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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 153, 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 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/153\hfil 
 Inverse problem for the fractional telegraph equation]
{Inverse coefficient problem for the semi-linear fractional telegraph equation}

\author[H. Lopushanska, V. Rapita \hfil EJDE-2015/153\hfilneg]
{Halyna Lopushanska, Vitalia Rapita}

\address{Halyna Lopushanska \newline
Department of Differential Equations\\
Ivan Franko National University of Lviv\\
Lviv, Ukraine}
\email{lhp@ukr.net}

\address{Vitalia Rapita \newline
Department of Differential Equations\\
Ivan Franko National University of Lviv\\
Lviv, Ukraine}
\email{vrapita@gmail.com}

\thanks{Submitted April 22, 2015. Published June 11, 2015.}
\subjclass[2010]{35S15}
\keywords{Fractional derivative; inverse boundary value problem;
\hfill\break\indent
over-determination integral condition; Green's function; integral equation}

\begin{abstract}
 We establish the unique solvability for an inverse problem for semi-linear
 fractional telegraph equation
 $$
 D^\alpha_t u+r(t)D^\beta_t u-\Delta u=F_0(x,t,u,D^\beta_t u), \quad
 (x,t) \in \Omega_0\times (0,T]
 $$
 with regularized fractional derivatives $D^\alpha_t u, D^\beta_t u$ of orders
 $\alpha\in (1,2)$, $\beta\in (0,1)$ with respect to time on bounded cylindrical
 domain. This problem consists in the determination of a pair of functions:
 a classical solution $u$ of the first boundary-value problem for such equation,
 and an unknown continuous coefficient $r(t)$ under the over-determination 
 condition
 $$
 \int_{\Omega_0}u(x,t)\varphi(x)dx=F(t), \quad    t\in [0,T]
 $$
 with given functions $\varphi$ and $F$.
\end{abstract}

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

\section{Introduction}\label{Sect1}

The existence and uniqueness theorems for fractional Cauchy problems were proved
in \cite{Anh,djr1,Duan,mon,Koch,Kocubei1,LLP,LL,L,Lopushansky,voroshylov}
and other works. The conditions of classical solvability of the first 
boundary-value problem for equation
\[
D^{\beta}_t u(x,t)-A(x,D)u(x,t)=F_0(x,t)
\]
with regularized fractional derivative (see, for example, \cite{djr1})
and some elliptic differential second order operator $A(x,D)$
were obtained in \cite{Luchko} and \cite{E}.

Equations with fractional derivatives are applied in studying of anomalous
diffusion and various processes in physics, mechanics, chemistry and engineering.
The telegraph fractional equations
in theory of thermal stresses is considered, for example, in \cite{Povstenko}.
Inverse problems to such equations arise in many branches of science and engineering.
Some inverse boundary-value problems to diffusion-wave equation with different
unknown functions or parameters were investigated, for example, in
\cite{Aleroev,Cheng,El-Borai,Hatano,L-1,Nakagawa,Rundell,Zhang}.
In particular, the article \cite{Aleroev} was devoted to determination of
a source term for a time fractional diffusion equation with an integral
type over-determination condition.

In this note we prove the existence and uniqueness of a
classical solution $(u,r)$ of the inverse boundary-value problem
\begin{gather} \label{eq}
D^\alpha_t u+r(t)D^\beta_t u-\Delta u=F_0(x,t,u,D^\beta_t u), \quad
 (x,t) \in \Omega_0\times (0,T], \\
 \label{b2}
u(x,t)=0,\quad (x,t) \in \partial \Omega_0 \times [0,T], \\
 \label{b3}
u(x,0)=F_1(x),\quad x\in \bar{\Omega}_0, \\
 \label{b4}
u_t(x,0)= F_2(x), \quad  x\in \bar{\Omega}_0, \\
 \label{b5}
\int_{\Omega_0} u(x,t)\varphi_0(x)dx=F(t), \quad   t\in [0,T]
\end{gather}
for regularized telegraph equation, where $\alpha\in (1,2)$, $\beta\in (0,1)$,
$\Omega_0$ is a bounded domain in $\mathbb{R}^N$, $N\ge 3$,
with a boundary $\partial \Omega_0$ of class $C^{1+s}$, $s\in (0,1)$,
$F_0$, $F_1$, $F_2$, $F$, $\varphi_0$ -- given functions.
We shall use  Green's functions to prove the solvability of this problem.

\section{Definitions and auxiliary results}\label{Sect2}

We shall use the notation: $\Omega_1=\partial \Omega_0$,
$Q_i=\Omega_i\times (0,T]$, $i=0,1$, $Q_2=\Omega_0$,
$\mathfrak{D}({R}^m)$ is a space of indefinitely differentiable
functions with compact supports in ${R}^m$, $m=1,2,\dots$,
$\mathfrak{D}(\overline{Q}_0)=\{v\in C^\infty(\overline{Q}_0):
(\frac{\partial}{\partial t})^k v|_{t=T}=0,\; k=0,1,\dots\}$,

$\mathfrak{D}'({R}^m)$ and $\mathfrak{D}'(\overline{Q}_0)$ are spaces of
linear continuous functionals (generalized functions \cite[p. 13-15]{Sh}) over
$\mathfrak{D}({R}^m)$ and $\mathfrak{D}(\overline{Q}_0)$, respectively,
$(f,\varphi)$ stands for the value of $f\in \mathfrak{D}'({R}^m)$ on the
test function $\varphi\in \mathfrak{D}({R}^m)$ and also the value of
$f\in \mathfrak{D}'(\overline{Q}_0)$ on $\varphi\in \mathfrak{D}(\overline{Q}_0)$.

We denote by $f\ast g$ the convolution of generalized functions $f$ and $g$,
use the function
\[
f_{\lambda}(t)=\begin{cases}
\frac{\theta(t)t^{\lambda-1}}{\Gamma(\lambda)} &\text{for }
 \lambda>0\\
f'_{1+\lambda}(t) & \text{for }\lambda\leq 0,
\end{cases}
\]
 where $\Gamma(z)$ is the Gamma-function, and $\theta(t)$ is the Heaviside function.
 Note that 
$$
f_\lambda \ast f_\mu=f_{\lambda+\mu}\,.
$$
Also note that the Riemann-Liouville derivative $v^{(\alpha)}_t(x,t)$ of order
$\alpha>0$ is defined by
\[
v^{(\alpha)}_t(x,t)=f_{-\alpha}(t)\ast v(x,t)
\]
and
\begin{align*}
D^\alpha_t v(x,t)
&=\frac{1}{\Gamma(1-\alpha)}\Big[\frac{\partial}{\partial t}\int_0^t
\frac{v(x,\tau)}{(t-\tau)^{\alpha}}d\tau-\frac{v(x,0)}{t^\alpha}\Big] \\
&=v^{(\alpha)}_t(x,t)-f_{1-\alpha}(t)v(x,0),\quad\text{for }\alpha\in (0,1),
\end{align*}
while
\begin{align*}
D^\alpha_t v(x,t)
&=\frac{1}{\Gamma(2-\alpha)}
\Big[\frac{\partial}{\partial t}\int_0^t\frac{
v_{\tau}(x,\tau)}{(t-\tau)^{\alpha-1}}d\tau
 -\frac{v_t(x,0)}{(t-\tau)^{\alpha-1}}\Big] \\
&=v^{(\alpha)}_t(x,t)-f_{1-\alpha}(t)v(x,0)-f_{2-\alpha}(t)v_t(x,0)\quad
\text{for } \alpha\in (1,2).
\end{align*}
We denote $D^1_t v=\frac{\partial v}{\partial t}$.


