''.

\item[(e)] The ``fractional erf function''
$1-W(-\cdot,-\frac{\alpha}{2},1)$ is a positive and strictly increasing function
in $\mathbb{R}^+$ such that $0<1-W(-x,-\frac{\alpha}{2},1)< 1$, for all $x>0$.

\item[(f)] For every $\alpha\in (0,1)$, and $ \beta>0$,
$$
\lim_{x \to \infty } W(-x,-\frac{\alpha}{2},\beta)=0.
$$
\end{itemize}
\end{proposition}

For items (a) and (b), see \cite{Wr2:1940}.  A proof of (c) is given in
\cite{Pskhu-libro}, and a proof of (d) and (e) can be founded in \cite{RoSa1}.
For (f), see \cite{FM:1999} 

We will consider the following two regions related to problem \eqref{(P)}:
\begin{gather*}
 \Omega_0=\{ (x,t) : s_1(t)<x<s _2(t), \; 0<t\leq T \}, \\
\partial_p \Omega_0=\{ (s_1(t),t): 0<t\leq T \} \cup \{ (s_2(t),t):
 0<t \leq T \} \cup \{(x,0): a\leq x \leq b\},
\end{gather*}
 where the latter is called parabolic boundary.

\begin{definition} \label{def4} \rm
A function $u=u(x,t)$ is a solution of problem
\eqref{(P)} if
\begin{itemize}
	\item $u$ is defined in $[a_0,b_0]\times [0,T]$, where
$a_0:=\min\{ s_1(t): t \in [0,T]\}$ and
$b_0:=\max\{ s_2(t): t \in [0,T]\}$.

	\item $u\in CW_{\Omega_0}:=C(\Omega_0)\cap W^1_t((0,T))\cap C^2_x(\Omega_0)$, 	
where
\[
W^1_t((0,T)):=\{f(x,\cdot)\in C^1((0,T))\cap L^1(0,T) \text{for every fixed }
x\in [a_0,b_0] \}.
\]

\item $u$ is continuous in $\Omega_0 \cup \partial_p \Omega_0$ except perhaps at
$(a,0)$ and $(b,0)$ where
\begin{gather*}
0\leq \liminf_{(x,t)\to (a,0)} u(x,t)\leq \limsup_{(x,t)\to (a,0)} u(x,t)<+\infty, \\
0 \leq \liminf_{(x,t)\to (b,0)} u(x,t)\leq \limsup_{(x,t)\to (b,0)} u(x,t)<+\infty.
\end{gather*}

\item $u$ satisfies the conditions in \eqref{(P)}.
\end{itemize}
 \end{definition}


\begin{remark} \label{rmk1} \rm
 We require  $u$ to be defined in $[a_0,b_0]\times [0,T]$ since the fractional
derivative $_0D^\alpha_t u(x,t)$ involves values $u_t(x,\tau) $ for all $\tau$
in $[0,t]$.
\end{remark}

\subsection*{Problem 1}
Consider $s_1(t)= 0$, $s_2(t)=t^{\alpha/2}, g(t)=1$ and $h(t)=0$ in problem
\eqref{(P)}. Note that condition \eqref{(P)}(4) is not considered because $a=b=0$.
Taking $\beta=1$ and $\rho=\frac{\alpha}{2}$ in Proposition \ref{WrightProp}(c),
using Proposition \ref{WrightProp}(b) and the principle of superposition
(valid due to Proposition \ref{PropiedadesDervCaputo}), we can state that function
	$$
u(x,t)=a+b[1-W(-\frac{x}{t^{\alpha/2}},-\frac{\alpha}{2}, 1)]
$$
is a solution to the following initial-boundary problem associated with the FDE,
	\begin{equation}\label{PprimCuad}
\begin{gathered}
  {}_0D^{\alpha}_t u(x,t)= u_{xx}(x,t), \quad 0<x, \; 0<t\leq T, \; 0<\alpha<1, \\
  u(0,t)=a, \quad 0<t\leq T,\\
  u(x,0)=a+b,\quad   0<x.
 \end{gathered}
\end{equation}
Clearly $a=1$. Evaluating $u$ at the curve $s_2(t)=t^{\alpha/2}$, $0<t<T$, we obtain
$$
u(t^{\alpha/2},t)=1+b[1-W(-1,-\frac{\alpha}{2}, 1)].
$$

