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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 182, pp. 1--14.\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-2017/182\hfil Inverse source Cauchy problem]
{Inverse source Cauchy problem for a time fractional diffusion-wave
equation with distributions}

\author[A. Lopushansky, H. Lopushanska \hfil EJDE-2017/182\hfilneg]
{Andrzej Lopushansky, Halyna Lopushanska}

\address{Andrzej Lopushansky \newline
Rzesz\'ow University\\
Rejtana str., 16A, 35-310 Rzesz\'ow, Poland}
\email{alopushanskyj@gmail.com}

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

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted April 24, 2017. Published July 18, 2017.}
\subjclass[2010]{35S15}
\keywords{Distribution; fractional derivative; inverse problem;
\hfill\break\indent Green vector-function}

\begin{abstract}
 We study the inverse Cauchy problem for a time fractional diffusion-wave
 equation with distributions in right-hand sides. The problem is to find a
 solution (continuous in time in generalized sense) of the direct problem and
 an unknown continuous time-dependent part of a source. The unique
 solvability of the problem is established.
\end{abstract}

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

\section{Introduction}\label{Sect1}

Elliptic and parabolic initial and boundary value problems for differential and
pseudo-differential equations having distributions in right-hand sides are
investigated by many authors; see \cite{Berezansky,Gorod1,Los,m-m} and
references therein.

Equations with fractional derivatives and inverse problems to them appear
in different branches of science and engineering.
The conditions of classical solvability of the Cauchy and boundary value problems
to equations with the regularized time fractional derivative were obtained
in \cite{anh,djr1,duan,mon,Koch,Luchko,M,E,voroshylov,voroshylov1} and other works.
The inverse boundary value problems to a time fractional diffusion equation
with different unknown functions or parameters were investigated, for example,
in  \cite{Aleroev,Cheng,El-Borai,H-N,Ism,Janno,Jim,N,N-bound,Zhang}.
Most papers are devoted to the inverse problems with an unknown right-hand
side (see, for example, \cite{Aleroev,El-Borai,Ism,N,Zhang}). Mainly such
problems were studied under regular data.

In this article, for the equation
\begin{equation} \label{e}
u^{(\beta)}_t-\Delta u=g(t)F_0(x), \quad (x,t) \in \mathbb{R}^n\times (0,T]:=Q
\end{equation}
with the Riemann-Liouville fractional derivative of order $\beta\in (m-1,m)$,
$m,n\in \mathbb{N}$, we study the inverse Cauchy problem
\begin{gather} \label{in}
\frac{\partial^{j-1}}{\partial t^{j-1}} u(x,0)=F_j(x),\quad
 x\in \mathbb{R}^n,\; j=1,2,\dots,m, \\
 \label{ov}
(u(\cdot,t),\varphi_0(\cdot))=F(t), \quad  t\in [0,T]
\end{gather}
where $F_j$ ($j=0,1,\dots,m$) are given distributions, $F$ is given continuous
function, the symbol $(u(\cdot,t),\varphi_0(\cdot))$ stands for the value
of an unknown distribution $u$ on given test function $\varphi_0$ for every
$t\in [0,T]$, $g(t)$ is an unknown continuous function on $[0,T]$.
We prove the existence and uniqueness of a solution $(u,g)$ of the problem
in the cases $m=1,2$.

Note that the inverse boundary value problems of finding a pair $(u,g)$
under smooth given data in right-hand sides and similar (integral)
over-determination conditions were studied, for example, in \cite{Aleroev,Ism}.
The over-determination condition of kind \eqref{ov}, but with the scalar
product $(u,\varphi_0)$ in abstract Hilbert space was used in \cite{El-Borai}.

Conditions of the unique classical solvability of the Cauchy problem \eqref{in}
for the diffusion-wave equation with the Caputo partial derivative of order
$\beta\in (0,2)$ and the Cauchy type problem for the diffusion-wave equation
with the Riemann-Liouville partial derivative of order $\beta\in (0,2)$ where
obtained in \cite{voroshylov,voroshylov1}, respectively.
The method of the Green function was used to prove the solvability of these
problems. The representations of components of the Green vector-functions for
 mentioned problems by means of the H-functions of Fox \cite{K,S}, were
obtained also in \cite{duan}.

\section{Notation, definitions and auxiliary results}\label{Sect2}

We use the following:
$\mathcal{D}(\mathbb{R}^n)$ is the space of indefinitely differentiable functions
with compact supports in $\mathbb{R}^n$,
$C^{\infty,(0)}(\bar{Q})=\{v\in C^\infty(\overline{Q}):
(\frac{\partial}{\partial t})^k v|_{t=T}=0,\;\; k\in \mathbb{Z}_+\}$,
$\mathcal{D}(\overline{Q})$ is the space of functions from
$C^{\infty,(0)}(\bar{Q})$ having compact supports with respect to the
space variables,
$\mathcal{D}'(\mathbb{R}^n)$ and $\mathcal{D}'(\overline{Q})$
are the spaces of linear continuous functionals (distributions \cite[p. 13-15]{Sh})
on $\mathcal{D}(\mathbb{R}^n)$ and $\mathcal{D}(\overline{Q})$, respectively,
${\mathcal E}'(\mathbb{R}^n)=[C^\infty(\mathbb{R}^n)]'$ is the space of
distributions with compact supports, the symbol $(f,\varphi)$ stands for the
value of the distribution $f$ on the test function $\varphi$,
$$
\mathcal{D}'_C(\bar{Q})=\{v\in \mathcal{D}'(\bar{Q}):
(v(\cdot,t),\varphi(\cdot))\in C[0,T]\quad \forall
\varphi\in \mathcal{D}(\mathbb{R}^n)\}.
$$
We denote $(g\hat{\ast}\varphi)(x)=\big(g(\xi),\varphi(x+\xi))\big)$, by
$f{\ast}g$ the convolution of the distributions $f$ and $g$:
\[
(f\ast g,\varphi)=(f,g\hat{\ast} \varphi)\quad\text{for any test function } \varphi,
\]
by $f{\times}g$ the direct product of the distributions $f$ and $g$:
\[
(f\times g,\varphi)=(f(x),(g(t),\varphi(x,t))\quad\text{for any test function }
\varphi(x,t).
\]
We shall use the function
\[
f_{\lambda}(t)=\frac{\theta(t)t^{\lambda-1}}{\Gamma(\lambda)}
\text{ for }\lambda>0 \quad \text{and} \quad
f_{\lambda}(t)=f'_{1+\lambda}(t)\text{ for } \lambda\leq 0,
\]
where $\Gamma(\lambda)$ is the Gamma-function, $\theta(t)$ is the
Heaviside function. Note that
\[
f_\lambda \ast f_\mu=f_{\lambda+\mu}, \quad
f_\lambda \hat{\ast} f_\mu=f_{\lambda+\mu}.
\]