Let $C(Q_0)$, $C(\overline{Q}_0)$, $C[0,T]$ be spaces of continuous
functions on $Q_0$,  $\overline{Q}_0$ and $[0,T]$, respectively,
$C^\gamma(\Omega_0)$ ($C^\gamma(\bar{\Omega}_0)$) be a space of bounded continuous
functions on $\Omega_0$ ($\bar{\Omega}_0$) satisfying H\"{o}lder continuity
condition,
$C^\gamma(Q_i)$ ($C^\gamma(\bar{Q}_i)$), $i=0,1$, be a space of bounded
continuous functions on $Q_i$ ($\bar{Q}_i$) which for all $t\in (0,T]$
satisfies H\"{o}lder continuity condition with respect to space variables,
$C^\gamma(Q_0\times \mathbb{R}^2)$ be a space of bounded continuous functions
$f(x,t,v,z)$ on $Q_0\times \mathbb{R}^2$ which for all $t\in (0,T]$, $v,z\in \mathbb{R}$
satisfies H\"{o}lder continuity condition with respect to space variables
$x\in \Omega_0$,
\begin{gather*}
C_\beta(Q_0)=\{v\in C(Q_0):  D^\beta_t v\in C(Q_0)\},\\
C^\gamma_\beta(Q_0)=\{v\in C^\gamma(Q_0): D^\beta_t v\in C^\gamma(Q_0)\},\\
C_{2,\alpha}(Q_0)=\{v\in C(Q_0):\Delta v, D^\alpha_t v\in C(Q_0)\}, \\
C_{2,\alpha}({\bar Q}_0)=\{v\in C_{2,\alpha}({Q}_0):  v,v_t\in C({\bar Q}_0)\}.
\end{gather*}
We use the following assumptions:
\begin{itemize}
\item[(A1)] $F_0\in C^\gamma(Q_0\times \mathbb{R}^2)$, $\gamma\in (0,1)$, 
$|F_0(x,t,v,w)|\le A_0+B_0[|v|^q+|w|^p]$,
\[
|F_0(x,t,v_1,w_1)-F_0(x,t,v_2,w_2)|\le D_0[|v_1-v_2|^q+|w_1-w_2|^p]
\]
 for all $v,w,v_1,v_2,w_1,w_2\in \mathbb{R}$
 where $p,q\in [0,2]$, $A_0,B_0,D_0$ are some positive constants,

\item[(A2)]  $F_1\in C^\gamma(\bar{\Omega}_0)$, $F_1|_{\Omega_1}=0$,

\item[(A3)]  $F_2\in C^\gamma(\bar{\Omega}_0)$,

\item[(A4)]  $F, D^\beta F, D^\alpha F\in C[0,T]$, there exists
 $f:=\inf_{t\in [0,T]}|D^\beta F(t)|>0$,

\item[(A5)] $\varphi_0\in C^2(\bar{\Omega}_0)$, $\varphi_0|_{\Omega_1}=0$.

\end{itemize}

\begin{definition}\label{def1} \rm
A pair of functions
$(u,r)\in C_{2,\alpha}({\bar Q}_0)\times C[0,T]$
satisfying  \eqref{eq} on $Q_0$ and the conditions
\eqref{b2}-\eqref{b5} is called a solution of the problem
\eqref{eq}-\eqref{b5}.
\end{definition}

From \eqref{b3}, \eqref{b4} and \eqref{b5}, it follows the necessary agreement
conditions
\begin{equation} \label{n}
\int_{\Omega_0}F_1(x)\varphi_0(x)dx=F(0),\quad
\int_{\Omega_0}F_2(x)\varphi_0(x)dx=F'(0).
\end{equation}
We introduce the operators
\begin{gather*}
(Lv)(x,t)\equiv v^{(\alpha)}_t (x,t)-\Delta v(x,t),\quad (x,t)\in \overline{Q}_0,
 \quad v\in \mathfrak{D}'(\overline{Q}_0),
\\
(L^{\rm reg}v)(x,t)\equiv D^{\alpha}_t
v(x,t)-\Delta v(x,t),\quad (x,t)\in \overline{Q}_0,\quad  v\in
C_{2,\alpha}(\bar{Q}_0).
\end{gather*}

\begin{definition}\label{def2} \rm
A vector-function
$(G_0(x,t,y,\tau),G_1(x,t,y),G_2(x,t,y))$ is called a Green's vector-function
 of the problem
\begin{gather} \label{eq2}
(L^{\rm reg}u)(x,t)=g_0(x,t), \quad (x,t) \in Q_0, \\
 \label{b12}
u(x,t)=0,\quad (x,t) \in \bar{Q}_1, \\
 \label{in2}
u(x,0)=g_1(x),\quad u_t(x,0)=g_2(x),\quad x\in \bar{\Omega}_0,
\end{gather}
if for rather regular $g_0$, $g_1$, $g_2$ the function
\begin{equation}\label{ro}
u(x,t)=\int_0^{t} d\tau\int_{{\Omega}_0}
G_0(x,t,y,\tau)g_0(y,\tau)dy
+\sum_{j=1}^2\int_{\Omega_0}G_j(x,t,y)g_j(y)dy,
\end{equation}
for $(x,t) \in \overline{Q}_0$,
is the classical solution (in $C_{2,\alpha}(\bar{Q}_0)$)
of the first boundary-value problem \eqref{eq2}--\eqref{in2}.
\end{definition}

It follows from  Definition \ref{def2} that
\begin{gather*}
(L G_0)(x,t,y,\tau)=\delta(x-y,t-\tau),\quad (x,t), (y,\tau) \in {Q}_0, \\
(L^{\rm reg} G_j)(x,t,y)=0, \quad (x,t)\in { Q}_0,\; y\in
\Omega_0,\; j=1,2, \\
G_1(x,0,y)=\delta(x-y), \quad \frac{\partial}{\partial t}G_1(x,0,y)=0, \\
G_2(x,0,y)=0,\quad \frac{\partial}{\partial t}G_2(x,0,y)=\delta(x-y),\quad
x,y\in {\Omega}_0
\end{gather*}
where $\delta$ is Dirac delta-function.