Since $1-W(-1,-\frac{\alpha}{2}, 1)\neq 0$ by Proposition \ref{WrightProp} item (d),
we can take $b=-\frac{1}{1-W(-1,-\frac{\alpha}{2}, 1)}$, and state that
\begin{equation}\label{sol-Pb1}
	u(x,t)=1-\frac{1}{1-W(-1,-\frac{\alpha}{2}, 1)}
[1-W(-\frac{x}{t^{\alpha/2}},-\frac{\alpha}{2}, 1)]
\end{equation}
is a solution to the problem	
\begin{equation}\label{PprimCuad2}
\begin{gathered}
 {}_0D^{\alpha}_t u(x,t)= u_{xx}(x,t), \quad 0<x<t^{\alpha/2}, \; 0<t\leq T, \;
 0<\alpha<1, \\
 u(0,t)=1, \quad 0<t\leq T,\\
 u(t^{\alpha/2},t)=0, \quad  0<t\leq T.
 \end{gathered}
\end{equation}

\subsection*{Problem 2}
Consider $s_1(t)= -t^{\alpha/2}$, $s_2(t)=t^{\alpha/2}$,
in problem \eqref{(P)}. Functions $g$ and $h$ will be determined latter.
It is reasonable to try to find a solution related to the solution \eqref{sol-Pb1}.
But now we have to deal with negative values of the variable $x$.

From \cite{FM-TheFundamentalSolution} and \cite{Eidelman-Kochubei-LIBRO}, we can
state that function
 \begin{equation}\label{u_1(x,t)}
 u_1(x,t)=\int^{\infty}_{-\infty}\frac{1}{2 t^{\alpha/2}}
M_{\alpha/2}(\frac{|x-\xi|}{t^{\alpha/2}}) f(\xi) d\xi\,
 \end{equation}
is a solution to the Cauchy problem
\begin{equation}\label{(PC)}
\begin{gathered}
_0D^{\alpha}_t u(x,t)= u_{xx}(x,t), \quad x\in \mathbb{R}, \; 0<t\leq T, \; 0<\alpha<1, \\
 u(x,0)=f(x), \quad x\in \mathbb{R}.
 \end{gathered}
\end{equation}
for any piecewise continuous and bounded function $f$.
By considering
\[
f(x)=\begin{cases}
1 & \text{if } x>0 \\
 -1 & \text{if } x<0,
\end{cases}
\]
(as it was done in \cite{LiuXu:2009} and \cite{RoTa:2014}) it results that
 \begin{equation}\label{c_1(x,t)}
 u_1(x,t)=\int^{\infty}_{0}\frac{1}{2 t^{\alpha/2}}
\big[ M_{\alpha/2}\big(\frac{|x-\xi|}{t^{\alpha/2}}\big)
- M_{\alpha/2}\big(\frac{|x+\xi|}{t^{\alpha/2}}\big) \big]d\xi.
 \end{equation}
In particular, for $x<0$ we have that $|x-\xi|=\xi-x$ for all $\xi>0$,
and using Proposition \ref{WrightProp} items 2 and 5, it results that
\begin{align*}
 u_1(x,t)
&= \int^{\infty}_{0}\frac{1}{2 t^{\alpha/2}}
\big[ M_{\alpha/2}\big(\frac{\xi-x}{t^{\alpha/2}}\big)
 - M_{\alpha/2}\big(\frac{|x+\xi|}{t^{\alpha/2}}\big) \big]d\xi \\
&= \int^{\infty}_{0}\frac{1}{2 t^{\alpha/2}} M_{\alpha/2}
 \big(\frac{\xi-x}{t^{\alpha/2}}\big)d\xi
 - \int^{-x}_{0}\frac{1}{2 t^{\alpha/2}}M_{\alpha/2}\big(\frac{-(x+\xi)}{t^{\alpha/2}}\big)
 d\xi\\
&\quad - \int^{\infty}_{-x}\frac{1}{2 t^{\alpha/2}}M_{\alpha/2}
 \big(\frac{x+\xi}{t^{\alpha/2}}\big)d\xi\\
&= \int^{\infty}_{0}\frac{1}{2 t^{\alpha/2}} M_{\alpha/2}
 \big(\frac{\xi-x}{t^{\alpha/2}}\big)d\xi
 - \int^{-x}_{0}\frac{1}{2 t^{\alpha/2}}M_{\alpha/2}
 \big(\frac{-(x+\xi)}{t^{\alpha/2}}\big)d\xi \\
&\quad - \int^{\infty}_{-x}\frac{1}{2 t^{\alpha/2}}M_{\alpha/2}
 \big(\frac{x+\xi}{t^{\alpha/2}}\big)d\xi\\
&= W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)-1.
 \end{align*}
Then, the function
$$
u(x,t)=\begin{cases}
1-W(-\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1), & 0<x<t^{\alpha/2}\\
W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)-1, & -t^{\alpha/2}< x<0
\end{cases}
$$
is a solution to \eqref{(P)}(1) for every $x\neq 0$,
$ -t^{\alpha/2}< x<t^{\alpha/2}$. Moreover, due to the linearity of the Caputo
derivative we can state that
$$
u(x,t)=\begin{cases}
a [1-W(-\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)]+b, & 0<x<t^{\alpha/2}\\
a[W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)-1]+b, & -t^{\alpha/2}< x<0\end{cases}
$$
is a solution to \eqref{(P)}(1) for every $x\neq 0$, $ -t^{\alpha/2}< x<t^{\alpha/2}$.

