# Stability and Regularity the MGT-Fourier Model with Fractional Coupling

Filomena Barbosa Rodrigues Mendes
Department of Engenhary Electric, Federal University of Technological of
Paraná, Brazil
Fredy M. Sobrado Suárez and Richard S. W. Sanguino Bejarano
Department of Mathematics, Federal University of Technological of Paraná,
Brazil

###### Abstract

In this work we study the stability and regularity for the abstract
differential system made by of the solutions of the Moore-Gilson-Thompson
equation coupled with the classical Fourier heat equation with fractional
coupling. The fractional coupling is given by $A^{\phi}$ where $A$ is a
strictly positive selfadjoint operator on a complex Hilbert space and the
parameter $\phi$ can vary between $0$ and $1$. When we set $\phi=1$ we have
the MGT-Fourier physical model, previous research: 2013[1] and 2022[8]
respectively showed that the semigroup associated with the MGT-Fourier model
$S(t)=e^{t\mathbb{B}}$ is exponentially stable and analytic. The model
abstract of this research is given by: (1)–(3), we show directly that the
semigroup $S(t)$ is exponentially stable for $\phi\in[0,1]$, we also show
directly that for $\phi=1$, $S(t)$ is analytic and using spectral analysis we
show lack of analyticity of $S(t)$ when $\phi\in[0,1)$, in the last part of
our research we address the study of the Gevrey classes of $S(t)$ and show
that for $\phi\in(\frac{1}{2},1)$ there are two families of sharp Gevrey
classes: $s_{1}>\frac{1}{2-2\phi}$ for $\phi\in(1,\frac{3}{4}]$ and
$s_{2}>\frac{1}{2\phi-1}$ for $\phi\in[\frac{3}{4},1)$ and it is also shown
that when the parameter $\phi\in[0,\frac{1}{2}]$ the semigroup $S(t)$ does not
admit Gevrey classes. For the study of the existence, stability and
regularity, semigroup theory is used together with the techniques of the
frequency domain, multipliers and spectral analysis of the fractional operator
$A^{\phi}$ for $\phi\in[0,1]$.

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

.

keyword: Asymptotic behaviour, Stability, Regularity, Gevrey Sharp-class,
Analiticity, MGT-Fourier System.

##  1 Introduction

Various researchers year after year have been devoting their attention to the
study of asymptotic behavior the model Moore-Gilson-Thompson equation
appearing in the context of acoustic wave propagation in viscous thermally
relaxing fluids, in [12], they studied the model abstract:

$\tau u_{ttt}+\alpha u_{tt}+c^{2}Au+bAu_{t}.$

where $A$ be a seftadjoint positive operator on $\mathbb{H}$ with a dense
domain $\mathfrak{D}(A)\subset\mathbb{H}$, show exponential stability requires
$\gamma\equiv\alpha-\frac{\tau c^{2}}{b}>0$. In the complementary region of
the parameters the system is unstable ($\gamma<0$) or marginally stable (
$\gamma=0$). Quite interestingly, it can be used as a model for the vibrations
in a standard linear viscoelastic solid, for it can obtained by
differentiating in time the equation of viscoelasticity with an exponential
kernel (see [7] for more details). The same equation also arises as a model
for the temperature evolution in a type III heat conduction with a relaxation
parameter (see [21]). In 2013 Alves et at. [1] studied the exponential decay
of the MGT-Fourier physical model given by

$\left\\{\begin{array}[]{c}\alpha u_{ttt}+u_{tt}-a^{2}\Delta
u-a^{2}\beta\Delta u_{t}+\eta\Delta\theta=0.\\\
\theta_{t}-\Delta\theta-\alpha\eta\Delta u_{tt}-\eta\Delta
u_{t}=0.\end{array}\right.$

where $x\in\Omega$, $t\in(0,\infty)$. The function $u=u(x,t)$ represents the
vibration of flexible structures and $\theta=\theta(x,t)$ the difference of
temperature between the actual state and a reference temperature. The
constants $\alpha$ and $\beta$ are positive, $\eta$ is the coupling constant.
$\Omega$ is a bounded open connected set in $\mathbb{R}^{n}(n\geq 1)$ havind a
smooth boundary $\Gamma=\partial\Omega$. The initial conditions are given by
(3) and the boundary conditions

