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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 201, pp. 1--7.\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-2017/201\hfil Nonlocal initial boundary value problems]
{Nonlocal initial boundary value  problem for the time-fractional 
 diffusion equation}

\author[M. Sadybekov, G. Oralsyn \hfil EJDE-2017/201\hfilneg]
{Makhmud Sadybekov, Gulaiym Oralsyn}

\address{Makhmud Sadybekov \newline
 Institute of Mathematics and Mathematical Modeling,
125 Pushkin str., 050010 Almaty, Kazakhstan}
\email{sadybekov@math.kz}

\address{Gulaiym Oralsyn (corresponding author)\newline
Institute of Mathematics and Mathematical Modeling,
125 Pushkin str., 050010 Almaty, Kazakhstan.\newline
Al-Farabi Kazakh National University, Almaty, Kazakhstan}
\email{g.oralsyn@list.ru}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted August 22, 2016. Published August 31, 2017.}
\subjclass[2010]{26A33, 31A10}
\keywords{Time-fractional diffusion equation; fundamental solution; 
\hfill\break\indent time-fractional heat potential; layer potentials;
 nonlocal boundary condition}

\begin{abstract}
 In this article we discuss a method for constructing trace formulae for
 the heat-volume potential of the time-fractional diffusion equation to
 lateral surfaces of cylindrical domains and use these conditions to construct
 as well as to study a nonlocal initial boundary value problem for the
 time-fractional diffusion equation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

Let us consider the one-dimensional potential
\begin{equation}
u(t)=\int_{0}^{1}-\frac{1}{2}|t-\tau|f(\tau)d\tau\quad\text{in } \Omega=(0,1),
 \label{eq1}
\end{equation}
where $f$ is an integrable function in $\Omega$.
The kernel of the one-dimensional potential is a fundamental solution 
of the second order differential equation; that is,
\begin{equation}
-\partial_{t}^{2}E(t-\tau)=\delta(t-\tau),\label{eq2}
\end{equation}
where  $E(t-\tau)=-\frac{1}{2}|t-\tau|$ and $\delta$ is the Dirac distribution.
Hence the potential \eqref{eq1} satisfies the equation
\begin{equation}
-\partial_{t}^{2}u(t)=f(t),\; t\in\Omega.\label{eq3}
\end{equation}
An interesting  question having several important applications (in general) 
is what boundary condition can be put on $u$ on
the boundary of $\Omega$ so that equation \eqref{eq3} complemented by this
boundary condition would have a unique solution in $\Omega$ still given 
by the same formula \eqref{eq1} (with the same kernel).
This amounts to finding the trace of the one-dimensional Newton 
potential \eqref{eq1} to the boundary of $\Omega$.

Simply, by using integration by parts, one obtains
that boundary conditions for the potential \eqref{eq1}  are
\begin{equation}
u'(0)+u'(1)=0,
-u'(1)+u(0)+u(1)=0.\label{eq4}
\end{equation}

Hence if we solve equation \eqref{eq3} with the boundary
conditions \eqref{eq4}, then we find a unique solution of this
boundary value problem in the form \eqref{eq1}. This problem
becomes more interesting for PDE. The trace of the Newton
potential on a boundary surface appeared in Kac's work
\cite{Kac1}, where he called it and the subsequent spectral
analysis as ``the principle of not feeling the boundary''. This
was further expanded in Kac's book \cite{Kac2} with several
further applications to the spectral theory and the asymptotics of
the Weyl eigenvalue counting function. Some results towards
answering these questions can be found in papers of Kac
\cite{Kac1,Kac2}, Saito \cite{Sa}, as well as in systematic studies 
of Kal'menov and
Suragan \cite{KS3, KS2,KS4,KS5,ST}, see also Kal'menov and
Otelbaev \cite{KO} for the more general analysis. The analogues of
the problem for the Kohn Laplacian and its powers on the
Heisenberg group have been recently investigated by Ruzhanksy and
Suragan in \cite{Ruzhansky-Suragan:Kohn-Laplacian} as well 
as in \cite{Ruzhansky-Suragan:Layers} for general stratified Lie groups.