We would like to extend this solution to the values of $x=0$, but if we extend
$u$ to this values, it results that
\begin{equation}\label{u}
u(x,t)=\begin{cases}
a [1-W(-\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)]+b, & 0\leq x< t^{\alpha/2}\\
a[W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)-1]+b, & -t^{\alpha/2}<x<0,
\end{cases}
\end{equation}
is a $\mathcal{C}^1_x(D_T)$ function, but $u_{xx}(0^+,t)\neq u_{xx}(0^-,t)$.

To obtain a $\mathcal{C}^2_x(\Omega_0)$ solution according to definition \ref{def4}
we  apply the following Lemma proved in \cite{GoReRoSa:2015}.

\begin{lemma}\label{lemma}
Let $v(x,t)$ be a solution of the FDE such that
 $F(x,t)=\int^{\infty}_{x}v(\xi,t)d\xi$ is  well defined  for every
$(x,t) \in \mathbb{R} \times \mathbb{R}^+$,
$\lim_{x \to \infty}\frac{\partial v}{\partial x}(x,t)=0$,
$|\frac{\partial}{\partial \tau}v(\xi,\tau)|\leq g(\xi)$ in 
$L^1(x,\infty)$ and
\[
 \frac{\frac{\partial}{\partial \tau}v(\xi,\tau)}{(t-\tau)^\alpha}
\in L^1((x,\infty)\times(0,t)).
\]
Then $\int^{\infty}_{x}v(\xi,t)d\xi$ is a solution to the FDE.
\end{lemma}

From Proposition \ref{WrightProp} and using estimates made in \cite{GoReRoSa:2015},
it yields that $v(x,t)=W(-\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)$ is under the
assumptions of Lemma \ref{lemma} for every $x\geq 0$.
Then
\[
\int^{\infty}_{x} v(\xi,t)d\xi=t^{\alpha/2}W(-\frac{x}{t^{\alpha/2}},
 -\frac{\alpha}{2},1+\frac{\alpha}{2})
\]
 satisfies \eqref{(P)}(1). By using the linearity of the Caputo derivative
and the principle of superposition we can state that
\begin{equation}\label{w_pos}
w_{\rm pos}(x,t)=a[x+ t^{\alpha/2}W(-\frac{x}{t^{\alpha/2}},
-\frac{\alpha}{2},1+\frac{\alpha}{2})]+bx
\end{equation}
is a solution to the FDE such that $\frac{\partial}{\partial x}w_{\rm pos}(x,t)=u(x,t)$
for every $x\geq 0$, $t>0$.