$u=0,\qquad u_{t}=0,\qquad\theta=0,\qquad{\rm on}\quad\partial\Omega=\Gamma.$

In more recent research from 2022 [8] they studied the model abstract MGT-
Fourier system:

$\left\\{\begin{array}[]{c}u_{ttt}+\alpha u_{tt}+\beta A^{\rho}u_{t}+\gamma
A^{\rho}u=\eta A\theta,\\\ \theta_{t}+\kappa A\theta=-\eta Au_{tt}-\eta\alpha
Au_{t},\end{array}\right.$

in the subcritical regime $\beta>\frac{\gamma}{\alpha}$, where the operator
$A$ is strictly positive selfadjoint. For any fixed $\rho\in[1,2]$. More
recent research in this direction can be found at [4, 9, 17, 18, 23, 26].

In the last decade, the number of investigations focused on the regularity of
the semigroups $S(t)$ associated with coupled systems has increased
considerably. The interest is basically centered on Differentiability, Gevrey
class and/or Analyticity, among the models considered systems with fractional
couplings, with different types of dissipation (internal, localized) and also
fractional dissipations. One of the motivations for this new research is that
the analytic semigroups associated with linear systems imply that this type of
semigroup admits Gevrey classes and these in turn imply the differentiability
of the semigroup. And depending on the regularity, the semigroup will have the
best asymptotic behavior (Exponential Decay). More recent research in this
direction can be found at [2, 3, 5, 6, 11, 14, 15, 16, 21, 22, 23, 24, 27].

The paper is organized as follows. In section 2, we study the well-posedness
of the system (1)-(3) through semigroup theory. We leave our main
contributions for the last two sections, in the third we dedicate to stability
and the fourth to regularity. in the section (3) we showed that the semigroup
$S(t)=e^{\mathbb{B}t}$ is exponentially stable as $\phi\in[0,1]$. In the last
section (4) of the regularity we studied the analyticity when $\phi=1$,
missing of analyticity when $\phi\in[0,1)$ and we determine two families of
Gevrey classes: $s_{1}>\frac{1}{2-2\phi}$ when
$\phi\in(\frac{1}{2},\frac{3}{4}]$ and $s_{2}>\frac{1}{2\phi-1}$ when
$\phi\in[\frac{3}{4},1)$, it is also shown that both families of Gevrey
classes are Sharp. And when the parameter $\phi$ assumes values in the
interval $[0,\frac{1}{2}]$ it is shown that $S(t)$ does not support a Gevrey
class. We prioritize direct proofs in which we use semigroup-theory
characterization together with frequency domain methods, multipliers, and
spectral analysis of the fractional operator $A^{\phi}$ for $\phi\in[0,1]$.

##  2  Well-posedness of the system

In this section, we will use the semigroup theory to assure the existence and
uniqueness of strong solutions for the system (1)–(3) where the operator $A$
is defined to follow, let
$A\colon\mathfrak{D}(A)\subset\mathbf{H}\to\mathbf{H}$ be a strictly positive
selfadjoint operator with compact inverse on a complex Hilbert space
$\mathbf{H}$, with a continuous embedded domain
$\mathfrak{D}(A)\hookrightarrow\mathbf{H}$. Consider the abstract linear
system

$\displaystyle\alpha u_{ttt}+u_{tt}+a^{2}Au+a^{2}\beta Au_{t}-\eta
A^{\phi}\theta=0,$ (1) $\displaystyle\theta_{t}+A\theta+\alpha\eta
A^{\phi}u_{tt}+\eta A^{\phi}u_{t}=0.$ (2)

The initial conditions are given by

$u(x,0)=u_{0}(x),\quad u_{t}(x,0)=u_{1}(x),\quad
u_{tt}(x,0)=u_{2}(x),\quad\theta(x,0)=\theta_{0}(x),\;{\rm in}\;\Omega.$ (3)

###### Remark 1

It is known that this operator $A$ strictly positive, selfadjoint, has compact
inverse and it has compact resolvent. And the operator $A^{\tau}$ is self-
adjoint positive for all $\tau\in{\mathbb{R}}$, bounded for $\tau\leq 0$, and
the embedding

$\mathfrak{D}(A^{\tau_{1}})\hookrightarrow\mathfrak{D}(A^{\tau_{2}}),$

is continuous for $\tau_{1}>\tau_{2}$. Here, the norm in
$\mathfrak{D}(A^{\tau})$ is given by
$\|u\|_{\mathfrak{D}(A^{\tau})}:=\|A^{\tau}u\|$, $u\in\mathfrak{D}(A^{\tau})$,
where $\langle{\cdot},{\cdot}\rangle$ and $\|\cdot\|$ denotes the inner
product and norm in the complex Hilbert space
$\mathbf{H}=\mathfrak{D}(A^{0})$. Some of the most used spaces at work are
$\mathfrak{D}(A^{\frac{1}{2}})$ and $\mathfrak{D}(A^{-\frac{1}{2}})$.

