# A Note on the Regularity of Thermoelastic Plates with Fractional Rotational
Inertial Force

Fredy Maglorio Sobrado Suárez
Department of Mathematics, Federal University of Technological of Paraná,
Brazil

###### Abstract

The present work intends to complement the study of the regularity of the
solutions of the thermoelastic plate with rotacional forces. The rotational
forces involve the spectral fractional Laplacian, with power parameter
$\tau\in[0,1]$ ( $\gamma(-\Delta)^{\tau}u_{tt}$). Previous research regarding
regularity showed that, as for the analyticity of the semigroup
$S(t)=e^{\mathbb{B}t}$ for the Euler-Bernoulli Plate($\tau=0$) model, the
first result was established by Liu and Renardy, [12] in the case of hinged
and clamped boundary conditions, for the case $\tau=1$ (Plate Kirchoff-Love)
Lasiecka and Triggiani showed, that the semigroup is not differentiable [6,
10] and more recently in 2020 Tebou et al.[5] showed that for
$\tau\in(0,\frac{1}{2})$, $S(t)$ is of class Gevrey
$s>\frac{2-\tau}{2-4\tau}$. Our main contribution here is to show that $S(t)$
is of Gevrey class $s>\frac{3-\tau}{2-2\tau}$ when the parameter $\tau$ lies
in the interval $[\frac{1}{2},1)$ and also show that $S(t)$ is not analytic
for $\tau\in(0,1]$ both results for Hinged plate/ Dirichlet temperature
boundary conditions.

††Email address<EMAIL_ADDRESS>(Fredy Maglorio Sobrado Suárez)

.

keyword: Euler-Bernoulli Plate, Plate Kirchoff-Love, Gevrey Class, Fractional
Rotational Inertial force, Analiticity, Thermoelastic Plates.

##  1 Introduction

Consider that $\Omega$ is a bounded open subset of $\mathbb{R}^{n}$, $n\geq
1$, with sufficiently smooth boundary. The system we study is given by the
following coupled plate equations:

$\displaystyle
u_{tt}+\gamma(-\Delta)^{\tau}u_{tt}+\Delta^{2}u+\alpha\Delta\theta=0,\quad$
$\displaystyle x\in\Omega,$ $\displaystyle t>0,$ (1)
$\displaystyle\theta_{t}-\kappa\Delta\theta-\beta\Delta u_{t}=0,\quad$
$\displaystyle x\in\Omega,$ $\displaystyle t>0,$ (2)

satisfying the boundary conditions

$u=\Delta u=0,\quad\theta=0,\quad x\in\partial\Omega,\ t>0,$ (3)

and prescribed initial data

$\displaystyle u(x,0)=u_{0}(x),\ u_{t}(x,0)=u_{1}(x),\
\theta(x,0)=\theta_{0}(x),\quad x\in\Omega.$ (4)

Here $u$, denote the transversal displacements of the plates. The coefficients
and the inertia rotational is given by $\gamma$ positive number. The exponent
$\tau$ lies in the interval $[0,1]$, the positive numbers $\alpha$ and $\beta$
are the coupling coefficient.

The system we study is given by the following coupled plate equations:

$\displaystyle
u_{tt}+\gamma(-\Delta)^{\tau}u_{tt}+\Delta^{2}u+\alpha\Delta\theta=0,\quad$
$\displaystyle x\in\Omega,$ $\displaystyle t>0,$ (5)
$\displaystyle\theta_{t}-\kappa\Delta\theta-\beta\Delta u_{t}=0,\quad$
$\displaystyle x\in\Omega,$ $\displaystyle t>0,$ (6)

and prescribed initial data

$\displaystyle u(x,0)=u_{0}(x),\ u_{t}(x,0)=u_{1}(x),\
\theta(x,0)=\theta_{0}(x),\quad x\in\Omega.$ (7)

Here $u$, denote the transversal displacements of the plates. The inertia
rotational coefficients given by $\gamma$ positive number. The exponent $\tau$
lies in the interval $[0,1]$, $\alpha$ and $\beta$ constant positive are the
coupling coefficient.

Various researchers year after year have been devoting their attention to the
study of the asymptotic behavior and regularity of the solutions of the
thermoelastic plate system, especially when considering the rotational
inertial force in the system given by: $\gamma(-\Delta)^{\tau}u_{tt}$. We
emphasize that this mathematical model, when the parameter $\tau$ takes the
values $0$ and $1$, are called Euler-Bernoulli and Kirchoff-Love thermoelastic
plates, respectively. We know that regarding the asymptotic behavior the best
decay rate is in exponential rate, the first results for the Euler-Bernoulli
model, it is well known that the underlying semigroup is both analytic and
exponentially stable in the case of hinged boundary conditions and clamped
boundary condition. However, for the Kirchhoff model, only the exponential
stability of the semigroup is true; Lasiecka and Triggiani showed that the
semigroup is not only not analytic, it is not even differentiable [6, 10] for
any of the hinged or clamped boundary conditions.

Later, many other works followed, establishing the exponential stability of
thermoelastic plates (Euler-Bernoulli and Kirchoff-Love) with various boundary
conditions, e.g. [1, 2, 13, 23]. Regarding the analyticity of the semigroup
for the Euler-Bernoulli model, the first result was established by Liu and
Renardy, [19] in the case of bounded and articulated boundary conditions.
Subsequently, Liu and Liu, [11], and Lasiecka and Triggiani [6, 7, 8, 9]
demonstrated other analyticity results under various boundary conditions.

In more recent research from 2020 Tebou et al. [5] studied thermoelastic
plates considering the fractional rotational inertial force
($\gamma(-\Delta)^{\tau}u_{tt}$ for the parameter $\tau\in[0,1]$. In $\Omega$,
limited open subset of $\mathbb{R}^{n}$, $n\geq 1$, with smooth enough
boundary In this research the authors prove that the semigroup associated to
the system is the Gevrey class $s$ for each $s>\frac{2-\tau}{2-4\tau}$ for
both: the Hinged plate/Dirichlet temperature boundary conditions and Clamped
plate/Dirichlet temperature boundary conditions when the parameter $\tau$ lies
in the interval $(0,\frac{1}{2})$, also show that the semigroup $S(t)$ is
exponentially stable for Hinged boundary conditions, for $\tau$ in the
interval $[0,1]$ and finish their investigation, constructing a
counterexample, that, under hinged boundary conditions, the semigroup is not
analytic, for all $\tau$ in the interval $(0,1)$. To determine the Gevrey
class of $S(t)$ use the domain method of the frequency, the appropriate
decompositions of the components of the system and the use of Lions’
interpolation inequalities. More recent research in this direction can be
found at [15, 17, 18, 24].

The rest of this article is organized as follows: in section 2, we study the
well-posedness of the system (9)-(12) through semigroup theory. We leave our
main contributions for the third section, which is subdivided into two
subsections. In 3.1 we showed that the semigroup $S(t)=e^{\mathbb{B}t}$ is not
analytic when $t\in(0,1]$ and in the last subsection 3.2 we showed that the
underlying semigroup is Gevrey class $s$ for each $s>\frac{3-\tau}{2-2\tau}$
for Hinged plate/Dirichlet temperature boundary condition when the parameter
$\tau$ is in the interval $[\frac{1}{2},1]$. We end this investigation with
one observation of the exponential decay of $S(t)$ for $\tau\in[0,1]$.

####  1.0.1 Well-Posedness of the System of plates thermoelastic

In this section we will use the semigroup theory for assure the existence and
uniqueness of strong solutions for the system (9)-(12). Before this, we are
going to recall some preliminary results.

######  Theorem 1 (See Theorem 1.2.4 in [14])

Let $\mathbb{B}$ be a linear operator with domain $D(\mathbb{B})$ dense in a
Hilbert space $\mathbb{X}$. If $\mathbb{B}$ is dissipative and
$0\in\rho(\mathbb{B})$, the resolvent set of $\mathbb{B}$, then $\mathbb{B}$
is the generator of a $C_{0}$-semigroup of contractions on $\mathbb{X}$.