\begin{lemma}[\cite{L}] \label{lem1}
The following relations hold:
\[
G_j(x,t,y)=\int_0^t f_{j-\alpha}(\tau)G_0(x,t,y,\tau)d\tau,\quad
(x,t)\in \bar{Q}_0, \; y\in \Omega_0,\; j=1,2.
\]
\end{lemma}

\begin{lemma}\label{lem2}
A Green's vector-function of the first boundary-value problem
\eqref{eq2}--\eqref{in2} exists.
\end{lemma}

The above lemma is proved following the strategy  in \cite[Lemma 2]{L-1}.

\begin{theorem}\label{th1}
If $g_0\in C^\gamma(Q_0)$, $\gamma\in (0,1)$,
$g_j\in C^\gamma(\bar{\Omega}_0)$, $j=1,2$, $g_1|_{\Omega_1}=0$
then there exists a unique solution $u\in C_{2,\alpha}(\bar{Q}_0)$ of
\eqref{eq2}--\eqref{in2}. It is defined by
\begin{equation}\label{sys1}
u(x,t)=\big({\mathfrak{G}_0}g_0\big)(x,t)+\big({\mathfrak{G}_1}g_1)(x,t)
+\big({\mathfrak{G}_2}g_2)(x,t),\quad (x,t)\in \bar{Q}_0
\end{equation}
 where
\begin{gather*}
(\mathfrak{G}_0 g_0)(x,t)
=\int_0^{t} d\tau\int_{\Omega_0}G_0(x,t,y,\tau)g_0(y,\tau)dy, \\
(\mathfrak{G}_j g_j)(x,t)=\int_{\Omega_0}G_j(x,t,y)g_j(y)dy, \quad j=1,2.
\end{gather*}
\end{theorem}

\begin{proof}
Taking Lemmas \ref{lem1} and \ref{lem2} into account,
as in \cite{mon,Fr,ivas,voroshylov}
for the Cauchy problems,
we show that the function \eqref{sys1} belongs to $C_{2,\alpha}(\bar{Q}_0)$
and satisfies the problem \eqref{eq2}--\eqref{in2}.
We use the estimates founded in \cite{mon,Koch,L,voroshylov}:
\begin{gather*}%\label{oc1}
|G_0(x,t,y,\tau)|\le \frac{C^*_0}{(t-\tau)|x-y|^{N-2}},\quad
|x-y|^2<4(t-\tau)^\alpha,\\
|G_j(x,t,y)|\le\frac{C^*_j t^{j-1-\alpha}}{|x-y|^{N-2}},\quad j=1,2,\;
 |x-y|^2<4t^\alpha, \\
\begin{aligned}
|G_0(x,t,y,\tau)|
&\le \frac{{\hat C_0} (t-\tau)^{\alpha-1}}{|x-y|^{N}}
\Big(\frac{|x-y|^2}{4(t-\tau)^\alpha}\Big)^{1+\frac{N}{2(2-\alpha)}}
e^{-c(\frac{|x-y|^2}{4a_0 (t-\tau)^\alpha})^{\frac{1}{2-\alpha}}}\\
&\le \frac{C_0 (t-\tau)^{\alpha-1}}{|x-y|^N},\quad|x-y|^2>4(t-\tau)^\alpha
\end{aligned} \\
\begin{aligned}
G_j(x,t,y)
&\le \frac{{\hat C}_j t^{j-1}}{|x-y|^{N}}
\Big(\frac{|x-y|^2}{4t^\alpha}\Big)^{\frac{N}{2(2-\alpha)}}
e^{-c(\frac{|x-y|^2}{4t^\alpha})^{\frac{1}{2-\alpha}}}\\
&\le \frac{C_j t^{j-1}}{|x-y|^N}, \quad j=1,2,\; |x-y|^2>4t^\alpha
\end{aligned}
\end{gather*}
where $c, {C}^*_j, C_j, {\hat C}_j\quad (j=0,1,2)$ are positive constants;
\begin{gather*}
|G_j(x+\Delta x,t+\Delta t,y,\tau)-G_j(x,t,y,\tau)|\le A_{j}(x,t,y,\tau)
[|\Delta x|+|\Delta t|^{\alpha/2}]^{\gamma},
\\
\begin{aligned}
&|D^\beta_t G_j(x+\Delta x,t+\Delta t,y,\tau)-D^\beta_t G_j(x,t,y,\tau)|\\
&\le A_{\beta,j}(x,t,y,\tau)[|\Delta x|+|\Delta t|^{\alpha/2}]^{\gamma}\\
& \forall (x,t),(x+\Delta x,t+\Delta t)\in \bar{Q}_0,(y,\tau)\in \bar{Q}_j,
 \; j=0,1,2
\end{aligned}
\end{gather*}
with some $0<\gamma<1$ where non-negative functions $A_{j}(x,t,y,\tau)$,
$A_{\beta,j}(x,t,y,\tau)$ have the same kind of estimates as  $G_j(x,t,y,\tau)$,
$D^\beta_t G_j(x,t,y,\tau)$, $j=0,1,2$, have, respectively, and
$G_j(x,t,y,\tau)=G_j(x,t,y)$, $A_{j}(x,t,y,\tau)=A_j(x,t,y)$,
$A_{\beta,j}(x,t,y,\tau)=A_{\beta,j}(x,t,y)$ for $j=1,2$.
Note that for the general boundary-value problem to a parabolic equation with
partial derivatives the last properties of a Green's vector-function were
obtained in \cite{ivas}.
\end{proof}

\section{Existence and uniqueness theorems for the inverse problem}\label{Sect3}

Now we prove the existence of a solution for the inverse problem
 \eqref{eq}--\eqref{b5}.
It follows from the theorem \ref{th1} that under assumptions
 (A1), (A2), (A3) for a given  $r\in C[0,T]$ the solution
$u\in C_{2,\alpha}(\bar{Q}_0)$ of the first boundary-value problem
\eqref{eq}--\eqref{b4} satisfies
\begin{equation}\label{u1}
\begin{aligned}
u(x,t)&=-\int_{0}^{t}r(\tau)d\tau\int_{\Omega_0}G_0(x,t,y,\tau)
 D^\beta_\tau u(y,\tau)dy\\
&\quad +h_0(x,t,u,D^\beta_t u)+h(x,t),\quad (x,t)\in \Omega_0
\end{aligned}
\end{equation}
 where
\begin{equation}\label{h1}
\begin{gathered}
h_0(x,t,u,D^\beta_t u)=\int_{0}^{t}d\tau\int_{\Omega_0}{G}_0(x,t,y,\tau)
F_0(y,\tau,u(y,\tau),D^\beta_\tau u(y,\tau))dy,\\
h(x,t)=\sum_{j=1}^2\int_{\Omega_0}{G}_j(x,t,y)F_j(y)dy,\quad (x,t)\in \Omega_0.
\end{gathered}
\end{equation}
Conversely, any solution $u\in C^\gamma_\beta(Q_0)$ of  \eqref{u1}
belongs to $C_{2,\alpha}(\bar{Q}_0)$ and is the solution of
\eqref{eq}--\eqref{b5}.