For negative values of $x$ we enunciate an analogous lemma.

\begin{lemma}\label{lema-bis}
Let $v(x,t)$ be a solution of the FDE such that
$F(x,t)=\int_{-\infty}^{x}v(\xi,t)d\xi$ is  well defined  for every
$(x,t) \in \mathbb{R} \times \mathbb{R}^+$,
$\lim_{x \to -\infty}\frac{\partial v}{\partial x}(x,t)=0$,
$|\frac{\partial}{\partial \tau}v(\xi,\tau)|\leq g(\xi)$  in $L^1(-\infty,x)$ and
\[
 \frac{\frac{\partial}{\partial \tau}v(\xi,\tau)}{(t-\tau)^\alpha}
\in L^1((-\infty,x)\times(0,t)).
\]
 Then $\int_{-\infty}^{x}v(\xi,t)d\xi$ is a solution to the FDE.
\end{lemma}

\begin{proof}
The required assumptions allows us to apply Fubini's theorem. Then
\begin{align*}
{}_0 D^{\alpha}_t F(x,t)
&=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}\frac{\frac{\partial}{\partial \tau}
 F(x,\tau)}{(t-\tau)^\alpha}d\tau \\
&=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}\frac{1}{(t-\tau)^\alpha}
\Big(\frac{\partial}{\partial \tau}\int_{-\infty}^{x}v(\xi,\tau)d\xi\Big)d\tau\\
&=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}\frac{1}{(t-\tau)^\alpha}
 \int_{-\infty}^{x}\frac{\partial}{\partial \tau}v(\xi,\tau)d\xi d\tau \\
&=\int_{-\infty}^{x}\frac{1}{\Gamma(1-\alpha)}
 \int^{t}_{0}\frac{\frac{\partial}{\partial \tau} v(\xi,\tau)}{(t-\tau)^\alpha}d\tau \\
&=\int_{-\infty}^{x}\, _0 D^{\alpha}_t v(\xi,t)d\xi \\
&=\int_{-\infty}^{x} \lambda^2\frac{\partial^2v}{\partial x^2}(\xi,t)d\xi \\
&= \lambda^2\frac{\partial v}{\partial x}(\xi,t)\Big|^{x}_{-\infty}
= \frac{\partial^2}{\partial x^2}F(x,t) .
\end{align*}
\end{proof}
Applying Lemma \ref{lema-bis} to function
$v(x,t)=W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1)$ for $x<0$, and reasoning as
before, it yields that
\begin{equation}\label{w_neg}
w_{\rm neg}(x,t)=a[-x+ t^{\alpha/2}W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},
1+\frac{\alpha}{2})]+bx
\end{equation}
is a solution to the FDE such that $\frac{\partial}{\partial x}w_{\rm neg}(x,t)=u(x,t)$
for every $x<0$, $t>0$.

Combining \eqref{w_pos} with \eqref{w_neg} we have
\begin{equation}\label{w}
w(x,t)= \begin{cases}
a[x+ t^{\alpha/2}W(-\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1+\frac{\alpha}{2})]+bx,
& 0\leq x< t^{\alpha/2}\\[4pt]
a[-x+ t^{\alpha/2}W(\frac{x}{t^{\alpha/2}}, -\frac{\alpha}{2},1+\frac{\alpha}{2})]+bx,
& -t^{\alpha/2}<x<0,
\end{cases}
\end{equation}
is a solution to \eqref{(P)}(1). Clearly $w$ is a $\mathcal{C}^2_x(\Omega_0)$ function
and by varying the parameters $a$ and $b$ we obtain different solutions associated
with different boundary conditions. For example, if $b=\frac{1}{2}$ and
$a= \frac{1}{2(1+W(-1, -\frac{\alpha}{2},1+\frac{\alpha}{2}))}$ it yields that \eqref{w}
 is a solution to the moving-boundary problem
\begin{equation}\label{(P2)}
\begin{gathered}
 {}_0D^{\alpha}_t u(x,t)=\lambda^2\, u_{xx}(x,t), \quad
 -t^{\alpha/2}<x<t^{\alpha/2}, \; 0<t\leq T, \; 0<\alpha<1, \\
  u(-t^{\alpha/2},t)=0, \quad  0<t\leq T,\\
  u(t^{\alpha/2},t)=t^{\alpha/2}, \quad  0<t\leq T.
\end{gathered}
\end{equation}

