# Inverse problems for a generalized fractional diffusion equation with
unknown history

Jaan Janno111Department of Cybernetics. Tallinn University of Technology.
Ehitajate tee 5, 19086 Tallinn, Estonia. E-mail<EMAIL_ADDRESS>
###### Abstract

Inverse problems for a diffusion equation containing a generalized fractional
derivative are studied. The equation holds in a time interval $(0,T)$ and it
is assumed that a state $u$ (solution of diffusion equation) and a source $f$
are known for $t\in(t_{0},T)$ where $t_{0}$ is some number in $(0,T)$.
Provided that $f$ satisfies certain restrictions, it is proved that product of
a kernel of the derivative with an elliptic operator as well as the history of
$f$ for $t\in(0,t_{0})$ are uniquely recovered. In case of less restrictions
on $f$ the uniqueness of the kernel and the history of $f$ is shown. Moreover,
in a case when a functional of $u$ for $t\in(t_{0},T)$ is given the uniqueness
of the kernel is proved under unknown history of $f$.

Keywords: fractional diffusion equation, inverse problem, generalized
fractional derivative

##  1 Introduction

In many cases a relative irregularity in a direct problem increases
informativeness in an inverse problem. For example, the smoother the kernel of
a Volterra equation of the first kind, the bigger the degree of the ill-
posedness of the equation [1]. Another example is the reconstruction of a
penetrable obstacle from far field pattern of a scattered single incident
wave. The uniqueness has been proved in case of some obstacles with corners
[2, 3], but in the general case it is open.

As a third example, let us mention a problem to reconstruct a space-dependent
source function of a fractional diffusion equation from a measured state at a
final time value $t=T$ over the space. The solution of this problem is unique
[4]. If an initial state of the process is relatively less smooth than the
source, then the final data contain sufficient information to recover
simultaneously the source and an order of the fractional derivative $\alpha$
[5]. The proof of uniqueness is decomposed into 2 stages: 1) uniqueness of
$\alpha$ and 2) uniqueness of the source. Similar results were obtained for
inverse problems for fractional wave equations [6] and problems with unknown
final time $T$ [7].

In [8] an inverse problem to reconstruct a time- and space dependent source
$f$ of a generalized fractional diffusion equation by means of
overdetermination in a left neighborhood $(t_{0},T)$ of the final time value
$T$ was considered. It was shown that given values of $f$ and the state $u$
for $t\in(t_{0},T)$, the history of $f$ is uniquely recovered for
$t\in(0,t_{0})$. Let us suppose that we have a possibility to choose $f$ for
$t\in(t_{0},T)$ and ask: can we do it so that the measured $u$ in $(t_{0},T)$
contains sufficient information to recover simultaneously the history of $f$
and parameters of the equation? In this paper we will show that this is
possible provided $f$ is a non-analytic with respect to $t$ function of
special form, namely such that $f\equiv 0$ for $t\in(t_{0},t_{1})$ and
$f\not\equiv 0$ for $t\in(t_{1},T)$, where $t_{1}$ is some number between
$t_{0}$ and $T$. A proof of uniqueness consists of 2 stages. Firstly, thanks
to the assumed irregularity of $f$, we can exclude the unknown history and
prove the uniqueness of the parameters. After that we prove the uniqueness of
the history of source. In addition, we will study the uniqueness of a problem
to identify a kernel contained in the generalized fractional diffusion
equation from a measured functional of the state for $t\in(t_{0},T)$ in case
the source is unknown for $t\in(0,t_{0})$.

Let us give an overview of other studies in the field of inverse source
problems for fractional diffusion equations. Several papers [9, 10, 11, 12]
deal with reconstruction of time-dependent components of the sources making
use of different measurements over time. The paper [13] is concerned with the
reconstruction of the source function that depends on time and part of spatial
variables from boundary measurements over the time. In [14, 15, 16]
determination source functions in a form of separated variables under
overdetermination in subdomains or portions of boundary is studied.

##  2 Physical background. Formulation of problems and overview of results

Fractional (and generalized fractional) diffusion equations are widely used to
model anomalous diffusion processes in biology, social sciences, engineering
sciences etc. [17, 18, 19]. There are several ways to derive such equations.
Some methods are based on continuous time random walk [20] or particle flows
[18]. A more classical approach [21] presumes a constitutive relation with
memory $Q(t,x)=-{\partial\over\partial t}\int_{0}^{t}M(t-\tau)\nabla
u(\tau,x)d\tau,$ where $t>0$ is the time, $x\in\mathbb{R}^{d}$,
$d\in\mathbb{N}$, is a space variable, $Q$ is the flux, $u$ is a state
variable and $M$ is a memory kernel. Plugging this relation into the
conservation equation ${\partial\over\partial t}u+{\rm div}\,Q=f$, where $f$
is the source, we reach the following diffusion equation:

$\displaystyle{\partial\over\partial t}u(t,x)-{\partial\over\partial
t}M*\Delta u(t,x)=f(t,x),$ (1)

where $*$ stands for the time convolution, i.e.

$v_{1}*v_{2}(t)=\int_{0}^{t}v_{1}(t-\tau)v_{2}(\tau)d\tau,$

and $\Delta$ is the Laplacian.

The operator ${\partial\over\partial t}M*$ is a generalized fractional
derivative with the kernel $M$. In the case of the usual fractional diffusion,
$M(t)=c{t^{\alpha-1}\over\Gamma(\alpha)}$, $0<\alpha<1$, $c>0$, Then
${\partial\over\partial t}M*=cD^{1-\alpha}$, where $D^{1-\alpha}$ is the
Riemann-Liouville fractional derivative of the order $1-\alpha$ and the
equation (1) takes the form ${\partial\over\partial t}u-cD^{1-\alpha}\Delta
u=f$. Such an equation is self-similar, i.e. rescaling the time does not
change the type of memory. However, several anomalous diffusion processes are
not self-similar. Then other memory functions are used. One example is the
kernel of distributed fractional derivative

$\displaystyle M(t)=\int_{0}^{1}{t^{\alpha-1}\over\Gamma(\alpha)}dp(\alpha),$
(2)

where $p(\alpha)$ is a nonnegative Borel measure [22, 23]. This includes as
particular cases the kernels of multiterm fractional derivatives
$M(t)=\sum_{l=1}^{n}\varkappa_{l}{t^{\alpha_{l}-1}\over\Gamma(\alpha_{l})}$,
$\varkappa_{l}>0$, $1>\alpha_{1}>\ldots>\alpha_{n}>0$, and the kernels of the
form
$M(t)=\int_{0}^{1}\varkappa(\alpha){t^{\alpha-1}\over\Gamma(\alpha)}d\alpha$,
$\varkappa\neq 0$, $\varkappa\geq 0$. Another example is the kernel of
tempered fractional derivative [24, 25]

$\displaystyle M(t)=ce^{-\lambda
t}{t^{\alpha-1}\over\Gamma(\alpha)},\;0<\alpha<1,\,\lambda>0,\,c>0.$ (3)

The kernels (2) and (3) belong to the following space of integrable at $t=0$
completely monotonic functions:

$\displaystyle\begin{split}&\mathcal{CM}=\big{\\{}M\,:\,M\in
L_{1,loc}(0,\infty)\cap C^{\infty}(0,\infty),\;\\\
&\qquad\quad(-1)^{n}M^{(n)}(t)\geq
0,\;t>0,\;n=0,1,2,\ldots\big{\\}}.\end{split}$ (4)

Moreover, they satisfy the condition $\lim_{t\to 0^{+}}M(t)=\infty$.

If $M\in\mathcal{CM}$ and $\lim_{t\to 0^{+}}M(t)=\infty$ then the Sonine
equation

$\displaystyle M*K(t)=1,\quad t>0,$ (5)

has the unique solution in $K\in L_{1,loc}(0,\infty)$ and this solution has
the properties $K\in\mathcal{CM}$, $\lim_{t\to 0^{+}}K(t)=\infty$ ([26],
Theorem 3) In such a case the equation (1) can be transformed to the following
equation that contains the explicit Laplacian:

$\displaystyle K*{\partial\over\partial t}u(t,x)-\Delta u(t,x)=K*f(t,x).$ (6)

If $M(t)=c{t^{\alpha-1}\over\Gamma(\alpha)}$, $0<\alpha<1$, then the solution
of (5) is $K(t)={1\over c}{t^{-\alpha}\over\Gamma(1-\alpha)}$ and (6) involves
the Caputo fractional derivative of the order $\alpha$. To the author’s
opinion, the form (6) of the fractional generalized diffusion equation
prevails in mathematical literature. If $M$ (and therefore also $K$) are
known, then it is possible to switch from (1) to (6) and vice versa. But if
$M$ is unknown then it is preferable to work with the equation (1).

If in addition to the diffusion a linear reaction occurs then the term $\Delta
u$ in (1) is complemented with an addend $au(x)$, where $-a$ is the speed of
the reaction [18, 27]. Moreover, in models of anisotropic anomalous diffusion
more general elliptic operators of the
form$\sum_{l,m=1}^{d}a_{lm}{\partial^{2}\over\partial x_{l}\partial
x_{m}}+\sum_{l=1}^{d}a_{l}{\partial\over\partial x_{l}}$ occur [28]. In this
paper we will consider the following equation:

$\displaystyle{\partial\over\partial t}u(t,x)-{\partial\over\partial
t}M*\mathscr{A}[\mathbf{a}]u(t,x)=f(t,x),$ (7)

where

$\displaystyle\hskip-25.60747pt\mathscr{A}[\mathbf{a}]v(x)\\!=\\!\sum_{l,m=1}^{d}\\!\\!a_{lm}(x){\partial^{2}\over\partial
x_{l}\partial x_{m}}v(x)\\!+\\!\sum_{l=1}^{d}a_{l}(x){\partial\over\partial
x_{l}}v(x)\\!+\\!a(x)v(x),\;$ (8)
$\displaystyle\hskip-19.91692pt\mathbf{a}=\big{(}(a_{lm})|_{l,m=1,\ldots,d},(a_{l})|_{l=1,\ldots,d},a\big{)}.$

In equations modelling nonlocal in space diffusion processes fractional or
generalized fractional elliptic operators are involved [23, 29, 30]. Another
equation that we will consider in our paper is the following one:

$\displaystyle{\partial\over\partial t}u(t,x)-{\partial\over\partial
t}M*A_{\varrho,a}u(t,x)=f(t,x),$ (9)

where $A_{\varrho,a}=-\int_{0}^{1}(-\Delta-aI)^{\beta}d\varrho(\beta)$ is a
distributed fractional elliptic operator defined in Subsection 4.2.

Let $u$ solve either (7) or (9) in $(0,T)\times\Omega$, where $T>0$,
$\Omega\subset\mathbb{R}^{d}$. Assume that $f$ and $u$ are sufficiently
smooth, $u(t,\cdot)$ belongs to a set where the space operator (i.e.
$\mathscr{A}[\mathbf{a}]$ or $A_{\varrho,a}$, respectively) is injective and
$M\in\mathcal{CM}_{\dagger}$, where

$\displaystyle\begin{split}&\mathcal{CM}_{\dagger}=\\{M\in\mathcal{CM}\,:\,\mbox{the
Laplace transform of $M$}\\\ &\quad\mbox{cannot be meromorphically extended to
the whole complex plane}\\}.\end{split}$ (10)

Moreover, suppose that $f|_{(t_{0},T)\times\Omega}$ and
$u|_{(t_{0},T)\times\Omega}$ are given, where $t_{0}$ is some number in
$(0,T)$. Then history of $f$ and $u$ is uniquely recovered in
$(0,t_{0})\times\Omega$ [8].

This result is valid in case of the kernels (2) and (3), because they belong
to $\mathcal{CM}_{\dagger}$ [8].

In the present paper we will find conditions for $f|_{(t_{0},T)\times\Omega}$
such that in addition to the history of $f$ and $u$ parameters of the equation
(i.e. $M$ and elliptic operators) are uniquely recovered by
$f|_{(t_{0},T)\times\Omega}$ and $u|_{(t_{0},T)\times\Omega}$.

We will work with inverse problems for an abstract equation${d\over
dt}[u(t)-AM*u(t)]=f(t)$, $t\in(0,T)$, in a Banach space $X$, where $A$ is a
sectorial operator. In next section we will deduce some auxiliary results and
in Subsection 4.1 we will prove Theorem 1 that establishes the uniqueness for
a problem to determine simultaneously $f$, $u$ and $MA$ provided
$f|_{(t_{0},T)}$ and $u|_{(t_{0},T)}$ are given and $f|_{(t_{0},T)}$ satisfies
certain restrictions. In Subsection 4.2 we will apply this result to equations
${\partial\over\partial
t}[u(t,x)-\mathscr{A}[\mathbf{a}]M*u(t,x)](t,x)=f(t,x)$ and
${\partial\over\partial t}[u(t,x)-A_{\varrho,a}M*u(t,x)]=f(t,x)$ that are
weaker forms of (7) and (9), respectively. There Corollaries 1 \- 3 provide
the uniqueness of parameters of $\mathscr{A}[\mathbf{a}]$ and $A_{\varrho,a}$,
too. In Subsection 5.1 we will study uniqueness of a problem to determine
simultaneously $f$, $u$ and $M$ from the same data in case of less
restrictions on $f|_{(t_{0},T)}$ (Theorem 2) and in Subsection 5.2 we will
prove the uniqueness for a problem to recover $M$ provided $f$ and a
functional of $u$ are given in $(t_{0},T)$ (Theorem 3). In the latter case the
unique reconstruction of history of $f$ and $u$ may in general not be
possible. The final Section 6 contains conclusions.

##  3 Preliminaries

Let $X$ be a complex Banach space, $X^{*}$ its dual and $\mathcal{B}(X)$ the
space of linear bounded operators in $X$. We will denote the norms in $X$ and
$\mathcal{B}(X)$ by $\|\cdot\|$ and the pairing $X^{*}\times
X\mapsto\mathbb{C}$ by $\langle\cdot,\cdot\rangle$.

Further, let $A\,:\,D(A)\mapsto X$ be a linear closed densely defined operator
in $X$. Let us denote by $X_{A}$ the set $D(A)$ endowed with the graph norm
$\|{\rm x}\|_{A}=\|{\rm x}\|+\|A{\rm x}\|$. Since $A$ is closed, $X_{A}$ is a
Banach space with respect to this norm.