Taking $v=\alpha u_{t}+u$, the system (1)-(2) can be rewritten as

$\displaystyle v_{tt}+a^{2}Av-a^{2}(\alpha-\beta)Au_{t}-\eta
A^{\phi}\theta=0,$ (4) $\displaystyle\theta_{t}+A\theta+\eta A^{\phi}v_{t}=0.$
(5)

This system (4)-(5), for $\phi=1$ is the model of a system of the coupled
viscoelastic equation coupled with the heat equation given by Fourier’s law.

Taking $u_{t}=v$ and $v_{t}=w$, the initial boundary value problem (1)-(3) can
be reduced to the following abstract initial value problem for a first-order
evolution equation

$\frac{d}{dt}U(t)=\mathbb{B}U(t),\quad U(0)=U_{0},$ (6)

where $U(t)=(u,v,w,\theta)^{T}$, $U_{0}=(u_{0},u_{1},u_{2},\theta_{0})^{T}$
and the operator
$\mathbb{B}\colon\mathfrak{D}(\mathbb{B})\subset\mathbb{H}\to\mathbb{H}$ is
given by

$\mathbb{B}U:=\Big{(}v,w,-\dfrac{1}{\alpha}\big{[}a^{2}Au+a^{2}\beta Av-\eta
A^{\phi}\theta+w\big{]},-\big{[}\eta A^{\phi}v+\alpha\eta
A^{\phi}w+A\theta\big{]}\Big{)}^{T},$ (7)

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

$\mathbb{H}:=[\mathfrak{D}(A^{\frac{1}{2}})]^{2}\times[\mathfrak{D}(A^{0})]^{2},$

it is a Hilbert space with the inner product

$\displaystyle\langle{U_{1}},{U_{2}}\rangle_{\mathbb{H}}$ $\displaystyle:=$
$\displaystyle
a^{2}\alpha(\beta-\alpha)\langle{A^{\frac{1}{2}}v_{1}},{A^{\frac{1}{2}}v_{2}}\rangle+a^{2}\langle{A^{\frac{1}{2}}u_{1}+\alpha
A^{\frac{1}{2}}v_{1}},{A^{\frac{1}{2}}u_{2}+\alpha
A^{\frac{1}{2}}v_{2}}\rangle$ $\displaystyle+\langle{v_{1}+\alpha
w_{1}},{v_{2}+\alpha w_{2}}\rangle+\langle{\theta_{1}},{\theta_{2}}\rangle.$

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

$\|U\|_{\mathbb{H}}^{2}:=a^{2}\alpha(\beta-\alpha)\|A^{\frac{1}{2}}v\|^{2}+a^{2}\|A^{\frac{1}{2}}u+\alpha
A^{\frac{1}{2}}v\|^{2}+\|v+\alpha w\|^{2}+\|\theta\|^{2}.$ (8)

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

$\mathfrak{D}(\mathbb{B}):=\Big{\\{}U\in\mathbb{H}\colon(v,w)\in[\mathfrak{D}(A^{\frac{1}{2}})]^{2},\;a^{2}u+a^{2}\beta
v-\eta A^{\phi-1}\theta\in\mathfrak{D}(A),\\\ \eta A^{\phi-1}v+\alpha\eta
A^{\phi-1}w+\theta\in\mathfrak{D}(A)\Big{\\}}.$ (9)

To show that the operator $\mathbb{B}$ is the generator of a
$C_{0}$-semigroup, we invoke a result from Liu-Zheng’ [20].

######  Theorem 2 (see Theorem 1.2.4 in [20])

Let $\mathbb{B}$ be a linear operator with domain $\mathfrak{D}(\mathbb{B})$
dense in a Hilbert space $\mathbb{H}$. 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 $\mathbb{H}$.