To re-write the system (9)-(12) in an abstract form, considering the operator
self-adjoint, positive and has inverse compact on a complex Hilbert space
$H=L^{2}(\Omega)$ $A:D(A)\subset L^{2}(\Omega)\rightarrow L^{2}(\Omega)$,
where

$\displaystyle A:=-\Delta,\quad D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega).$ (8)

Therefore, the operator $A^{\theta}$ is self-adjoint positive for all
$\theta\in{\mathbb{R}}$, bounded for $\theta\leq 0$, and the embedding

$\displaystyle D(A^{\theta_{1}})\hookrightarrow D(A^{\theta_{2}}),$

is continuous for $\theta_{1}>\theta_{2}$. Here, the norm in $D(A^{\theta})$
is given by $\|u\|_{D(A^{\theta})}:=\|A^{\theta}u\|$, $u\in D(A^{\theta})$,
where $\|\cdot\|$ denotes the norm in the Hilbert space $L^{2}(\Omega)$. Some
of these spaces are $D(A^{0})=L^{2}(\Omega)$, $D(A^{1/2})=H_{0}^{1}(\Omega)$
and $D(A^{-1/2})=H^{-1}(\Omega)$.

It is known that the above operator is self-adjoint, positive and has inverse
compact. Using this notation to find a solution of the the system (5)-(7) is
equivalent to find $u,v$ in some subset of $D(A)$ such that satisfy the
equations

$\displaystyle u_{tt}+\gamma A^{\tau}u_{tt}+A^{2}u-\alpha A\theta=0,$ (9)
$\displaystyle\theta_{t}+\kappa A\theta+\beta Au_{t}=0,$ (10)

satisfying the boundary conditions (Hinged plate/Dirichlet temperature):

$u=\Delta u=0,\quad\theta=0,\quad x\in\partial\Omega,\ t>0,$ (11)

and the initial data

$\displaystyle u(x,0)=u_{0},\ u_{t}(x,0)=u_{1},\ \theta(x,0)=\theta_{0},\qquad
x\in\Omega.$ (12)

For $\gamma$ positive we can extend the operator $I+\gamma A^{\tau}$ in the
following sense:

$(I+\gamma A^{\tau})\colon D(A^{\tau/2})\to\;D(A^{-\tau/2})$

defined by

$\langle{(I+\gamma A^{\tau})z_{1}},{z_{2}}\rangle_{D(A^{-\tau/2})\times
D(A^{\tau/2})}=\langle{z_{1}},{z_{2}}\rangle+\gamma\langle{A^{\tau/2}z_{1}},{A^{\tau/2}z_{2}}\rangle,$
(13)

for $z_{1},z_{2}\in D(A^{\tau/2})$, where $\langle{\cdot},{\cdot}\rangle$
denotes the inner product in the Hilbert space $D(A^{0})$. Note that this
operator is an isometric operator when we consider the equivalent norm in the
space $D(A^{\tau/2})$:
$\left({\|z\|^{2}+\gamma\|A^{\tau/2}z\|^{2}}\right)^{1/2}:=\|z\|^{1/2}_{D(A^{\frac{\tau}{2}})}.$

Now, we will use a semigroup approach to study the well-posedness of the
system (9)-(12). Tomando $v=u_{t}$ and applying the operators $(I+\gamma
A)^{-1}$, to equations (9) and considering $U=(u,v,\theta)$ and
$U_{0}=(u_{0},v_{0},\theta_{0})$, the system (5)-(7), can be written in the
following abstract framework

$\frac{d}{dt}U(t)=\mathbb{B}U(t),\quad U(0)=U_{0},$ (14)

where the operator $\mathbb{B}$ is given by

$\mathbb{B}U:=\Big{(}v,(I+\gamma A^{\tau})^{-1}\left\\{{-A^{2}u+\alpha
A\theta}\right\\},-\kappa A\theta-\beta Av\Big{)},$ (15)

for $U=(u,v,\theta)$. This operator will be defined in a suitable subspace of
the phase space

$\mathbb{H}:=D(A)\times D(A^{\frac{\tau}{2}})\times D(A^{0}),$

which in view of (13), it is a Hilbert space with the inner product

$\displaystyle\langle{U_{1}},{U_{2}}\rangle$ $\displaystyle:=$
$\displaystyle\beta\langle{Au_{1}},{Au_{2}}\rangle_{D(A^{0})\times
D(A^{0})}+\beta\langle{(I+\gamma
A^{\tau})v_{1}},{v_{2}}\rangle_{D(A^{-\frac{\tau}{2}})\times
D(A^{\frac{\tau}{2}})}$
$\displaystyle\times\alpha\langle{\theta_{1}},{\theta_{2}}\rangle_{D(A^{0})\times
D(A^{0})},$

for $U_{i}=(u_{i},v_{i},\theta_{i})\in\mathbb{H}$, $i=1,2$ and induced norm

$\|U\|_{\mathbb{H}}^{2}:=\beta\|Au\|^{2}+\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+\alpha\|\theta\|^{2}.$
(16)

In these conditions, we define the domain of $\mathbb{B}$ as

$\mathcal{D}(\mathbb{B}):=\Big{\\{}U\in\mathbb{H}\colon v\in
D(A),-Au+\alpha\theta\in D(A^{1-\frac{\tau}{2}}),-\kappa\theta-\beta v\in
D(A)\Big{\\}}.$ (17)

To show that the operator $\mathbb{B}$ is the generator of a $C_{0}$\-
semigroup we invoke a result from Liu-Zheng’ book Theorem(1).

Let us see that the operator $\mathbb{B}$ satisfies the conditions of
Theorem(1). Clearly, we see that $D(\mathbb{B})$ is dense in $\mathbb{H}$.
Taking the inner product of $\mathbb{B}U$ with $U$ we have

$\text{Re}\langle{\mathbb{B}U},{U}\rangle=-\kappa\alpha\|A^{\frac{1}{2}}\theta\|^{2},\quad\forall\
U\in D(\mathbb{B}),$ (18)

that is, the operator $\mathbb{B}$ is dissipative.
To complete the conditions of the above theorem, it remains to show that
$0\in\rho(\mathbb{B})$. Let $F=(f,g,h)\in\mathbb{H}$, let us see that the
stationary problem $\mathbb{B}U=F$ has a solution $U=(u,v,\theta)$. From the
definition of the operator $\mathbb{B}$ given in (15) this system can be
written as

$\displaystyle v=f,\qquad$ $\displaystyle\quad\quad-A^{2}u+\alpha
A\theta=(I+\gamma A^{\tau})g\quad\quad-\kappa A\theta-\beta Av=h.$ (19)

Therefore, it is not difficult to see that there exist only one solution $u$
and $\theta$ of the system

$-A^{2}u+\alpha A\theta=(I+\gamma A^{\tau})g\in D(A^{\frac{\tau}{2}})\\\
-\kappa A\theta=h+\beta Af\in D(A^{0}),$ (20)

from where we have that

$\|U\|_{\mathbb{H}}\leq C\|F\|_{\mathbb{H}},$

wich in particular implies that
$\|\mathbb{B}^{-1}F\|_{\mathbb{H}}\leq\|F\|_{\mathbb{H}}$, so we have that $0$
belongs to the resolvent set $\rho(\mathbb{B})$. Consequently, from Theorem 1
we have $\mathbb{B}$ is the generator of a contractions semigroup. As a
consequence of the previous Theorem(1), we obtain

###### Theorem 2

Given $U_{0}\in\mathbb{H}$ there exists a unique weak solution $U$ to the
problem (14) satisfying

$U\in C([0,+\infty),\mathbb{H}).$

Futhermore, if $U_{0}\in D(\mathcal{B}^{k}),\;k\in\mathbb{N}$, then the
solution $U$ of (14) satisfies

$U\in\bigcap_{j=0}^{k}C^{k-j}([0,+\infty),D(\mathcal{B}^{j}).$

In what follows, $C$ and $C_{\delta}$ will denote a positive constant that
assumes different values in different places.