The Riemann-Liouville derivative $v^{(\beta)}(t)$ of order $\beta>0$ is defined
by the formula
$$
v^{(\beta)}(t)=f_{-\beta}(t)\ast v(t),
$$
the Djrbashian-Caputo fractional derivative (regularized fractional derivative)
\[
D^\beta v(t)=\frac{1}{\Gamma(m-\beta)}\int_0^t (t-\tau)^{m-\beta-1}
\frac{d^m}{d \tau^m}v(\tau)d\tau\quad\text{for } m-1<\beta<m,\; m\in \mathbb{N},
\]
and therefore,
\[
 D^{\beta} v(t)=v^{(\beta)}(t)-\sum_{j=0}^{m-1}f_{j+1-\beta}(t)v^{(j)}(0)\quad
\text{for }\beta\in (m-1,m).
\]
Assume that
\[
 C_{2,\beta}(Q)=\{v\in C(Q)\,|\,\Delta v, D^\beta_t v\in C(Q)\},
\]
 and for $\beta\in (m-1,m)$,
\begin{gather*}
C_{2,\beta}({\bar Q})=\{v\in C_{2,\beta}(Q):\frac{\partial^j v}{\partial t^j}
\in C({\bar Q}),\; j=\overline{0,m-1}\}, \\
(Lv)(x,t)\equiv v^{(\beta)}_t(x,t)-\Delta v(x,t), \\
(L^{reg}v)(x,t)\equiv D^{\beta}_t v(x,t)-\Delta v(x,t), \\
({\widehat L}v)(x,t)\equiv f_{-\beta}(t)\hat{\ast} v(x,t)-\Delta v(x,t),\quad
(x,t)\in Q, \\
\mathcal{X}(\overline{Q})=\{v\in C^{\infty,(0)}(\overline{Q})
: {\widehat L}v\in \mathcal{D}(\bar{Q})\}.
\end{gather*}

\begin{lemma}\label{lemG}
The Green formula holds:
\begin{gather*}
\begin{aligned}
&\int_{Q}v(x,\tau)(\widehat{L}\psi)(x,\tau)dx\,d\tau \\
&=\int_{Q}({L}^{reg}v)(x,\tau)\psi(x,\tau)dx\,d\tau
+\sum_{j=1}^{m}\int_{\mathbb{R}^n}\frac{\partial^{j-1}}{\partial \tau^{j-1}}
 v(x,0)\big(f_{j-\beta}(\tau),\psi(x,\tau)\big)dx,
\end{aligned}\\
\beta\in (m-1,m),\quad m\in \mathbb{N},\; v\in C_{2,\beta}(\bar{Q}),\;
 \psi\in \mathcal{X}(\bar{Q})\,.
\end{gather*}
\end{lemma}

\begin{proof}
Integrating by parts, for any $v\in C_{2,\beta}(\bar{Q})$,
$\psi\in \mathcal{X}(\bar{Q})$ we have
\begin{align*}
&\int_{Q}D^\beta_t v(x,t)\psi(x,t)\,dx\,dt\\
&=\frac{1}{\Gamma(m-\beta)}\int_{Q}\Big(\int_0^t
(t-\tau)^{m-1-\beta}\frac{\partial^m}{\partial \tau^m} v(x,\tau)\big)d\tau\Big)
 \psi(x,t)\,dx\,dt\\
&=\frac{1}{\Gamma(m-\beta)}\int_{\mathbb{R}^n}dx
 \int_0^T\Big(\int_\tau^T (t-\tau)^{m-1-\beta}\psi(x,t)dt\Big)
 \frac{\partial^m}{\partial \tau^m} v(x,\tau)d\tau\\
&=\frac{1}{\Gamma(m-\beta)}\int_{\mathbb{R}^n}dx\int_0^T\Big(\int_0^{T-\tau}
\eta^{m-1-\beta}\psi(x,\eta+\tau)d\eta\Big)\frac{\partial^m}{\partial \tau^m}
  v(x,\tau)d\tau\\
&=\int_{\mathbb{R}^n}dx\int_0^T\big(f_{m-\beta}(\tau)\hat{\ast}\psi(x,\tau)\big)
 \frac{\partial^m}{\partial \tau^m} v(x,\tau)d\tau\\
&=\int_{\mathbb{R}^n}\Big((f_{m-\beta}(\tau)\hat{\ast}\psi(x,\tau))
 \frac{\partial^{m-1}}{\partial \tau^{m-1}}v(x,\tau)\Big)\Big|_{\tau=0}^{\tau=T}dx\\
&\quad -\int_{\mathbb{R}^n}dx\int_0^T(f_{m-\beta}(\tau)\hat{\ast}\psi(x,\tau))_{\tau}
 \frac{\partial^{m-1}}{\partial \tau^{m-1}}v(x,\tau)d\tau \\
&=-\int_{\mathbb{R}^n}(f_{m-\beta}\hat{\ast}\psi)(x,0)
 \frac{\partial^{m-1}}{\partial \tau^{m-1}}v(x,0)dx\\
&\quad +\int_{\mathbb{R}^n}dx\int_0^T(f_{m-1-\beta}(\tau)\hat{\ast}\psi(x,\tau))
 \frac{\partial^{m-1}}{\partial \tau^{m-1}}v(x,\tau)d\tau \\
&=-\sum_{j=0}^{m-1}\;\int_{\mathbb{R}^n}\big(f_{m-j-\beta}\hat{\ast}\psi\big)(x,0)
 \frac{\partial^{m-j-1}}{\partial \tau^{m-j-1}}v(x,0)dx\\
&\quad +\int_{\mathbb{R}^n}dx\int_0^T\big(f_{-\beta}(\tau)\hat{\ast}\psi(x,\tau)
 \big)v(x,\tau)d\tau \\
&=-\sum_{j=1}^{m}\;\int_{\mathbb{R}^n}\frac{\partial^{j-1}}
 {\partial \tau^{j-1}}v(x,0)\big(f_{j-\beta}(t),\psi(x,t)\big)dx
 +\int_{Q}\big(f_{-\beta}(t)\hat{\ast}\psi(x,t)\big)v(x,t)\,dx\,dt.
\end{align*}
\end{proof}

We use the following assumptions:
\begin{itemize}
\item[(A1)]  $F_j\in \mathcal{E}'(\mathbb{R}^n),\quad j=\overline{0,m}$,


\item[(A2)]  $F,F^{(\beta)}\in C[0,T]$,
$\varphi_0 \in \mathcal{D}(\mathbb{R}^n)$, $(F_0,\varphi_0)\ne 0$.
\end{itemize}

\begin{definition} \label{def1} \rm
The pair $(u,g)\in \mathcal{D}'_C(\bar{Q})\times C[0,T]$ is called a solution
of the problem \eqref{e}-\eqref{ov} if
\begin{equation}\label{iden}
\begin{aligned}
&\int_0^T\big(u(\cdot,t),(\hat{L}\psi)(\cdot,t)\big)dt \\
&=\int_0^T g(t)\big(F_0(\cdot),\psi(\cdot,t)\big)dt
+\sum_{j=1}^m \big(F_j(y)\times f_{j-\beta}(t),\psi(y,t)\big)\quad
\forall \psi \in X(\bar{Q})
\end{aligned}
\end{equation}
and  condition \eqref{ov} holds.
\end{definition}

Note that \eqref{iden} is obtained as the generalization of the above Green
formula.
Then from \eqref{in} and \eqref{ov} it follows the compatibility conditions
\begin{equation} \label{n}
(F_j,\varphi_0)=F^{(j-1)}(0),\quad j=\overline{1,m}.
\end{equation}

\begin{definition} \label{def2} \rm
The vector-function
$(G_0(x,t),G_1(x,t),\dots, G_m(x,t))$ is called a Green vector-function of
the Cauchy problem \eqref{in} to the equation
$$
(Lu)(x,t)=F(x,t), \quad (x,t) \in Q,
$$
and also of such problem to the equation
\begin{equation} \label{eq2}
(L^{\rm reg}u)(x,t)=F(x,t), \;\;\; (x,t) \in Q,
\end{equation}
if under rather regular $F$, $F_1,\dots, F_m$ the function
\begin{equation}\label{ro}
\begin{aligned}
u(x,t)&=\int_0^t d\tau\int_{\mathbb{R}^n}G_0(x-y,t-\tau)F(y,\tau)dy\\
&\quad +\sum_{j=1}^m\int_{\mathbb{R}^n} G_j(x-y,t)F_j(y)dy,\quad
(x,t) \in \overline{Q}
\end{aligned}
\end{equation}
is a classical (from $C_{2,\beta}(\bar{Q})$)
solution of the problem \eqref{eq2}, \eqref{in}.
\end{definition}