Suppose that $M\in L_{1,loc}(0,\infty)$ and consider the following equation:

$\displaystyle v(t)-AM*v(t)=g(t),\;\;t\in[0,T]$ (11)

in the space $X$.

A family of operators $S\,:\,[0,\infty)\mapsto{\mathcal{B}}(X)$ is called the
resolvent of (11) if $S(t)$ is strongly continuous in $[0,\infty)$ and

$\displaystyle S(t){\rm x}\in D(A),\;AS(t){\rm x}=S(t)A{\rm
x},\;t\in[0,\infty),\;\;\mbox{for}\;\;{\rm x}\in D(A),$ $\displaystyle
S(t){\rm x}-M*AS(t){\rm x}={\rm x},\;t\in[0,\infty),\;\;\mbox{for}\;\;{\rm
x}\in D(A).$ (12)

We formulate a lemma regarding the equation (11).

###### Lemma 1

([31], Proposition 1.2) Let (11) have a resolvent. Then the following
assertions are valid.

(i) (uniqueness) If $v\in C([0,T];X)$ satisfy $M*v\in C([0,T];X_{A})$ and
solve $v(t)-AM*v(t)=0$, $t\in[0,T]$, then $v=0$.

(ii) If $g\in W_{1}^{1}((0,T);X)$ then the function

$\displaystyle v(t)=S(t)g(0)+S*g^{\prime}(t)$ (13)

belongs to $C([0,T];X)$, satisfies $M*v\in C([0,T];X_{A})$ and is a solution
of (11).

(iii) If $g\in W_{1}^{1}((0,T);X_{A})$ then $v$ defined by (13) belongs to
$C([0,T];X_{A})$ and is a solution of the stronger equation

$v(t)-M*Av(t)=g(t),\;\;t\in[0,T].$

Next we are going to establish sufficient conditions for the existence of the
resolvent of (11) and its analyticity. To this end we introduce additional
notation. Denote

$\Sigma(\omega,\theta)=\\{z\in\mathbb{C}\setminus\\{\omega\\}\,:\,|\arg(z-\omega)|<\theta\\},\;\omega\in\mathbb{R},\,\theta\in(0,\pi].$

Let $\omega\in\mathbb{R},\,\theta\in(0,\pi/2]$. The say that the operator $A$
belongs to a class $\mathcal{S}(\omega,\theta)$ if

$\displaystyle\rho(A)\supset\Sigma(\omega,{\pi\over 2}+\theta),\;\;\exists
C>0\,:\,\|(\lambda
I-A)^{-1}\|\leq{C\over|\lambda-\omega|},\,\lambda\in\Sigma(\omega,{\pi\over
2}+\theta).$

We mention that $\mathcal{S}(\omega,\theta)$ is the class of sectorial
operators that generate analytic semigroups [32].

Let $\psi\,:\,[0,\infty)\mapsto Y$, where $Y$ is a complex Banach space and
$e^{-\sigma t}\psi\in L_{1}((0,\infty);X)$ for some $\sigma\in\mathbb{R}$. By
$\widehat{\psi}$ we denote the Laplace transform of $\psi$, i.e.

$\widehat{\psi}(s)=\int_{0}^{\infty}e^{-st}\psi(t)dt,\;\;s\in\mathbb{C}\,:\,{\rm
Re}\,s>\sigma.$

###### Lemma 2

Let $M\in\mathcal{CM}$. Then $\widehat{M}(s)$ is analytically extendable to
$\Sigma(0,\pi)$. Moreover, for any $\theta\in(0,\pi)$ there exists a
nonincreasing function $\mu_{M,\theta}\,:\,(0,\infty)\mapsto(0,\infty)$ such
that $\lim\limits_{\rho\to\infty}\mu_{M,\theta}(\rho)=0$ and
$|\widehat{M}(s)|\leq\mu_{M,\theta}(|s|)$, $|\arg s|\leq\theta$.

Proof. Since $M$ is completely monotonic, due to the Bernstein’s theorem,
there exists a nonnegative Borel measure $q$ such that
$M(t)=\int_{0}^{\infty}e^{-t\tau}dq(\tau)$. The inclusion $M\in
L_{1,loc}(0,\infty)$ implies
$\int_{\delta}^{\infty}{1\over\tau}dq(\tau)<\infty$ for any $\delta>0$. We
have

$\displaystyle\widehat{M}(s)=\int_{0}^{\infty}e^{-st}\int_{0}^{\infty}e^{-t\tau}dq(\tau)=\int_{0}^{\infty}{1\over
s+\tau}dq(\tau),\quad{\rm Re}\,s>0.$ (14)

Let $\theta\in(0,\pi)$. For any $s$ such that $|\arg s|\leq\theta$ and
$\tau>0$ we have

${\tau\over|s+\tau|}={\tau\over\sqrt{|s|^{2}+2|s|\tau\cos\arg
s+\tau^{2}}}\leq{\tau\over\sqrt{|s|^{2}+2|s|\tau\cos\theta+\tau^{2}}}\leq
c_{\theta},$

where $c_{\theta}=1$ if $\theta\leq{\pi\over 2}$ and $c_{\theta}=\sqrt{1\over
1-\cos^{2}\theta}$ if $\theta>{\pi\over 2}$. Moreover, if $0<\tau<{|s|\over
2}$ then $|s+\tau|>{|s|\over 2}$. Therefore, we obtain

$\displaystyle\int_{0}^{\infty}{1\over|s+\tau|}dq(\tau)\leq{2\over|s|}\int_{0}^{|s|\over
2}dq(\tau)+c_{\theta}\int_{|s|\over
2}^{\infty}{1\over\tau}dq(\tau)<\infty,\;\;|\arg s|\leq\theta.$

Therefore, $\int_{0}^{\infty}{1\over s+\tau}dq(\tau)$ defines an analytic
function of $s$ in $\Sigma(0,\theta)$. Since $\theta\in(0,\pi)$ is arbitrary,
$\int_{0}^{\infty}{1\over s+\tau}dq(\tau)$ is analytic $\Sigma(0,\pi)$. This
with (14) proves that $\widehat{M}(s)$ is analytically extendable to
$\Sigma(0,\pi)$. Further, let us define
$\mu_{M,\theta}^{0}(\varrho)=\int_{0}^{\infty}P(\varrho,\tau)dq(\tau)$, where

$P(\varrho,\tau)=\left\\{\begin{array}[]{ll}{1\over\sqrt{\varrho^{2}+2\varrho\tau\cos\theta+\tau^{2}}}&\mbox{if}\;\;\tau>{\varrho\over
2}\\\\[8.61108pt] {2\over\varrho}&\mbox{if}\;\;0<\tau<{\varrho\over
2}.\end{array}\right.$

Then $|\widehat{M}(s)|\leq\mu_{M,\theta}^{0}(|s|)$ if $|\arg s|\leq\theta$.
The function $P(\varrho,\tau)$ approaches $0$ as $\varrho\to\infty$ for any
$\tau>0$ and is bounded by the integrable function
$B(\tau)=\left\\{\begin{array}[]{ll}\max\\{c_{\theta};1\\}{1\over\tau}&\mbox{if}\;\tau>1\\\
1&\mbox{if}\;0<\tau<1\end{array}\right.$ for $\varrho>2$. Due to the dominated
convergence theorem, $\lim\limits_{\rho\to\infty}\mu_{M,\theta}^{0}(\rho)=0$.
Finally, we define
$\mu_{M,\theta}(\rho)=\sup\limits_{\lambda>\rho}\mu_{M,\theta}^{0}(\lambda)$.
The proof is complete. $\Box$

Similarly to the resolvent of (11) we can define a resolvent of the perturbed
equation $v(t)-AM*v(t)-b*v(t)=g(t),$ where $b\in L_{1,loc}(0,\infty)$. This is
a family of operators $S\,:\,[0,\infty)\mapsto{\mathcal{B}}(X)$ that is
strongly continuous in $[0,\infty)$, $S(t){\rm x}\in D(A),\;AS(t){\rm
x}=S(t)A{\rm x},\;t\in[0,\infty)\;\;\mbox{for}\;\;{\rm x}\in D(A)$ and
$S(t){\rm x}-M*AS(t){\rm x}-b*S(t){\rm x}={\rm
x},\,t\in[0,\infty)\;\;\mbox{for}\;\;{\rm x}\in D(A).$

###### Lemma 3

Let $A_{0}\in\mathcal{S}(0,\theta)$, $\theta\in(0,{\pi\over 2}]$,
$M\in\mathcal{CM}$, $b\in L_{1,loc}(0,\infty)$. Then the equation

$\displaystyle v(t)-A_{0}M*v(t)-b*v(t)=g(t)$ (15)

has a resolvent $S(t)$. It can be represented by the series

$\displaystyle S(t)=\sum_{n=0}^{\infty}G_{n}(t),$ (16)

where $G_{n}\,:\,[0,\infty)\mapsto{\mathcal{B}}(X)$,
$k\in\\{0\\}\cup\mathbb{N}$, is a family of operators whose Laplace transforms
can be analytically extended to the set $\Sigma(0,{\pi\over 2}+\theta)$ and
have the formulas

$\displaystyle\hskip-28.45274pt\widehat{G}_{n}(s)=(1+\widehat{r}(s))(\widehat{r}(s))^{n}{1\over
s}{(I-\widehat{M}(s)A_{0})^{-1}}$
$\displaystyle\times(\widehat{M}(s)A_{0}(I-\widehat{M}(s)A_{0})^{-1})^{n},\quad
s\in\Sigma(0,{\pi\over 2}+\theta),$ (17)

and $r$ is the solution of the integral equation $r=b+b*r$. Moreover,

$\displaystyle\|(I-\widehat{M}(s)A_{0})^{-1}\|\leq{C_{2}},\;s\in\Sigma(0,{\pi\over
2}+\theta)$ (18)

with some constant $C_{2}$.

Proof. The assumption $A_{0}\in\mathcal{S}(0,\theta)$, $\theta\in(0,{\pi\over
2}]$, implies that $A_{0}$ generates a semigroup that is analytic and bounded
in $\Sigma(0,\theta)$ [32]. Due to this property and the assumption
$M\in\mathcal{CM}$, Corollary 2.4 of [31] implies that (15) with $b=0$ has a
resolvent $S_{0}(t)$ that is bounded and analytic in $\Sigma(0,\theta)$. This
fact with Theorem 2.1 of [31] implies the relations
${1\over\widehat{M}(s)}\in\rho(A_{0})$, $s\in\Sigma(0,{\pi\over 2}+\theta)$,
and the estimate (18). Further, the existence of the resolvent $S(t)$ of (15)
follows from the mentioned properties of $S_{0}(t)$ and $b\in
L_{1,loc}(0,\infty)$ from Theorem 2.3 of [31]. The formulas (16) and (17) are
contained in a proof of the latter theorem. $\Box$

###### Lemma 4

Let $A\in\mathcal{S}(\omega,\theta)$ for some $\omega\in\mathbb{R}$,
$\theta\in(0,{\pi\over 2}]$ and $M\in\mathcal{CM}$. Then (11) has a resolvent
$S(t)$ that can be analytically extended to the sector $\Sigma(0,\theta)$.

Proof. Let us define $A_{0}=A-\omega I$ and $b=\omega M$. Then (11) takes the
form of (15), so the existence of the resolvent $S(t)$ of (11) follows from
Lemma 3. It remains to show that $S(t)$ is analytically extendable to
$\Sigma(0,\theta)$.

Let us estimate $\widehat{G}_{n}(s)$ given by (17). For the term
$M(s)A_{0}(I-\widehat{M}(s)A_{0})^{-1}$ due to (18) we have

$\displaystyle\|\widehat{M}(s)A_{0}(I-\widehat{M}(s)A_{0})^{-1}\|=\|(I-\widehat{M}(s)A_{0})^{-1}-I\|$
$\displaystyle\quad\leq C_{2}+1,\quad s\in\Sigma(0,{\pi\over 2}+\theta).$ (19)

The Laplace transform $\widehat{r}(s)$ of the function $r$ that solves the
equation $r=b+b*r$ with $b=\omega M$ has the formula

$\widehat{r}(s)={\omega\widehat{M}(s)\over 1-\omega\widehat{M}(s)}.$

Let $\theta^{\prime}\in(0,\theta)$. Then $\cos\theta^{\prime}>0$ and for any
$\eta>0$ and $s\in\Sigma(\eta,{\pi\over 2}+\theta^{\prime})$ we have
$|s|\geq\eta\cos\theta^{\prime}$. Therefore, in view of Lemma 2, it holds
$|\omega\widehat{M}(s)|\leq|\omega|\mu_{M,\theta^{\prime}}(\eta\cos\theta^{\prime})$
for $s\in\Sigma(\eta,{\pi\over 2}+\theta^{\prime})$, $\eta>0$. Since
$\mu_{M,\theta^{\prime}}(\eta\cos\theta^{\prime})\to 0$ as $\eta\to\infty$,
there exists a sufficiently big $\eta=:\omega_{2}$ such that

$\displaystyle|\omega\widehat{M}(s)|<1\quad\mbox{and}\quad|\widehat{r}(s)|\leq{1\over
2(C_{2}+1)}\quad\mbox{for}\;s\in\Sigma(\omega_{2},{\pi\over
2}+\theta^{\prime}).$ (20)

Using (18) - (20) in (17) we obtain $\|\widehat{G}_{n}(s)\|\leq{C_{3}\over
2^{n}|s|},\quad s\in\Sigma(\omega_{2},{\pi\over 2}+\theta^{\prime})$ with some
constant $C_{3}$ and since
${1\over|s|}\leq({1\over\cos\theta^{\prime}}+1){1\over|s-\omega_{2}|}$,
$s\in\Sigma(\omega_{2},{\pi\over 2}+\theta^{\prime})$, we have

$\displaystyle\|\widehat{G}_{n}(s)\|\leq{C_{4}\over 2^{n}|s-\omega_{2}|},\quad
s\in\Sigma(\omega_{2},{\pi\over 2}+\theta^{\prime}),$ (21)

with a constant $C_{4}$. Let us define an extension of $G_{n}(t)$ to complex
arguments $z$ by the following integral:

$\displaystyle G_{n}(z)={1\over 2\pi
i}\int_{\gamma_{R}}e^{zs}\widehat{G}_{n}(s)ds,$ (22)

where ${\rm Re}z>0$, $R={1\over|z|}$,
$\gamma_{R}=\gamma_{R}^{1}\cup\gamma_{R}^{2}\cup\gamma_{R}^{3}$,
$\gamma_{R}^{1}=\\{z=\omega_{2}+\rho e^{-i({\pi\over
2}+\theta)},\;\infty>\rho>R\\}$,
$\gamma_{R}^{2}=\\{z=\omega_{2}+Re^{i\tau},\;-{\pi\over
2}-\theta<\tau<{\pi\over 2}+\theta\\}$, $\gamma_{R}^{3}=\\{z=\omega_{2}+\rho
e^{i({\pi\over 2}+\theta)},\;R<\rho<\infty\\}$. By means of a standard
procedure (see e.g. [31], p. 52) and (21) we deduce that this integral
converges absolutely for $z\in\Sigma(0,\theta^{\prime})$, and satisfies the
estimate

$\displaystyle\|G_{n}(z)\|\leq{C_{5,\theta_{1}}C_{4}\over
2^{n}}e^{\omega_{2}{\rm Re}z},\quad z\in\Sigma(0,\theta_{1}),$ (23)

where $\theta_{1}$ is any number in between $0$ and $\theta^{\prime}$ and
$C_{5,\theta_{1}}$ is a constant depending on $\theta_{1}$. Therefore,
$G_{n}(z)$ is analytic in $\Sigma(0,\theta^{\prime})$. Finally, we define the
extension of $S(t)$ to $\Sigma(0,\theta^{\prime})$ by

$S(z)=\sum_{n=0}^{\infty}G_{n}(z).$

Due to (23), this series converges uniformly in compact subsets of
$\Sigma(0,\theta_{1})$. Therefore, $S(z)$ is analytic in
$\Sigma(0,\theta_{1})$. Since $\theta_{1}\in(0,\theta^{\prime})$ is arbitrary
and $\theta^{\prime}\in(0,\theta)$ arbitrary, $S(z)$ is analytic in
$\Sigma(0,\theta)$. $\Box$

###### Lemma 5

Let $\theta\in(0,{\pi\over 2}]$ and
$S(z)\,:\,\Sigma(0,\theta)\mapsto{\mathcal{B}}(X)$ be analytic. Moreover, let
$f\in L_{1,loc}((0,\infty);X)$ and $t_{0}>0$. Then

$h(z)=\int_{0}^{t_{0}}S(z-\tau)f(\tau)d\tau$

is analytic in $\Sigma(t_{0},\theta)$.

Proof. Let $t_{1}>t_{0}$, $z,z_{1}\in\Sigma(t_{1},\theta)$, $\phi\in X^{*}$.
We have

$\displaystyle{\langle\phi,h(z_{1})\rangle-\langle\phi,h(z)\rangle\over
z_{1}-z}=\int_{0}^{t_{0}}H(z,z_{1},\tau)d\tau,$ $\displaystyle
H(z,z_{1},\tau)=\left\langle\phi,{1\over
z_{1}-z}(S(z_{1}-\tau)-S(z-\tau))f(\tau)\right\rangle.$

For a.e. $\tau\in(0,t_{0})$ we have $\lim\limits_{z_{1}\to
z}H(z,z_{1},\tau)=\left\langle\phi,S^{\prime}(z-\tau)f(\tau)\right\rangle$. On
the other hand, $H(z,z_{1},\tau)$ can be represented in the form
$H(z,z_{1},\tau)=\left\langle\phi,{1\over
z_{1}-z}\int_{\gamma_{z,z_{1}}}S^{\prime}(y-\tau)f(\tau)dy\right\rangle$,
where $\gamma_{z,z_{1}}$ is the straight path from $z$ to $z_{1}$. Thus, in
case $|z_{1}-z|\leq\varepsilon$, $\tau\in(0,t_{0})$,

$\displaystyle|H(z,z_{1},\tau)|\leq\|\phi\|\max_{y\in\overline{\Sigma(t_{1}-t_{0},\theta)}\cap\\{y\,:\,{\rm
Re}\,y\leq{\rm Re}\,z+\varepsilon\\}}\|S^{\prime}(y)\|\|f(\tau)\|.$

This shows that $H(z,z_{1},\tau)$ is bounded by an integrable in $(0,t_{0})$
function of $\tau$ for $z,z_{1}$ such that $|z_{1}-z|\leq\varepsilon$. By
dominated convergence theorem,
${\langle\phi,h(z_{1})\rangle-\langle\phi,h(z)\rangle\over
z_{1}-z}\to\int_{0}^{t_{0}}\left\langle\phi,S^{\prime}(z-\tau)f(\tau)\right\rangle
d\tau$ as $z_{1}\to z$. This shows that $\langle\phi,h(z)\rangle$ is
differentiable in $\Sigma(t_{1},\theta)$, hence analytic in
$\Sigma(t_{1},\theta)$. Since $\phi\in X^{*}$ and $t_{1}>t_{0}$ are arbitrary,
$h(z)$ is analytic in $\Sigma(t_{0},\theta)$. $\Box$

Finally, we prove a lemma concerning an abstract weak analogue of the
equations (7) and (9).

###### Lemma 6

Let $A\in\mathcal{S}(\omega,\theta)$ for some $\omega\in\mathbb{R}$,
$\theta\in(0,{\pi\over 2}]$, $M\in\mathcal{CM}$, $f\in L_{1}((0,T);X)$ and
$u_{0}\in X$. Then there exists a unique function $u\in C([0,T];X)$ that
satisfies $M*u\in C([0,T];X_{A})$, $u-AM*u\in W_{1}^{1}((0,T);X)$, has the
initial value $u(0)=u_{0}$ and solves the equation

$\displaystyle{d\over dt}\left[u(t)-AM*u(t)\right]=f(t),\quad
a.e.\;t\in(0,T).$ (24)

Proof. Let us consider the equation

$\displaystyle v(t)-AM*v(t)=1*f(t)+v_{0},\;\;t\in[0,T].$ (25)

By Lemma 4, this equation has the resolvent $S$ and by Lemma 1 (ii), the
function

$\displaystyle v(t)=S(t)u_{0}+S*f(t)$ (26)

belongs to $C([0,T];X)$, satisfies $M*v\in C([0,T];X_{A})$ and solves (25).
Since the right-hand side of (25) belongs to $W_{1}^{1}((0,T);X)$, we have
$v-AM*v\in W_{1}^{1}((0,T);X)$. Moreover, ${d\over
dt}\left[v(t)-AM*v(t)\right]=f(t)$, a.e. $t\in(0,T)$ and from (26) we deduce
$v(0)=u_{0}$. Therefore, $u=v$ is a function that meets the requirements of
the lemma. This proves the existence assertion. To prove the uniqueness,
suppose that $u\in C([0,T];X)$ satisfies $M*u\in C([0,T];X_{A})$, $u-AM*u\in
W_{1}^{1}((0,T);X)$, $u(0)=0$ and solves ${d\over
dt}\left[u(t)-AM*u(t)\right]=0$, a.e. $t\in(0,T)$. The latter relation implies
$u(t)-AM*u(t)=c$, $t\in[0,T]$, where $c\in X$. Due to Lemma 1 (ii),
$u(t)=S(t)c$. Using the condition $u(0)=0$ we get $c=0$. Lemma 1 (i) implies
$u=0$. This proves the uniqueness assertion. $\Box$

##  4 Inverse problem to determine the operator, kernel, source and state

###  4.1 Abstract inverse problem

Let $u$ solve (24). We pose the following inverse problem: given $u(t)$,
$f(t)$, $t\in(t_{0},T)$, where $t_{0}$ is some number between $0$ and $T$,
determine $MA$ and $f(t)$, $u(t)$ in the whole interval $(0,T)$. In the sequel
we prove the uniqueness for this problem.

###### Theorem 1

Let $M_{1}\in\mathcal{CM}_{\dagger}$, $M_{2}\in\mathcal{CM}$,
$A_{j}\,:\,D(A_{j})\mapsto X$, $j=1,2$ \- linear closed densely defined
operators in a complex Banach space $X$,
$A_{j}\in\mathcal{S}(\omega_{j},\theta_{j})$ for some
$\omega_{j}\in\mathbb{R}$, $\theta_{j}\in(0,{\pi\over 2}]$ and $A_{1}$ \-
injective. Let $f_{j}$, $u_{j}$, $j=1,2$, satisfy

$\displaystyle\hskip-14.22636ptf_{j}\in L_{1}((0,T);X),\;u_{j}\in
C([0,T];X),\;$ (27) $\displaystyle\hskip-14.22636ptM_{j}*u_{j}\in
C([0,T];X_{A_{j}}),\;u_{j}-A_{j}M_{j}*u_{j}\in W_{1}^{1}((0,T);X),$ (28)
$\displaystyle\hskip-14.22636pt{d\over
dt}\left[u_{j}(t)-A_{j}M_{j}*u_{j}(t)\right]=f_{j}(t),\;a.e.\;t\in(0,T),$ (29)
$\displaystyle\hskip-14.22636ptf_{1}(t)=f_{2}(t)=:f(t)\;a.e.\;t\in(t_{0},T)\;\mbox{for
some $t_{0}\in(0,T)$},\;$ (30)

where $f$ fulfills the condition

$\displaystyle\hskip-14.22636ptf(t)=0\;a.e.\;t\in(t_{0},t_{1})\;\mbox{for some
$t_{1}\in(t_{0},T)$.}$ (31)

Moreover, assume that either

$\displaystyle\left.\begin{split}&\mbox{$D(A_{1})\cap D(A_{2})$ is not empty
and dense in $X_{A_{j}}$, $j=1,2$,}\\\
&f(t)=\sum_{i=1}^{\infty}\psi_{i}(t){\rm x}_{i},\;t\in(t_{1},T),\;\psi_{i}\in
L_{1}(t_{1},T),\;\\\ &{\rm
ess\,supp}\,\psi_{i}\subset[t_{i},t_{i+1}),\;t_{1}<t_{2}<\ldots<T,\;\\\
&t_{i}=\min{\rm ess\,supp}\,\psi_{i},\;{\rm span}\\{{\rm
x}_{i}\,:\,i\in\mathbb{N}\\}-\mbox{dense in $X$}\end{split}\right\\}$ (32)

or

$\displaystyle\left.\begin{split}&X\;\mbox{is a Hilbert
space},\;\;A_{j}\;\mbox{is selfadjoint},\,\;\;\\\
&\sigma(A_{j})\;\mbox{consists of a countable number of eigenvalues
$\lambda_{jk}$, $k\in\mathbb{N}$},\\\ &\mbox{the eigenvectors ${\rm x}_{k}$,
$k\in\mathbb{N}$, of $A_{1}$ and $A_{2}$ coincide, i.e. }\\\ &\mbox{$A_{j}{\rm
x}_{k}=\lambda_{jk}{\rm x}_{k}$, $k\in\mathbb{N}$},\\\ &\mbox{the system
$({\rm x}_{k})_{k\in\mathbb{N}}$ is orthonormal and complete in $X$},\;\;\\\ \
&\langle f(t),{\rm x}_{k}\rangle\not\equiv
0\;\mbox{in}\;(t_{1},T),\>\;k\in\mathbb{N}.\end{split}\right\\}$ (33)

Finally, let

$\displaystyle u_{1}(t)=u_{2}(t),\;t\in(t_{0},T).$ (34)

Then $D(A_{1})=D(A_{2})$, there exists $c>0$ such that $cM_{1}=M_{2}$,
$A_{1}=cA_{2}$, $f_{1}=f_{2}$ and $u_{1}=u_{2}$.

In (33), $\langle\cdot,\cdot\rangle$ is interpreted as a scalar product in
$X$. We split most of the proof of this theorem in lemmas.

###### Lemma 7

Let $M_{j}\in\mathcal{CM}$, $j=1,2$, $A_{j}\,:\,D(A_{j})\mapsto X$, $j=1,2$ \-
linear closed densely defined operators in a complex Banach space $X$ and
$A_{j}\in\mathcal{S}(\omega_{j},\theta_{j})$ for some
$\omega_{j}\in\mathbb{R}$, $\theta_{j}\in(0,{\pi\over 2}]$. Moreover, let
$f_{j}$, $u_{j}$, $j=1,2$, satisfy (27) - (30), (34) and $f$ involved in (30)
fulfill the condition (31). Then

$\displaystyle\int_{t_{1}}^{t}S_{1}(t-\tau)f(\tau)d\tau=\int_{t_{1}}^{t}S_{2}(t-\tau)f(\tau)d\tau,\quad
t\in[t_{1},T],$ (35)

where $S_{j}$ is the resolvent of the equation (11) with $M=M_{j}$ and
$A=A_{j}$.

Proof. We have $u_{j}(t)-A_{j}M_{j}*u_{j}(t)=1*f_{j}(t)+u_{j}(0)$,
$t\in[0,T]$. Due to Lemma 1 we obtain

$\displaystyle u_{j}(t)=S_{j}(t)u_{j}(0)+S_{j}*f_{j}(t),\quad
t\in[0,T],\;\;j=1,2.$ (36)

Let $t\in[t_{0},T]$. Subtracting the equation (36) with $j=2$ from the one
with $j=1$ and taking (30) and (34) into account, we deduce

$\displaystyle 0=Y(t)+Z(t),\quad t\in[t_{0},T],$ (37)

where

$\displaystyle\hskip-19.91692ptY(t)=S_{1}(t)u_{1}(0)-S_{2}(t)u_{2}(0)$
$\displaystyle\qquad+\int_{0}^{t_{0}}\left[S_{1}(t-\tau)f_{1}(\tau)d\tau-
S_{2}(t-\tau)f_{2}(\tau)\right]d\tau,$
$\displaystyle\hskip-19.91692ptZ(t)=\int_{t_{0}}^{t}\left[S_{1}(t-\tau)-S_{2}(t-\tau)\right]f(\tau)d\tau.$

Since $f(t)=0$ a.e. $t\in(t_{0},t_{1})$, we have $Z(t)=0$,
$t\in[t_{0},t_{1}]$. Using this in (37) we get $Y(t)=0$, $t\in[t_{0},t_{1}]$.
In view of Lemmas 4 and 5, $Y(t)$ is analytic for $t>t_{0}$. By analytic
continuation we obtain $Y(t)=0$, $t\geq t_{0}$. Therefore, (37) reduces to
$Z(t)=0$, $t\in[t_{0},T]$. This by $f(t)=0$ a.e. $t\in(t_{0},t_{1})$, implies
(35). $\Box$

Lemma 7 provides an independent equation (35) that involves $M_{j}$ and
$A_{j}$ but not $u_{j}|_{(0,t_{0})}$ and $f_{j}|_{(0,t_{0})}$. Under certain
assumptions, this equation implies $M_{1}A_{1}=M_{2}A_{2}$. We will show it in
the next lemma.