\section{Asymptotic behaivor as $t$ tends to infinity}

 Hereinafter we call $L^\alpha$ to the operator associated with the FDE,
$ L^\alpha:= \frac{\partial ^2}{\partial x^2 }- D^\alpha$.
The following two results have been proved in \cite{Luchko3} and \cite{Ro:2016}
 respectively.

\begin{proposition}\label{FracExtremumPrinciple}
Let $f \in W_t^1((0,T])\cup \mathcal{C}([0,T])$ be a function that attains its maximum
at the point $t_0 \in (0,T]$. Then for every $\alpha \in (0,1)$ it results that
$ D^\alpha f(t_0)\geq 0$.
\end{proposition}

\begin{remark} \label{rmk2} \rm
Note that this extremum principle is not valid either if $\alpha > 1$
or if the fractional derivative is taken in the Riemann-Liouville sense.
\end{remark}

\begin{proposition}\label{L[u]<0}
If $u$ is a function with $L^\alpha [u]>0$ (resp. $L^\alpha [u]<0$) in $\Omega_0$,
then $u$ does not attain its maximum (resp. minimum) at $\Omega_0$.
\end{proposition}

Let us adapt here the next theorem obtained in \cite{Luchko1} to the moving-boundary
problem \eqref{(P)}.

\begin{theorem}\label{Ppio del maximo debil-negativo}
Let $u\in CW_{\Omega_0}$ be a solution of \eqref{(P)}. Then either
$u(x,t)\geq 0$ for all $(x,t) \in \overline{\Omega_0}$ or
$u$ attains its negative minimum on $\partial_p \Omega_0$.
\end{theorem}

\begin{proof}
If $u\geq 0$ in $\overline{\Omega}_0$ the prove is finished.
Now, suppose that there exists a point $(x_0,t_0)$, such that
$s_1(t_0)<x_0<s_2(t_0)$, $ 0<t_0\leq T$ and
$u(x_0,t_0)<\min_{\partial_p \Omega_0}{u(x,t)}=m\leq 0$.
Let $\epsilon=m-u(x_0,t_0)>0$ be and consider the auxiliary function
\begin{equation}\label{1-0}
w(x,t)=u(x,t)-\frac{\epsilon}{2}\frac{T-t}{T}, \quad (x,t) \in \overline{\Omega}_0.
\end{equation}
Note that
\begin{equation}\label{1-1}
w(x,t)\geq u(x,t)-\frac{\epsilon}{2} \quad \forall (x,t) \in \overline{\Omega}_0,
\end{equation}
and that for every $(x,t) \in \partial_p \Omega_0$, it results that
\begin{align*}
w(x_0,t_0)&=u(x_0,t_0)-\frac{\epsilon}{2}\frac{T-t_0}{T}
 \leq u(x_0,t_0)\\
&=m- \epsilon \leq u(x,t) - \epsilon \leq w(x,t)+\frac{\epsilon}{2}-\epsilon \\
&=w(x,t)-\frac{\epsilon}{2}.
\end{align*}
Consequently $ w(x,t)>w(x_0,t_0)$ for all $(x,t) \in \partial_p \Omega_0 $ and we
can state that
\begin{equation}\label{1-2}
w \text{ must attain its minimum at } \Omega_0.
\end{equation}
On the other hand, from Proposition \ref{PropiedadesDervCaputo} it follows that
$$
L^\alpha[w]= L^\alpha[u]-\frac{\epsilon}{2}\frac{\Gamma(2)}{\Gamma(2-\alpha)}\frac{t^{1-\alpha}}{T}<0
\quad \forall  (x,t)\in \Omega_0 .
$$
Now, applying Proposition \ref{L[u]<0} to $w$, it results that $w$ can not
attain its minimum in $\Omega_0$, which contradicts \eqref{1-2}.
\end{proof}

\begin{theorem}\label{conv unif}
 Let $u$ be a solution of the fractional moving-boundary problem \eqref{(P)}
such that:
\begin{itemize}
\item[(a)]  $s_1$ is a decreasing continuous functions in $\mathbb{R}_0^+$ such that
 $\lim_{t\to \infty}s_1(t)=a_0$,