It follows from the definition \ref{def2} that
\[
(L G_0)(x,t)=\delta(x,t), \quad (x,t)\in {Q}
\]
where $\delta$ is the Dirac delta-function,
\[
(L^{\rm reg} G_j)(x,t)=0, \quad (x,t)\in {Q},\quad
\frac{\partial^{j-1}}{\partial t^{j-1}}G_j(x,0)=\delta(x),\quad
x\in \mathbb{R}^n,\; j=\overline{1,m}.
\]
Let
\begin{gather*}
(\widehat{\mathcal G}_0\varphi)(y,\tau)=\int_\tau^T dt\int_{\mathbb{R}^n}
\varphi(x,t)G_0(x-y,t-\tau)dx, \quad (y,\tau)\in \bar{Q}, \\
(\widehat{\mathcal G}_j\varphi)(y)
=\int_0^T dt\int_{\mathbb{R}^n}\varphi(x,t)G_j(x-y,t)dx, \quad y\in \mathbb{R}^n.
\end{gather*}


\begin{lemma}\label{lem1}
For any $\psi\in \mathcal{X}(\bar{Q})$ the following equalities hold:
\begin{gather}\label{b}
(\widehat{\mathcal G}_0({\widehat L}\psi))(y,\tau)=\psi(y,\tau),\quad
  (y,\tau)\in \bar{Q}, \\
\label{c}
(\widehat{\mathcal G}_j({\widehat L}\psi))(y)=\big(f_{j-\beta}(\tau),\psi(y,
\tau)\big),\quad y\in {\rm R}^n,\quad j=\overline{1,m}.
\end{gather}
\end{lemma}

\begin{proof}
Substituting the classical solution \eqref{ro} of the Cauchy problem \eqref{eq2},
 \eqref{in} in the Green formula (instead of $v$) one obtains
\begin{align*}
&\int_{Q}\Big(\int_0^t d\tau\int_{\mathbb{R}^n }G_0(x-y,t-\tau)F(y,\tau)dy\Big)
(\hat{L}\psi)(x,t)\,dx\,dt\\
&+\sum_{j=1}^m\;\int_{Q}\Big(\int_{\mathbb{R}^n}G_j(x-y,t)F_{j}(y)dy\Big)
(\hat{L}\psi)(x,t)\,dx\,dt\\
&=\int_{Q}F(x,t)\psi(x,t)\,dx\,dt+
\sum_{j=1}^m\;\int_{\mathbb{R}^n}F_j(x)\big(f_{j-\beta}(t),\psi(x,t)\big)dx;
\end{align*}
that is
\begin{align*}
&\int_{Q}\Big(\int_{\tau}^Tdt\int_{\mathbb{R}^n
}G_0(x-y,t-\tau) (\hat{L}\psi)(x,t)dx\Big)F(y,\tau)dy d\tau\\
&+\sum_{j=1}^m\;\int_{\mathbb{R}^n}
\Big(\int_{Q}G_j(x-y,t)(\hat{L}\psi)(x,t)\,dx\,dt\Big)F_j(y)dy\\
&=\int_{Q}\psi(y,\tau)F(y,\tau)dyd\tau+
\sum_{j=1}^m\;\int_{\mathbb{R}^n}\big(f_{j-\beta}(t),\psi(y,t)\big)F_j(y)dy.
\end{align*}
We obtain the formulas \eqref{b} and \eqref{c} after an arbitrariness
of $F,F_1,\dots,F_m$.
\end{proof}

\begin{lemma}\label{lemA}
For $m=1,2$, any $\varphi\in \mathcal{D}(\bar{Q})$ there exists
$\psi\in \mathcal{X}({\bar Q})$ such that
$$
(\widehat{L}\psi)(x,t)=\varphi(x,t), \qquad (x,t)\in \bar{Q}.
$$
\end{lemma}

\begin{proof}
We show that
$$
\psi(y,\tau)=\int_{\tau}^T dt\int_{\mathbb{R}^n}
G_0(x-y,t-\tau)\varphi(x,t)dx,\quad (y,\tau)\in \bar{Q}
$$
is the unknown function. Indeed,  it will follow from Lemma \ref{lem3}
(relatively $G_0$) that $\psi\in C^{\infty,(0)}(\bar{Q})$ for any
$\varphi\in \mathcal{D}(\bar{Q})$ in the cases $m=1,2$ and, in addition, we have
\begin{align*}
(\widehat{L}\psi)(y,\tau)
&=\widehat{L}\big(G_0(x,t),\varphi(x+y,t+\tau)\big)\\
&=\big(G_0(x,t),(\widehat{L}\varphi)(x+y,t+\tau)\big) \\
&=\big((LG_0)(x,t),\varphi(x+y,t+\tau)\big)\\
&=\big(\delta(x,t),\varphi(x+y,t+\tau)\big) \\
&= \varphi(y,\tau), \;\;(y,\tau)\in \bar{Q}.
\end{align*}
\end{proof}

\begin{corollary}\label{cor1}
The following hold:
\begin{equation}\label{a}
G_j(x,t)=f_{j-\beta}(t)\ast G_0(x,t),\quad (x,t)\in Q, \quad
j=\overline{1,m}, \quad m=1,2.
\end{equation}
\end{corollary}

\begin{proof}
Using \eqref{b} and the analogue of the Fubini theorem, we obtain
\begin{align*}
\big(f_{j-\beta}(\tau),\psi(y,\tau)\big)
&=\big(f_{j-\beta}(\tau),(\widehat{\mathcal G}_0({\widehat L}\psi))(y,\tau)\big) \\
&=\Big(f_{j-\beta}(\tau),\int_{\tau}^T\int_{\mathbb{R}^n}
G_0(x-y,t-\tau)(\hat{L}\psi)(x,t)\,dx\,dt\Big)\\
&=\int_{Q}\big(f_{j-\beta}(t)\ast G_0(x-y,t)\big)(\hat{L}\psi)(x,t)\,dx\,dt.
\end{align*}
From \eqref{c} we obtain
\[
\big(f_{j-\beta}(\tau),\psi(y,\tau)\big)
=(\widehat{\mathcal G}_j({\widehat L}\psi))(y)
=\int_{Q}G_j(x-y,t)(\hat{L}\psi)(x,t)\,dx\,dt,\;\;j=\overline{1,m}.
\]
Therefore, for any $\psi\in \mathcal{X}(\bar{Q})$, $j=\overline{1,m}$ we have
\[
\int_{Q} \Big(G_j(x-y,t)-f_{j-\beta}(t)\ast G_0(x-y,t)\Big)
(\widehat{L}\psi)(x,t)\,dx\,dt=0, \;\;y\in \mathbb{R}^n.
\]
By Lemma \ref{lemA}, for all $\varphi\in \mathcal{D}(\bar{Q})$,
$j=\overline{1,m}$, $m=1,2$ we obtain
$$
\int_{Q} \Big(G_j(x-y,t)-f_{j-\beta}(t)\ast G_0(x-y,t)\Big)\varphi(x,t)\,dx\,dt=0,
\quad y\in \mathbb{R}^n,
$$
and the desirable formula \eqref{a} follows from the Du Bois-Reymond lemma.
\end{proof}

\begin{lemma}\label{lem2}
There exists a Green function $G_0(x,t)$ of the Cauchy problem \eqref{e},
 \eqref{in}.
\end{lemma}