The main purpose of this paper is to construct trace formulae for
the heat-volume potentials of the time-fractional diffusion
equation to piecewise smooth lateral surfaces of cylindrical
domains and use these conditions to construct as well as to study
a nonlocal initial boundary value problem for the time-fractional
diffusion equation. Consider
\begin{gather}\label{tfde}
\lozenge_{\alpha,t}u=\partial_{t}^{\alpha}u-\Delta u=f\quad\text{in }
 \Omega\times(0,T), \\
\label{initcond}
u(0,x)=0,\quad x\in\Omega,
\end{gather}
where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with the boundary
$\partial\Omega\in C^{1+\gamma},\,0<\gamma<1$, 
$\Delta=\sum_{i=1}^{n}\partial^{2}_{x_{i}}$ is the Laplacian and
$$
\partial_{t}^{\alpha}u(t,x)=\frac{1}{\Gamma(1-\alpha)}
\int_{0}^{t}(t-\tau)^{-\tau}u'_{\tau}(\tau,x)d\tau
$$
is the fractional Caputo time derivative of order $0<\alpha\leq1$. 
Here $\Gamma$ is the gamma function. We shall note that for
$\alpha=1$ the fractional derivative coincides with the standard
time derivative.

For the convenience of the reader let us now briefly recapture the main results
 of this paper:

 We establish trace formulae for the time-fractional heat potential
 operator
 $$\int_{0}^{t}d\tau\int_{\Omega}
 E(x-y,t-\tau)f(\tau,y)dy
$$ to the surface $\partial \Omega\times
 (0,T)$, where $\partial \Omega$ is the boundary of the bounded
 domain $\Omega\subset \mathbb{R}^{n}$. Then we use this to
 introduce a version of Kac's boundary value problem, that is Kac's
 principle of ``not feeling the boundary'' for the time-fractional
 heat operator $\lozenge_{\alpha,t}$.


In Section \ref{SEC:prelim} we very briefly review the main
concepts of potential theory for the fractional diffusion equation
and fix the notation. In Section \ref{SEC:3} we derive trace
formulae and give the analogues of Kac's boundary value problem
for the time-fractional diffusion equation in Theorem
\ref{THM:main}.

\section{Preliminaries}
\label{SEC:prelim}

In this section we very briefly review some important concepts of
the time-fractional diffusion equation and fix the notation. For
the general background details on potential theory of the
time-fractional diffusion equation we refer to 
\cite{Kem2011,KemRou2009,KemRou2010,AKM}. The fundamental solution
of the time-fractional diffusion equation \eqref{tfde} is
given by
\begin{equation}
E(x,t)=\theta(t)\pi^{-d/2}t^{\alpha-1}
|x|^{-d}H_{12}^{20}
\Big(\frac{1}{4}|x|^{2}t^{-\alpha}|^{(\alpha,\alpha)}_{(-d/2,1),(1,1)}\Big),
\end{equation}
where $H$ is the Fox $H$-function (see e.g. \cite{Kilbas2004}) and $\theta$
 is the Heaviside step function.
It is constructed by taking the
Laplace-transform in the time and the Fourier-transform in the spatial variable of
the time-fractional diffusion equation
$$
\lozenge_{\alpha,t}E(x,t):=(\partial_{t}^{\alpha}-\Delta_{x})E(x,t)=\delta(x,t),
$$
where $\delta(x,t)$ is the Dirac distribution at the origin, and
by using the inverse Fourier-transform of the Mittag-Leffler
function. Heat volume potential,  single and double layer
potentials of the time-fractional diffusion equation,
respectively, can be defined by
\begin{gather}
(\lozenge^{-1}_{\alpha,t}\rho)(x,t)=\int_{0}^{t}\int_{\Omega}
E(x-y,t-\tau)\rho(y,\tau)dyd\tau, \\
(S\rho)(x,t)=\int_{0}^{t}\int_{\partial\Omega}E(x-y,t-\tau)\rho(y,\tau)dyd\tau, \\
(D\rho)(x,t)=\int_{0}^{t}\int_{\partial\Omega}
\partial_{n}E(x-y,t-\tau)\rho(y,\tau)dyd\tau,
\end{gather}
where $\partial_{n}$ is the outer normal derivative on the boundary 
$\partial\Omega$ of the bounded domain $\Omega$.
Here we also recall Green's formula (see, for example, \cite{KemRou2009}) 
for the the time-fractional diffusion operator
\begin{equation}\label{greenformula}
\int_{0}^{T}\int_{\Omega}\left(\lozenge_{{\alpha,\tau}}
uP_{T}v-P_{T}u\lozenge_{\alpha,\tau}v\right)dxd\tau
=\int_{0}^{T}\int_{\partial\Omega}(u\partial_{n}P_{T}v-\partial_{n}u P_{T}v)dSd\tau,
\end{equation}
where $P_{T}$ is a time involution operator on the interval
$(0,T)$ and is defined by setting
$$
P(T)v(\tau)=v(T-\tau).
$$