##  2 Regularity they Euler-Bernoulli to Kirchhoff-Love Thermoelastic Plates

In this section we discuss the regularity of the semigroup
$S(t)=e^{\mathbb{B}t}$, in two subsections: First we analyze the lack of
analyticity of $S(t)$ for $0<\tau\leq 1$, then we study the Gevrey class of
$S(t)$ for $\frac{1}{2}\leq\tau<1$.

The following theorem characterizes the analyticity of $S(t)$:

######  Theorem 3 (see [14])

Let $S(t)=e^{\mathbb{B}t}$ be $C_{0}$-semigroup of contractions on a Hilbert
space $\mathbb{H}$. Suppose that

$\rho(\mathbb{B})\supseteq\\{i\lambda/\lambda\in{\mathbb{R}}\\}\equiv
i{\mathbb{R}}$

Then $S(t)$ is analytic if and only if

$\limsup\limits_{|\lambda|\to\infty}\|\lambda(i\lambda
I-\mathbb{B})^{-1}\|_{\mathcal{L}(\mathbb{H})}<\infty.$ (21)

holds.

Note that to show the condition (21) it is enough to show that: Let
$\delta>0$. There exists a constant $C_{\delta}>0$ such that the solutions of
the system (9)-(12) for $|\lambda|>\delta$ satisfy the inequality

$|\lambda|\|U\|_{\mathbb{H}}^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad\Longleftrightarrow\qquad|\lambda|\|U\|_{\mathbb{H}}\leq
C_{\delta}\|F\|_{\mathbb{H}}.$ (22)

First, note that if $\lambda\in\rho(\mathbb{B})\subset{\mathbb{R}}$ and
$F=(f,g,h)\in\mathbb{H}$ then the solution
$U=(u,w,\theta)\in\hbox{D}(\mathbb{B})$ of the stationary system $(i\lambda
I-\mathbb{B})U=F$ can be written by

$\displaystyle i\lambda u-v$ $\displaystyle=$ $\displaystyle f\quad{\rm
in}\quad D(A)$ (23) $\displaystyle i\lambda(I+\gamma A^{\tau})v+A^{2}u-\alpha
A\theta$ $\displaystyle=$ $\displaystyle(I+\gamma A^{\tau})g\quad{\rm in}\quad
D(A^{-\frac{\tau}{2}})$ (24) $\displaystyle i\lambda\theta+\kappa
A\theta+\beta Av$ $\displaystyle=$ $\displaystyle h\quad{\rm in}\quad
D(A^{0})$ (25)

we have to

$\kappa\alpha\|A^{\frac{1}{2}}\theta\|^{2}=\text{Re}\langle{(i\lambda-\mathbb{B})U},{U}\rangle=\text{Re}\langle{F},{U}\rangle\leq\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$
(26)

Next we will show two lemmas that will be fundamental to achieve our results.

###### Lemma 4

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (9)-(12) for $|\lambda|>0$, satisfy

$\limsup\limits_{|\lambda|\to\infty}\|(i\lambda
I-\mathbb{B})^{-1}\|_{\mathcal{L}(\mathbb{H})}<\infty\qquad{\rm for}\qquad
0\leq\tau\leq 1.$ (27)

Proof: To show the (27), it suffices to show that, given $\delta>0$ there
exists a constant $C_{\delta}>0$ such that the solutions of the system
(9)-(12) for $|\lambda|>\delta$ satisfy the inequality

$\|(i\lambda
I-\mathbb{B})^{-1}F\|_{\mathbb{H}}^{2}=\|U\|^{2}_{\mathbb{H}}=\beta\|Au\|^{2}+\beta\|v\|_{D(A^{\frac{\tau}{2}})}^{2}+\alpha\|\theta\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$ (28)

As $0\leq\frac{1}{2}$, applying continuous immersions and the estimate (26) we
will have $\alpha\|\theta\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}$, therefore it remains to show
that

$\beta[\|Au\|^{2}+\|v\|^{2}_{D(A^{\frac{\tau}{2}})}]\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$

Taking the duality product between equation(24) and $\beta u$ and using the
equation (23), taking advantage of the self-adjointness of the powers of the
operator $A$, we obtain

$\beta\|Au\|^{2}=\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+\beta\gamma\langle{A^{\frac{\tau}{2}}v},{A^{\frac{\tau}{2}}f}\rangle\\\
+\beta\langle{v},{f}\rangle+\alpha\beta\langle{\theta},{Au}\rangle+\beta\langle{g},{u}\rangle+\beta\gamma\langle{A^{\frac{\tau}{2}}g},{A^{\frac{\tau}{2}}u}\rangle.$
(29)

On the other hand, taking the duality product between equation(25) and
$A^{-1}(I+\gamma A^{\tau})v$, using (24) and taking advantage of the self-
adjointness of the powers of the operator $A$, we obtain

$\displaystyle\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}$ $\displaystyle=$
$\displaystyle\langle{A^{-1}\theta},{i\lambda(I+\gamma
A^{\tau})v}\rangle-\kappa\langle{\theta},{(I+\gamma
A^{\tau})v}\rangle+\langle{A^{-1}h},{v}\rangle+\gamma\langle{h},{A^{\tau-1}v}\rangle$
$\displaystyle=$
$\displaystyle-\langle{\theta},{Au}\rangle+\alpha\|\theta\|^{2}+\langle{A^{-1}\theta},{g}\rangle+\gamma\langle{\theta},{A^{\tau-1}g}\rangle$
$\displaystyle-\kappa\langle{A^{\frac{1}{2}}\theta},{A^{-\frac{1}{2}}(I+\gamma
A^{\tau})v}\rangle+\langle{A^{-1}h},{v}\rangle+\gamma\langle{h},{A^{\tau-1}v}\rangle$

Applying Cauchy-Schwarz and Young’s inequalities, for $\varepsilon>0$ exists
$C_{\varepsilon}>0$ such that

$\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}\leq
C_{\varepsilon}\|\theta\|^{2}+\varepsilon\|Au\|^{2}+C_{\delta}\|A^{-1}\theta\|\|g\|+\gamma\|\theta\|\|A^{\tau-1}g\|+C_{\varepsilon}\|A^{\frac{1}{2}}\theta\|^{2}\\\
+\varepsilon[\|A^{-\frac{1}{2}}v\|^{2}+\|A^{\tau-\frac{1}{2}}v\|^{2}]+\|A^{-1}h\|\|v\|+\gamma\|h\|\|A^{\tau-1}v\|,$
(30)

then, as $-\frac{1}{2}\leq\tau-\frac{1}{2}\leq\frac{\tau}{2}$, the continuous
embedding $D(A^{\theta_{2}})\hookrightarrow D(A^{\theta_{1}})$,
$\theta_{2}>\theta_{1}$, from norms $\|F\|_{\mathbb{H}}$ and
$\|U\|_{\mathbb{H}}$ and estimative (26), we have

$\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}\leq\varepsilon\|Au\|^{2}+C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$
(31)

Using (31) in (29), applying Cauchy-Schwarz and Young’s inequalities and from
norms $\|F\|_{\mathbb{H}}$ and $\|U\|_{\mathbb{H}}$, we obtain

$\beta\|Au\|^{2}\leq C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$ (32)

Now, using (32) in (31), we have

$\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$ (33)

Therefore, from the estimates (32), (33) and (26), we conclude the proof of
(28), thus finishing the proof of this lemma.