It follows from the equation \eqref{eq} and assumption (A5) that
\begin{align*}
&\int_{\Omega_0}D^\alpha_t u(x,t)\varphi_0(x)dx+r(t)
 \int_{\Omega_0}D^\beta_t u(x,t)\varphi_0(x)dx\\
&= \int_{\Omega_0}u(x,t)\Delta \varphi_0(x)dx
 +\int_{\Omega_0}F_0(x,t,u(x,t),D^\beta_t u(x,t))\varphi_0(x)dx, \quad t\in (0,T].
\end{align*}
Using \eqref{b5} we obtain
\begin{align*}
& D^\alpha F(t)+r(t)D^\beta F(t)\\
& =\int_{\Omega_0}u(x,t)\Delta \varphi_0(x)dx
+\int_{\Omega_0}F_0(x,t,u(x,t),D^\beta_t u(x,t))\varphi_0(x)dx
\end{align*}
(here $D^\alpha F(t)=D^\alpha_t F(t)$ and so on); that is,
by using (A4),
\begin{equation}\label{p}
\begin{aligned}
r(t)&=\Big[\int_{\Omega_0}u(x,t)\Delta \varphi_0(x)dx+
\int_{\Omega_0}F_0(x,t,u(x,t),D^\beta_t u(x,t))\varphi_0(x)dx\\
&\quad -D^\alpha F(t)\Big] [D^\beta F(t)]^{-1},\quad t\in (0,T].
\end{aligned}
\end{equation}
Note that, for $u\in C^\gamma_\beta(Q_0)$, the function $r(t)$, defined
by \eqref{p}, belongs to $C[0,T]$.
By substituting the right-hand side of \eqref{p} into \eqref{u1} in
place of $r(t)$ we obtain
\begin{equation}\label{u2}
\begin{aligned}
u(x,t)
&=\int_{0}^{t} [D^\beta F(\tau)]^{-1}d\tau\int_{\Omega_0}G_0(x,t,y,\tau)\\
&\quad\times \Big[\int_{\Omega_0}F_0(z,\tau,u(z,\tau),D^\beta_\tau u(z,\tau))\varphi_0(z)dz\\
&\quad +\int_{\Omega_0}u(z,\tau)\Delta \varphi_0(z)dz
 -D^\alpha F(\tau)\Big]D^\beta_\tau u(y,\tau)dy\\
&\quad +h_0(x,t,u(x,t),D^\beta_t u(x,t))+h(x,t),\quad (x,t)\in \bar{Q}_0,
\end{aligned}
\end{equation}
where the functions $h_0,h$ are defined by \eqref{h1}.
We have reduced the problem \eqref{eq}--\eqref{b5} to system \eqref{p}, \eqref{u2}.
Conversely, a pair $(u,r)\in C^\gamma_\beta(Q_0)\times C[0,T]$ satisfying
\eqref{p} and \eqref{u2} is a solution of the problem \eqref{eq}-\eqref{b5}.

Thus, under assumptions  (A1)--(A5) and \eqref{n}, a pair of functions
 $(u,r)\in C_{2,\alpha}({\bar Q}_0)\times C[0,T]$ is the solution of
 \eqref{eq}-\eqref{b5} if and only if the function $u\in C^\gamma_\beta(Q_0)$
is the solution of \eqref{u2}, with $r\in C[0,T]$ defined by \eqref{p}.


\begin{theorem}\label{th2}
Under the assumptions  {\rm (A1)--(A5)} and the condition \eqref{n},
 there exists  $T^*\in (0,T]$ ($Q^*_0=\Omega_0\times (0,T^*]$)
and a solution $(u,r)\in C_{2,\alpha}({\bar Q}^*_0)\times C[0,T^*]$ of
 \eqref{eq}-\eqref{b5}. The function $u$ is the solution of \eqref{u2}, and
$r(t)$ is defined by \eqref{p}.
\end{theorem}

\begin{proof}
From the previous conclusion, it is sufficient to prove the
solvability of the equation \eqref{u2} in $C^\gamma_\beta(Q_0)$.
We shall use the Schauder principle.
 Let $\|r\|_{C[0,T]}=\max_{t\in [0,T]}|r(t)|$, and
\begin{align*}
\|v\|_{C^\gamma_\beta(Q_0)}
=\max\Big\{&\sup_{(x,t)\in Q_0}|v(x,t)|,
\sup_{(x,t)\in Q_0}|D^\beta_t v(x,t)|, \\
&\sup_{(x,t)\in Q_0,|\Delta x|<1}\frac{|v(x+\Delta x,t)-v(x,t)|}{|\Delta x|^\gamma},\\
&\sup_{(x,t)\in Q_0,|\Delta x|<1}\frac{|D^\beta_t v(x+\Delta x,t)
-D^\beta_t v(x,t)|}{|\Delta x|^\gamma}\Big\},
\end{align*}
Let $R$ be some positive number, and
\[
M_R=M_R(Q_0)=\{v\in C^\gamma_\beta(Q_0): \|v\|_{C^\gamma_\beta(Q_0)}\le R\}.
\]
On ${M}_R$ we consider the operator
\begin{align*}
(Pv)(x,t)
&:=-\int_{0}^{t} [D^\beta F(\tau)]^{-1}d\tau\int_{\Omega_0}G_0(x,t,y,\tau)\big[
\int_{\Omega_0}v(z,\tau)\Delta \varphi_0(z)dz\\
&\quad +F_0(z,\tau,v(z,\tau),D^\beta_t v(z,\tau))\varphi_0(z)dz
 -D^\alpha_t F(\tau)\big]D^\beta_\tau v(y,\tau)dy\\
&\quad +h_0(x,t,v,D^\beta_t v)+h(x,t),\quad (x,t)\in \Omega_0,\; v\in M_R
\end{align*}
 with the functions $h_0,h$  defined by \eqref{h1}.

At the beginning we show the existence of $R>0$, $T^*\in (0,T]$ and
therefore $M^*_R=M_R(Q^*_0)$ such that $P:M^*_R\to M^*_R$.