\item[(b)] $s_2$ is an increasing continuous functions in $\mathbb{R}_0^+$
 such that $\lim_{t\to \infty}s_2(t)=b_0$ and $s_1(t)<s_2(t)$ for every $t>0$,

\item[(c)] $f$ is a non-negative continuous function defined in $[a,b]$,

\item[(d)] $g$ and $h$ are non-negative continuous functions defined in
$\mathbb{R}^+$ such that $\lim_{t\to \infty}g(t)=\lim_{t\to \infty}h(t)=0$.
\end{itemize}
Then $\lim_{t\to \infty}u(x,t)=0$ uniformly in $[a_0,b_0]$.
\end{theorem}

\begin{proof}
Consider the function $\varphi(x)=\exp\{ 2 b_0\}-\exp\{x\}$ defined in $[a_0,b_0]$.
 Clearly $\varphi $ is a decreasing positive function in $[a_0,b_0]$, with
$\varphi_{\rm min}=\varphi (b_0)$, $\varphi_{\rm max}=\varphi (a_0)$.
Let $\Psi\colon [a_0,b_0]\times \mathbb{R}^+_0$ be the non-negative function defined by
\begin{equation}\label{Psi}
\Psi(x,t)=\epsilon \varphi (x) + \frac{A}{\varphi_{\rm min}}\varphi(x)E_{\alpha}(-\gamma t^\alpha)
\end{equation}
where $\epsilon$, $A$ and $\gamma$ will be determined later, and $E_\alpha$ is the
Mittag-Leffler function with parameter $\alpha$.
Applying the $L^\alpha$ operator to $\Psi$ and using Proposition
\ref{PropiedadesML}(c), it yields that
\begin{equation}\label{L^al Psi}
L^\alpha\Psi(x,t)=-\epsilon \exp\{ x\} + \frac{A}{\varphi_{\rm min}}E_{\alpha}(-\gamma t^\alpha)
[- \exp\{ x\}+\gamma \varphi (x) ].
\end{equation}
Applying Proposition \ref{PropiedadesML}(d) it results that
\begin{equation}\label{LPsi<0}
L^\alpha\Psi(x,t)<0 \text{ for every } (x,t)  \text{ in } [a_0,b_0]\times \mathbb{R}_0^+
 \text{ if }  \gamma< \frac{1}{\exp\{2b_0-a_0 \}-1}.
\end{equation}
Now, let $z=\Psi - u$ be. Taking into account the non negativity of the
 Mittag-Leffler function and that $g$ and $h$ are non--negative functions,
such that $\lim_{t\to \infty}g(t)=\lim_{t\to \infty}h(t)=0$, there exist
$t_1$ and $t_2$ such that the next inequalities hold
\begin{gather}\label{1}
\begin{aligned}
 z(s_1(t),t)&=\epsilon \varphi(s_1(t)) +\frac{A\varphi(s_1(t))}{\varphi_{\rm min}}E_{\alpha}
(-\gamma t^\alpha)-g(t) \\
&\geq\epsilon \varphi_{\rm min}-g(t)>0 \quad \text{if }   t>t_1.
\end{aligned}\\
\label{2}
\begin{aligned}
 z(s_2(t),t)&=\epsilon \varphi(s_2(t)) +\frac{A\varphi(s_2(t))}{\varphi_{\rm min}}
E_{\alpha}(-\gamma t^\alpha)-h(t) \\
&\geq\epsilon \varphi_{\rm min}-h(t)>0 \quad \text{if }  t>t_2.
\end{aligned}
\end{gather}
Consider $\tilde{t}=\max\{ t_1,t_2\}$,
$M_{\tilde{t}}=\max\{ u(x,t); s_1(t)\leq x\leq s_2(t), 0\leq t\leq \tilde{t} \}$
and take $A $ such that
\begin{equation}\label{AmayorM}
A \geq \frac{M_{\tilde{t}}}{E_{\alpha}(-\gamma \tilde{t}^\alpha)}.
\end{equation}
 Then
\begin{equation}\label{3}
z(x,\tilde{t})\geq 0, \quad s_1(\tilde{t})<x<s_2(\tilde{t}).
\end{equation}