$\Box$

###### Lemma 5

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (9)-(12) for $|\lambda|>\delta$, satisfy

$|\lambda|[\beta\|Au\|^{2}+\alpha\|\theta\|^{2}]\leq\beta|\lambda|\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$
(34)

Proof: Estimative of $\beta|\lambda|\|Au\|^{2}$

Taking the duality product between equation(24) and $\beta\lambda u$ and using
the equation (23), taking advantage of the self-adjointness of the powers of
the operator $A$, we obtain

$\displaystyle\beta\lambda\|Au\|^{2}$ $\displaystyle=$
$\displaystyle\beta\lambda\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+\beta\langle{\lambda(I+\gamma
A^{\tau})v},{f}\rangle+\alpha\beta\langle{\theta},{A(-iv-if)}\rangle$
$\displaystyle+\beta\langle{(I+\gamma A^{\tau})g},{-if-iv}\rangle$
$\displaystyle=$
$\displaystyle\beta\lambda\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+i\beta\langle{Au},{Af}\rangle-i\beta\langle{\theta},{f}\rangle-i\beta\langle{(I+\gamma
A^{\tau})g},{f}\rangle$
$\displaystyle+i\alpha\beta\langle{\theta},{Av}\rangle+i\alpha\beta\langle{\theta},{f}\rangle+i\beta\langle{(I+\gamma
A^{\tau})g},{f}\rangle+i\beta\langle{(I+\gamma A^{\tau})g},{v}\rangle$
$\displaystyle=$
$\displaystyle\beta\lambda\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+i\beta\langle{Au},{Af}\rangle-i\beta\langle{\theta},{f}\rangle+i\alpha\beta\langle{\theta},{Av}\rangle+i\alpha\beta\langle{\theta},{f}\rangle$
$\displaystyle+i\beta\langle{(I+\gamma A^{\tau})g},{v}\rangle.$

On the other hand, taking the duality product between equation(25) and
$\frac{\alpha}{\kappa}A^{-1}(\lambda\theta)$ and using the equation (24),
taking advantage of the self-adjointness of the powers of the operator $A$, we
obtain

$\displaystyle\alpha\lambda\|\theta\|^{2}=-i\dfrac{\alpha}{\kappa}\lambda^{2}\|A^{-\frac{1}{2}}\theta\|^{2}-\dfrac{\alpha\beta}{\kappa}\langle{v},{\lambda\theta}\rangle+\dfrac{\alpha}{\kappa}\langle{h},{A^{-1}(\lambda\theta)}\rangle,$
(36)

from:

$\displaystyle-\dfrac{\alpha\beta}{\kappa}\langle{v},{\lambda\theta}\rangle$
$\displaystyle=$ $\displaystyle
i\alpha\beta\langle{Av},{\theta}\rangle+i\dfrac{\alpha\beta^{2}}{\kappa}\|A^{\frac{1}{2}}v\|^{2}+i\dfrac{\alpha\beta}{\kappa}\langle{v},{h}\rangle$
(37)
$\displaystyle\dfrac{\alpha}{\kappa}\langle{h},{A^{-1}(\lambda\theta}\rangle$
$\displaystyle=$
$\displaystyle-i\alpha\langle{h},{\theta}\rangle-i\dfrac{\alpha\beta}{\kappa}\langle{h},{v}\rangle+i\dfrac{\alpha^{2}}{\kappa}\|A^{-\frac{1}{2}}h\|^{2},$
(38)

Adding (2) with (36) and in the result using the identities (37) and (38), we
get

$\lambda[\beta\|Au\|^{2}+\alpha\|\theta\|^{2}]=\beta\lambda\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+i\beta\langle{Au},{Af}\rangle\\\
-i\beta\langle{\theta},{f}\rangle+i\alpha\beta\langle{\theta},{Av}\rangle+i\alpha\beta\langle{\theta},{f}\rangle+i\beta\langle{(I+\gamma
A^{\tau})g},{v}\rangle\\\
-i\dfrac{\alpha}{\kappa}\lambda^{2}\|A^{-\frac{1}{2}}\theta\|^{2}+i\alpha\beta\langle{Av},{\theta}\rangle+i\dfrac{\alpha\beta^{2}}{\kappa}\|A^{\frac{1}{2}}v\|^{2}+i\dfrac{\alpha\beta}{\kappa}\langle{v},{h}\rangle\\\
-i\alpha\langle{h},{\theta}\rangle-i\dfrac{\alpha\beta}{\kappa}\langle{h},{v}\rangle+i\dfrac{\alpha^{2}}{\kappa}\|A^{-\frac{1}{2}}h\|^{2}.$
(39)

Of identity
$i\alpha\beta[\langle{\theta},{Av}\rangle+\langle{Av},{\theta}\rangle]=i2\alpha\beta{\rm
Re}\langle{\theta},{Av}\rangle$, taking real part of (39), we have

$\lambda[\beta\|Au\|^{2}+\alpha\|\theta\|^{2}]=\beta\lambda\|v\|^{2}_{D(A^{\frac{\tau}{2}})}+i\beta\langle{Au},{Af}\rangle\\\
-i\beta\langle{\theta},{f}\rangle+i\alpha\beta\langle{\theta},{f}\rangle+i\beta\langle{(I+\gamma
A^{\tau})g},{v}\rangle\\\
+i\dfrac{\alpha\beta}{\kappa}\langle{v},{h}\rangle-i\alpha\langle{h},{\theta}\rangle-i\dfrac{\alpha\beta}{\kappa}\langle{h},{v}\rangle.$
(40)

Applying Cauchy-Schwarz and Young inequalities and norms $\|F\|_{\mathbb{H}}$
and $\|U\|_{\mathbb{H}}$, we finish proving this lemma.

$\Box$

###### Lemma 6

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (9)-(12) for $|\lambda|>\delta$, satisfy

$\|A^{\frac{1}{2}}v\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm
for}\qquad\dfrac{1}{2}\leq\tau\leq 1.$ (41)

Proof: Performing the product of duality between equation (25) and
$A^{-\tau}(I+\gamma A^{\tau})v$ and again using the property that for all
$\eta\in\mathbb{R}$, $A^{\eta}$ is self-adjoint, we have

$\displaystyle\beta\langle{Av},{A^{-\tau}(I+\gamma A^{\tau})v}\rangle$
$\displaystyle=$ $\displaystyle\langle{A^{-\tau}\theta},{i\lambda(I+\gamma
A^{\tau})v}\rangle-\kappa\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}-\tau}(I+\gamma
A^{\tau})v}\rangle$ (42) $\displaystyle+\langle{h},{A^{-\tau}(I+\gamma
A^{\tau})v}\rangle,$

then, using (24) in (42), we have

$\displaystyle\beta\|A^{\frac{1-\tau}{2}}v\|^{2}+\beta\gamma\|A^{\frac{1}{2}}v\|^{2}$
$\displaystyle=$ $\displaystyle\langle{A^{-\tau}\theta},{-A^{2}u+\alpha
A\theta+(I+\gamma
A^{\tau})g}\rangle-\kappa\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}-\tau}v}\rangle$
(43)
$\displaystyle-\kappa\gamma\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}}v}\rangle+\langle{h},{A^{-\tau}v}\rangle+\gamma\langle{h},{v}\rangle.$

Considering continuous immersion $D(A^{\theta_{2}})\hookrightarrow
D(A^{\theta_{1}}),\;\theta_{2}>\theta_{1}$ for
$\frac{1-\tau}{2}\leq\frac{1}{2}$ and Cauchy-Schwarz inequality, we have

$\displaystyle\beta\gamma\|A^{\frac{1}{2}}v\|^{2}$ $\displaystyle\leq$
$\displaystyle|-\langle{A^{\frac{1}{2}}\theta},{A^{\frac{3}{2}-\tau}u}\rangle|+\alpha\|A^{\frac{1-\tau}{2}}\theta\|^{2}+|\langle{\theta},{A^{-\tau}g}\rangle|+|\gamma\langle{\theta},{g}\rangle|$
(44)
$\displaystyle|\kappa\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}-\tau}v}\rangle|+|-\kappa\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}}v}\rangle|+|\langle{h},{A^{-\tau}v}\rangle|+\gamma|\langle{h},{v}\rangle|.$

Considering $\frac{3}{2}-\tau\leq 1\Leftrightarrow\frac{1}{2}\leq\tau\leq 1$,
for $\varepsilon>0$ exists $C_{\varepsilon}>0$ independent de $\lambda$, and
applying Young inequality, we have

$\beta\gamma\|A^{\frac{1}{2}}v\|^{2}\leq
C\\{\|A^{\frac{1}{2}}\theta\|^{2}+\|AU\|^{2}+\|\theta\|\|g\|+\|h\|\|A^{-\tau}v\|+\|h\|\|v\|\\}\\\
+\varepsilon\|A^{\frac{1}{2}}v\|^{2}+C_{\varepsilon}\|A^{\frac{1}{2}}\theta\|^{2}$
(45)

Finally using estimative (26), norms $\|F\|_{\mathbb{H}}$ and
$\|U\|_{\mathbb{H}}$ and applying Lemma(4). The proof of this lemma is
finished.