\begin{proof}
From \cite{duan,mon} we have the representation
\begin{equation}\label{g2}
G_0(x,t)=\frac{\pi^{-n/2}t^{\beta-1}}{|x|^n}H^{2,0}_{1,2}
\Big(\frac{|x|^2}{4 t^\beta};
(\beta,\beta); (1,1), (n/2,1)\Big).
\end{equation}
The function $H_{p,q}^{m,n}(\cdot)$ is defined in Fox \cite{K,S}
(with the same arguments in a different format)
\[
H_{p,q}^{m,n}\Big(z; (a_1,\gamma_1), \dots, (a_p,\gamma_p);
(b_1,\beta_1), \dots, (b_q,\beta_q)\Big)
=\int_{\mathbb{C}}\mathcal{H}(s)z^{-s}\,ds\,,
\]
where
\[
\mathcal{H}(s) = \frac{ \prod_{j=1}^m (b_j+\beta_js)
\prod_{i=1}^n \Gamma(1-a_i-\gamma_i s)}
{\prod_{i=n+1}^q\Gamma(a_i+\gamma_i s) \prod_{j=m+1}^p\Gamma(1-b_j-\beta_j s)}\,,
\]
$z^{-s}=\exp[s(\log|z|+i \arg{z})]$, $z\neq 0$, $i^2=1$,
$\Gamma(x)$ is the usual Gamma function,
$\mathbb{C}$ is the boundless contour that separates the poles
$b_{jl}=\frac{-b_j-l}{\beta_j}$, $1\leq j\leq m$, $l=0,1,\dots$,
of the function  $\Gamma (b_j+\beta_js)$ to the left and the poles
$a_{ik}=\frac{1-a_i-k}{\gamma_i}$, $1\leq i\leq m$, $k=0,1,\dots$,
of the function  $\Gamma (1-a_i-\gamma_i s)$ to the right
(it is assumed that they does not coincide).

 Let
\begin{gather*}
a^*=\sum_{i=1}^n\gamma_i-\sum_{i=n+1}^p\gamma_i
+\sum_{i=1}^m\beta_i-\sum_{i=m+1}^q\beta_i, \\
\Delta^*=\sum_{i=1}^q\beta_i-\sum_{i=1}^p\gamma_i.
\end{gather*}
For the function \eqref{g2} we have $a^*=2-\beta\ne 0$, $\Delta^*=2-\beta\ne 0$,
and by \cite[Thm. 1.1]{K} the function $G_0(x,t)$ exists for all $x\ne 0$, $t>0$.
\end{proof}

\begin{corollary}\label{cor2}
The Green vector-function of the Cauchy problem \eqref{e}, \eqref{in} with $m=1,2$
exists.
\end{corollary}

\begin{proof}
By Corollary \ref{cor1}
$$
G_j(x,t)=(f_{m-\beta}(t)\ast G_0(x,t))^{(m-j)}_t,\quad
j=\overline{1,m},\;m=1,2.
$$
Using \eqref{g2}, the formula of fractional differentiation of the
H-function \cite[Thm. 2.7]{K}
\begin{align*}
&f_{\varrho}(z)\ast \Big[z^\omega H^{m,n}_{p,q}\Big(cz^\sigma;
(a_1,\gamma_1),\dots, (a_p,\gamma_p); (b_1,\beta_1), \dots, (b_q,\beta_q)
\Big)\Big]\\
& =z^{\omega+\varrho}H^{m,n+1}_{p+1,q+1}\Big(cz^\sigma;
(-\omega,\sigma), (a_1,\gamma_1),\dots ,(a_p,\gamma_p);
(b_1,\beta_1),\dots, \\
&\quad (b_q,\beta_q), (-\omega-\varrho,\sigma)\Big)
\end{align*}
for \quad $a^*>0$, $\varrho>0$,
$\sigma\min_{1\le j\le m}[\frac{Re b_j}{\beta_j}]+Re \omega>-1$
and properties of this function (see, for example, \cite{duan,K}),
in the case $a^*=2-\beta>0$ (that is for $m=1,2$) we obtain
\begin{align*}
&f_{m-\beta}(t)\ast G_0(x,t) \\
&=f_{m-\beta}(t)\ast \Big[\frac{\pi^{-n/2}t^{\beta-1}}{|x|^n}H^{2,0}_{1,2}
\Big(\frac{|x|^2}{4t^\beta};
(\beta,\beta); (1,1), (n/2,1)\Big)\Big]\\
&=f_{m-\beta}(t)\ast \Big[\frac{\pi^{-n/2}t^{\beta-1}}{|x|^n}H^{0,2}_{2,1}
\Big(\frac{4t^\beta}{|x|^2}; (0,1),(1-n/2,1);
(1-\beta,\beta)\Big)\Big]\\
&=\frac{\pi^{-n/2}t^{m-1}}{|x|^n}H^{0,3}_{3,2}\Big(\frac{4t^\beta}{|x|^2};
\Big((1-\beta,\beta),(0,1),(1-n/2,1);
(1-\beta,\beta), (1-m,\beta) \Big) \\
&=\frac{\pi^{-n/2}t^{m-1}}{|x|^n}H^{0,2}_{2,1}\Big(\frac{4t^\beta}{|x|^2};
(0,1),(1-n/2,1); (1-m,\beta))\Big).
\end{align*}

By the formula of differentiation \cite[Prop. 2.8]{K} of the H-function
\begin{align*}
&(\frac{d}{dz})^k\Big[z^\omega H^{m,n}_{p,q}\Big(cz^\sigma;
(a_1,\alpha_1),\dots , (a_p,\alpha_p);
(b_1,\beta_1),\dots ,(b_q,\beta_q) \Big) \Big]\\
&=z^{\omega-k}H^{m,n+1}_{p+1,q+1}\Big(cz^\sigma;
(-\omega,\sigma), (a_1,\alpha_1), \dots , (a_p,\alpha_p);
(b_1,\beta_1), \dots , \\
&\quad (b_q,\beta_q),(k-\omega,\sigma)
\Big)
\end{align*}
for $\omega,c\in \mathbb{C}$, $\sigma>0$, $k\in \mathbb{Z}_+$, we have
\begin{align*}
&(f_{m-\beta}(t)\ast G_0(x,t))^{(m-j)}_t\\
&=\frac{\pi^{-n/2}t^{m-1}}{|x|^n}H^{0,3}_{3,2}\Big(\frac{4t^\beta}{|x|^2};
(1-m,\beta),(0,1), (1-n/2,1) ;
(1-m,\beta), (1-j,\beta) \Big) \\
&=\frac{\pi^{-n/2}t^{j-1}}{|x|^n}H^{0,2}_{2,1}\Big(\frac{4t^\beta}{|x|^2};
(0,1),(1-n/2,1);
(1-j,\beta) \Big)\\
&=\frac{\pi^{-n/2}t^{j-1}}{|x|^n}H^{2,0}_{1,2}\Big(\frac{|x|^2}{4t^\beta};
(j,\beta); (1,1), (n/2,1)\Big).
\end{align*}

So, we find the representations
\begin{equation}\label{gj}
G_j(x,t)=\frac{\pi^{-n/2}t^{j-1}}{|x|^n}H^{2,0}_{1,2}
\Big(\frac{|x|^2}{4t^\beta};
(j,\beta);
(1,1), (n/2,1) \Big),
\end{equation}
for $j=\overline{1,m}$, $ m=1,2$.