Consider the region $\Omega_{\tilde{t}}=\{ (x,t) : s_1(t)< x< s_2(t), \,
 \tilde{t}<t \}$. Note that inequalities \eqref{1}, \eqref{2} and \eqref{3}
imply that $z\geq 0$ in $\partial_p\Omega_{\tilde{t}}$. With the aim to prove that
$z\geq 0$ in $\overline{\Omega_{\tilde{t}}}$, fix $T>\tilde{t}$ large enough
and suppose that there exists a point
$(x_0,t_0)\in \{ (x,t) : s_1(t)< x< s_2(t), \, \tilde{t}<t\leq T \} $ where
$z$ attains its negative minimum.
Clearly
\begin{equation}\label{z1}
z(x_0,t)\geq z(x_0,t_0) \quad \text{ for all }  \tilde{t} \leq t\leq T.
\end{equation}
From Proposition \ref{PropiedadesML}(d) and \eqref{AmayorM} it results that
\begin{equation}\label{z2}
z(x_0,t)\geq 0 \quad \text{for all }  t< \tilde{t}.
\end{equation}
Inequalities \eqref{z1} and \eqref{z2} imply that the function $z(x_0,\cdot)$
attains its absolute minimum at $t_0$ in the interval $[0,T]$.
Applying Proposition \ref{FracExtremumPrinciple} it holds that
 $D^\alpha z(x_0,t_0) \leq 0$. Then
\begin{align*}
 L^\alpha u (x_0,t_0)
&=L^\alpha \Psi (x_0,t_0)-L^\alpha z (x_0,t_0) \\
&=L^\alpha \Psi (x_0,t_0)
-\frac{\partial^2}{\partial x^2} z (x_0,t_0)+D^\alpha z(x_0,t_0)<0
\end{align*}
which is a contradiction.
Therefore $z\geq 0$ in $\overline{\Omega_{\tilde{t}}}$, or equivalently
\begin{equation}\label{4}
u(x,t)\leq \Psi (x,t) \quad \forall (x,t) \in \overline{\Omega_{\tilde{t}}}.
\end{equation}
The non-negativity of functions $f, g$ and $h$ imply the non-negativity of $u$
in $\partial\Omega_0$. Then Theorem \ref{Ppio del maximo debil-negativo} yields
\begin{equation}\label{4-b}
0\leq u(x,t) \quad \forall (x,t) \in \overline{\Omega_{\tilde{t}}}.
\end{equation}
Extending $u$ by $0$ outside $\Omega_0$, using \eqref{4} and \eqref{4-b} we
can state that
$$
0\leq u(x,t)\leq \epsilon \varphi_{\rm max}+\frac{A\varphi_{\rm max}}{\varphi_{\rm min}}
E_{\alpha}(-\gamma t^\alpha) \quad \text{for all } x\in [a_0,b_0],  \text{ and all }
t>\tilde{t}.
$$
By Proposition \ref{PropiedadesML}(b),  there exists $t_3>0$
(we can take $t_3>\tilde{t}$) such that
\begin{equation}\label{u<eps}
0\leq u(x,t)\leq 2\epsilon \varphi_{\rm max} \quad \text{for all } x\in [a_0,b_0],
 \text{ and all } t>t_3.
\end{equation}
Taking into account that inequality \eqref{u<eps} holds for every $\epsilon>0$
it results that $\lim_{t\to \infty}u(x,t)=0$ uniformly in $[a_0,b_0]$.
\end{proof}