###### Lemma 8

Let the assumptions of Lemma 7 be satisfied for $M_{j}$, $A_{j}$, $j=1,2$. Let
$f\in L_{1}((t_{1},T);X)$ and (35) be valid where $S_{j}$ is the resolvent of
the equation (11) with $M=M_{j}$ and $A=A_{j}$. Moreover, assume that either
(32) or (33) holds. Then $D(A_{1})=D(A_{2})$ and there exists $c>0$ such that
$cM_{1}=M_{2}$ and $A_{1}=cA_{2}$.

Proof. Firstly, we consider the case (32). Then from (35) we have

$\displaystyle\int_{t_{1}}^{t}\left[S_{1}(t-\tau)-S_{2}(t-\tau)\right]\sum_{i=1}^{\infty}\psi_{i}(\tau){\rm
x}_{i}d\tau=0,\quad t\in[t_{1},T].$ (38)

Let us show by induction that

$\displaystyle S_{1}(t){\rm x}_{m}=S_{2}(t){\rm x}_{m},\quad t\geq
0,\;\;m=1,2,\ldots.$ (39)

To this end, we suppose that either $m_{0}=0$ or $m_{0}\in\mathbb{N}$ and (39)
holds for $m=1,\ldots,m_{0}$. The aim is to show that (39) is valid for
$m=m_{0}+1$, too. Due to such a supposition, from (38) in view of properties
of $\psi_{i}$ we have

$\displaystyle\int_{t_{m_{0}+1}}^{t}\\!\\!\left[S_{1}(t-\tau)-S_{2}(t-\tau)\right]\sum_{i=m_{0}+1}^{\infty}\psi_{i}(\tau){\rm
x}_{i}d\tau=0,\;\;t\in[t_{m_{0}+1},T].$

For $t\in[t_{m_{0}+1},t_{m_{0}+2}]$ this yields
$\int_{t_{m_{0}+1}}^{t}\psi_{m_{0}+1}(\tau)[S_{1}(t-\tau)\break-
S_{2}(t-\tau)]{\rm x}_{m_{0}+1}d\tau=0$. Let $\phi\in X^{*}$. Then

$\displaystyle\int_{t_{m_{0}+1}}^{t}\\!\\!\psi_{m_{0}+1}(\tau)\langle\phi,\left[S_{1}(t-\tau)-S_{2}(t-\tau)\right]{\rm
x}_{m_{0}+1}\rangle d\tau=0,$ $\displaystyle\quad\quad\quad
t\in[t_{m_{0}+1},t_{m_{0}+2}].$

By (32), $\psi_{m_{0}+1}(t)$ does not vanish identically in an arbitrarily
small right neighborhood of $t_{m_{0}+1}$. Applying Titchmarsh convolution
theorem, we get

$\langle\phi,\left[S_{1}(t)-S_{2}(t)\right]{\rm
x}_{m_{0}+1}\rangle=0,\;\;t\in[0,t_{m_{0}+2}-t_{m_{0}+1}].$

Since $\phi\in X^{*}$ is arbitrary and the embedding $X\subseteq X^{**}$ in
injective, we have $S_{1}(t){\rm x}_{m_{0}+1}=S_{2}(t){\rm x}_{m_{0}+1}$,
$t\in[0,t_{m_{0}+2}-t_{m_{0}+1}]$, and by analytic continuation, (39) for
$m=m_{0}+1$ follows.

Since ${\rm span}\\{{\rm x}_{i}\,:\,i\in\mathbb{N}\\}$ is dense in $X$, (39)
implies $S_{1}(t)=S_{2}(t)=:S(t)$, $t\geq 0$. Due to (12) we obtain

$\displaystyle S(t){\rm x}={\rm x}+M_{j}(t)*A_{j}S(t){\rm x},\quad t\geq
0,\;{\rm x}\in D(A_{j}),\;j=1,2.$

This yields

$\displaystyle v_{\rm x}(t):=S*[M_{1}(t)A_{1}-M_{2}(t)A_{2}]{\rm x}=0,\quad
t\geq 0,\;{\rm x}\in D(A_{1})\cap D(A_{2}).$

From the left equality and Lemma 1 we deduce that $v_{\rm x}$ is the solution
of the following equation:

$\displaystyle v_{\rm x}(t)-A_{j}M_{j}*v_{\rm
x}(t)=1*[M_{1}(t)A_{1}-M_{2}(t)A_{2}]{\rm x},\quad t\geq 0,$

for both $j=1,2$. Therefore, $v_{\rm x}(t)=0$ implies
$1*[M_{1}(t)A_{1}-M_{2}(t)A_{2}]{\rm x}=0$. This yields

$\displaystyle M_{1}(t)A_{1}{\rm x}=M_{2}(t)A_{2}{\rm x},\quad t>0,\;{\rm
x}\in D(A_{1})\cap D(A_{2}).$ (40)

Let us define $c={M_{2}(1)\over M_{1}(1)}$. Then setting $t=1$ in (40) we
deduce

$\displaystyle A_{1}{\rm x}=cA_{2}{\rm x},\;{\rm x}\in D(A_{1})\cap D(A_{2}).$
(41)

Using (41) in (40) we prove $cM_{1}=M_{2}$. Further, let ${\rm x}$ be an
arbitrary element of $D(A_{2})$. Since $D(A_{1})\cap D(A_{2})$ is dense in
$X_{A_{2}}$, there exist ${\rm x}_{n}\in D(A_{1})\cap D(A_{2})$,
$n\in\mathbb{N}$, such that ${\rm x}_{n}\to{\rm x}$ and $A_{2}{\rm x}_{n}\to
A_{2}{\rm x}$ as $n\to\infty$. By (41), $A_{1}{\rm x}_{n}\to cA_{2}{\rm x}$ as
$n\to\infty$. Since $A_{1}$ is closed, we have ${\rm x}\in D(A_{1})$ and
$A_{1}{\rm x}=cA_{2}{\rm x}$. This shows that $D(A_{1})\supseteq D(A_{2})$ and
$A_{1}{\rm x}=cA_{2}{\rm x}$, ${\rm x}\in D(A_{2})$. Similarly we show that
$D(A_{2})\supseteq D(A_{1})$ and $A_{2}{\rm x}={1\over c}A_{1}{\rm x}$, ${\rm
x}\in D(A_{1})$. Consequently, $D(A_{1})=D(A_{2})$ and $A_{1}=cA_{2}$.

Secondly, we consider the case (33). From (35) we have

$\displaystyle\sum_{k=1}^{\infty}\int_{t_{1}}^{t}\\!f_{k}(\tau)\langle\left[S_{1}(t\\!-\\!\tau)-S_{2}(t\\!-\\!\tau)\right]{\rm
x}_{k},{\rm x}_{l}\rangle d\tau=0,\;t\in[t_{1},T],\;l\in\mathbb{N},$ (42)

where $f_{k}(t)=\langle f(t),{\rm x}_{k}\rangle$. The function $S_{j}(t){\rm
x}_{k}$ solves the equation

$\displaystyle S_{j}(t){\rm x}_{k}-M_{j}*A_{j}S_{j}(t){\rm x}_{k}={\rm
x}_{k},\quad t\geq 0.$

Since $A_{j}$ commutes with $S_{j}$ and $A_{j}{\rm x}_{k}=\lambda_{jk}{\rm
x}_{k}$, this implies

$\displaystyle S_{j}(t){\rm x}_{k}-\lambda_{jk}M_{j}*S_{j}(t){\rm x}_{k}={\rm
x}_{k},\quad t\geq 0.$

Thus,

$\displaystyle\langle S_{j}(t){\rm x}_{k},{\rm
x}_{l}\rangle-\lambda_{jk}M_{j}*\langle S_{j}(t){\rm x}_{k},{\rm
x}_{l}\rangle=\delta_{kl},\quad t\geq 0,$ (43)

where $\delta_{kl}$ is the Kronecker delta. The relation (43) is a scalar
Volterra equation of the second kind. In case $k\neq l$ it is homogeneous.
Therefore, $\langle S_{j}(t){\rm x}_{k},{\rm x}_{l}\rangle=0$, $t\geq 0$,
$k\neq l$, and from (42) we deduce

$\displaystyle\int_{t_{1}}^{t}f_{k}(\tau)\langle\left[S_{1}(t-\tau)-S_{2}(t-\tau)\right]{\rm
x}_{k},{\rm x}_{k}\rangle d\tau=0,\;\;t\in[t_{1},T],\;k\in\mathbb{N}.$ (44)

Let us choose some $k\in\mathbb{N}$. By (33), $f_{k}(t)$ does not vanish in
$(t_{1},T)$. Denote $\tau_{k}=\min{\rm ess\,supp}\,f_{k}$. Then (44) reduces
to$\int_{\tau_{k}}^{t}f_{k}(\tau)\langle\left[S_{1}(t-\tau)-S_{2}(t-\tau)\right]{\rm
x}_{k},{\rm x}_{k}\rangle d\tau=0$, $t\in[\tau_{k},T]$. Moreover, $f_{k}(t)$
is not identically zero in an arbitrarily small right neighborhood of
$\tau_{k}$. Applying Titchmarsh convolution theorem, we obtain
$\langle\left[S_{1}(t)-S_{2}(t)\right]{\rm x}_{k},{\rm x}_{k}\rangle=0$,
$t\in[0,T-\tau_{k}]$. By analytic continuation,

$\displaystyle\langle\left[S_{1}(t)-S_{2}(t)\right]{\rm x}_{k},{\rm
x}_{k}\rangle=0,\quad t\geq 0.$

Using this relation and (43) in case $l=k$ we deduce

$\displaystyle(\lambda_{1k}M_{1}-\lambda_{2k}M_{2})*\langle S_{1}(t){\rm
x}_{k},{\rm x}_{k}\rangle=0,\quad t\geq 0.$

Since $S_{1}(0){\rm x}_{k}={\rm x}_{k}$ and $S_{1}$ is strongly continuous,
$\langle S_{1}(t){\rm x}_{k},{\rm x}_{k}\rangle$ differs from zero in a right
neighborhood of $0$. Using the Titchmarsh convolution theorem again, we obtain

$\displaystyle\lambda_{1k}M_{1}(t)=\lambda_{2k}M_{2}(t),\quad t>0.$

We have chosen an arbitrary $k\in\mathbb{N}$. Therefore, this relation is
valid for any $k\in\mathbb{N}$. Define $c={M_{2}(1)\over M_{1}(1)}$. Arguing
as in the previous case we deduce $cM_{1}=M_{2}$ and
$\lambda_{1k}=c\lambda_{2k}$, $k\in\mathbb{N}$. Let us define the set

$\displaystyle D=\left\\{{\rm x}\in
X\,:\,\sum_{k=1}^{\infty}|\lambda_{2k}|^{2}|\langle{\rm x},{\rm
x}_{k}\rangle|^{2}<\infty\right\\}.$

For any ${\rm x}\in D(A_{1})$ we have $\langle A_{1}{\rm x},{\rm
x}_{k}\rangle=\langle{\rm x},A_{1}{\rm x}_{k}\rangle=\lambda_{1k}\langle{\rm
x},{\rm x}_{k}\rangle$. Therefore, $\|A_{1}{\rm
x}\|^{2}=\sum_{k=1}^{\infty}|\lambda_{1k}|^{2}|\langle{\rm x},{\rm
x}_{k}\rangle|^{2}=c^{2}\sum_{k=1}^{\infty}|\lambda_{2k}|^{2}|\langle{\rm
x},{\rm x}_{k}\rangle|^{2}<\infty$. Thus, ${\rm x}\in D$ and we have
$D(A_{1})\subseteq D$. Next let us choose some ${\rm x}\in D$. Since
$\sum_{k=1}^{\infty}|\lambda_{2k}|^{2}|\langle{\rm x},{\rm
x}_{k}\rangle|^{2}<\infty$, the element ${\rm
y}:=\sum_{k=1}^{\infty}\lambda_{2k}\langle{\rm x},{\rm x}_{k}\rangle x_{k}$ is
well-defined in $X$. Since ${\rm x}_{k}\in D(A_{1})$, $k\in\mathbb{N}$, it
holds $\hat{\rm x}_{n}:=\sum_{k=1}^{n}\langle{\rm x},{\rm x}_{k}\rangle
x_{k}\in D(A_{1})$. We deduce $\hat{\rm x}_{n}\to{\rm x}$ and $A_{1}\hat{\rm
x}_{n}=\break\sum_{k=1}^{n}\lambda_{1k}\langle{\rm x},{\rm x}_{k}\rangle{\rm
x}_{k}=c\sum_{k=1}^{n}\lambda_{2k}\langle{\rm x},{\rm x}_{k}\rangle{\rm
x}_{k}\to c{\rm y}$ as $n\to\infty$. Since $A_{1}$ is closed, we have ${\rm
x}\in D(A_{1})$. This shows that $D\subseteq D(A_{1})$. Consequently,
$D(A_{1})=D$. Similarly we prove $D(A_{2})=D$. Finally, the relation
$A_{1}=cA_{2}$ follows from the equality $A_{1}{\rm
x}=\sum_{k=1}^{\infty}\lambda_{1k}\langle{\rm x},{\rm x}_{k}\rangle{\rm
x}_{k}=c\sum_{k=1}^{\infty}\lambda_{2k}\langle{\rm x},{\rm x}_{k}\rangle{\rm
x}_{k}=cA_{2}{\rm x}$. $\Box$

In order to complete the proof of Theorem 1, we will need one more lemma.