For every function $G_j$, $j=\overline{1,m}$, $m=1,2$ we have
$a^*=\Delta^*=2-\beta>0$.
So, by \cite[Thm. 1.1]{K} these functions exist for all $x\ne 0$, $t>0$.
\end{proof}

Let $\mathcal{D}^k({\rm R}^n)$ be the space of functions from $C^k({\rm R}^n)$
having compact supports,
$\|\varphi\|_{D^k({\rm R}^n)}
=\max_{|\kappa|\le k}\max_{x \in supp\varphi}|D^{\kappa}\varphi(x)|$
where $\kappa=(\kappa_1,\dots,\kappa_n)$, $\kappa_j\in {\rm Z}_+$,
$j\in \{1,\dots,n\}$, $|\kappa|=\kappa_1+\dots+\kappa_n$,
$D^{\kappa}\varphi(x)=\frac{\partial^{|\kappa|}}{\partial x_1^{\kappa_1}
\dots \partial x_n^{\kappa_n}}\varphi(x)$,
\[
 (\widehat{G_j}\varphi)(y,t)=\int_{{\rm R}^n} G_j(x-y,t)\varphi(x)\,dx,\quad
(y,t)\in \bar{Q},\; j=\overline{0,m}.
\]


\begin{lemma}\label{lem3}
For $\beta\in (m-1,m)$, $m=1,2$, all $k \in {\rm Z}_+$, multi-index
$\kappa$, $|\kappa|=k$, $\varphi\in  \mathcal{D}({\rm R}^n)$ the following
bounds hold:
\begin{gather*}
\big| D^\kappa_y(\widehat{G_0}\varphi)(y,t)\big|
\le c_k t^{\beta-1}\|\varphi\|_{\mathcal{D}^k({\rm R}^n)}, \quad (y,t)\in Q,\\
\big| D^\kappa_y(\widehat{G_j}\varphi)(y,t)\big|
\le c_kt^{j-1}\|\varphi\|_{\mathcal{D}^k({\rm R}^n)},\quad
(y,t)\in \bar{Q}, \; j=\overline{1,m}.
\end{gather*}
\end{lemma}

Hereinafter $c_k,\widehat{c}_k,d_k,\widehat{d}_k,C_k,C$ ($k\in {\rm Z}_+$)
are positive constants.

\begin{proof}
We use the bounds of components of the Green vector-function.
We obtain them from the properties of the H-functions.
It is known \cite{K} that
$$
H^{q,0}_{p,q}\Big(z; (a_1,\gamma_1), \dots , (a_p,\gamma_p);
(b_1,\alpha_1), \dots , (b_q,\alpha_q)\Big)
\le C|z|^{\frac{\mu^*+1/2}{\Delta^*}}e^{-c|z|^{\frac{1}{\Delta^*}}},
$$
as $|z|\to \infty$ in the case $a^*>0$ where
$\mu^*=\sum_{i=1}^q b_i-\sum_{i=1}^p a_i+\frac{p-q}{2}$,
$c=(2-\beta)\beta^{\frac{\beta}{2-\beta}}$.

Using the representations \eqref{g2} and \eqref{gj} we obtain
$$
\mu^*_0=\frac{n+1}{2}-\beta,\quad
\mu^*_j=\frac{n+1}{2}-j,\quad j=\overline{1,m},\; m=1,2.
$$
So, in the case $|x|>t^{\beta/2}$ we obtain
\begin{gather*}
|G_0(x,t)|\le C\frac{t^{\beta-1}}{|x|^n}
\Big(\frac{|x|^2}{t^\beta}\Big)^{1+\frac{n-\beta}{2(2-\beta)}}
e^{-c\big(\frac{|x|^2}{t^\beta}\big)^{\frac{1}{2-\beta}}},\\
|G_j(x,t)|\le C\frac{t^{j-1}}{|x|^n}\Big(\frac{|x|^2}{t^\beta}
 \Big)^{\frac{n+2-2j}{2(2-\beta)}}
e^{-c\big(\frac{|x|^2}{t^\beta}\big)^{c{1}{2-\beta}}},\quad
j=\overline{1,m}.
\end{gather*}

Using \cite[Thm 1.11]{K}, we obtain the following bounds in the case
$|x|<t^{\beta/2}$:
\begin{gather*}
|G_0(x,t)|
\le C\frac{t^{\beta-1}}{|x|^n}\Big(\frac{|x|^2}{t^\beta}\Big)^{\min\{1,\frac{n}{2}\}}
=C\begin{cases} \frac{|x|^{2-n}}{t}, & n>2\\
\frac{1}{t}\big(1+\big|ln\frac{|x|^2}{t^\beta}\big|\big), & n=2 \\
 t^{\frac{\beta}{2}-1}, & n=1\end{cases},\\
|G_j(x,t)|
\le C\frac{t^{j-1}}{|x|^n}\Big(\frac{|x|^2}{t^\beta}\Big)^{\min\{1,\frac{n}{2}\}}
=C\begin{cases} |x|^{2-n}t^{j-1-\beta}, & n>2 \\
t^{j-1-\beta}\big(1+\big|ln\frac{|x|^2}{t^\beta}\big|\big), & n=2\\
 t^{j-1-\frac{\beta}{2}}, & n=1,
\end{cases}
\end{gather*}
for $j=\overline{1,m}$.
 Then in the case $n>2$ for all multi-index $\alpha$, $|\alpha|=k$,
$\varphi\in \mathcal{D}(\mathbb{R}^n)$ we have
\begin{align*}
&\Big|\int_{\mathbb{R}^n} G_0(x-y,t-\tau) D^\alpha\varphi(x) dx\Big| \\
&\le \int_{\{x\in {\mathbb{R}^n}: |x-y|^{2}<(t-\tau)^{\beta}\}}
G_0(x-y,t-\tau)|D^\alpha\varphi(x)|dx  \\
&\quad +\int_{\{x\in {\mathbb{R}^n}:
|x-y|^{2}>(t-\tau)^{\beta}\}}
G_0(x-y,t-\tau)|D^\alpha\varphi(x)|dx \\
&\le C\int_{\{x\in {\mathbb{R}^n}: |x-y|^{2}<(t-\tau)^{\beta}\}}
\frac{|D^\alpha\varphi(x)|} {(t-\tau)|x-y|^{n-2}}dx \\
&\quad +C \int_{\{x\in {\mathbb{R}^n}:
|x-y|^2>(t-\tau)^{\beta}\}} \frac{(t-\tau)^{\beta-1}}{|x-y|^n}\cdot
\Big(\frac{|x-y|^2}{(t-\tau)^\beta}\Big)^{1+\frac{n-\beta}{2(2-\beta)}} \\
&\quad \times e^{-c\Big(\frac{|x-y|^2}{(t-\tau)^\beta}\Big)^{\frac{1}{2-\beta}}}
 |D^\alpha\varphi(x)| dx \\
&\le C_1 \Big[\frac{1}{(t-\tau)}\int_0^{(t-\tau)^{\beta/2}}r dr
 +\int_{t^{\beta/2}}^{d} r^{1+\frac{n-\beta}
 {2-\beta}}(t-\tau)^{-1-\frac{(n-\beta)\beta}{2(2-\beta)}} \\
&\quad\times  e^{-c\big(\frac{r^2}{(t-\tau)^\beta}\big)^{\frac{1}{2-\beta}}}dr
\Big] \|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)} \\
&\le C_2 \Big[(t-\tau)^{\beta-1}+(t-\tau)^{\beta-1}\int_{1}^{+\infty}
 z^{\frac{n}{2}-\beta}e^{-cz}\,dz
\Big] \|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)} \\
&\le c_0 (t-\tau)^{\beta-1}\|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)},\quad
y\in \mathbb{R}^n,\quad 0\le \tau<t\le T,
\end{align*}
where $d=\operatorname{diam}\operatorname{supp} \varphi$,
\begin{align*}
&\Big|\int_{\mathbb{R}^n}G_j(x-y,t)D^\alpha\varphi(x)dx\Big| \\
&\le C\Big[\int_{\{x\in \mathbb{R}^n: |x-y|^2<t^{\beta}\}}
 \frac{t^{j-1-\beta}}{|x-y|^{n-2}}dx \\
&\quad +\int_{\{x\in \mathbb{R}^n:
|x-y|^2>t^{\beta}\}}\frac{t^{j-1}}{|x-y|^n}\cdot
\Big(\frac{|x-y|^2}{4t^\beta}\Big)^{\frac{n+2-2j}{2(2-\beta)}}
e^{-c\Big(\frac{|x-y|^2}{4t^\beta}\Big)^{\frac{1}{2-\beta}}}dx\Big]
 \|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)} \\
&\le C_3\Big[t^{j-1-\beta}\int_{0}^{t^{\beta/2}}r\,dr
+\int_{t^{\beta/2}}^{d_0} r^{-1+\frac{n+2-2j}{2-\beta}}
 t^{j-1-\frac{(n+2-2j)\beta}{2(2-\beta)}}
 e^{-c\big(\frac{r^2}{t^\beta}\big)^{\frac{1}{2-\beta}}}dr
\Big] \|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)} \\
&\le C_4 \Big[t^{j-1}+t^{j-1}\int_{1}^{+\infty}z^{\frac{n}{2}-j}e^{-cz}\,dz
\Big] \|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)} \\
&\le c_j t^{j-1}\|\varphi\|_{\mathcal{D}^k(\mathbb{R}^n)},\quad (y,t)\in \bar{Q},
\end{align*}
$j=\overline{1,m}$, and similarly for $n=1,2$. Integrating by parts we
finish the proof.
\end{proof}