\begin{remark} \label{rmk3} \rm
The following initial-boundary-value problem was considered in \cite{Luchko3}:
\begin{equation}\label{(PVIC)}
\begin{gathered}
{}_0D^{\alpha}_t u(x,t)=\lambda^2 u_{xx}(x,t), \quad 0<x<L, \; 0<t, \; 0<\alpha<1, \\
 u(0,t)=0, \quad 0<t,\\
 u(L,t)=0, \quad 0<t,\\
 u(x,0)=f(x), \quad 0\leq x\leq L.
\end{gathered}
\end{equation}
There a classical solution was obtained of the form
\begin{equation}\label{formal sol}
u(x,t)=\sum_{i=1}^\infty (f,X_i)E_{\alpha}(-\lambda_i t^\alpha)X_i(x),
\end{equation}
where $X_i$, $i=1, 2,dots $ are the eigenfunctions corresponding to the
eigenvalues $\lambda_i$ of the eigenvalue problem
\begin{gather*}
X''(x)=\lambda X(x)\\
X(0)=X(L)=0,
\end{gather*}
and $f\in \mathcal{C}^1([0,L])\cup \mathcal{C}^2(0,L)$, $f'' \in L^2(0,L)$ and $f(0)=f(L)=0$.
 Note that if we ask $f$ to be a non-negative function then
Theorem \ref{conv unif} gives the uniformly convergence to zero in
$[0,L]$ when $t\to \infty$ of \eqref{formal sol}.
\end{remark}

\begin{corollary}\label{coro1}
Let $u$ be a solution of the fractional initial-boundary-value problem
\begin{equation}\label{(PVIC-2)}
\begin{gathered}
 {}_0D^{\alpha}_t u(x,t)=\lambda^2 u_{xx}(x,t), \quad 0<x<L, \; 0<t, \; 0<\alpha<1, \\
  u(0,t)=g(t), \quad 0<t,\\
  u(L,t)=h(t), \quad 0<t,\\
  u(x,0)=f(x), \quad 0\leq x\leq L,
\end{gathered}
\end{equation}
 such that:
\begin{itemize}
\item[(a)] $g$ is a continuous functions defined in $\mathbb{R}^+$ such that
$\lim_{t\to \infty}g(t)=g_0$ and $g(t)\geq g_0$ for every $t \in \mathbb{R}^+$,

\item[(b)] $h$ is a continuous functions defined in $\mathbb{R}^+$ such that
$\lim_{t\to \infty}h(t)=h_0$ and $h(t)\geq h_0$ for every $t \in \mathbb{R}^+$,

\item[(c)] $f$ is a continuous function defined in $[0,L]$ such that
$f(x)-[\frac{h_0-g_0}{L} x+g_0]\geq 0$ for every $x\in [0,L]$.
\end{itemize}
Then $\lim_{t\to \infty}u(x,t)=\frac{h_0-g_0}{L}x+g_0$ for every
$ x \in [0,L]$.
\end{corollary}

The proof of the above corollary, follows by applying Theorem \ref{conv unif}
 to the function  $w(x,t)=u(x,t)-[\frac{h_0-g_0}{L}x+g_0]$.


\begin{remark} \label{rmk4} \rm
The hypotheses $g(t)\geq m_1$ and $h(t)\geq m_2$ for every $t \in \mathbb{R}^+$
in Corollary \ref{coro1} are essential because we need to guarantee the
non-negativity of the solution in the hole parabolic boundary $\Omega_0$
to apply Theorem \ref{Ppio del maximo debil-negativo}, which is used in the
proof of Theorem \ref{conv unif}. This simple observation is interesting
because it makes gains relevance to the memory of the operator that we are
considering.

Note that if we consider the classical parabolic operator, the weak maximum
principle is valid for every region $\Omega_{\tilde{t}}$ with $\tilde{t}\neq 0$,
but for the ``fractional weak maximum principle''
(Theorem \ref{Ppio del maximo debil-negativo}) this is not valid. This principle
only holds in the hole region $\Omega_0$, that is, the region including
everything that happened from the initial time.
\end{remark}


\subsection*{Acknowledgements}
The present work has been sponsored by the Projects PIP N$^\circ$ 0534 from
CONICET Univ. Austral, and by AFOSR-SOARD Grant FA9550-14-1-0122.
I appreciate the valuable suggestions by the anonymous referee which
improved this article.

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\end{document}