###### Lemma 9

Let $M\in\mathcal{CM}_{\dagger}$, $A\,:\,D(A)\mapsto X$ \- an injective
operator in a complex Banach space $X$ and $\varphi$, $u$ satisfy $\varphi\in
L_{1}((0,T);X)$, $u\in C([0,T];X)$, $M*u(t)\in D(A)$, a.e. $t\in(0,T)$,
$u-AM*u\in W_{1}^{1}((0,T);X)$ and solve the equation

$\displaystyle{d\over dt}\left[u(t)-AM*u(t)\right]=\varphi(t),\quad
a.e.\;t\in(0,T).$ (45)

Moreover, assume that for some $t_{0}\in(0,T)$ it holds $\varphi(t)=0$ a.e.
$t\in(t_{0},T)$ and $u(t)=0$, $t\in(t_{0},T)$. Then $\varphi=0$ and $u=0$.

If $u$ is a sufficiently good function then from (45) we obtain ${d\over
dt}M*\langle\phi,Au(t)\rangle=\langle\phi,u^{\prime}(t)-\varphi(t)\rangle$,
a.e. $t\in(0,T)$, $\forall\phi\in X^{*}$ and since the right hand side of this
equation vanishes for a.e. $t\in(t_{0},T)$, the assertion $u=0$ directly
follows from Theorem 1 of [8] and the injectivity of $A$. This by (45) implies
$\varphi=0$, too. In our case have to provide a bit longer proof, that however
uses ideas of [8].

Proof. By (45) we have

$\displaystyle u(t)-AM*u(t)=1*\varphi(t)+{\rm y},\;t\in[0,T],$

with some ${\rm y}\in X$. Using the assumptions $\varphi(t)=0$ a.e.
$t\in(t_{0},T)$ and $u(t)=0$, $t\in(t_{0},T)$ we obtain

$\displaystyle AM*u(t)=-\int_{0}^{t_{0}}\varphi(\tau)d\tau-{\rm y},\quad
t\in[t_{0},T].$ (46)

Let us extend $u(t)$ by zero for $t>T$ and define

$F(t)=M*u(t),\;\;t\geq 0.$

By $u(t)=0$, $t>t_{0}$, we have

$\displaystyle F(t)=\int_{0}^{t_{0}}M(t-\tau)u(\tau)d\tau,\quad t>t_{0}.$ (47)

In view of Bernstein’s theorem, the completely monotonic function $M(t)$ has
an analytic extension to $\Sigma(0,{\pi\over 2})$. Therefore, by Lemma 5 we
deduce that $F(t)$ is analytic for $t\in\Sigma(t_{0},{\pi\over 2})$. On the
other hand, from (46) we have $AF(t)={\rm x}$, $t\in[t_{0},T]$, where ${\rm
x}=-\int_{0}^{t_{0}}\varphi(\tau)d\tau-{\rm y}$ is a constant element. Thus,
$F(t)=A^{-1}{\rm x}$, $t\in[t_{0},T]$. Using analytic continuation we obtain

$\displaystyle F(t)=A^{-1}{\rm x},\quad t\geq t_{0}.$ (48)

Next we express $F$ as follows:

$F(t)=F_{1}(t)+A^{-1}{\rm x},\quad t\geq 0.$

Due to (48), $F_{1}$ is compactly supported. Let $\phi$ be an arbitrary
element of $X^{*}$ and apply the Laplace transform to the relation
$\langle\phi,M*u\rangle=\langle\phi,F_{1}\rangle+\langle\phi,A^{-1}{\rm
x}\rangle$. We obtain

$\widehat{M}(s)\langle\phi,\widehat{u}(s)\rangle=\langle\phi,\widehat{F}_{1}(s)\rangle+{\langle\phi,A^{-1}{\rm
x}\rangle\over s},\quad{\rm Re}s>0.$

If $u$ is not a zero function, then there exists $\tilde{\phi}\in X^{*}$ such
that $\langle\tilde{\phi},\widehat{u}(s)\rangle$ is not a zero function and we
get

$\displaystyle\widehat{M}(s)={\langle\tilde{\phi},\widehat{F}_{1}(s)\rangle+{\langle\tilde{\phi},A^{-1}{\rm
x}\rangle\over s}\over\langle\tilde{\phi},\widehat{u}(s)\rangle},\quad{\rm
Re}s>0\,:\,\langle\tilde{\phi},\widehat{u}(s)\rangle\neq 0.$ (49)

Since $u$ and $F_{1}$ are compactly supported,
$\langle\tilde{\phi},\widehat{u}(s)\rangle$ and
$\langle\tilde{\phi},\widehat{F}_{1}(s)\rangle$ are entire functions. This
implies that the right-hand side of (49) can be meromorphically extended to
the whole complex plane. But due to $M\in\mathcal{CM}_{\dagger}$, this is not
the case for the left-hand side. We reached the contradiction. Consequently,
$u=0$. This with (45) implies $\varphi=0$. $\Box$

Proof of Theorem 1. Lemma 8 implies $D(A_{1})=D(A_{2})$ and $\exists c>0$:
$cM_{1}=M_{2}$, $A_{1}=cA_{2}$. Denote $M=M_{1}$, $A=A_{1}$, $u=u_{1}-u_{2}$
and $\varphi=f_{1}-f_{2}$. Then $M$, $A$, $u$ and $\varphi$ satisfy the
assumptions of Lemma 9. Consequently, $u_{1}-u_{2}=0$, $f_{1}-f_{2}=0$. $\Box$

We mention that input functions similar to $f$ in (32) have been used earlier
in other inverse problems to recover parameters of equations, e.g. in problems
to determine coefficients [35] and unknown manifolds [36].

###  4.2 Applications

In this subsection we will consider some inverse problems that are posed for
equations governing subdiffusion in a

$\displaystyle\begin{split}&\mbox{bounded domain
$\Omega\subset\mathbb{R}^{d}$, $d\geq 1$, with a boundary $\partial\Omega$
}\\\ &\mbox{that of the class $C^{2}$ in case $d>1$. }\end{split}$ (50)

Firstly, let us consider inverse problems for equations with operators of the
form (8), where
$\mathbf{a}=\big{(}(a_{lm})|_{l,m=1,\ldots,d},(a_{l})|_{l=1,\ldots,d},a\big{)}\in{\mathcal{Q}}$,

$\displaystyle{\mathcal{Q}}=\Big{\\{}\mathbf{a}\in(C(\overline{\Omega}))^{d^{2}+d+1}\,:\,a_{lm}=a_{ml},\,l,m=1,\ldots,d,$
$\displaystyle\qquad\mbox{$\exists\epsilon>0$:
$\sum\limits_{l,m=1}^{d}a_{lm}(x)\xi_{l}\xi_{m}\geq\epsilon|\xi|^{2}$,
$x\in\overline{\Omega}$, $\xi\in\mathbb{R}^{d}$}\Big{\\}}.$

Let us define

$\displaystyle D_{p}:=\\{z\in
W_{p}^{2}(\Omega)\,:\,z|_{\partial\Omega}=0\\},\quad 1<p<\infty,$ (51)

and an operator $A[p;\mathbf{a}]\,:\,D(A[p;\mathbf{a}])=D_{p}\mapsto
L_{p}(\Omega)$ such that

$A[p;\mathbf{a}]v=\mathscr{A}[\mathbf{a}]v\mbox{ for $v\in D_{p}$}.$

The operator $A[p;\mathbf{a}]$ closed and densely defined in $L_{p}(\Omega)$.
Moreover, it belongs to $\mathcal{S}(\omega,\theta)$ for some
$\omega\in\mathbb{R}$, $\theta\in(0,{\pi\over 2}]$ [32].

###### Corollary 1

Let $M_{1}\in\mathcal{CM}_{\dagger}$, $M_{2}\in\mathcal{CM}$,
$\mathbf{a}_{j}\in{\mathcal{Q}}$, $j=1,2$, $p\in(1,\infty)$,
$0\not\in\sigma(A[p,\mathbf{a}_{1}])$. Let $f_{j}$, $u_{j}$, $j=1,2$, satisfy
$f_{j}\in L_{1}((0,T);L_{p}(\Omega))$, $u_{j}\in C([0,T];L_{p}(\Omega))$,
$M_{j}*u_{j}\in C([0,T];D_{p})$, $u_{j}-A[p;\mathbf{a}_{j}]M_{j}*u_{j}\in
W_{1}^{1}((0,T);L_{p}(\Omega))$ and solve the equations

$\displaystyle{\partial\over\partial
t}\left[u_{j}(t,x)-A[p;\mathbf{a}]M_{j}*u_{j}(t,x)\right]=f_{j}(t,x),\;a.e.\;(t,x)\in(0,T)\times\Omega.$
(52)

Moreover, assume that $f_{1}(t,\cdot)=f_{2}(t,\cdot)=:f(t,\cdot)$ a.e.
$t\in(t_{0},T)$, for some $t_{0}\in(0,T)$, $f(t,\cdot)=0$, a.e.
$t\in(t_{0},t_{1})$ for some $t_{1}\in(t_{0},T)$ and

$\displaystyle\left.\begin{split}&f(t,x)=\sum_{i=1}^{\infty}\psi_{i}(t)w_{i}(x),\;t\in(t_{1},T),\;\psi_{i}\in
L_{1}(t_{1},T),\;\\\ &{\rm
ess\,supp}\,\psi_{i}\subset[t_{i},t_{i+1}),\;t_{1}<t_{2}<\ldots<T,\;\\\
&t_{i}=\min{\rm ess\,supp}\,\psi_{i},\;\;{\rm
span}\\{w_{i}\,:\,i\in\mathbb{N}\\}-\mbox{dense in
$L_{p}(\Omega)$}.\end{split}\right\\}$ (53)

Finally, let $u_{1}(t,\cdot)=u_{2}(t,\cdot)$, $t\in(t_{0},T)$. Then there
exists $c>0$ such that $cM_{1}=M_{2}$, $\mathbf{a}_{1}=c\mathbf{a}_{2}$,
$f_{1}=f_{2}$, and $u_{1}=u_{2}$.

Proof. Theorem 1 implies that $f_{1}=f_{2}$, $u_{1}=u_{2}$ and there exists
$c>0$ such that $cM_{1}=M_{2}$ and $A[p;\mathbf{a}_{1}]=cA[p;\mathbf{a}_{2}]$.
Let us denote the components of $\mathbf{a}_{j}$, $j=1,2$, as follows:

$\mathbf{a}_{j}=\big{(}(a_{lm,j})|_{l,m=1,\ldots,d},(a_{l,j})|_{l=1,\ldots,d},a_{,j}\big{)}.$

Let us choose some $x^{*}\in\Omega$ and $v^{*}\in D_{p}$ such that
$v^{*}(x)=1$ for $x$ in some neighborhood of $x^{*}$. Then
$A[p;\mathbf{a}_{j}]v^{*}(x)|_{x=x^{*}}=a_{,j}(x^{*})$. Thus,
$a_{,1}(x^{*})=ca_{,2}(x^{*})$. Secondly, let
$v^{l}(x)=v^{*}(x)(x_{l}-x_{l}^{*})$, $l=1,\ldots,d$. Then
$A[p;\mathbf{a}_{j}]v^{l}(x)|_{x=x^{*}}=a_{l,j}(x^{*})$. Thus,
$a_{l,1}(x^{*})=ca_{l,2}(x^{*})$, $l=1,\ldots,d$. Finally, working similarly
with $v^{lm}(x)=v^{*}(x)(x_{l}-x_{l}^{*})(x_{m}-x_{m}^{*})$ we deduce
$a_{lm,1}(x^{*})=ca_{lm,2}(x^{*})$, $l,m=1,\ldots,d$. Since $x^{*}\in\Omega$
is arbitrary we obtain $\mathbf{a}_{1}=c\mathbf{a}_{2}$. $\Box$

Secondly, we consider problems with distributed fractional elliptic operators.
To this end we firstly introduce an elliptic operator $B_{a}$ by the formula

$B_{a}v(x)=\Delta v(x)+a(x)v(x),\quad x\in\Omega,$

where $a\in C(\overline{\Omega})$. It maps the set $D_{2}$ defined by (51) to
$L_{2}(\Omega)$. Assume that $a\leq 0$ and introduce the eigenvalues and
orthonormal in $L_{2}(\Omega)$ eigenfunctions of $-B_{a}$:

$\displaystyle\begin{split}&-\Delta
v_{k}(x)-a(x)v_{k}(x)=\mu_{k}v_{k}(x),\;x\in\Omega,\;v_{k}|_{\partial\Omega}=0,\quad
k\in\mathbb{N},\\\ &\qquad 0<\mu_{1}\leq\mu_{2}\leq\ldots,\;\;\langle
v_{k},v_{l}\rangle=\delta_{kl},\end{split}$ (54)

where $\langle\cdot,\cdot\rangle$ is the scalar product in $L_{2}(\Omega)$.
Next, let us define the distributed fractional powers of $B_{a}$ by the
formula

$\displaystyle
A_{\varrho,a}v:=-\int_{0}^{1}(-B_{a})^{\beta}d\varrho(\beta)v=-\sum_{k=1}^{\infty}\lambda_{k}\langle
v,v_{k}\rangle v_{k},$
$\displaystyle\quad\lambda_{k}=\int_{0}^{1}\mu_{k}^{\beta}d\varrho(\beta),$

where $\varrho$ is a nonnegative Borel measure such that ${\rm
supp}\,\varrho\cap(0,1]\neq\emptyset$.

The domain of $A_{\varrho,a}$ is

$D(A_{\varrho,a})=\left\\{v\in
L_{2}(\Omega)\,:\,\sum_{k=1}^{\infty}\lambda_{k}^{2}|\langle
v,v_{k}\rangle|^{2}<\infty\right\\}.$

Obviously, $\lambda_{k}$ and $v_{k}$ are the eigenvalues and eigenfunctions of
$-A_{\varrho,a}$, i.e.

$-A_{\varrho,a}v_{k}=\lambda_{k}v_{k},\quad k\in\mathbb{N},$

and $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$.