\begin{theorem}\label{th1}
Assume that {\rm (A1)} with $m=1$ ($m=2$) holds, $g\in C[0,T]$. Then there exists
a unique solution $u\in \mathcal{D}'_C(\bar{Q})$
of the Cauchy problem \eqref{e}, \eqref{in} with $m=1$ ($m=2$, respectively).
It is defined by
\begin{equation}\label{sys1}
\Big(u(\cdot,t),\varphi(\cdot)\Big)=h_\varphi(t)\quad
 \forall \varphi\in \mathcal{D}({\rm R}^n), \; t\in [0,T]
\end{equation} where
$$
h_{\varphi}(t)=\int_{0}^{t}g(\tau)\Big(F_0(\cdot),
(\widehat{G}_0\varphi)(\cdot,t-\tau)\Big)d\tau
+\sum_{j=1}^m\Big(F_j(\cdot),(\widehat{G}_j\varphi)(\cdot,t)\Big),\quad t\in [0,T].
$$
\end{theorem}

\begin{proof}
We say that the distribution $F$ has the order of the singularity $s(F)\le k$,
$k\in \mathbb{Z}_+$ if there exist the functions
$g_{\kappa}\in L_{1,loc}({\rm R}^n)$, $|\kappa|\le k$ such that
\begin{equation}\label{F}
\big(F,\varphi\big)=\sum_{|\kappa|\le k}\; \int_{{\rm R}^n}
g_{\kappa}(y)D^{\kappa}\varphi(y)dy\quad
\forall \varphi\in \mathcal{D}({\rm R}^n).
\end{equation}

A distribution from ${\mathcal E}'({\rm R}^n)$ has a finite order of the
singularity. So, $s(F_j)\le k_j$ with some $k_j\in \mathbb{Z}_+$,
$j=\overline{0,m}$. Using this fact and Lemma \ref{lem3},
we show that the function \eqref{sys1} belongs to $\mathcal{D}'_C(\bar{Q})$.
Namely, it follows from \eqref{F} for $F_j$, $j=\overline{0,m}$ that there
exist positive constants $B_j$ such that
$$
\big|\big(F_j,\varphi\big)\big|
\le B_j\|\varphi\|_{\mathcal{D}^{k_j}(\mathbb{R}^n)}\quad \forall
\varphi\in \mathcal{D}(\mathbb{R}^n),\quad j=\overline{0,m}.
$$
Then by Lemma \ref{lem3},
\begin{gather*}
\begin{aligned}
\big|\big(F_0(y),(\widehat{G_0}\varphi)(y,t-\tau)\big)\big|
&\le B_0\|(\widehat{G_0}\varphi)(\cdot,t-\tau)\|_{\mathcal{D}^{k_0}(\mathbb{R}^n)} \\
&\le \widehat{c}_0 \|\varphi\|_{\mathcal{D}^{k_0}(\mathbb{R}^n)}
 (t-\tau)^{\beta-1},\quad 0\leq \tau < t \leq T,
\end{aligned}\\
\int_0^t|g(\tau)|\,\big|\big(F_0(y),(\widehat{G_0}\varphi)
(y,t-\tau)\big)\big|d\tau
\le \widehat{d}_0 \|\varphi\|_{\mathcal{D}^{k_0}(\mathbb{R}^n)}\, t^\beta,\\
\begin{aligned}
\big|\big(F_j(\cdot),({\widehat{G}}_j\varphi)(\cdot,t)\big)\big|
&\le B_j\|({\widehat{G}}_j\varphi)(\cdot,t)\|_{\mathcal{D}^{k_j}(\mathbb{R}^n)}\\
&\le \widehat{d}_j \|\varphi\|_{\mathcal{D}^{k_j}(\mathbb{R}^n)}t^{j-1},
\quad t\in [0,T],\;j=\overline{1,m}.
\end{aligned}
\end{gather*}
So, $h_\varphi\in C[0,T]$ for any $\varphi\in \mathcal{D}(\mathbb{R}^n)$.

Using Lemma \ref{lem1}, we show that the function \eqref{sys1} satisfies
\eqref{iden}. Indeed, for all $\psi\in \mathcal{X}(\overline{Q})$,
\begin{align*}
\big(u,\widehat{L}\psi\big)
&=\int_0^T\Big(u(\cdot,t),(\widehat{L}\psi)(\cdot,t)\Big)dt\\
&=\int_0^T\Big(\int_0^t
g(\tau)\Big(F_0(y),({\widehat{G}}_0(\widehat{L}\psi))(y,t-\tau)\Big)d\tau\Big)dt \\
&\quad +\sum_{j=1}^m\int_0^T\Big(F_j(y),({\widehat{
G}}_j(\widehat{L}\psi))(y,t)\Big)dt\\
&=\Big(F_0(y),\int_0^T dt\int_0^t
g(\tau)({\widehat{G}}_0(\widehat{L}\psi))(y,t-\tau)d\tau\Big)\\
&\quad +\sum_{j=1}^m\Big(F_j(y),\;\int_0^T({\widehat{G}}_j(\widehat{L}\psi))(y,t)dt
 \Big)\\
&=\Big(F_0(y),\int_0^T g(\tau)d\tau\int_\tau^T({\widehat{G}}_0
 (\widehat{L}\psi))(y,t-\tau)dt\Big)\\
&\quad +\sum_{j=1}^m\Big(F_j(y),\int_0^T\big({\widehat{G}}_j(\widehat{L}\psi)
 \big)(y,t)dt\Big)\\
&=\Big(F_0(y)\cdot g(\tau),\big({\widehat{\mathcal G}}_0
 (\widehat{L}\psi)\big)(y,\tau)\Big)+
\sum_{j=1}^m\Big(F_j,{\widehat{\mathcal G}}_j(\widehat{L}\psi)\Big).
\end{align*}

From Lemma \ref{lem1} we obtain \eqref{iden}. By Definition \ref{def1}
the function \eqref{sys1} is the solution of \eqref{e}, \eqref{in}.