For $v\in M_R$, $(x,t)\in \bar{Q}_0$ we find the estimates
\begin{align*}
&|h_0(x,t,v,D^\beta_t v)|\\
&=\Big|\int_0^t d\tau\int_{\Omega_0}G_0(x,t,y,\tau)F_0(y,\tau,v(y,\tau),
 D^\beta_t v(y,\tau))dy\Big|\\
&\le \int_0^t d\tau\Big[\int_{\substack{(y,\tau)\in {\Omega}_0:\\
|y-x|<2(t-\tau)^{\alpha/2}}}
|G_0(x,t,y,\tau)|\, |F_0(y,\tau,v(y,\tau),D^\beta_t v(y,\tau))|dy\\
&\quad +\int_{\substack{(y,\tau)\in {\Omega}_0:\\
|y-x|>2(t-\tau)^{\alpha/2}}}
|G_0(x,t,y,\tau)|\, |F_0(y,\tau,v(y,\tau),D^\beta_t v(y,\tau))|dy\Big]
\\
&\le \int_0^t d\tau
\Big[\int_{\substack{(y,\tau)\in {\Omega}_0:\\ |y-x|<2(t-\tau)^{\alpha/2}}}
\frac{C^*}{(t-\tau)|y-x|^{N-2}}
\big[A_0+|v(y,\tau|^q+|D^\beta_t v(y,\tau))|^p\big]dy
\\
&\quad +\int_{\substack{(y,\tau)\in {\Omega}_0:\\
|y-x|>2(t-\tau)^{\alpha/2}}} \frac{C^*}{(t-\tau)^{1-\alpha}|y-x|^{N}}
 \big[A_0+|v(y,\tau|^q+|D^\beta_t v(y,\tau))|^p\big]\Big]dy\\
&\le C\int_0^t \Big[\frac{1}{(t-\tau)}\int_0^{2(t-\tau)^{\alpha/2}} \!\! r dr
 +\frac{1}{(t-\tau)^{1-\alpha}}
\int_{2(t-\tau)^{\alpha/2}}^{\operatorname{diam}\Omega_0}\frac{dr}{r}\Big]d\tau
 \Big[A_0+B_0(R^q+R^p)\Big]\\
&\le \hat C\int_0^t \Big[(t-\tau)^{\alpha-1}+(t-\tau)^{\alpha-1}\ln
\frac{\operatorname{diam} \Omega_0}{(t-\tau)^{\alpha/2}}\Big]d\tau
\Big[A_0+B_0(R^q+R^p)\Big]\\
&\le k_0 t^{\alpha_1} \big[A_0+B_0(R^q+R^p)\big]
\end{align*}
 where  $C^*, C, \hat C, \hat k, k_0$ are positive constants, and
$\alpha_1=\alpha-\varrho$, with $\varrho$ an arbitrary number in $(0,1)$.
Also we have
\begin{align*}
&\Big|\int_{\Omega_0}G_j(x,t,y)F_j(y)dy\Big|\\
&\le \Big[\int_{\substack{(y,\tau)\in {\Omega}_0:\\ |y-x|<2t^{\alpha/2}}}
G_j(x,t,y)dy
+\int_{\substack{(y,\tau)\in {\Omega}_0:\\
|y-x|>2t^{\alpha/2}}} G_j(x,t,y)dy\Big] \|F_j\|_{C(\bar{\Omega}_0)}\\
&\le c_0\Big[\int_{\substack{(y,\tau)\in {\Omega}_0:\\
 |y-x|<2t^{\alpha/2}}}  \frac{t^{j-1-\alpha}}{|x-y|^{N-2}}dy\\
&\quad +\int_{\substack{(y,\tau)\in {\Omega}_0:\\
|y-x|>2t^{\alpha/2}}} \frac{t^{j-1}}{|y-x|^{N}}
\big(\frac{|y-x|^2}{4t^\alpha}\big)^{\frac{N}{2(2-\alpha)}}
e^{-c(\frac{|y-x|^2}{4t^\alpha})^{\frac{1}{2-\alpha}}}dy\Big]
\|F_j\|_{C(\bar{\Omega}_0)}\\
& \le {\hat k}_j t^{j-1}\Big[1+\int_{2t^{\alpha/2}}
^{\operatorname{diam} \Omega_0}r^{\frac{N}{2-\alpha}-1}t^{-\frac{\alpha N}{2(2-\alpha)}}e^{-{\hat c}\Big(\frac{r^2}{t^\alpha}\Big)^{\frac{1}{2-\alpha}}}dr
\Big]\cdot \|F_j\|_{C(\bar{\Omega}_0)}\\
&\le k_j t^{j-1}\|F_j\|_{C(\bar{\Omega}_0)}, \quad j=1,2
\end{align*}
 where $c_0,{\hat c}, {\hat k}_j, k_j$ ($j=1,2$) are positive constants.
Therefore,
\[
|h(x,t)|\le \sum_{j=1,2}k_jt^{j-1}\|F_j\|_{C(\bar{\Omega}_0)},\quad
(x,t)\in \bar{Q}_0.
\]
Similarly,
\begin{align*}
&\Big|\int_{\Omega_0}F_0(y,\tau,v(y,\tau),D^\beta_t v(y,\tau))\varphi_0(z)dy\Big| \\
&\le \big[A_0+B_0(R^q+R^p)\big]\int_{\Omega_0}
|\varphi_0(z)|dz\quad \forall (y,\tau)\in \bar{Q}_0
\end{align*}
 and therefore
\begin{align*}
&\Big|\int_{0}^{t} [D^\beta F(\tau)]^{-1}d\tau\int_{\Omega_0}
 G_0(x,t,y,\tau)F_0(z,\tau,v(z,\tau),D^\beta_t v(z,\tau))\varphi_0(z)dz\Big| \\
&\le \frac{k_0}{f} t^{\alpha_1}\big[A_0+B_0(R^q+R^p)\big]\int_{\Omega_0}
|\varphi_0(z)|dz\quad \forall (y,\tau)\in \bar{Q}_0.
\end{align*}
Then, given $R\ge 1$, we obtain the estimate
$$
|(Pv)(x,t)|\le \frac{k_0}{f} t^{\alpha_1}(c_1 R+c_2 R^2)+H_0(t)
\le q_0 t^{\alpha_1} R^2+H_0,\quad (x,t)\in {\bar Q}_0
$$
where
\begin{gather*}
c_1=\int_{\Omega_0}dx\cdot \|D^\alpha F\|_{C[0,T]},\\
c_2=2B_0\Big(f+\int_{\Omega_0}
|\varphi_0(z)|dz\Big)+\int_{\Omega_0}|\Delta \varphi_0(z)|dz,\\
H_0(t)=A_0 k_0 t^{\alpha_1}\Big(1+\frac{1}{f}\int_{\Omega_0}
|\varphi_0(z)|dz\Big)+k_1\|F_1\|_{C({\bar \Omega}_0)}
+k_2 t\|F_2\|_{C({\bar \Omega}_0)}\le H_0,\\
q_0=\frac{k_0 (c_1+c_2)}{f}.
\end{gather*}
In  the same way for $v\in M_R$, $(x,t)\in Q_0$, $|\Delta x|<1$  we obtain
\[
\frac{|(Pv)(x+\Delta x,t)-(Pv)(x,t)|}{|\Delta x|^\gamma}
\le q_1 t^{\alpha_1} R^2+H_1,\quad (x,t)\in Q_0.
\]
Here and later $q_j,c_j,H_j$ (j=1,2,\dots) are positive numbers.