Since ${\rm span}\\{v_{k}\,:\,k\in\mathbb{N}\\}$ is dense in $L_{2}(\Omega)$
and forms a subset of $D(A_{\varrho,a})$, $D(A_{\varrho,a})$ is dense in
$L_{2}(\Omega)$. Further,

$(\lambda I-A_{\varrho,a})v=\sum_{k=1}^{\infty}(\lambda+\lambda_{k})\langle
v,v_{k}\rangle v_{k}.$

This shows that or any $\lambda\in\Sigma(0,\pi)$, the inverse $(\lambda
I-A_{\varrho,a})^{-1}$ exists, belongs to $\mathcal{B}(L_{2}(\Omega))$ and has
the formula

$(\lambda
I-A_{\varrho,a})^{-1}v=\sum_{k=1}^{\infty}{1\over\lambda+\lambda_{k}}\langle
v,v_{k}\rangle v_{k},\;\;\lambda\in\Sigma(0,\pi).$

Since $\rho(A_{\varrho,a})$ is not empty, $A_{\varrho,a}$ is closed. Moreover,
for any $\theta^{\prime}\in(0,\pi)$ and $\lambda\in\Sigma(0,\theta^{\prime})$
we have

$\displaystyle|\lambda+\lambda_{k}|^{2}=|\lambda|^{2}+2|\lambda|\lambda_{k}\cos\arg\lambda+\lambda_{k}^{2}\geq|\lambda|^{2}+2|\lambda|\lambda_{k}\cos\theta^{\prime}+\lambda_{k}^{2}$
$\displaystyle=|\lambda|^{2}\sin^{2}\theta^{\prime}+(|\lambda|\cos\theta^{\prime}+\lambda_{k})^{2}\geq|\lambda|^{2}\sin^{2}\theta^{\prime}.$

Hence

$\|(\lambda
I-A_{\varrho,a})^{-1}v\|=\sqrt{\sum_{k=1}^{\infty}{1\over|\lambda+\lambda_{k}|^{2}}|\langle
v,v_{k}\rangle|^{2}}\leq{1\over|\lambda|\sin\theta^{\prime}}\,\|v\|.$

Consequently, $A_{\varrho,a}\in\mathcal{S}(0,\theta^{\prime})$ for any
$\theta^{\prime}\in(0,{\pi\over 2})$.

###### Corollary 2

Let $M_{1}\in\mathcal{CM}_{\dagger}$, $M_{2}\in\mathcal{CM}$, $a_{j}\in
C(\overline{\Omega})$, $a_{j}\leq 0$, $j=1,2$, and measures $\varrho_{j}$,
$j=1,2$, satisfy either

$\displaystyle\left.\begin{split}&\varrho_{j}=\sum_{l=1}^{n_{j}}\varkappa_{l,j}\delta_{\beta_{l,j}},\;j=1,2,\;\quad\mbox{where}\\\
&n_{j}\in\mathbb{N},\,\;0<\beta_{1,j}<\beta_{2,j}<\ldots<\beta_{n_{j},j}\leq
1,\;\varkappa_{l,j}>0,\\\ &\mbox{$\delta_{\beta_{l,j}}$ - the point measure
concentrated at $\beta_{l,j}$}\end{split}\right\\}$ (55)

or

$\displaystyle\left.\begin{split}&\mbox{$\varrho_{j}$ - absolutely
continuous},\;j=1,2,\\\
&\varrho_{j}(U)=\int_{U}\varkappa_{j}(\beta)d\beta,\;\;\mbox{$U$ - any Borel
subset of $(0,1)$},\;\\\ &\mbox{there exists $\delta>0$ such that
$\varkappa_{j}$ is analytic in $(0,1+\delta)$}.\end{split}\right\\}$ (56)

Let $f_{j}$, $u_{j}$, $j=1,2$, satisfy $f_{j}\in L_{1}((0,T);L_{2}(\Omega))$,
$u_{j}\in C([0,T];L_{2}(\Omega))$, $M_{j}*u_{j}\in
C([0,T];X_{A_{\varrho_{j},a_{j}}})$,
$u_{j}-A_{\varrho_{j},a_{j}}M_{j}*u_{j}\in W_{1}^{1}((0,T);L_{2}(\Omega))$ and
solve the equations

$\displaystyle{\partial\over\partial
t}\left[u_{j}(t,x)-A_{\varrho_{j},a_{j}}M_{j}*u_{j}(t,x)\right]=f_{j}(t,x),\;a.e.\;(t,x)\in(0,T)\times\Omega.$
(57)

Moreover, assume that $f_{1}(t,\cdot)=f_{2}(t,\cdot)=:f(t,\cdot)$ a.e.
$t\in(t_{0},T)$, for some $t_{0}\in(0,T)$, $f(t,\cdot)=0$, a.e.
$t\in(t_{0},t_{1})$ for some $t_{1}\in(t_{0},T)$ and

$\displaystyle\left.\begin{split}&f(t,x)=\sum_{i=1}^{\infty}\psi_{i}(t)w_{i}(x),\;t\in(t_{1},T),\;\psi_{i}\in
L_{1}(t_{1},T),\;\\\ &{\rm
ess\,supp}\,\psi_{i}\subset[t_{i},t_{i+1}),\;t_{1}<t_{2}<\ldots<T,\;\\\
&t_{i}=\min{\rm ess\,supp}\,\psi_{i},\;\;{\rm
span}\\{w_{i}\,:\,i\in\mathbb{N}\\}-\mbox{dense in
$L_{2}(\Omega)$}.\end{split}\right\\}$ (58)

Finally, let $u_{1}(t,\cdot)=u_{2}(t,\cdot)$, $t\in(t_{0},T)$. Then there
exists $c>0$ such that $cM_{1}=M_{2}$, $\varrho_{1}=c\varrho_{2}$,
$a_{1}=a_{2}$, $f_{1}=f_{2}$ and $u_{1}=u_{2}$.

Proof. To apply Theorem 1 we have to verify that $D(A_{\varrho_{1},a_{1}})\cap
D(A_{\varrho_{2},a_{2}})$ is not empty and dense in
$X_{A_{\varrho_{j},a_{j}}}$, $j=1,2$. Other assumptions of this theorem are
evident. Let $0<\mu_{1,j}\leq\mu_{2,j}\leq\ldots$ be the Dirichlet eigenvalues
of $-\Delta-a_{j}I$ in $\Omega$ and $v_{1,j}$, $v_{2,j},\ldots$ the
corresponding orthonormal in $L_{2}(\Omega)$ eigenfunctions. Let us denote
$\lambda_{k,j}=\int_{0}^{1}\mu_{k,j}^{\beta}d\varrho_{j}(\beta)$. Then for any
$v\in D_{2}$, where $D_{2}$ is given by (51), we have

$\displaystyle\|v\|_{X_{A_{\varrho_{j},a_{j}}}}^{2}=\sum_{k=1}^{\infty}\lambda_{k,j}^{2}|\langle
v,v_{k,j}\rangle|^{2}=\sum_{k=1}^{\infty}\left(\int_{0}^{1}\mu_{1,j}^{\beta}\left[{\mu_{k,j}\over\mu_{1,j}}\right]^{\beta}d\varrho_{j}(\beta)\right)^{2}$
$\displaystyle\times|\langle
v,v_{k,j}\rangle|^{2}\leq\sum_{k=1}^{\infty}\left(\int_{0}^{1}\mu_{1,j}^{\beta}d\varrho_{j}(\beta){\mu_{k,j}\over\mu_{1,j}}\right)^{2}|\langle
v,v_{k,j}\rangle|^{2}$
$\displaystyle=\mu_{1,j}^{-2}\left(\int_{0}^{1}\mu_{1,j}^{\beta}d\varrho_{j}(\beta)\right)^{2}\,\sum_{k=1}^{\infty}\mu_{k,j}^{2}|\langle
v,v_{k,j}\rangle|^{2},\;j=1,2.$

The quantity $\sqrt{\sum_{k=1}^{\infty}\mu_{k,j}^{2}|\langle
v,v_{k,j}\rangle|^{2}}$ is a norm of $v$ in $D_{2}$. Therefore,
$\|v\|_{X_{A_{\varrho_{j},a_{j}}}}<\infty$ $j=1,2$. This proves $D_{2}\subset
D(A_{\varrho_{1},a_{1}})\cap D(A_{\varrho_{2},a_{2}})$. In particular, ${\rm
span}\,\\{v_{k,j},:\,k\in\mathbb{N}\\}\subset D_{2}$, $j=1,2$. Next let us
choose some $j\in\\{1;2\\}$ and $v\in D(A_{\varrho_{j},a_{j}})$. Then
$\tilde{v}_{n}:=\sum_{k=1}^{n}\lambda_{k,j}\langle v,v_{k,j}\rangle v_{k,j}\in
D_{2}$. It holds $\|v-\tilde{v}_{n}\|_{X_{A_{\varrho_{j},a_{j}}}}\to 0$ as
$n\to\infty$. This proves that $D_{2}$ is dense in
$X_{A_{\varrho_{j},a_{j}}}$. Thus, $D(A_{\varrho_{1},a_{1}})\cap
D(A_{\varrho_{2},a_{2}})$ is dense in $X_{A_{\varrho_{j},a_{j}}}$, $j=1,2$.

From Theorem 1 we obtain that $f_{1}=f_{2}$, $u_{1}=u_{2}$,
$D(A_{\varrho_{1},a_{1}})=D(A_{\varrho_{2},a_{2}})$ and there exists $c>0$
such that $cM_{1}=M_{2}$, $A_{\varrho_{1},a_{1}}=cA_{\varrho_{2},a_{2}}$.
Since $(\lambda_{k,j},v_{k,j})|_{k\in\mathbb{N}}$, are the pairs of
eigenvalues and eigenfunctions of $-A_{\varrho_{j},a_{j}}$, from the latter
relation we have $\lambda_{k,1}=c\lambda_{k,2}$, $v_{k,1}=v_{k,2}$,
$k\in\mathbb{N}$. Therefore, denoting $v_{k}:=v_{k,1}=v_{k,2}$, we have

$\displaystyle-\Delta v_{k}-a_{j}v_{k}=\mu_{k,j}v_{k},\quad
k\in\mathbb{N},\;j=1,2.$ (59)

Setting here $k=1$, subtracting the equations for $j=1$ and $j=2$ and dividing
by $v_{1}(x)$ ($v_{1}$ preserves the sign [33], p. 214), we deduce

$\displaystyle
a_{1}(x)=a_{2}(x)-\eta,\;\;x\in\Omega,\quad\eta:=\mu_{1,1}-\mu_{1,2}.$ (60)

Returning to (59) and using (60) we obtain

$\displaystyle\mu_{k,1}=\mu_{k,2}+\eta,\;\;k\in\mathbb{N}.$ (61)

Let us proceed to the relation $\lambda_{k,1}=c\lambda_{k,2}$. This by (61)
implies

$\displaystyle\int_{0}^{1}(\mu_{k}+\eta)^{\beta}d\varrho_{1}(\beta)=c\int_{0}^{1}\mu_{k}^{\beta}d\varrho_{2}(\beta),\;\;k\in\mathbb{N},$
(62)

where $\mu_{k}=\mu_{k,2}$. We will analyze this equation separately in cases
(55) and (56).

In case (55) we have

$\displaystyle\sum_{l=1}^{n_{1}}\varkappa_{l,1}(\mu_{k}+\eta)^{\beta_{l,1}}=c\sum_{l=1}^{n_{2}}\varkappa_{l,2}\mu_{k}^{\beta_{l,2}},\;\;k\in\mathbb{N}.$
(63)

Let us rewrite this in the following form:

$\displaystyle\underbrace{\sum_{l=1}^{n_{1}}\varkappa_{l,1}\mu_{k}^{\beta_{l,1}}}_{W_{k,1}}\,-\,\underbrace{c\sum_{l=1}^{n_{2}}\varkappa_{l,2}\mu_{k}^{\beta_{l,2}}}_{W_{k,2}}=R_{k},\quad
k\in\mathbb{N},$ (64) $\displaystyle
R_{k}=\sum_{l=1}^{n_{1}}\varkappa_{l,1}\left[\mu_{k}^{\beta_{l,1}}-(\mu_{k}+\eta)^{\beta_{l,1}}\right].\;\;$

Since $\mu_{k}\to\infty$ as $k\to\infty$ and
$0<\beta_{1,j}<\beta_{2,j}<\ldots<\beta_{n_{j},j}$, $j=1,2$, we have
$W_{k,1}\sim\varkappa_{n_{1},1}\mu_{k}^{\beta_{n_{1},1}}\to\infty$ and
$W_{k,2}\sim c\varkappa_{n_{2},2}\mu_{k}^{\beta_{n_{2},2}}\to\infty$ as
$k\to\infty$. On the other hand, the sequence $R_{k}$ is bounded, because

$\left|\mu_{k}^{\beta_{l,1}}-(\mu_{k}+\eta)^{\beta_{l,1}}\right|=\left|\int^{\mu_{k}}_{\mu_{k}+\eta}\beta_{l,1}\tau^{\beta_{l,1}-1}d\tau\right|\leq\beta_{l,1}|\eta|$

for $k$ such that $\mu_{k}\geq 1$ and $\mu_{k}+\eta\geq 1$. The difference
$W_{k,1}-W_{k,2}$ can be bounded in the process $k\to\infty$ only if
$\beta_{n_{1},1}=\beta_{n_{2},2}$ and
$\varkappa_{n_{1},1}=c\varkappa_{n_{2},2}$. Thus, from (64) we obtain

$\displaystyle\underbrace{\sum_{l=1}^{n_{1}-1}\varkappa_{l,1}\mu_{k}^{\beta_{l,1}}}_{\tilde{W}_{k,1}}\,-\,\underbrace{c\sum_{l=1}^{n_{2}-1}\varkappa_{l,2}\mu_{k}^{\beta_{l,2}}}_{\tilde{W}_{k,2}}=R_{k},\quad
k\in\mathbb{N}.$

Now
$\tilde{W}_{k,1}\sim\varkappa_{n_{1}-1,1}\mu_{k}^{\beta_{n_{1}-1,1}}\to\infty$
and $\tilde{W}_{k,2}\sim
c\varkappa_{n_{2}-1,2}\mu_{k}^{\beta_{n_{2}-1,2}}\to\infty$ as $k\to\infty$.
Using again the fact that $R_{k}$ is bounded we deduce
$\beta_{n_{1}-1,1}=\beta_{n_{2}-1,2}$ and
$\varkappa_{n_{1}-1,1}=c\varkappa_{n_{2}-1,2}$. Continuing such arguments we
prove that

$\displaystyle
n_{1}=n_{2}=:n,\;\;\varkappa_{l,1}=c\varkappa_{l,2},\;\beta_{l,1}=\beta_{l,2},\;l=1,\ldots
n.$ (65)

This by (55) implies $\varrho_{1}=c\varrho_{2}$. From (63) by means of (65) we
have

$\displaystyle\sum_{l=1}^{n}\varkappa_{l,1}(\mu_{1}+\eta)^{\beta_{l,1}}=\sum_{l=1}^{n}\varkappa_{l,1}\mu_{1}^{\beta_{l,1}}.$
(66)

The function
$\Psi(\xi)=\sum_{l=1}^{n}\varkappa_{l,1}(\mu_{1}+\xi)^{\beta_{l,1}}$ is
strictly increasing in $(-\mu_{1},\infty)$. Therefore, (66) implies $\eta=0$.
From (60) we obtain $a_{1}=a_{2}$.