\section{Trace formula and initial boundary value problem}
\label{SEC:3}

Let $\Omega\subset \mathbb{R}^{d},\, d\geq2$, be a bounded domain
with Lyapunov boundary $\partial\Omega\in C^{1+\lambda},\,
0<\lambda<1$, and $f\in C(\overline{(0,T)\times\Omega})$ such that
$f(\cdot,t)$ is H\"older continuous uniformly in $t\in[0,T]$ and
$\operatorname{supp} f(\cdot,t)\subset\Omega,\, t\in[0,T]$. Consider the
following time-fractional generalization of the heat potential
(time-fractional heat potential)
\begin{equation}\label{heatpot}
u(x,t):=\lozenge_{\alpha,t}^{-1}f=\int_{0}^{t}d\tau\int_{\Omega}
E(x-y,t-\tau)f(\tau,y)dy,\; x\in\Omega,\;t\in(0,T),
\end{equation}
where $E$ is a fundamental solution of $\lozenge_{\alpha,t}$. Here
our aim is to find a boundary condition for $u$ on the boundary
$\partial\Omega$ of a bounded domain $\Omega$ such that with this
boundary condition the equation
\begin{equation}\label{CauchyonG}
\begin{gathered}
\lozenge_{\alpha,t} u(x,t)= f(x,t),\quad{\rm in}\quad \Omega\times(0,T),\\ 
u(x,0)=0,\quad x\in \Omega,
\end{gathered}
\end{equation}
has a unique classical solution
and this solution is the time-fractional heat potential
\eqref{heatpot}.
This amounts to finding the trace of
the integral operator in \eqref{heatpot}
on $\partial\Omega$.

A starting point for us will be that
if $f\in C(\overline{\Omega\times(0,T)})$ such that $f(\cdot,t)$ 
is H\"older continuous uniformly in $t\in[0,T]$ and $\operatorname{supp}
f(\cdot,t)\subset\Omega$, $t\in[0,T]$, then
$u$ defined by \eqref{heatpot} is well defined
and
satisfies the initial problem \eqref{CauchyonG}
(see \cite[Theorem 2.4]{Kem2011a}).

Our main result for the time-fractional heat potential operator is
the following variant of Kac's formula (see the discussion in the
introduction of \cite{RRS} and
\cite{Ruzhansky-Suragan:Kohn-Laplacian}) for a case of setting of
the time-fractional diffusion equation.

\begin{theorem} \label{THM:main} 
For each $f\in C(\overline{\Omega\times(0,T)})$
 such that $f(\cdot,t)$ is H\"older continuous uniformly in $t\in[0,T]$ and
 $\operatorname{supp} f(\cdot,t)\subset\Omega,\, t\in[0,T]$, the
 time-fractional heat potential $u=\lozenge_{\alpha,t}^{-1}f$ satisfies the
 following nonlocal boundary condition:
\begin{equation}\label{8}
\begin{aligned}
&-\frac{u(x,t)}{2}+
 \int_{0}^{t}d\tau\int_{\partial \Omega}
 \partial_{n}E(x-y,t-\tau) u(y,\tau)dS_{y}  \\
&-\int_{0}^{t}d\tau\int_{\partial \Omega}E(x-y,t-\tau)\partial_{n}u(y,\tau)dS_{y}=0,
\end{aligned}
 \end{equation}
 for all $x\in \partial \Omega$ and $t\in(0,T)$.
 Conversely, if $u$ is a solution of the time-fractional diffusion
 equation
 \begin{equation}\label{9}
 \lozenge_{\alpha,t}  u=f,
 \end{equation}
 satisfying the initial condition
 \begin{equation}\label{10}
 u|_{t=0}=0,\quad \text{on } \Omega,
 \end{equation} 
and the boundary condition \eqref{8},
 then it is given as the time-fractional heat potential
 $u=\lozenge_{\alpha,t}^{-1}f$ by formula \eqref{heatpot} and it is
 unique.
\end{theorem}