$\Box$

###### Lemma 7

Let $\varrho(\mathbb{B})$ be the resolvent set of operator $\mathbb{B}$. Then

$i\hskip 0.5pt\mathbb{R}\subset\varrho(\mathbb{B}).$ (46)

Proof: Let us prove that $i{\mathbb{R}}\subset\rho(\mathbb{B})$ by using an
argument by contradiction, so we suppose that
$i{\mathbb{R}}\not\subset\rho(\mathbb{B})$. As $0\in\rho(\mathbb{B})$ and
$\rho(\mathbb{B})$ is open, we consider the highest positive number
$\lambda_{0}$ such that the
$]-i\lambda_{0},i\lambda_{0}[\subset\rho(\mathbb{B})$ then $i\lambda_{0}$ or
$-i\lambda_{0}$ is an element of the spectrum $\sigma(\mathbb{B})$. We Suppose
$i\lambda_{0}\in\sigma(\mathbb{B})$ (if $-i\lambda_{0}\in\sigma(\mathbb{B})$
the proceeding is similar). Then, for $0<\delta<\lambda_{0}$ there exist a
sequence of real numbers $(\lambda_{n})$, with
$\delta\leq\lambda_{n}<\lambda_{0}$, $\lambda_{n}\rightarrow\lambda_{0}$, and
a vector sequence $U_{n}=(u_{n},v_{n},\theta_{n})\in D(\mathbb{B})$ with
unitary norms, such that

$\displaystyle\|(i\lambda_{n}I-\mathbb{B})U_{n}\|_{\mathbb{H}}=\|F_{n}\|_{\mathbb{H}}\rightarrow
0,$

as $n\rightarrow\infty$. From estimative (28), we have

$\|U_{n}\|^{2}_{\mathbb{H}}=\beta\|Au_{n}\|^{2}+\beta\|v_{n}\|^{2}_{D(A^{\frac{\tau}{2}})}+\alpha\|\theta_{n}\|^{2}\leq
C_{\delta}\|F_{n}\|_{\mathbb{H}}\|U_{n}\|_{\mathbb{H}}\rightarrow 0.$

Therefore, we have $\|U_{n}\|_{\mathbb{H}}\rightarrow 0$ but this is absurd,
since $\|U_{n}\|_{\mathbb{H}}=1$ for all $n\in{\mathbb{N}}$. Thus,
$i{\mathbb{R}}\subset\rho(\mathbb{B})$. This completes the proof this lemma.

$\Box$

###  2.1 Lack of analyticity of $S(t)=e^{\mathbb{B}t}$

Since the operator $A$ defined on (8) is positive, self-adjoint and it has
compact resolvent, its spectrum is constituted by positive eigenvalues
$(\sigma_{n})$ such that $\sigma_{n}\rightarrow\infty$ as
$n\rightarrow\infty$. For $n\in{\mathbb{N}}$ we denote with $e_{n}$ an unitary
$D(A^{\frac{\tau}{2}})$-norm eigenvector associated to the eigenvalue
$\sigma_{n}$, that is,

$Ae_{n}=\sigma_{n}e_{n},\quad\|e_{n}\|_{D(A^{\frac{\tau}{2}})}=1,\quad
n\in{\mathbb{N}}.$ (47)

###### Theorem 8

The semigroup $S(t)=e^{\mathbb{B}t}$ is not analytic for $\tau\in(0,1]$.

Proof: We apply Theorem (3) to show this result. Consider the eigenvalues and
eigenvectors of the operator $A$ as in (47). Let
$F_{n}=(0,-e_{n},0)\in\mathbb{H}$. The solution
$U_{n}=(u_{n},v_{n},\theta_{n})$ of the system
$(i\lambda_{n}I-\mathbb{B})U_{n}=F_{n}$ satisfies $v_{n}=i\lambda_{n}u_{n}$
and the following equations

$\displaystyle\lambda^{2}_{n}(I+\gamma A^{\tau})u_{n}-A^{2}u_{n}+\alpha
A\theta_{n}$ $\displaystyle=$ $\displaystyle(I+\gamma A^{\tau})e_{n},$
$\displaystyle i\lambda\theta_{n}+\kappa A\theta_{n}+i\lambda_{n}\beta Au_{n}$
$\displaystyle=$ $\displaystyle 0.$

Let us see whether this system admits solutions of the form

$u_{n}=\mu\tilde{e}_{n},\quad\theta_{n}=\nu\tilde{e}_{n},$

for some complex numbers $\mu_{n}$ and $\nu_{n}$. Then, the numbers $\mu_{n}$,
$\nu_{n}$ should satisfy the algebraic system

$\displaystyle\\{\lambda^{2}_{n}(1+\gamma\sigma_{n}^{\tau})-\sigma_{n}^{2}\\}\mu_{n}+\alpha\sigma_{n}\nu_{n}$
$\displaystyle=$ $\displaystyle(1+\gamma\sigma_{n}^{\tau}),$ (48)
$\displaystyle
i\lambda\beta\sigma_{n}\mu_{n}+(i\lambda+\kappa\sigma_{n})\nu_{n}$
$\displaystyle=$ $\displaystyle 0.$ (49)

In this point, we introduce the numbers

$\lambda_{n}^{2}:=\dfrac{\sigma_{n}^{2}}{1+\gamma\sigma_{n}^{\tau}}.$

Thus, if we introduce the notation $x_{n}\approx y_{n}$ meaning that
$\displaystyle\lim_{n\rightarrow\infty}\frac{|x_{n}|}{|y_{n}|}$ is a positive
real number, we have that

$\displaystyle|\lambda_{n}|\approx|\sigma_{n}|^{\frac{2-\tau}{2}}.$

With these considerations we have that to solving this system (48)–(49), we
find that

$|\mu_{n}|=\Big{|}\dfrac{\lambda_{n}+\gamma\lambda_{n}\sigma_{n}^{\tau}+i\kappa[\gamma\sigma_{n}^{1+\tau}+\sigma_{n}]}{\alpha\beta\lambda_{n}\sigma_{n}^{2}}\Big{|}\approx|\sigma_{n}|^{\frac{3\tau-4}{2}}\approx|\lambda_{n}|^{\frac{3\tau-4}{2-\tau}}.$

Therefore, the solution $U_{n}$ of the system
$(i\lambda_{n}-\mathbb{B})U_{n}=F_{n}$ satisfy

$\|U_{n}\|_{\mathbb{H}}\geq
K\|v_{n}\|_{D(A^{\frac{\tau}{2}})}=K|\lambda_{n}|\|u_{n}\|_{D(A^{\frac{\tau}{2}})}=K|\lambda_{n}||\mu_{n}|\|e_{n}\|_{D(A^{\frac{\tau}{2}})}=K|\lambda_{n}|^{\frac{2\tau-2}{2-\tau}}.$

Then

$|\lambda_{n}|\|U_{n}\|_{\mathcal{H}}\geq K|\lambda|^{\frac{\tau}{2-\tau}}.$

Therefore $\|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\to\infty$ for $\tau>0$
approaches infinity as $|\lambda_{n}|\to\infty$. Therefore the (21) condition
fails. consequently for $\tau>0$, to semigroup $S(t)$ is not analytic, in
particular $S(t)$ is not analytic for $\tau\in(\frac{1}{2},1]$. This completes
the proof of this theorem.

$\Box$

###  2.2 Gevrey Class

Before exposing our results, it is useful to recall the next definition and
result presented in [5] (adapted from [19], Theorem 4, p. 153]).