If $u_1,u_2$ are two solutions of the problem \eqref{e}, \eqref{in},
 then for $u=u_1-u_2$ from \eqref{iden} we obtain
$$
(u,\widehat{L}\psi)=0\quad \forall \psi\in \mathcal{X}(\overline{Q}).
$$
By using Lemma \ref{lemA} we obtain $(u,\varphi)=0$ for all
$\varphi\in \mathcal{D}(\bar{Q})$, and also $(u(\cdot,t),\varphi(\cdot))=0$
for all $\varphi\in \mathcal{D}(\mathbb{R}^n)$, $t\in [0,T]$, that is $u=0$ in
$\mathcal{D}'_C(\overline{Q})$.
\end{proof}


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

We pass to the problem \eqref{e}--\eqref{ov} with $m=1$ and $m=2$.
It follows from \eqref{e} that
$$
\big(u^{(\beta)}_t(\cdot,t),\varphi(\cdot)\big)
=\big(u(\cdot,t),\Delta\varphi(\cdot)\big)+(F_0,\varphi)g(t)\quad
 \forall \varphi\in \mathcal{D}(\mathbb{R}^n),
$$
in particular,
$$
\big(u^{(\beta)}_t(\cdot,t),\varphi_0(\cdot)\big)
=\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)+(F_0,\varphi_0)g(t).
$$
By the over-determination condition \eqref{ov} we obtain
$$
F^{(\beta)}(t)=\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)+(F_0,\varphi_0) g(t).
$$
Using the assumption (A2) we find the expression for $g(t)$ through $u$
\begin{equation}\label{pg}
g(t)=\big[F^{(\beta)}(t)-\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)\big]
[(F_0,\varphi_0)]^{-1},\quad t\in [0,T].
\end{equation}

It follows from Theorem \ref{th1} and the assumption (A2) that the right-hand
side of \eqref{pg} is a continuous function on $[0,T]$.
By substituting it in \eqref{sys1} (instead of $g(t)$) one obtains
\begin{align*}
\big(u(\cdot,t),\varphi(\cdot)\big)
&=\frac{1}{(F_0,\varphi_0)}\int_0^t
\big[F^{(\beta)}(\tau)-\big(u(\cdot,\tau),\Delta\varphi_0(\cdot)\big)\big]
\big(F_0(\cdot),({\widehat{G}}_0\varphi)(\cdot,t-\tau)\big)d\tau \\
&\quad +\sum_{j=1}^m\big(F_j(\cdot),({\widehat{G}}_j\varphi)(\cdot,t)\big)\quad
 \forall \varphi\in \mathcal{D}(\mathbb{R}^n),\; t\in[0,T],
\end{align*}
in particular,
\begin{align*}
&\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)\\
&=\frac{1}{(F_0,\varphi_0)}\int_0^t
\big[F^{(\beta)}(\tau)-\big(u(\cdot,\tau),\Delta\varphi_0(\cdot)\big)\big]
\big(F_0(\cdot),({\widehat{G}}_0\Delta\varphi_0)(\cdot,t-\tau)\big)d\tau\\
&\quad +\sum_{j=1}^m\big(F_j(\cdot),({\widehat{G}}_j\Delta\varphi_0)(\cdot,t)\big),
\quad  t\in[0,T].
\end{align*}
Denote
$r(u,t)=\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)$.
Then we have
$$
r(u,t)=-\int_0^t K(t,\tau)r(u,\tau)d\tau+v(t),\;\; t\in[0,T]
$$
where
\begin{gather}\label{k}
K(t,\tau)=\frac{\big(F_0(\cdot),({\widehat{G}}_0\Delta\varphi_0)
(\cdot,t-\tau)\big)}{(F_0,\varphi_0)}, \\
\label{vg}
v(t)=\int_0^t K(t,\tau)F^{(\beta)}(\tau)d\tau
+\sum_{j=1}^m\big(F_j(\cdot),({\widehat{G}}_j\Delta\varphi_0)(\cdot,t)\big),\;\; t\in[0,T].
\end{gather}

\begin{theorem}\label{th2}
Assume that {\rm (A1), (A2)} and \eqref{n} with $m=1$ ($m=2$) hold.
Then there exists a unique solution $(u,g)\in \mathcal{D}'_{C}(\bar{Q})\times C[0,T]$
of the problem \eqref{e}--\eqref{ov} with $m=1$ ($m=2$, respectively): $u$ is
defined by \eqref{sys1},
\begin{equation}\label{p}
g(t)=\big[F^{(\beta)}(t)-r(t)\big][(F_0,\varphi_0)]^{-1},\quad t\in [0,T]
\end{equation}
where $r(t)$ is the solution of the integral equation
\begin{equation}\label{ru}
r(t)=-\int_0^t K(t,\tau)r(\tau)d\tau+v(t),\;\; t\in[0,T]
\end{equation}
 with the integrable kernel \eqref{k}, and the function $v$ is defined
by \eqref{vg}.
\end{theorem}

\begin{proof}
As in the proof of Theorem \ref{th1} we obtain
\begin{gather*}
\begin{aligned}
\big|\big(F_0(\cdot),({\widehat{G}}_0\Delta\varphi_0)(\cdot,t-\tau)\big)\big|
&\le B_0\|({\widehat{G}}_0\Delta\varphi_0)(\cdot,t-\tau)
 \|_{\mathcal{D}^{k_0+2}(\mathbb{R}^n)}\\
&\le \widehat{d}_{0,2} \|\varphi_0\|_{\mathcal{D}^{k_j+2}(\mathbb{R}^n)}
 (t-\tau)^{\beta-1},
\end{aligned}\\
\begin{aligned}
\big|\big(F_j(\cdot),({\widehat{G}}_j\Delta\varphi_0)(\cdot,t)\big)\big|
&\le B_j\|({\widehat{G}}_j\Delta\varphi_0)(\cdot,t)\|_{\mathcal{D}^{k_j}
 (\mathbb{R}^n)}\\
&\le \widehat{d}_{j,2} \|\varphi_0\|_{\mathcal{D}^{k_j+2}(\mathbb{R}^n)}
 t^{j-1},\quad j=\overline{1,m},
\end{aligned}
\end{gather*}
where $\widehat{d}_{j,2}$, $j=\overline{0,m}$ are positive constants.
 So, the kernel \eqref{k} is integrable, the function \eqref{vg} is continuous
on $[0,T]$, and the equation \eqref{ru} has the unique solution $r\in C[0,T]$.

Let $r,g$ be defined by \eqref{ru}, \eqref{p}, respectively.
 Then on previous considerations the function \eqref{sys1} is the solution
of the Cauchy problem \eqref{e}--\eqref{in} with the known $g(t)$,
 $m=1$ ($m=2$) and satisfies the conditions \eqref{n}.
Define $F^*(t)=\big(u(\cdot,t),\varphi_0(\cdot)\big)$.
It satisfies the conditions
\begin{equation} \label{nn}
(F_j,\varphi_0)=F^{*(j-1)}(0),\quad j=\overline{1,m}.
\end{equation}
From the over-determination condition \eqref{ov} we obtain
\begin{equation}\label{gt}
g(t)=\big[F^{*(\beta)}(t)-\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)\big]
[(F_0,\varphi_0)]^{-1},\quad t\in [0,T].
\end{equation}
As in the previous reasoning we obtain that the function
$\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)$ satisfies the equation \eqref{ru},
and by uniqueness of a solution of this equation we obtain
$\big(u(\cdot,t),\Delta\varphi_0(\cdot)\big)=r(t)$ for all $t\in [0,T]$.
Then it follows from \eqref{gt} and \eqref{p} that
$F^{*(\beta)}(t)=F^{(\beta)}(t)$, $t\in [0,T]$. From the conditions
\eqref{n} and \eqref{nn} we obtain $F^{*(j-1)}(0)=F^{(j-1)}(0)$,
$j=\overline{0,m}$. Then $F^*(t)=F(t)$, $t\in [0,T]$. So, the function
\eqref{sys1}, where $r,g$ are defined by \eqref{ru}, \eqref{p},
respectively, is the solution of the problem \eqref{e}--\eqref{ov} with
$m=1$ ($m=2$).