\begin{corollary} \label{coro3.2}
 It follows from Theorem \ref{THM:main} that the kernel $E$, which
 is a fundamental solution of the time-fractional diffusion
 equation, is Green's function of the nonlocal initial boundary
 value problem \eqref{8}-\eqref{10} in $\Omega\times(0,T)$.
 Therefore, the initial nonlocal boundary value problem
 \eqref{8}-\eqref{10} can serve as an example of an explicitly
 solvable initial boundary value problem for the time-fractional
 diffusion equation for any $0<\alpha\leq 1$ (and independent of
 the shape of the domain $\Omega$).
\end{corollary}

\begin{proof}[Proof of Theorem \ref{THM:main}] 
By using Green's formula \eqref{greenformula}, for any $x\in\Omega$ 
and $t\in(0,T)$, we obtain
\begin{align*}
u(x,T-t)&=\int_{0}^{T-t}d\tau\int_{\Omega}
 E(x-y,T-t-\tau)f(y,\tau)dy \\
&=\int_{0}^{T-t}d\tau\int_{\Omega}
 E(x-y,T-t-\tau)\lozenge_{\alpha,\tau} u(y,\tau) dy \\
&=\int_{0}^{T}d\tau \int_{\Omega} E(x-y,T-t-\tau)\lozenge_{\alpha,\tau} u(y,\tau)dy \\
&=\int_{0}^{T}d\tau \int_{\Omega} \lozenge_{\alpha,\tau}E(x-y,\tau-t) u(y,T-\tau)dy \\
&\quad  + \int_{0}^{T}d\tau\int_{\partial \Omega}
 \partial_{n}E(x-y,T-t-\tau) u(y,\tau)dS_{y} \\
&\quad -\int_{0}^{T}d\tau\int_{\partial \Omega}E(x-y,T-t-\tau)
 \partial_{n}u(y,\tau)dS_{y} \\
& =u(y,T-t)+\int_{0}^{T}d\tau\int_{\partial \Omega}\partial_{n}E(x-y,T-t-\tau) 
u(y,\tau)dS_{y} \\
&\quad -\int_{0}^{T}d\tau\int_{\partial \Omega}E(x-y,T-t-\tau)
 \partial_{n}u(y,\tau)dS_{y},
\end{align*}
 for any $x\in\Omega$ and $t\in(0,T)$.
 That is, we have
 \begin{equation}\label{12}
\begin{aligned}
&\int_{0}^{T}d\tau\int_{\partial \Omega}\partial_{n}E(x-y,T-t-\tau) u(y,\tau)dS_{y}
 \\
&-\int_{0}^{T}d\tau\int_{\partial \Omega}E(x-y,T-t-\tau)
 \partial_{n}u(y,\tau)dS_{y}\equiv 0,
\end{aligned}
 \end{equation}
for any $x\in\Omega$ and $t\in(0,T)$.
 Since $\theta(T-t-\tau)=0$ for $T-t<\tau$, this means
 \begin{equation}\label{12b}
\begin{aligned}
&\int_{0}^{T-t}d\tau\int_{\partial \Omega}\partial_{n}E(x-y,
 T-t-\tau) u(y,\tau)dS_{y} \\
&\quad -\int_{0}^{T-t}d\tau\int_{\partial \Omega}E(x-y,T-t-\tau)
\partial_{n}u(y,\tau)dS_{y}\equiv 0,
\end{aligned}
 \end{equation}
 for any $x\in\Omega$ and $t\in(0,T)$. Therefore, denoting $T-t$ by
 $t$, we obtain
 \begin{equation}\label{13ba} 
\begin{aligned}
&\int_{0}^{t}d\tau\int_{\partial \Omega}\partial_{n}E(x-y,t-\tau) u(y,\tau)dS_{y} \\
&-\int_{0}^{t}d\tau\int_{\partial \Omega}E(x-y,t-\tau)\partial_{n}u(y,\tau)dS_{y}=0,
\end{aligned}
 \end{equation}
for all $t\in(0,T)$ and $x\in\Omega$. By using the properties of
 the (time-fractional) double and single layer potentials (see
 \cite[Theorem 1]{Kem2011} and \cite[Theorem 2.1]{Kem2011b}) as $x$
 approaches the boundary $\partial\Omega$ from the interior, from
 \eqref{13ba}, we obtain
 \begin{equation}\label{21}
\begin{aligned}
&-\frac{u(x,t)}{2}+\int_{0}^{t}d\tau\int_{\partial\Omega}
 \partial_{n}E
 (x-y,t-\tau)u(y,\tau)dS_{y}\\
&- \int_{0}^{t}d\tau\int_{\partial\Omega}
 E(x-y,t-\tau)\partial_{n}u(y,\tau)dS_{y}=0,
\end{aligned}
 \end{equation}
 for all $t\in(0,T)$ and $x\in\partial\Omega$.
This shows that \eqref{heatpot} is a solution of the initial boundary 
value problem \eqref{9}-\eqref{10}-\eqref{8}.