For every $g\in C^\gamma(Q_0)$ we have
\[
D^\beta_t\big({\mathfrak{G}_0}g\big)(x,t)=\int_{0}^{t}
\frac{(t-\tau)^{-\beta}d\tau}{\Gamma(1-\beta)}
\int_{\Omega_0}{G}_0(x,t,y,\tau)g(y,\tau)dy, \quad
(x,t)\in Q_0
\]
 and, as previously, we find that
\[
|D^\beta_t\big({\mathfrak{G}_0}g\big)(x,t)|
\le {\tilde q}_0 t^{\alpha_1-\beta}\|g\|_{C(Q_0)},\quad
(x,t)\in Q_0,\;\;{\tilde q}_0=const>0.
\]
Furthermore,
\begin{align*}
D^\beta_t\big({\mathfrak{G}_1}F_1\big)(x,t)
&=\frac{\partial}{\partial t}\int_{0}^{t}
\frac{(t-\tau)^{-\beta}d\tau}{\Gamma(1-\beta)}\int_{\Omega_0}{G}_1(x,\tau,y)
F_1(y)dy-f_{1-\beta}(t)F_1(x) \\
&=\int_{0}^{t}f_{1-\beta}(t-\tau)\big({\mathfrak{G}_1}F_1\big)_\tau(x,\tau)d\tau,
\end{align*}
\begin{align*}
D^\beta_t\big({\mathfrak{G}_2}F_2\big)(x,t)
&=\frac{\partial}{\partial t}\int_{0}^{t}
 \frac{(t-\tau)^{-\beta}d\tau}{\Gamma(1-\beta)}
 \int_{\Omega_0}{G}_2(x,\tau,y)F_2(y)dy-f_{2-\beta}(t)F_2(x)\\
&=\int_{0}^{t}f_{1-\beta}(t-\tau)\big({\mathfrak{G}_2}F_2\big)_\tau(x,\tau)d\tau.
\end{align*}
Since $\big({\mathfrak{G}_j}F_j\big)_\tau$ ($j=1,2$) are continuous functions on
$\bar{Q}_0$, we have
\[
 \big|D^\beta_t\big({\mathfrak{G}_j}F_j\big)(x,t)\big|\le
c_{2+j}t^{1-\beta},\quad (x,t)\in {\bar Q}_0,\; j=1,2.
\]
 So,
\[
|D^\beta_t(Pv)(x,t)|\le q_2 t^{\alpha_2}R^2+H_2,
\quad (x,t)\in {\bar Q}_0
\]
and similarly
\[
\frac{|D^\beta_t(Pv)(x+\Delta x,t)-D^\beta_t(Pv)(x,t)|}{|\Delta x|^\gamma}
\le q_3 t^{\alpha_2}R^2+H_3,
\quad (x,t)\in Q_0,\;|\Delta x|<1
\]
for all $v\in M_R$ with $\alpha_2=\alpha_1-\beta$ and
\begin{gather*}
H_2=c_5+c_6\|F_1\|_{C({\bar \Omega}_0)}+c_7 t\|F_2\|_{C({\bar \Omega}_0)},\\
H_3=c_8+c_9\|F_1\|_{C({\bar \Omega}_0)}+c_{10}\|F_2\|_{C({\bar \Omega}_0)}.
\end{gather*}
As a result we obtain
\begin{align*}
&\|Pv\|_{C^\gamma_\beta(Q_0)}\\
&\le \max_{t\in[0,T]}\max\{q_0 t^{\alpha_1}R^2+H_0,q_1 t^{\alpha_1}R^2
 +H_1,q_2t^{\alpha_2}R^2+H_2,q_3t^{\alpha_2}R^2+H_3\}\\
&\le \max_{t\in[0,T]}[qR^2 t^{\alpha_2}+H]\quad \forall v\in M_R, \; R\ge 1
\end{align*}
with $q=\max\{q_0T^\beta,q_1T^\beta,q_2,q_3\}$, $H=\max\{H_0,H_1,H_2,H_3\}$.

To show the inequality
\begin{equation}\label{o2}
\max\{qR^2 t^{\alpha_2}+H,1\}\le R\quad \forall t\in [0,T^*],\; v\in M^*_R
\end{equation}
consider the function $w(s)=qs^2 t^{\alpha_2}-s$, $s\ge 0$.
We find $w'(s)=2q t^{\alpha_2}s-1$ and prove that
$s_0=s_0(t)=[2q t^{\alpha_2}]^{-1}$ is the point of the minimum of $f(s)$.
Then the inequality
\[
\|Pv\|_{C^\gamma_\beta(Q^*_0)}\le R\quad \forall v\in M^*_R
\]
is satisfied for some $R\ge \max\{1,H\}$, $T^*=\min\{t^*,T\}$ if
$q t^{\alpha_2}s_0^2-s_0\le -H$  and $2q t^{\alpha_2}\max\{1,H\}\le 1$
for all $t\in [0,t^*]$.
We have $q t^{\alpha_2}s_0-s_0=-\frac{1}{4q t^{\alpha_2}}$.
There exists $t^*>0$ such that
$4q t^{\alpha_2}H\le 1$ and $2q t^{\alpha_2}\max\{1,H\}\le 1$ for all 
$t\in [0,t^*]$.
Then
\[
t^*=\min\{[\frac{1}{4q H}]^{1/\alpha_2}, [\frac{1}{2q\max\{1,H\}}]^{1/\alpha_2}\}.
\]
Note that \eqref{n} and (A4) imply $\|F_1\|_{C(\bar{\Omega}_0)}>0$.
We have proven the existence of $R\ge 1$, $T^*>0$ such that $P:M^*_R\to M^*_R$.