Secondly, we consider the case (56). Then from (62) we have

$\displaystyle\int_{0}^{1}\varkappa_{1}(\beta)(\mu_{k}+\eta)^{\beta}d\beta=c\int_{0}^{1}\varkappa_{2}(\beta)\mu_{k}^{\beta}d\beta,\;\;k\in\mathbb{N}.$
(67)

This yields

$\displaystyle\int_{0}^{1}[\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)]\mu_{k}^{\beta}d\beta=\overline{R}_{k},\;\;k\in\mathbb{N},$
(68)
$\displaystyle\overline{R}_{k}=\int_{0}^{1}\varkappa_{1}(\beta)[\mu_{k}^{\beta}-(\mu_{k}+\eta)^{\beta}]d\beta.$

Like in the previous case, we can show that the sequence $\overline{R}_{k}$ is
bounded. Let us choose an arbitrary $\beta_{*}\in(0,1)$ and suppose that
$\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)$ differs from zero and preserves
the sign in the interval $(\beta_{*},1)$. Then there exists
$\beta_{**}\in(\beta_{*},1)$ and $\epsilon\in(0,\beta_{**}-\beta_{*})$ such
that $|\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)|>\epsilon$,
$|\beta-\beta_{**}|<\epsilon$. Thus for $k$ such that $\mu_{k}\geq 1$ we
obtain

$\displaystyle\left|\int_{0}^{1}[\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)]\mu_{k}^{\beta}d\beta\right|\geq\left|\int_{\beta_{*}}^{1}[\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)]\mu_{k}^{\beta}d\beta\right|$
$\displaystyle-\left|\int_{0}^{\beta_{*}}[\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)]\mu_{k}^{\beta}d\beta\right|\geq\int_{\beta_{**}-\epsilon}^{\beta_{**}+\epsilon}|\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)|\mu_{k}^{\beta}d\beta$
$\displaystyle-\left|\int_{0}^{\beta_{*}}[\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)]\mu_{k}^{\beta}d\beta\right|\geq
2\epsilon^{2}\mu_{k}^{\beta_{**}-\epsilon}-C_{*}\mu_{k}^{\beta_{*}},$

where
$C_{*}=\int_{0}^{\beta_{*}}|\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)|d\beta$.
Since $\beta_{**}-\epsilon>\beta_{*}$, we have
$\left|\int_{0}^{1}[\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)]\mu_{k}^{\beta}d\beta\right|\to\infty$
as $k\to\infty$. This is in the contradiction with the fact that the right-
hand side of (68) is bounded. Therefore,
$\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)$ cannot preserve the sign in the
interval $(\beta_{*},1)$. Since $\beta_{*}\in(0,1)$ is arbitrary, the zeros of
$\varkappa_{1}(\beta)-c\varkappa_{2}(\beta)$ have the accumulation point
$\beta=1$. The functions $\varkappa_{j}(\beta)$ are analytic in
$(0,1+\delta)$, therefore we obtain
$\varkappa_{1}(\beta)=c\varkappa_{2}(\beta)$, $\beta\in(0,1+\delta)$. This
implies $\varrho_{1}=c\varrho_{2}$. From (67) we deduce

$\int_{0}^{1}\varkappa_{1}(\beta)(\mu_{1}+\eta)^{\beta}d\beta=\int_{0}^{1}\varkappa_{1}(\beta)\mu_{1}^{\beta}d\beta.$

The function
$\Phi(\xi)=\int_{0}^{1}\varkappa_{1}(\beta)(\mu_{1}+\xi)^{\beta}d\beta$ is
strictly increasing. Therefore, $\eta=0$ and from (60) we obtain
$a_{1}=a_{2}$. $\Box$

If a priori $a_{1}=a_{2}$ then the eigenfunctions of $A_{\varrho_{1},a_{1}}$
and $A_{\varrho_{2},a_{2}}$ coincide and we can formulate a uniqueness result
also in case $f$ satisfies (33). Let us formulate such a statement without a
proof.

###### Corollary 3

Let $M_{1}\in\mathcal{CM}_{\dagger}$, $M_{2}\in\mathcal{CM}$, $a\leq 0$, and
either (55) or (56) hold. Let $f_{j}$, $u_{j}$, $j=1,2$, satisfy $f_{j}\in
L_{1}((0,T);L_{2}(\Omega))$, $u_{j}\in C([0,T];L_{2}(\Omega))$,
$M_{j}*u_{j}\in C([0,T];X_{A_{\varrho_{j},a}})$,
$u_{j}-A_{\varrho_{j},a}M_{j}*u_{j}\in W_{1}^{1}((0,T);L_{2}(\Omega))$ and
solve

$\displaystyle{\partial\over\partial
t}\left[u_{j}(t,x)-A_{\varrho_{j},a}M_{j}*u_{j}(t,x)\right]=f_{j}(t,x),\;a.e.\;(t,x)\in(0,T)\times\Omega.$
(69)

Moreover, assume that $f_{1}(t,\cdot)=f_{2}(t,\cdot)=:f(t,\cdot)$ a.e.
$t\in(t_{0},T)$, for some $t_{0}\in(0,T)$, $f(t,\cdot)=0$, a.e.
$t\in(t_{0},t_{1})$ for some $t_{1}\in(t_{0},T)$ and $f$ satisfies either (58)
or

$\displaystyle\begin{split}&\mbox{$\langle f(t),v_{k}\rangle\not\equiv 0$ in
$(t_{1},T)$, $k\in\mathbb{N}$, where $v_{k}$ are the Dirichlet}\\\
&\mbox{eigenfunctions of $\Delta+aI$ (i.e. solutions of
\eqref{bqeigen}).}\end{split}$

Finally, let $u_{1}(t,\cdot)=u_{2}(t,\cdot)$, $t\in(t_{0},T)$. Then there
exists $c>0$ such that $cM_{1}=M_{2}$, $\varrho_{1}=c\varrho_{2}$,
$f_{1}=f_{2}$ and $u_{1}=u_{2}$.

##  5 Inverse problems to determine the kernel

###  5.1 Inverse problem to determine the kernel, source and state

Now we consider a problem to identify $M$, $u$ and $f$ using prescribed values
of $u$ and $f$ in the interval $(t_{0},T)$ in case $A$ is known. Then we are
able prove the uniqueness in case of less restrictions on $f$.

###### Theorem 2

Let $M_{1}\in\mathcal{CM}_{\dagger}$, $M_{2}\in\mathcal{CM}$,
$A\,:\,D(A)\mapsto X$\- linear closed densely defined injective operator in a
complex Banach space $X$, $A\in\mathcal{S}(\omega,\theta)$ for some
$\omega\in\mathbb{R}$, $\theta\in(0,{\pi\over 2}]$. Let $f_{j}$, $u_{j}$,
$j=1,2$, satisfy the assumptions (27) - (30) with $A_{1}=A_{2}=A$ and $f$
involved in (30) fulfill the condition (31). Assume that $f(t)\not\equiv 0$ in
$(t_{1},T)$. Finally, let (34) hold. Then $M_{1}=M_{2}$, $f_{1}=f_{2}$ and
$u_{1}=u_{2}$.

Proof. Applying Lemma 7 in case $A_{1}=A_{2}=A$ , we
have$\int_{t_{1}}^{t}S_{1}(t-\tau)f(\tau)d\tau=\int_{t_{1}}^{t}S_{2}(t-\tau)f(\tau)d\tau$,
$t\in[t_{1},T]$, where $S_{j}$ is the resolvent of (11) with $M=M_{j}$. Let us
extend $f(t)$ by $0$ for $t<t_{0}$. Then we obtain $w_{1}(t)=w_{2}(t)$,
$t\in[0,T]$, where $w_{j}(t)=S_{j}*f(t)$. Due to Lemma 1 (ii), the function
$w_{j}$ satisfies the equation $w_{j}(t)-AM_{j}*w_{j}(t)=1*f(t)$, $t\in[0,T]$.
Therefore, denoting $w:=w_{1}=w_{2}$, we obtain $A(M_{1}-M_{2})*w(t)=0$,
$t\in[0,T]$. Since $A$ is injective, we have

$\displaystyle(M_{1}-M_{2})*w(t)=0,\;\;t\in[0,T].$ (70)

The function $w(t)$ is not identically zero, because otherwise the left-hand
side of the equation $w(t)-AM_{j}*w(t)=1*f(t)$ vanishes and this contradicts
the assumption $f\not\equiv 0$. Therefore, there exists $\phi\in X^{*}$ such
that $\langle\phi,w(t)\rangle\not\equiv 0$. Denote $t_{w}=\min\,{\rm
ess\,supp}\langle\phi,w\rangle$. Then from (70) we deduce

$\displaystyle\int_{t_{w}}^{t}\langle\phi,w(\tau)\rangle(M_{1}(t-\tau)-M_{2}(t-\tau))d\tau=0,\;\;t\in[t_{w},T].$

The function $\langle\phi,w(t)\rangle$ does not vanish identically in an
arbitrarily small right neighborhood of $t_{w}$. Titchmarsh convolution
theorem implies $M_{1}(t)=M_{2}(t)$, $t\in(0,T-t_{w}]$ and by analytic
continuation we obtain $M_{1}=M_{2}$.

The assertions $f_{1}=f_{2}$ and $u_{1}=u_{2}$ can be deduced by means of
Lemma 9 like in the proof of Theorem 1. $\Box$

Theorem 2 applies to inverse problems for generalized fractional diffusion
equations with operators defined in Subsection 4.2. Let $\Omega$ meet the
requirements (50) and either

$\displaystyle\hskip-14.22636ptX=L_{p}(\Omega)\;\mbox{for some
$p\in(1,\infty)$},\;\;{\mathbf{a}}\in\mathcal{Q},\;\;A=A[p;\mathbf{a}],\;0\not\in\sigma(A[p;\mathbf{a}])$

or

$\displaystyle\hskip-14.22636ptX=L_{2}(\Omega),\;\mbox{$\varrho$ - a
nonnegative Borel measure},\;$ $\displaystyle a\in
C(\overline{\Omega}),\;a\leq 0,\;A=A_{\varrho,a}.$

In both cases $A$ satisfies the assumptions of Theorem 2.

###  5.2 Inverse problem to determine the kernel in case of unknown history

In this section we consider the case when a functional of $u(t)$ is given
after $t=t_{0}$. This information may not suffice to reconstruct the history
of $f$ and $u$ before $t=t_{0}$ but it still enables to recover the kernel
$M$.

Firstly let us introduce the square of $A$ and formulate a lemma. Let
$A\,:\,D(A)\mapsto X$ \- linear closed densely defined operator in a complex
Banach space $X$, $A\in\mathcal{S}(\omega,\theta)$ for some
$\omega\in\mathbb{R}$, $\theta\in(0,\pi/2]$. From the fact that $\rho(A)$ is
not empty, we can deduce that $D(A^{2}):=\\{{\rm x}\in D(A)\,:\,A{\rm x}\in
D(A)\\}\neq\emptyset$. Let us denote by $X_{A^{2}}$ the set $D(A^{2})$ endowed
with the norm

$\displaystyle\|{\rm x}\|_{X_{A^{2}}}=\|{\rm x}\|+\|A{\rm x}\|+\|A^{2}{\rm
x}\|.$ (71)

The closedness of $A$ implies that $X_{A^{2}}$ is a Banach space with respect
to the norm (71).

###### Lemma 10

Let $M_{j}\in\mathcal{CM}$, $j=1,2$, $A\,:\,D(A)\mapsto X$ \- linear closed
densely defined operator in a complex Banach space $X$,
$A\in\mathcal{S}(0,\theta)$ for some $\theta\in(0,{\pi\over 2}]$, $g\in
W^{1,1}((0,T);X_{A^{2}})$ and $w_{j}$, $j=1,2$, satisfy $w_{j}\in
C([0,T];X_{A^{2}})$ and solve

$w_{j}(t)-M_{j}*Aw_{j}(t)=g(t),\;\;t\in[0,T].$

Moreover, let $\Phi\in X^{*}$, $\langle\Phi,Ag(0)\rangle\neq 0$ and
$\langle\Phi,w_{1}(t)\rangle=\langle\Phi,w_{2}(t)\rangle$, $t\in(0,T)$. Then
$M_{1}=M_{2}$.

Proof. In Theorem 5.1 of [34], the equality $M_{1}(t)=M_{2}(t)$, $t\in(0,T)$,
was proved in case $A$, $w_{j}$, $g$ satisfy the assumptions of Lemma 10 but
$M_{j}$, $j=1,2$, belong to a class of kernels that is more general than
$\mathcal{CM}$. Since we assume $M_{j}\in\mathcal{CM}$, the relation
$M_{1}(t)=M_{2}(t)$, $t>0$, follows by analytic continuation. $\Box$

###### Theorem 3

Let $M_{j}\in\mathcal{CM}$, $j=1,2$, $A\,:\,D(A)\mapsto X$ \- linear closed
densely defined operator in a complex Banach space $X$,
$A\in\mathcal{S}(0,\theta)$ for some $\theta\in(0,{\pi\over 2}]$ and $\Phi\in
X^{*}$. Let $f_{j}$, $u_{j}$, $j=1,2$, satisfy the assumptions (27) - (30)
with $A_{1}=A_{2}=A$ and $f$ involved in (30) fulfill the condition (31).
Moreover, assume that

$\displaystyle\begin{split}&\exists
m\in\\{0\\}\cup\mathbb{N}\,:\,f|_{(t_{1},T)}\in
W_{1}^{m+1}((t_{1},T);X_{A^{2}}),\\\
&\langle\Phi,f^{(j)}(t_{1}^{+})\rangle=0,\,j=0,\ldots,m-1\;\;\mbox{(if $m\geq
1$),}\;\;\langle\Phi,Af^{(m)}(t_{1}^{+})\rangle\neq 0.\end{split}$ (72)

Finally, let $\langle\Phi,u_{1}(t)\rangle=\langle\Phi,u_{2}(t)\rangle$,
$t\in(t_{0},T)$. Then $M_{1}=M_{2}$.