###### Definition 9

Let $t_{0}\geq 0$ be a real number. A strongly continuous semigroup $S(t)$,
defined on a Banach space $\mathbb{H}$, is of Gevrey class $s>1$ for
$t>t_{0}$, if $S(t)$ is infinitely differentiable for $t>t_{0}$, and for every
compact set $K\subset(t_{0},\infty)$ and each $\mu>0$, there exists a constant
$C=C(\mu,K)>0$ such that

$||S^{(n)}(t)||_{\mathcal{L}(\mathcal{H})}\leq C\mu^{n}(n!)^{s},\text{ for all
}\quad t\in K,n=0,1,2...$ (50)

######  Theorem 10 ([19])

Let $S(t)$ be a strongly continuous and bounded semigroup on a Hilbert space
$\mathbb{H}$. Suppose that the infinitesimal generator $\mathbb{B}$ of the
semigroup $S(t)$ satisfies the following estimate, for some $0<\phi<1$:

$\lim\limits_{|\lambda|\to\infty}\sup|\lambda|^{\phi}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathcal{L}(\mathbb{H})}<\infty.$ (51)

Then $S(t)$ is of Gevrey class $s$ for $t>0$, for every $s>\dfrac{1}{\phi}$.

Our main result in this subsection is as follows:

###### Theorem 11

Let $S(t)=e^{\mathbb{B}t}$ strongly continuos-semigroups of contractions on
the Hilbert space $\mathbb{H}$, the semigroups $S(t)$ is of Grevrey class $s$
for every $s>\frac{3-\tau}{2-2\tau}$ for $\tau\in[\dfrac{1}{2},1)$, as there
exists a positive constant $C$ such that we have the resolvent estimative:

$|\lambda|^{\frac{2-2\tau}{3-\tau}}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathcal{L}(\mathbb{H})}\leq
C,\quad\lambda\in\rho(\mathbb{B})\subset{\mathbb{R}}.$ (52)

Proof: Note that the estimate

$|\lambda|^{\frac{2-2\tau}{3-\tau}}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathbb{H}}=|\lambda|^{\frac{2-2\tau}{3-\tau}}\|U\|_{\mathbb{H}}\leq
C_{\delta}\|F\|_{\mathbb{H}}$ (53)

implies the inequality (52). Therefore from now on we will show (53), for this
purpose let us estimate the term $|\lambda|\|A^{\frac{\tau}{2}}v\|^{2}$, we
assume $\lambda\in\rho(\mathbb{B})\subset{\mathbb{R}}$ with $|\lambda|>1$, we
shall borrow some ideias from [12]. Set $v=v_{1}+v_{2}$, where $v_{1}\in D(A)$
and $v_{2}\in D(A^{\frac{\tau}{2}})$, with

$i\lambda(I+\gamma A^{\tau}A)v_{1}+Av_{1}=(I+\gamma A^{\tau})g,\hskip
9.38945pti\lambda(I+\gamma A^{\tau})v_{2}=-A^{2}u+\alpha A\theta+Av_{1}.$ (54)

Firstly, applying the product duality the first equation in (54) by $v_{1}$,
we have

$i\lambda\|v_{1}\|^{2}+i\lambda\gamma\|A^{\frac{\tau}{2}}v_{1}\|^{2}+\|A^{\frac{1}{2}}v_{1}\|^{2}=\langle{g},{v_{1}}\rangle+\gamma\langle{A^{\frac{\tau}{2}}g},{A^{\frac{\tau}{2}}v_{1}}\rangle.$
(55)

Taking first the imaginary part of (55) and in the sequence the real part and
applying Cauchy-Schwarz inequality, we have

$\displaystyle|\lambda|\|v_{1}\|^{2}+\gamma|\lambda|\|A^{\frac{\tau}{2}}v_{1}\|\leq|\rm{Im}\langle{g},{v_{1}}\rangle|+\gamma|\rm{Im}\langle{A^{\frac{\tau}{2}}g},{A^{\frac{\tau}{2}}v_{1}}\rangle|\leq
C\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}$ and
$\displaystyle\|A^{\frac{1}{2}}v_{1}\|^{2}\leq
C\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$

Equivalently

$\displaystyle\|v_{1}\|\leq
C\dfrac{[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}}{|\lambda|^{\frac{1}{2}}},\quad\|A^{\frac{\tau}{2}}v_{1}\|\leq
C\dfrac{[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}}{|\lambda|^{\frac{1}{2}}}$
(56) $\displaystyle\rm{and}$ $\displaystyle\|A^{\frac{1}{2}}v_{1}\|\leq
C[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}.$ (57)

From $A^{\frac{\tau}{2}}v_{2}=A^{\frac{\tau}{2}}v+A^{\frac{\tau}{2}}v_{1}$, we
have

$\displaystyle\|A^{\frac{\tau}{2}}v_{2}\|^{2}$ $\displaystyle\leq$
$\displaystyle
C\\{\|A^{\frac{\tau}{2}}v\|^{2}+\|A^{\frac{\tau}{2}}v_{1}\|^{2}\\}\leq
C\bigg{\\{}\dfrac{|\lambda|+1}{|\lambda|}\bigg{\\}}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$

From Lemma(4) and second inequality of (56). Let $\delta>0$. Exists a positive
constant $C_{\delta}$ such that for $|\lambda|\geq\delta$, we obtain

$\|A^{\frac{\tau}{2}}v_{2}\|^{2}\leq
C_{\delta}\|F\|_{\mathcal{H}}\|U\|_{\mathcal{H}}.$ (58)

From second equation of (54), we have

$\displaystyle|\lambda||(I+\gamma A^{\tau})[A^{-1}v_{2}]||$
$\displaystyle\leq$ $\displaystyle C\\{\|Au\|+\|\theta\|\\}+\|v_{1}\|,$

then, we find

$|\lambda|(\|A^{-1}v_{2}\|^{2}+\gamma\|A^{\frac{\tau}{2}-1}v_{2}\|^{2})^{\frac{1}{2}}\leq
C\\{\|Au\|+\|\theta\|\\}+\|v_{1}\|,$

applying Cauchy-Schwarz and Young inequalities and using first inequality of
(56) and estimative (26) for $|\lambda|^{\frac{1}{2}}>1$, for $0\leq\tau\leq
1$, we obtain

$\displaystyle|\lambda|\|A^{\frac{\tau}{2}-1}v_{2}\|$ $\displaystyle\leq$
$\displaystyle
C[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\bigg{[}\dfrac{|\lambda|^{\frac{1}{2}}+1}{|\lambda|^{\frac{1}{2}}}\bigg{]}$
$\displaystyle\|A^{\frac{\tau}{2}-1}v_{2}\|$ $\displaystyle\leq$
$\displaystyle
C|\lambda|^{-1}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}$ (59)

On the other hand, from $v_{2}=v-v_{1}$, we have

$\|A^{\frac{1}{2}}v_{2}\|\leq\|A^{\frac{1}{2}}v\|+\|A^{\frac{1}{2}}v_{1}\|.$
(60)

From Lemma(6) and (57), we have

$\|A^{\frac{1}{2}}v_{2}\|\leq
C_{\delta}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\quad{\rm
for}\quad\dfrac{1}{2}\leq\tau\leq 1.$ (61)

Now, by Lion’s interpolations inequality
($\frac{\tau}{2}-1<\frac{\tau}{2}\leq\frac{1}{2}$) and estimatives (59) and
(61), we derive

$\displaystyle\|A^{\frac{\tau}{2}}v_{2}||$ $\displaystyle\leq$ $\displaystyle
C\|A^{\frac{\tau}{2}-1}v_{2}\|^{\frac{1-\tau}{3-\tau}}\|A^{\frac{1}{2}}v_{2}\|^{\frac{2}{3-\tau}}\quad{\rm
for}\quad\dfrac{1}{2}\leq\tau\leq 1$ $\displaystyle\leq$ $\displaystyle
C|\lambda|^{-\frac{1-\tau}{3-\tau}}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\quad{\rm
for}\quad\dfrac{1}{2}\leq\tau\leq 1.$