The operator $P$ is continuous on
\[
{\tilde M}^*_R=\{v\in C_\beta(Q^*_0):
\|v\|_{C_\beta(Q^*_0)}=\max\{\|v\|_{C(Q^*_0)},\|D^\beta_t v\|_{C(Q^*_0)}\}\le R\};
\]
 thus, on $M^*_R$. Namely, for $v_1,v_2\in {\tilde M}^*_R$, $(x,t)\in Q^*_0$,
\begin{align*}
&|(Pv_1)(x,t)-(Pv_2)(x,t)|\\
&=\Big|\int_{0}^{t}[D^\beta F(\tau)]^{-1}d\tau\int_{\Omega_0}G_0(x,t,y,\tau)
 \Big[\int_{\Omega_0}v_2(z,\tau)\Delta \varphi_0(z)dz
 [D^\beta_\tau v_2(y,\tau)\\
&\quad -D^\beta_\tau v_1(z,\tau)]
 -\int_{\Omega_0}[v_1(z,\tau)-v_2(z,\tau)]
 \Delta \varphi_0(z)dz\cdot D^\beta_\tau v_1(y,\tau)\\
&\quad +\int_{\Omega_0}[F_0(z,\tau,v_1(z,\tau),D^\beta_\tau v_1(z,\tau)
 -F_0(z,\tau,v_2(z,\tau),D^\beta_\tau v_2(z,\tau)]\varphi_0(z)dz\Big]dy\\
&\quad +h_0(z,\tau,v_1(z,\tau),D^\beta_\tau v_1(z,\tau)
 -h_0(z,\tau,v_2(z,\tau),D^\beta_\tau v_2(z,\tau)\Big|
\\
&\le c_{11}\sup_{(x,t)\in Q^*_0}\int_{0}^{t}[D^\beta F(\tau)]^{-1}d\tau
 \int_{\Omega_0}\big|G_0(x,t,y,\tau)\big|dy\\
&\quad \times \Big[\|v_2\|_{C(Q^*_0)}\|D^\beta v_1-D^\beta v_2\|_{C(Q^*_0)}
 +\|D^\beta v_1\|_{C(Q^*_0)}\|v_1-v_2\|_{C(Q^*_0)}\\
&\quad +2D_0\|v_1-v_2\|^q_{C(Q^*_0)}+2D_0\|D^\beta v_1-D^\beta v_2\|^p_{C(Q^*_0)}\Big]\\
&\quad +2D_0\sup_{(x,t)\in Q^*_0}\int_{0}^{t}d\tau\int_{\Omega_0}\big|G_0(x,t,y,\tau)
  \big|dy \Big[\|v_1-v_2\|^q_{C(Q^*_0)}\\
&\quad +2D_0\|D^\beta v_1 -D^\beta v_2\|^p_{C(Q^*_0)}\Big] \\
&\le c_{12}(T^*)^{\alpha_1}R\|v_1-v_2\|_{C^\gamma_\beta(Q^*_0)}.
\end{align*}
Similarly, by using previous estimates,
\[
|D^\beta_t(Pv_1)(x,t)-D^\beta_t(Pv_2)(x,t)|
\le c_{13} (T^*)^{\alpha_2} R\|v_1-v_2\|_{C^\gamma_\beta(Q^*_0)}
\]
for all $v_1,v_2\in {\tilde M}^*_R, \;(x,t)\in \bar{Q^*}_0$.

The operator $P$ is compact on ${\tilde M}^*_R$ (and thus on $M^*_R$): it was
established earlier the uniform boundedness of the set
\[
P{\tilde M}^*_R:=\{(Pv)(x,t),\,(x,t)\in Q^*_0: v\in {\tilde M}^*_R\}
\]
in addition, it follows from the properties of Green's operators that for all
$\varepsilon>0$ there exists $\delta=\delta(\varepsilon)$ such that 
for all $(x,t)\in Q^*_0$, $|\Delta x|<\delta$, $|\Delta t|<\delta$ 
and for all $v\in {\tilde M}^*_R$
\begin{gather*}
\sup_{(x,t)\in Q^*_0}|(Pv)(x+\Delta x,t+\Delta t)-(Pv)(x,t)|<\varepsilon, \\
\sup_{(x,t)\in Q^*_0}|D^\beta_t(Pv)(x+\Delta x,t+\Delta t)-D^\beta_t(Pv)(x,t)|
<\varepsilon.
\end{gather*}
As a result, the operator $P$ is equicontinuous on $M^*_R$.
According to Schauder principle there exists a solution  $u\in M^*_R$ 
of the equation \eqref{u2}.
\end{proof}


\begin{theorem}\label{th3}
Assume that $F_0\in C^1(Q_0\times \mathbb{R}^2)$ and is bounded, 
$D^\beta F\in C[0,T]$ and $D^\beta F(t)\neq 0$, $t\in [0,T]$, $\varphi$ 
satisfies the assumption (A5) and $\varphi(x)\ne 0$, $x\in \Omega_0$.
Then a solution $(u,r)\in C_{2,\alpha}({\bar Q}_0)\times C[0,T]$ of 
the problem \eqref{eq}-\eqref{b5} is unique.
\end{theorem}

\begin{proof}
Take two solutions $(u_1,r_1),(u_2,r_2)\in C_{2,\alpha}({\bar Q}_0)\times C[0,T]$ 
of  \eqref{eq}-\eqref{b5} and substitute them into equation \eqref{eq}.
For $u=u_1-u_2$, $r=r_1-r_2$ we obtain the equation
\[
D_t^\alpha u=\Delta u-r_1(t)D_t^\beta u-r(t)D_t^\beta u_2
+F_0(x,t,u_1,D_t^\beta u_1)-F_0(x,t,u_2,D_t^\beta u_2).
\]
By Hadamard lemma
\[
F_0(x,t,u_1,D_t^\beta u_1)-F_0(x,t,u_2,D_t^\beta u_2)
=F_{01}(x,t)u+F_{02}(x,t)D_t^\beta u
\] 
with some known functions $F_{0j}$, $j=1,2$, which are continuous 
and bounded on $Q_0$, depend on $u_1,u_2$, $D_t^\beta u_1,D_t^\beta u_2$. 
Then the previous equation  becomes
\[
D_t^\alpha u+\big(r_1(t)-F_{02}(x,t)\big)D_t^\beta u
=\Delta u+F_{01}(x,t)u-r(t)D_t^\beta u_2,\quad (x,t)\in Q_0.
\]
It follows from the boundary condition that
\[u(x,t)=0,\quad x\in \bar{\Omega}_1,\quad t\in[0,T],\]
and from the initial conditions
\[
u(x,0)=0,\quad u_t(x,0)=0,\quad x\in \bar{\Omega}_0.
\]