Proof. We have

$\displaystyle u_{j}(t)=S_{j}(t)u_{j}(0)+S_{j}*f_{j}(t),\quad
t\in[0,T],\;\;j=1,2,$ (73)

where $S_{j}$ is the resolvent of the equation (11) with $M=M_{j}$. Hence,

$\displaystyle\langle\Phi,u_{j}(t)\rangle=\langle\Phi,S_{j}(t)u_{j}(0)\rangle+\langle\Phi,S_{j}*f_{j}(t)\rangle,\quad
t\in[0,T],\;\;j=1,2.$

Let $t\in[t_{0},T]$. Subtracting this relation with $j=2$ from the one with
$j=1$ and using the assumptions
$\langle\Phi,u_{1}(t)\rangle=\langle\Phi,u_{2}(t)\rangle$, $t\in(t_{0},T)$,
and $f_{1}(t)=f_{2}(t)=f(t)$, $t\in(t_{0},T)$, we deduce

$\displaystyle\tilde{Y}(t)+\tilde{Z}(t)=0,\quad t\in[t_{0},T],$

where

$\displaystyle\tilde{Y}(t)=\Big{\langle}\Phi,S_{1}(t)u_{1}(0)-S_{2}(t)u_{2}(0)$
$\displaystyle\qquad+\int_{0}^{t_{0}}S_{1}(t-\tau)f_{1}(\tau)d\tau-\int_{0}^{t_{0}}S_{2}(t-\tau)f_{2}(\tau)d\tau\Big{\rangle},$
$\displaystyle\tilde{Z}(t)=\Big{\langle}\Phi,\int_{t_{0}}^{t}(S_{1}(t-\tau)-S_{2}(t-\tau))f(\tau)d\tau\Big{\rangle}.$

The function $\tilde{Y}(t)$ is analytic in $(t_{0},T)$ and $\tilde{Z}(t)$
vanishes in $(t_{0},t_{1})$ because of the assumption $f(t)=0$,
$t\in(t_{0},t_{1})$. Arguing as in the proof of Lemma 7, we obtain
$\tilde{Z}(t)=0$, $t\in[t_{0},T]$. Thus in view of $f|_{(t_{0},t_{1})}=0$ we
have

$\displaystyle\Big{\langle}\Phi,\int_{0}^{t}S_{1}(\tau)f(t_{1}+t-\tau)d\tau\Big{\rangle}=\Big{\langle}\Phi,\int_{0}^{t}S_{2}(\tau)f(t_{1}+t-\tau)d\tau\Big{\rangle},\;\;$
$\displaystyle\qquad t\in[0,T-t_{1}].$

Differentiating this relation $m$ times and observing the assumptions
$\langle\Phi,f^{(j)}(t_{1}^{+})\rangle=0$, $j=0,\ldots,m-1$, we obtain
$\langle\Phi,w_{1}(t)\rangle=\langle\Phi,w_{2}(t)\rangle$, $t\in[0,T-t_{1}]$,
where

$w_{j}(t)=S_{j}(t)g(0)+\int_{0}^{t}S_{j}(t-\tau)g^{\prime}(\tau)d\tau$

and $g(t)=f^{(m)}(t_{1}+t)$. By $f|_{(t_{1},T)}\in
W_{1}^{m+1}((t_{1},T);X_{A^{2}})$ we have $g\in
W_{1}^{1}((0,T-t_{1}),X_{A^{2}})$. This implies that $w_{j}$ belongs to
$C([0,T-t_{1}],X_{A^{2}})$. Due to Lemma 1 (iii), $w_{j}$ solves
$w_{j}(t)-M_{j}*Aw_{j}(t)=g(t)$, $t\in[0,T-t_{1}]$. Moreover, by
$\langle\Phi,Af^{(m)}(t_{1}^{+})\rangle\neq 0$ we have
$\langle\Phi,Ag(0)\rangle\neq 0$. Assumptions of Lemma 10 are satisfied. We
obtain $M_{1}=M_{2}$. $\Box$

If $X$ is a space of functions defined in $\Omega$, then the functional $\Phi$
may represent for instance an integral observation of the form
$\langle\Phi,v\rangle=\int_{\Omega^{\prime}}v(x)dx$,
$\Omega^{\prime}\subseteq\Omega$ (then we can work in $L_{p}$ spaces as in the
previous examples) or a point observation $\langle\Phi,v\rangle=v(x_{0})$,
$x_{0}\in\overline{\Omega}$ (then we have to work in a space of continuous
functions). Let us consider the case of point observation and involve also
Neumann boundary conditions. The latter option enables to observe on the
boundary, too.

Assume that $\Omega$ meets the requirements (50) and either

$\displaystyle X=C_{0}(\overline{\Omega})=\\{v\in
C(\overline{\Omega})\,:\,v|_{\partial\Omega}=0\\},\;$ $\displaystyle
D(A)=\Big{\\{}v\in
C_{0}(\overline{\Omega})\,:\,v\in\bigcap\limits_{p>1}W_{p}^{2}(\Omega),\,\Delta
v\in C_{0}(\overline{\Omega})\Big{\\}},\quad$ $\displaystyle Av=\Delta
v,\,v\in D(A),\quad\langle\Phi,v\rangle=v(x_{0}),\,x_{0}\in\Omega$

or

$\displaystyle X=C(\overline{\Omega}),$ $\displaystyle D(A)=\Big{\\{}v\in
C(\overline{\Omega})\,:\,v\in\bigcap\limits_{p>1}W_{p}^{2}(\Omega),\,{\partial\over\partial\nu}v|_{\partial\Omega}=0,\,\Delta
v\in C(\overline{\Omega})\Big{\\}},$ $\displaystyle\mbox{$\nu$ - the outer
normal of $\partial\Omega$},$ $\displaystyle Av=\Delta v,\,v\in
D(A),\quad\langle\Phi,v\rangle=v(x_{0}),\,x_{0}\in\overline{\Omega}.$

In both these cases $A$ and $\Phi$ satisfy the assumptions of Theorem 3 (for
$A$, see e.g. [31], p. 139).

##  6 Concluding remarks

The assumption $f|_{(t_{0},t_{1})}=0$ is essential in the proof of Lemma 7. If
$f$ does not vanish identically in some right neighborhood of $t_{0}$, then we
cannot exclude $Y(t)$ from (37) and derive an independent equation for $M_{j}$
and $A_{j}$.

Let us give an interpretation for (35). To this end define the functions
$\tilde{u}_{j}\,:\,(0,T-t_{1})\mapsto X$ by
$\tilde{u}_{j}(t-t_{1})=\int_{t_{1}}^{t}S_{j}(t-\tau)f(\tau)d\tau$,
$t\in(t_{1},T)$. They satisfy the following equations:

${d\over
dt}\left[\tilde{u}_{j}(t)-A_{j}M_{j}*\tilde{u}_{j}(t)\right]=f(t+t_{1}),\;\;a.e.\;\;t\in(0,T-t_{1}).$

Then (35) implies that the solutions of these equations concide. In other
words, in Lemma 8 we proved the uniqueness for an inverse problem to recover
$MA$ in the equation ${d\over
dt}\left[\tilde{u}(t)-AM*\tilde{u}(t)\right]=f(t_{1}+t)$, $t\in(0,T-t_{1})$,
by means of fully given $\tilde{u}$.

Theorem 1 was proved in two stages: firstly we proved $M_{1}A_{1}=M_{2}A_{2}$
and thereupon $f_{1}=f_{2}$ and $u_{1}=u_{2}$. Similar decomposition of an
inverse problem into two stages was used in Theorem 2, too. Firstly, the
uniqueness for $M$ and then the uniqueness for source and state was
established there.

The function $f$ satisfying conditions of Theorems 1, 2 is not analytic in
$(t_{0},T)$. A singularity occurs in $[t_{1},T)$. This kind of irregularity
characterizes our method.

Theorem 3 also uses a decomposition of the inverse problem into two stages,
but the uniqueness was shown only in the first stage, i.e. for $M$. The
history of $f$ and $u$ remains unknown. We believe that a similar
incorporation of unknown history may be possible also in other inverse
problems to recover parameters of fractional diffusion equations, e.g.
problems to determine coefficients by means of boundary data.

Acknowledgement.
The study was supported by Estonian Research Council Grant PRG832.

## References

  * [1] B. Hofmann, U. Tautenhahn, On ill-posedness measures and space change in Sobolev scales. Zeitschrift für Analysis un ihre Anwendungen 16 (1997), 979 – 1000.
  * [2] J. Elschner, G. Hu, Corners and edges always scatter. Inverse Problems 31 (2015), 015003.
  * [3] E. Blåsten, L. Päivärinta, J. Sylvester, Corners Always Scatter. Communications in Mathematical Physics 331 (2014), 725 – 753.
  * [4] B. Jin, W. Rundell, A tutorial on inverse problems for anomalous diffusion processes. Inverse Problems 31 (2015), 035003.
  * [5] J. Janno, N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements. Inverse Problems 34 (2018), 025007.
  * [6] K. Liao, T. Wei, Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously. Inverse Problems 35 (2019), 115002.
  * [7] B. Jin, Y. Kian, Z. Zhou, Inverse problems for subdiffusion from observation at an unknown terminal time. SIAM Journal on Applied Mathematics 83 (2023), 1496 – 1517.
  * [8] N. Kinash, J. Janno, An inverse problem for a generalized fractional derivative with an application in reconstruction of time- and space-dependent sources in fractional diffusion and wave equations. Mathematics 7 (2019): 1138.
  * [9] M. I. Ismailov, M. Çiçek, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Applied Mathematical Modelling 40 (2016), 4891 – 4899.
  * [10] H. Lopushanska, A. Lopushansky, O. Myaus, Inverse problems of periodic spatial distributions for a time fractional diffusion equation. Electronic Journal of Differential Equations 2016 (2016), 14.
  * [11] K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. Journal of Mathematical Analysis and Applications 382 (2011), 426 – 447.
  * [12] M. Slodička, K. Šišková, An inverse source problem in a semilinear time-fractional diffusion equation. Computers and Mathematics with Applications 72 (2016), 1655 – 1669\.
  * [13] Y. Kian, M. Yamamoto, Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations. Inverse Problems 35 (2019), 115006.
  * [14] Y. Kian, Y. Liu, M. Yamamoto, Uniqueness of inverse source problems for general evolution equations. Communications in Contemporary Mathematics 25 (2023), 2250009.
  * [15] J. Janno, Y. Kian, Inverse source problem with a posteriori boundary measurement for fractional diffusion equations. Mathematical Methods in the Applied Sciences 46 (2023), 15868 – 15882.
  * [16] M. Yamamoto, Uniqueness for inverse source problems for fractional diffusion-wave equations by data during not acting time. Inverse Problems 39 (2023), 024004.
  * [17] A. V. Chechkin, V. Yu. Gonchar, R. Gorenflo, N. Korabel, I. M. Sokolov, Generalized fractional diffusion equations for accelerating subdiffusion and truncated Levy flights. Physical Review E 78 (2008), 021111.
  * [18] D. Frömberg, Reaction Kinetics under Anomalous Diffusion. PhD thesis, Humboldt-Universität zu Berlin, Berlin, 1981.https://edoc.hu-berlin.de/bitstream/handle/18452/17026/froemberg.pdf
  * [19] H. G. Sun, Y. Zhang, Y. D. Baleanu, W. Chen, W. Y. Q. Chen. A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation 64 (2018), 213 – 231.
  * [20] H. Scher, M. Lax. Stochastic transport in a disordered solid. I. Theory. Physical Review B 7 (1973), 4491 – 4502.
  * [21] Y. Povstenko, Thermoelasticity that uses fractional heat conduction equation. Journal of Mathematical Sciences 162 (2009), 296 – 305.
  * [22] F. Mainardi, A. Mura, G. Pagnini, R. Gorenflo, Time-fractional diffusion of distributed order. Journal of Vibration and Control 14 (2008), 1267 – 1290.
  * [23] I. M. Sokolov, A. V. Chechkin, J. Klafter, Distributed-order fractional kinetics. Acta Physica Polonica B 35 (2004), 1323 – 1341.
  * [24] J. Gajda, M. Magdziarz, Fractional Fokker-Planck equation with tempered $\alpha$\- stable waiting times. Langevin picture and computer simulation. Physical Review E 82 (2010), 011117.
  * [25] X. Wu, W. Deng, E. Barkai, Tempered fractional Feynman-Kac equation: Theory and examples. Physical Review E 93 (2016), 032151.
  * [26] G. Gripenberg, On Volterra equations of the first kind. Integral Equations and Operator Theory 3 (1980), 473 – 488.
  * [27] N. Kinash, J. Janno, Inverse problems for a generalized subdiffusion equation with final overdetermination. Mathematical Modelling and Analysis 24 (2019), 236 – 262.
  * [28] G. A. Mendes, E. K. Lenzi, R. S. Mendes, L. R. da Silva, Anisotropic fractional diffusion equation. Physica A 346 (2005), 271 – 283.
  * [29] M. Caputo, Diffusion with space memory modelled with distributed order space fractional differential equations. Annals of Geophysics 46 (2003), 223 – 234.
  * [30] J. Janno, Determination of time-dependent sources and parameters of nonlocal diffusion and wave equations from final data. Fractional Calculus and Applied Analysis 6 (2020), 1678 – 1701.
  * [31] J. Prüss, Evolutionary Integral Equations and Applications. Birkhäuser, Basel, 1993.
  * [32] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkhäuser, Basel, 1995.
  * [33] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Springer, NY, 1998.
  * [34] J. Janno, K. Kasemets, Identification of a kernel in an evolutionary integral equation occurring in subdiffusion. Journal of Inverse and Ill-posed Problems 25 (2017), 777 – 798\.
  * [35] Y. Kian, Simultaneous determination of different class of parameters for a diffusion equation from a single measurement. Inverse Problems 38 (2022), 075008.
  * [36] T. Helin, M. Lassas, L.Ylinen, Z. Zhang, Inverse problems for heat equation and space–time fractional diffusion equation with one measurement. Journal of Differential Equations 269 (2020), 7498 – 7528.