If $(u_1,g_1)$, $(u_2,g_2)$ are two solutions of the problem \eqref{e}--\eqref{ov}
then for $u=u_1-u_2$, $g=g_1-g_2$ we obtain the problem
\begin{gather*}
Lu(x,t)=F_0(x)g(t),\quad (x,t)\in Q,\\
u(x,0)=0,\;\;\;x\in \mathbb{R}^n,\\
\big(u(\cdot,t),\varphi_0(\cdot)\big)=0, \;\;\;  t\in [0,T].
\end{gather*}
As before, we find
\begin{gather*}
\big(u(\cdot,t),\varphi(\cdot)\big)=-\int_0^t
r(\tau)\big(F_0(\cdot),({\widehat{G}}_0\varphi)(\cdot,t-\tau)\big)d\tau\;\;\;\forall \varphi\in \mathcal{D}(\mathbb{R}^n),\\
g(t)=-\frac{r(t)}{(F_0,\varphi_0)},\quad t\in[0,T],
\end{gather*}
where $r(t)$ is a solution of the Volterra integral equation
$$
r(t)=-\int_0^t K(t,\tau)r(\tau)d\tau,\quad t\in [0,T].
$$
By uniqueness of a solution of this equation we obtain $r(t)=0$ for all
$t\in [0,T]$. Then, from the previous equalities, $g(t)=0$ for all
$t\in [0,T]$ and $u=0$ in $\mathcal{D}'_{C}(\bar{Q})$.
\end{proof}

In the same way as above, we can prove the existence and uniqueness of a solution
$(u,g)\in \mathcal{D}'_C(Q)\times C(0,T]$ of the inverse source Cauchy
problem to equation
$$
u^{(\alpha)}_t+a^2(-\Delta)^{\gamma/2} u=F_0(x)g(t),\quad(x,t)\in Q
$$
where $\alpha\in (0,2)$, $\min\{n,2,\gamma\}>(n-1)/2$, $\gamma>\alpha$,
$(-\Delta)^{\gamma/2}u$ is defined with the use of the Fourier transform as follows
$\mathcal{F}[(-\Delta)^{\gamma/2} u]=|\lambda|^{\gamma/2} \mathcal{F}[u]$ and
$$
\mathcal{D}'_C(Q)=\{v\in \mathcal{D}'(\bar{Q}):(v(\cdot,t),\varphi(\cdot))
\in C(0,T]\quad \forall \varphi\in \mathcal{D}(\mathbb{R}^n)\}.
$$


\subsection*{Acknowledgments}%\label{Sect5}
The authors are grateful to Prof. Mokhtar Kirane for useful discussions.

\begin{thebibliography}{12}

\bibitem{Aleroev}
T. S. Aleroev, M. Kirane, S. A. Malik;
\emph{Determination of a source term for a time fractional diffusion equation
with an integral type over-determination condition},
 Electronic J. of Differential Equations, 2013 (2013), no 270, 1--16.

\bibitem{anh} V. V. Anh and N. N. Leonenko;
\emph{Spectral analysis of fractional kinetic
equations with random datas}, J. of Statistical Physics, 104 (2001),
no 5/6, 1349-1387.

\bibitem{Berezansky} Ju. M. Berezansky;
\emph{Expansion on eigenfunctions of selfadjoint operators},
Kiev, Naukova dumka, 1965.

\bibitem{Cheng} J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki;
\emph{Uniqueness in an inverse problem for a one-dimentional fractional diffusion
equation}, Inverse Problems, 25 (2009), 1--16.

\bibitem{djr1} M. M. Djrbashian, A. B. Nersessyan;
\emph{Fractional derivatives and Cauchy problem for differentials of fractional
order}, Izv. AN Arm. SSR, Matematika, 3 (1968), no 1, 3-29.

\bibitem{duan} Jun Sh. Duan;
\emph{Time- and space-fractional partial differential equations},
J. Math. Phis., 46 (2005), 013504.

\bibitem{mon} S. D. Eidelman, S. D. Ivasyshen, A. N. Kochubei;
\emph{Analytic methods in the theory of differential and pseudo-differential
equations of parabolic type}, Basel-Boston-Berlin, Birkhauser Verlag, 2004.

\bibitem{El-Borai} M. M. El-Borai;
\emph{On the solvability of an inverse fractional abstract Cauchy problem},
LJRRAS, 4 (2001), 411--415.


\bibitem{Gorod1} V. V. Gorodetskii, V. A. Litovchenko;
\emph{The Cauchy Problem for pseudodifferential equations in spaces of
generalized functions of type $S'$}, Dop. NAS of Ukraine, 10 (1992), 6-9.

\bibitem{H-N} Y. Hatano, J. Nakagawa, Sh. Wang, M. Yamamoto;
\emph{Determination of order in fractional diffusion equation},
Journal of Math-for-Industry, 5A (2013), 51-57.

\bibitem{Ism} M. I. Ismailov;
\emph{Inverse source problem for a time-fractional diffusion equation with nonlocal
boundary conditions}, Applied Mathematical Modeling, 40 (2016), no 7/8, 4891-4899.

\bibitem{Janno} Ja. Janno;
\emph{Determination of the order of fractional derivative and a kernel in an
inverse problem for a generalized time fractional diffusion equation},
 Electronic J. of Differential Equations, 2016 (2016), no. 199,
http://ejde.math.txstate.edu or http://ejde.math.unt.edu.

\bibitem{Jim} B. Jim, W. Rundell;
\emph{A turorial on inverse problems for anomalous diffusion processes},
Inverse Problems, 31 (2015), 035003. -doi:10.1088/0266-5611/31/3/035003.

\bibitem{K} A. A. Kilbas, M. Sajgo;
\emph{H-Transforms: Theory and Applications}, Boca-Raton, Chapman
and Hall/CRC, 2004.

\bibitem{Koch} A. N. Kochubei;
\emph{A Cauchy problem for equations of fractional order}, Differential
Equations, 25 (1989), 967-974.

\bibitem{Los} V. M. Los, A. A. Murach;
\emph{Parabolic mixed problems in spaces of generalized smoothness},
Dop. of NAS of Ukrain. Math., Nature,  Tech. Sciences, 6 (2014), 23-31.

\bibitem{Luchko} Yu. Luchko;
\emph{Boundary value problem for the
generalized time-fractional diffusion equation of distributed order},
Fract. Calc. Appl. Anal., 12 (2009), no 4, 409-422.

\bibitem{M} M. I. Matijchuk;
\emph{Parabolic and elliptic problems in Dini spaces}, Chernivtsi, 2010.

\bibitem{E} M. M. Meerschaert, N. Erkan, P. Vallaisamy;
\emph{Fractional Cauchy problems on bounded domains}, Ann. Probab.,
37 (2009), 979-1007.

\bibitem{m-m} V. A. Mikhailets, A. A. Murach;
\emph{Hormander spaces, unterpolation, and elliptic problems},
Basel, Birkhauser, 2014.

\bibitem{N} J. Nakagawa, K. Sakamoto and M. Yamamoto;
\emph{Overview to mathematical analysis for fractional diffusion equation -
 new mathematical aspects motivated by industrial collaboration}, Journal
of Math-for-Industry, 2A (2010), 99--108.

\bibitem{N-bound} W. Rundell, X. Xu, L. Zuo;
\emph{The determination of an unknown
boundary condition in fractional diffusion equation}, Applicable
Analysis, 1 (2012), 1-16.

\bibitem{Sh} G. E. Shilov;
\emph{Mathematical Analyses. Second special curse}, Nauka,
 Moskow, 1965, Russian.

\bibitem{S} H. M. Srivastava, K. C. Gupta, S. P. Goyal;
\emph{The H-functions of one and two variables with applications},
New Dehli, South Asian Publishers, 1982.

\bibitem{voroshylov} A. A. Voroshylov, A. A. Kilbas;
\emph{Conditions of the existence of classical solution of the Cauchy
 problem for diffusion-wave equation with Caputo partial derivative},
Dokl. Ak. Nauk, 414 (2007), no 4, 1-4.

\bibitem{voroshylov1} A. A. Voroshylov, A. A. Kilbas;
\emph{Conditions of the existence of a classical solution of a Cauchy
type problem for the diffusion equation with the Riemann-Liouville
partial derivative},  Differential Equations, 44 (2008), no 6, 789-806.

\bibitem{Zhang} Y. Zhang, X. Xu;
\emph{Inverse source problem for a fractional diffusion equation},
Inverse Problems,  27 (2011), 1-12.

\end{thebibliography}

\end{document}