 Now let us prove its uniqueness. If the initial boundary value problem 
has two solutions
 $u$ and $u_1$, then the function $w = u-u_1$ satisfies
 \begin{equation}\label{Cauchyforw}
 \begin{gathered}
 \lozenge_{\alpha,t} w(x,t)= 0,\quad\text{in } \Omega\times(0,T), \\ 
w(x,0)=0,\quad x\in \Omega,
 \end{gathered}
 \end{equation}
 and the boundary condition \eqref{8}, i.e.
 \begin{equation}\label{21w}
\begin{aligned}
&-\frac{w(x,t)}{2}+\int_{0}^{t}d\tau\int_{\partial\Omega}
 \partial_{n}E
 (x-y,t-\tau)w(y,\tau)dS_{y}\\
&-  \int_{0}^{t}d\tau\int_{\partial\Omega}
 E(x-y,t-\tau)\partial_{n}w(y,\tau)dS_{y}=0,
\end{aligned}
 \end{equation}
 for all $t\in(0,T)$ and $x\in\partial\Omega$.

 Since $f=0$ in this case, instead of \eqref{13ba} we have the
 representation formula
 \begin{equation}\label{13w}
\begin{aligned} 
w(x,t)&=-\int_{0}^{t}d\tau\int_{\partial \Omega}\partial_{n}E(x-y,t-\tau)
 w(y,\tau)dS_{y} \\ 
&\quad +\int_{0}^{t}d\tau\int_{\partial \Omega}E(x-y,t-\tau)\partial_{n}
w(y,\tau)dS_{y},
\end{aligned}
 \end{equation}
 for all $t\in(0,T)$ and $x\in\Omega$.
 As above, by using the properties of the double and single layer
 potentials as $\Omega\ni x\rightarrow\partial\Omega$, we obtain
 \begin{equation}\begin{aligned} 
-w(x,t) &=-\frac{w(x,t)}{2}+\int_{0}^{t}d\tau\int_{\partial \Omega}
 \partial_{n}E(x-y,t-\tau) w(y,\tau)dS_{y} \\
&\quad -\int_{0}^{t}d\tau\int_{\partial \Omega}E(x-y,t-\tau)\partial_{n}
w(y,\tau)dS_{y},
\end{aligned}
 \end{equation}
 for any $x\in\partial\Omega$ and $t\in(0,T)$. Comparing this with
 \eqref{21w}, we arrive at $w(t,x)=0$, $x\in\partial\Omega$,
 $t\in(0,T)$, by uniqueness of the solution of the mixed
 Cauchy-Dirichlet problem (see \cite{Kem2011a}, see also \cite{AAK} 
for more general discussions) we get $w\equiv0$,
 i.e. $u=\lozenge_{\alpha,t}^{-1}f$. So we obtain the desired
 result.
\end{proof}

\subsection*{Acknowledgements}
The authors were supported in parts by the MES RK grants 0825/GF4 and 4075/GF4
as well as by the MES RK target grant 0085/PTSF-14.

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\end{document}