Let $G^*_0(x,t,y,\tau)$ be the main Green's function of the first 
boundary-value problem for the equation
\[
D_t^\alpha u+\big(r_1(t)-F_{02}(x,t)\big)D_t^\beta u=\Delta u+F_{01}(x,t)u.
\]
Then the function $u(x,t)$ satisfies the equation
\[
u(x,t)=-\int_{0}^{t}d\tau\int_{\Omega_0}G^*_0(x,t,y,\tau)
D_\tau^\beta u_2(y,\tau)r(\tau)dy,\quad (x,t)\in \bar{Q}_0.
\]
It follows from the over-determination condition \eqref{b5} that
\begin{equation}\label{p2}
\begin{aligned}
r(t)D^\beta F(t)
&=\int_{\Omega_0}\Big[u(x,t)\Delta \varphi_0(x)\\
&\quad +\Big(F_{01}(x,t)u(x,t)+F_{02}(x,t)D_t^\beta u(x,t)\Big)\varphi_0(x)\Big]dx,
\end{aligned}
\end{equation} 
and then $u(x,t)$ satisfies the equation
\begin{equation}\label{eqq}
\begin{aligned}
u(x,t)&=\int_{0}^{t}\frac{d\tau}{D^\beta F(\tau)}
 \int_{\Omega_0}K^*(x,t,\tau)\Big[\big(F_{01}(z,\tau)
 \varphi_0(z)+\Delta \varphi_0(z)\big)u(z,\tau)dz\\
&\quad +F_{02}(z,\tau)\varphi_0(z)D_\tau^\beta u(z,\tau)\Big]dz,\quad 
 (x,t)\in \bar{Q}_0
\end{aligned}
\end{equation} 
where 
\[
K^*(x,t,\tau)=\int_{\Omega_0}G^*_0(x,t,y,\tau)D_\tau^\beta u_2(y,\tau)dy
\]
 is continuous with respect to $x$ and integrable in time function.

If $F_{01}(z,\tau)=F_{02}(z,\tau)=0$, $(z,\tau)\in Q_0$ then by the uniqueness 
of the solution of this linear second type Volterra integral equation 
we obtain $u(x,t)=0$, $(x,t)\in Q_0$.
Then it  follows from \eqref{p2} that $r(t)D^\beta F(t)=0$, $t\in [0,T]$. 
Since $D^\beta F(t)\neq 0$ on $[0,T]$ (under the assumptions of this theorem),
it follows that $r(t)\equiv 0$ for $t\in [0,T]$.

Assume that
\begin{equation}\label{vq}
|F_{01}(z,\tau)|+|F_{02}(z,\tau)|\ne 0,\quad (z,\tau)\in Q_0.
\end{equation}
If $F_{02}(z,\tau)=0$, $(z,\tau)\in Q_0$ then by the uniqueness of the solution 
of the linear second type Volterra integral equation \eqref{eqq} with 
integrable kernel
\[
K^*(x,t,\tau)\big(F_{01}(z,\tau)\varphi_0(z)+\Delta \varphi_0(z)\big)
\]  
we obtain, as in previous case, that $u(x,t)=0$, $(x,t)\in Q_0$ and then, 
from \eqref{p2}, that $r(t)\equiv 0$, $t\in [0,T]$.

In the general case denote
\[
V(x,t)=\big(F_{01}(x,t)\varphi_0(x)+\Delta \varphi_0(x)\big)u(x,t)
+F_{02}(x,t)\varphi_0(x)D^\beta_t u(x,t).
\]
Then  \eqref{eqq} implies
\begin{align*}
V(x,t)&=\int_{0}^{t}\frac{d\tau}{D^\beta F(\tau)}\int_{\Omega_0}
 \big(F_{01}(x,t)\varphi_0(x)+\Delta \varphi_0(x)\big)K^*(x,t,\tau)V(z,\tau)dz\\
&\quad +F_{02}(x,t)\varphi_0(x)D^\beta_t \int_{0}^{t}\frac{d\tau}
 {D^\beta F(\tau)}\int_{\Omega_0}K^*(x,t,\tau)V(z,\tau)dz;
\end{align*}
that is,
\[
V(x,t)=\int_{0}^{t}d\tau\int_{\Omega_0}K(x,t,z,\tau)V(z,\tau)dz,
\quad (x,t)\in \bar{Q}_0,
\]
where
\begin{align*}
&K(x,t,z,\tau)\\
&=\frac{1}{D^\beta F(\tau)}\Big[F_{01}(x,t)\varphi_0(x)+\Delta \varphi_0(x)
+\frac{F_{02}(x,t)\varphi_0(x)(t-\tau)^{-\beta}}{\Gamma(1-\beta)}
\Big]K^*(x,t,\tau).
\end{align*}
By the uniqueness of the solution of this linear second type Volterra 
integral equation with integrable kernel $K(x,t,z,\tau)$ we obtain 
$V(x,t)=0$, $(x,t)\in Q_0$.

Note that \eqref{vq} implies
\[
|F_{01}(x,t)\varphi_0(x)+\Delta \varphi_0(x)|
+|F_{02}(x,t)\varphi_0(x)|\ne 0, \quad (x,t)\in Q_0.
\]
Note also that
\begin{align*}
D^\beta_t u(x,t)=0&\iff f_{-\beta}(t)\ast u(x,t)=0\\
&\iff f_{\beta}(t)\ast f_{-\beta}(t)\ast u(x,t)=0\\
&\iff u(x,t)=0,\;\; (x,t)\in Q_0,
\end{align*}
for the function $u(x,t)$ satisfying zero initial conditions.

Then it follows from the previous results that $u(x,t)=0$, $(x,t)\in Q_0$ and 
from \eqref{p2}, by the assumptions of this theorem, we obtain 
$r(t)\equiv 0$, $t\in [0,T]$.

In separate case $F_{01}(x,t)\varphi_0(x)+\Delta \varphi_0(x)=0$ for all 
$(x,t)\in Q_0$ we may put  $V_1(x,t)=F_{02}(x,t)\varphi_0(x)D^\beta_t u(x,t)$ and, 
as before, obtain the linear second type Volterra integral equation
\[
V_1(x,t)=\int_{0}^{t}d\tau\int_{\Omega_0}K_1(x,t,z,\tau)V_1(z,\tau)dz,\quad 
(x,t)\in \bar{Q}_0
\]
with integrable kernel
\[
K_1(x,t,z,\tau)=F_{02}(x,t)\varphi_0(x)K^*(x,t,\tau)(t-\tau)^{-\beta}\big/
\Gamma(1-\beta)D^\beta F(\tau).
\]
As before, from here we obtain $V_1(x,t)=0$, $(x,t)\in Q_0$ and, as in the 
general case, since $F_{02}(x,t)\varphi_0(x)\ne 0$ on $Q_0$, conclude that
$u(x,t)=0$, $(x,t)\in Q_0$ and $r(t)\equiv 0$, $t\in [0,T]$.
\end{proof}


The similar result holds for the inverse problem on determination of a pair 
of functions $(u,b)$: a solution $u$ of the first (or second) 
boundary-value problem for the equation
\[
D^\alpha_t u=\Delta u+b(t)u=F_0(x,t), \quad (x,t) \in Q_0
\]
and an unknown coefficient $b(t)$ under the same over-determinating 
condition \eqref{b2}.
We may study the cases $N=1,2$ in  the same way.

\subsection*{Acknowledgments}
The authors are grateful to Prof. Mokhtar Kirane for useful discussions.


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