On the other hand, as $\|A^{\frac{\tau}{2}}v\|\leq
C\\{\|A^{\frac{\tau}{2}}v_{1}\|+\|A^{\frac{\tau}{2}}v_{1}\|\\}$ and
$-\frac{1-\tau}{3-\tau}\geq-\frac{1}{2}$, we have

$\displaystyle\|A^{\frac{\tau}{2}}v\|$ $\displaystyle\leq$ $\displaystyle
C\\{|\lambda|^{-\frac{1-\tau}{3-\tau}}+|\lambda|^{-\frac{1}{2}}\\}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C|\lambda|^{-\frac{1-\tau}{3-\tau}}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\quad{\rm
for}\quad\dfrac{1}{2}\leq\tau\leq 1.$

Equivalently

$|\lambda|\|A^{\frac{\tau}{2}}v\|^{2}\leq
C_{\delta}|\lambda|^{\frac{1+\tau}{3-\tau}}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{1}{2}\leq\tau\leq 1.$ (62)

Using (62) in estimative (34) to Lemma(5), we have

$|\lambda|[\beta\|Au\|^{2}+\alpha\|\theta\|^{2}]\leq
C_{\delta}|\lambda|^{\frac{1+\tau}{3-\tau}}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm
for}\qquad\dfrac{1}{2}\leq\tau\leq 1.$ (63)

And as (62) it implies

$|\lambda|\beta\|v\|^{2}_{D(A^{\frac{\tau}{2}})}\leq
C_{\delta}|\lambda|^{\frac{1+\tau}{3-\tau}}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{1}{2}\leq\tau\leq 1.$ (64)

Finally, using (63) and (64). We finish the proof of this theorem. $\Box$

######  Remark 12 (Asymptotic Behavior)

Note that the semigroup with class Gevrey has more regular properties than a
differentiable semigroup, but is less regular than an analytic semigroup. The
Gevrey rate $s>\frac{1}{\phi}$ ‘measures’ the degree of divergence of its
power series. It should be noted that the Gevrey class or analyticity of the
particular model implies three important properties. The first is the property
of the smoothing effect on the initial data, that is, no matter how irregular
the initial data is, the model solutions are very smooth in positive time. The
second property is that systems are exponentially stable. Finally, the systems
enjoy the property of linear stability, which means that the type of the
semigroup is equal to the spectral limit of its infinitesimal operator. Thus,
as a consequence of the lemmas (4) and (7), we have that the semigroup
$S(t)=e^{\mathbb{B}t}$ is exponentially stable for all $\tau\in[0.1]$.

## References

  * [1] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system without mechanical dissipation, Rend. Istit. Mat. Univ. Trieste, 28(1997), pp 1-29.
  * [2] G. Avalos and I. Lasiecka, Exponential stability of a thermoelastic system with free boundary conditions without mechanical dissipation, SIAM J. Math. Anal., 29 (1998), pp 1-28.
  * [3] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann. 347 (2010), pp 455-478.
  * [4] K.J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Springer (2000).
  * [5] V. Keyantuo, L. Tebou and M. Warma, A Gevrey Class Semigroup for a Thermoelastic Plate Model with a Fractional Laplacian: Berween the Euler-Bernoulli and Kirchhoff Models. Discrete and Continuous Dynamical System, Vol 40. Number 5, May (2020), pp 2875-2889.
  * [6] I. Lasiecka and Triggiani, R.: Analyticity and lack thereof, of thermo-elastic semigroups. In: Control and Partial Differential Equations, ESAIM Proc. Soc. Math. Appl. Indust., Paris, vol. 4, (1998), pp 199-222.
  * [7] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), pp 457-482.
  * [8] I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the s.c semigroup arising in abstract thermo-elastic equations, Adv. Differential Equations, 3 (1998), pp 387-416.
  * [9] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with coupled hinged/Neumann B.C. Abstr. Appl. Anal., 3 (1998), pp 153-169.
  * [10] I. Lasiecka and R. Triggiani, Structural decomposition of thermo-elastic semigroups with rotational forces. Semigroup Forum 60 (2000), pp 16-66.
  * [11] K. S. Liu and Z. Y. Liu, Exponential stability and analyticity of abstract linear thermoelastic systems, Z. Angew. Math. Phys. 48 (1997) 885-904.
  * [12] Z. Liu and M. Renardy, A note on the equations of thermoelastic plate, Appl. Math. Lett., 8, (1995), pp 1-6.
  * [13] Z. Y. Liu and S. M. Zheng, Exponential stability of the Kirchoff plate with thermal or viscoelastic damping, Quarterly Appl. Math. , 55 (1997), 551-564.
  * [14] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall CRC Research Notes in Mathematics, Boca Raton, FL, 398 (1999).
  * [15] H.P. Oquendo and F.M.S. Suárez, Exact decay rates for coupled plates with partial fractional damping. Zeitschrift für angewandte Mathematik und Physik-ZAMP(online)V1, (2019), pp 70–88.
  * [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer, (1983).
  * [17] H. D. F. Sare, Z. Liu and R. Racke, Stability of abstract thermoelastic systems with inertial terms, Journal of Differential Equations, Volume 267, Issue 12, 5 december (2019), pp 7085–7134.
  * [18] B.T.S. Sozzo, J. E. M. Rivera, The Gevrey class of the Euler-Bernoulli beam model, Journal of Mathematical Analysis and Applications,505 (2022).
  * [19] S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Control- lability, Thesis (Ph.D.) University of Minnesota, (1989), 182 pp.
  * [20] L. Tebou, Stabilization of some coupled hyperbolic/parabolic equations. Discrete Contin. Dyn. Syst. Ser. B 14 (2010), pp 1601-1620.
  * [21] L. Tebon, Stabilization of some coupled hyperbolic/parabolic equations, Discrete and Continuous Dynamical Systems Series B 14, Number 4, November (2010).
  * [22] L. T. Tebou, Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control and Related Fields 2 (2012), pp 45-60.
  * [23] L. Tebou, Uniform analyticity and exponential decay of the semigroup associated with a thermoelastic plate equation with perturbed boundary conditions, C. R. Math. Acad. Sci. Paris, 351 (2013), pp 539-544.
  * [24] L. Tebou, Regularity and stability for a plate model involving frational rotational forces and damping, ZAMP Zeitschrift für angewandte Mathematik und Physik, 72:158 (2021).


# FDO Manager: Minimum Viable FAIR Digital Object Implementation

Oussama Zoubia  University Hospital of Cologne, Cologne, Germany  Zeyd
Boukhers  Fraunhofer Institute for Applied Information Technology FIT, Sankt
Augustin, Germany  University Hospital of Cologne, Cologne, Germany  Nagaraj
Bahubali Asundi  Fraunhofer Institute for Applied Information Technology FIT,
Sankt Augustin, Germany  Sezin Dogan  University Hospital of Cologne, Cologne,
Germany  Adamantios Koumpis  University Hospital of Cologne, Cologne, Germany
Christoph Lange  Fraunhofer Institute for Applied Information Technology FIT,
Sankt Augustin, Germany  RWTH Aachen University, Aachen, Germany  Oya Beyan
Fraunhofer Institute for Applied Information Technology FIT, Sankt Augustin,
Germany  University Hospital of Cologne, Cologne, Germany

###### Abstract

The concept of FAIR Digital Objects (FDOs) aims to revolutionise the field of
digital preservation and accessibility in the next few years. Central to this
revolution is the alignment of FDOs with the FAIR (Findable, Accessible,
Interoperable, Reusable) Principles, particularly emphasizing machine-
actionability and interoperability across diverse data ecosystems. This
abstract introduces the “FDO Manager”, a Minimum Viable Implementation,
designed to optimize the management of FDOs following these principles and the
FDO specifications111https://fairdo.org/specifications/. The FDO Manager is
tailored to manage research artefacts such as datasets, codes, and
publications, to foster increased transparency and reproducibility in
research. The abstract presents the implementation details of the FDO Manager,
its underlying architecture, and the metadata schemas it employs, thereby
offering a clear and comprehensive understanding of its functionalities and
impact on the research domain.

Keywords: FAIR Digital Objects, FDO Manager, Data Preservation,
Interoperability, Metadata Schemas, Research Artifact Management

##  1 Introduction

Since its introduction in 2018 [CGH+18], the concept of FAIR Digital Objects
(FDOs) has seen a remarkable rise in interest within the research and data
management communities. FDOs are key in enhancing data preservation and
accessibility, contributing to a robust ecosystem for managing digital
research resources [DSKW20]. This surge in popularity underscores the
increasing relevance of the FAIR (Findable, Accessible, Interoperable,
Reusable) Principles in the realm of scientific data management [WDA+16,
MNV+17].

The FDO Framework [BdSSSFG23] is built upon the FAIR Guiding Principles
[WDA+16], which are designed to enhance the FAIRness of digital objects,
focusing on concepts such as unique identification, rich metadata, etc.
However, implementing these specifications is challenging due to their purely
theoretical nature. The recurring requirement of metadata as FDOs, for
instance, can lead to complex scenarios where the implementation becomes a
task of balancing the depth of detail with practical usability.

To address these challenges, this abstract proposes the FDO Manager, which
aims to simplify these complexities, providing a pragmatic approach to
implementing FDOs that remains true to the FDO specifications while being
feasible and user-friendly for researchers and data curators. This involves
optimizing the management of research digital objects [BC23], making them more
accessible and usable in various research and data management contexts.

Our approach to FDO implementation is minimalistic yet viable. It offers an
easy-to-use method for employing FDOs, aligning well with the concepts
presented in Reproducible Research Publication Workflow (RRPW) [PBBC22], which
emphasizes the importance of FAIR principles in the research publication
process.

##  2 FDO Manager

The core concept driving the FDO Manager is to present a straightforward and
minimalist solution that not only conforms to the FAIR Principles but also
addresses the practical challenges of managing complex digital objects and
their metadata in a systematic and user-centric manner.

Figure 2 presents a refined view of the FDO Manager’s architecture, which
intricately outlines the interaction between key components within the FDO
ecosystem. The FDO Record, a central node in this structure, is uniquely
identified by a Persistent Identifier (PID), which directly points to the
associated Digital Object (DO), ensuring unambiguous reference and retrieval.

In parallel, the FDO Record is also connected and refers to its metadata
through a distinct PID, establishing a dual-PID system that differentiates
between the DO and its descriptive metadata. The metadata, which describes the
DO, is stored independently in a dedicated Metadata Registry, allowing for
streamlined management and retrieval.

These metadata entries, while connected to the FDO Records, are managed as
separate entities, reinforcing the metadata’s role as standalone FDOs. This
approach aligns with the FAIR principle of interoperability, allowing for the
metadata to be utilized across various systems and applications without direct
dependency on the DOs they describe.

Furthermore, the architecture includes an FDO Registry, a Catalog which serves
as a comprehensive index, registering each FDO Record. This ensures efficient
organization and accessibility of digital objects within the system. The
Operation Registry complements this by detailing the range of operations that
can be executed on a specific FDO Record, encapsulating the dynamic
functionalities the FDO Manager supports.

The FDO Manager was designed as a REST API, combining user-friendliness with
flexibility. The FDO Manager facilitates various operations (POST, GET, PUT,
…) for the efficient management of stored digital object records, including
retrieval, addition, modification and deletion of FDO records. In the pursuit
of interoperability, our approach leverages the widely recognized
Schema.org222https://schema.org/ standards to define the metadata schema for
the proposed FDO records. which ensures a standardized and universally
understood structure for metadata, enhancing the compatibility and exchange of
information within the broader digital ecosystem.

The storage mechanism in the FDO Manager goes beyond capturing individual
FDOs; it is designed to encapsulate the relationships between them as well.
This would ensure FDO records preservation, reproducibility, and overall
utility of digital content

In parallel with the FDO Manager, we have developed a user-friendly FDO
Manager Playground interface. This interface serves as a testing ground to
explore and assess the functionalities of the FDO Manager, allowing for a
comprehensive examination of various endpoints and thorough testing of input
and output processes. For a more in-depth understanding of both the FDO
Manager and the FDO Manager Playground functionalities, detailed information
can be accessed directly through our documentation
website.333https://fdda1.gitlab.io/fdom/

Figure 2 illustrates the proposed schema for FDO Manager classes, presenting
the structure, attributes and relationships of various classes. The schema
introduces four classes: CreativeWork, Service, Person and Organization. which
were derived from schema.org. Various types of relationships are shown,
including simple relationships such as the one between Person and
Organization. Conditional relationships are a notable feature in the model;
for instance, the CreativeWork class can be associated with either a Person or
an Organization, similar to a Service. Additionally, the CreativeWork class
shows a self-referential relationship; citation, allowing a CreativeWork,
whether a dataset, code, or publication, to cite another CreativeWork. These
relationships enable a flexible representation of entities, in addition to
prioritizing scalability and extensibility to accommodate the evolving digital
objects.

##  3 Conclusion

In conclusion, the FDO Manager emerges as a practical and efficient solution
for FDO management. Explored in this paper are its architecture,
implementation details, and alignment with FDO specifications and FAIR
Principles. With a focus on research artifacts, the FDO Manager streamlines
workflows, enhances persistence, and enforces discoverability, contributing
significantly to open science. Its adoption ensures that research artifacts
remain easily accessible, making a valuable contribution to transparent and
collaborative scientific practices.

## Author contributions

O.Z: Writing – Original Draft Preparation, Methodology, Software,
Visualization, Z.B: Writing – Review & Editing, Conceptualization,
Methodology, Supervision, Project Administration N.A: Conceptualization,
Methodology, S.D: Visualization, A.K: Conceptualization, Supervision, Project
Administration, C.L: Writing – Review, Funding Acquisition, O.B:
Conceptualization, Supervision, Project Administration, Funding Acquisition.

## References

  * [BC23] Zeyd Boukhers and Leyla Jael Castro.  Enhancing reproducibility in research through FAIR digital objects.  In Proceedings of the Conference on Research Data Infrastructure, volume 1, 2023.
  * [BdSSSFG23] Luiz Olavo Bonino da Silva Santos, Tiago Prince Sales, Claudenir M. Fonseca, and Giancarlo Guizzardi.  Towards a Conceptual Model for the FAIR Digital Object Framework.  IOS Press, December 2023.
  * [CGH+18] Sandra Collins, Francoise Genova, Natalie Harrower, Simon Hodson, Sarah Jones, Leif Laaksonen, Daniel Mietchen, Rūta Petrauskaitė, and Peter Wittenburg.  Turning FAIR into reality: Final report and action plan from the european commission expert group on FAIR data.  2018\.
  * [DSKW20] Koenraad De Smedt, Dimitris Koureas, and Peter Wittenburg.  FAIR digital objects for science: From data pieces to actionable knowledge units.  Publications, 8(2):21, 2020.
  * [MNV+17] Barend Mons, Cameron Neylon, Jan Velterop, Michel Dumontier, Luiz Olavo Bonino da Silva Santos, and Mark D Wilkinson.  Cloudy, increasingly FAIR; revisiting the FAIR data guiding principles for the european open science cloud.  Information services & use, 37(1):49–56, 2017.
  * [PBBC22] Limor Peer, Claudia Biniossek, Dirk Betz, and Thu-Mai Christian.  Reproducible research publication workflow: A canonical workflow framework and FAIR digital object approach to quality research output.  Data Intelligence, 4(2):306–319, 2022.
  * [WDA+16] Mark D Wilkinson, Michel Dumontier, IJsbrand Jan Aalbersberg, Gabrielle Appleton, Myles Axton, Arie Baak, Niklas Blomberg, Jan-Willem Boiten, Luiz Bonino da Silva Santos, Philip E Bourne, et al.  The FAIR guiding principles for scientific data management and stewardship.  Scientific data, 3(1):1–9, 2016.

##  Appendix A Appendix

Figure 1: The FDO Manager architecture

Figure 2: The FDO Manager Schema