Proof: Let us see that the operator $\mathbb{B}$ given in (7) satisfies the
conditions of this theorem(2). Clearly, we see that $\mathfrak{D}(\mathbb{B})$
is dense in $\mathbb{H}$. Taking the inner product of $\mathbb{B}U$ with $U$,
we have

$\text{Re}\langle{\mathbb{B}U},{U}\rangle_{\mathbb{H}}=-a^{2}(\beta-\alpha)\|A^{\frac{1}{2}}v\|^{2}-\|A^{\frac{1}{2}}\theta\|^{2}\leq
0,\quad\forall\ U\in\mathfrak{D}(\mathbb{B}),$ (10)

and since $\beta>\alpha$, the operator $\mathbb{B}$ is dissipative.

To complete the conditions of the above theorem, it remains to show that
$0\in\rho(\mathbb{B})$. Therefore we must show that $(0I-\mathbb{B})^{-1}$
exists and is bounded in $\mathbb{H}$. We will first prove that
$(0I-\mathbb{B})^{-1}$ exists, then it must be proved that $\mathbb{B}$ is
bijective. Here we are going to affirm that $\mathbb{B}$ is surjective, then
for all $F=(f^{1},f^{2},f^{3},f^{4})^{T}\in\mathbb{H}$ the stationary problem
$\mathbb{B}U=F$ has a solution for
$U=(u,v,w,\theta)^{T}\in\mathfrak{D}(\mathbb{B})$. From definition of the
operator $\mathbb{B}$ in (7), this system can be written as

$v=f^{1}\in\mathfrak{D}(A^{\frac{1}{2}}),\hskip
85.35826ptw=f^{2}\quad\in\quad\mathfrak{D}(A^{\frac{1}{2}})$ $a^{2}Au-\eta
A^{\phi}\theta+a^{2}\beta Av+w=-\alpha f^{3}\in\mathfrak{D}(A^{0}),\\\
A\theta+\eta A^{\phi}v+\alpha\eta
A^{\phi}w=-f^{4}\quad\in\quad\mathfrak{D}(A^{0}).$ (11)

equivalent $v=f^{1},\qquad w=f^{2}$ and

$\displaystyle a^{2}Au$ $\displaystyle=$ $\displaystyle-\eta
A^{\phi-1}f_{4}-\eta^{2}A^{2\phi-1}f^{1}-\alpha\eta^{2}A^{2\phi-1}f^{2}-\alpha
f^{3}-a^{2}\beta Af^{1}-f^{2},$ (12) $\displaystyle A\theta$ $\displaystyle=$
$\displaystyle-f^{4}-\eta A^{\phi}f^{1}-\alpha\eta A^{\phi}f^{2}.$ (13)

From these equations, this problem can be placed in a variational formulation:
$v,w\in[\mathcal{D}(A^{\frac{1}{2}})]^{2}$ and we write the last two equations
(12)-(13) in a variational form, using the sesquilinear form $b$ :

$b(u,\theta;\psi_{1},\psi_{2})=\langle g_{1},g_{2};\psi_{1},\psi_{2}\rangle$
(14)

where $g_{1}=-\eta
A^{\phi-1}f_{4}-\eta^{2}A^{2\phi-1}f^{1}-\alpha\eta^{2}A^{2\phi-1}f^{2}-\alpha
f^{3}-a^{2}\beta Af^{1}-f^{2},g_{2}=-f^{4}-\eta A^{\phi}f^{1}-\alpha\eta
A^{\phi}f^{2}$ and the sesquilinear form $b$ is given by

$b(u,\theta;\psi_{1},\psi_{2})=a^{2}\langle{A^{\frac{1}{2}}u},{A^{\frac{1}{2}}\psi_{1}}\rangle+\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}}\psi_{2}}\rangle.$
(15)

as $\mathfrak{D}(A)\hookrightarrow\mathfrak{D}(A^{\frac{1}{2}})\hookrightarrow
D(A^{0})\hookrightarrow\mathfrak{D}(A^{-\frac{1}{2}})\hookrightarrow\mathfrak{D}(A^{-1})$,
this sesquilinear form is coercive in the space
$[\mathfrak{D}(A^{\frac{1}{2}})]^{2}$. As
$g_{1},g_{2}\in[\mathfrak{D}(A^{-\frac{1}{2}})]^{2}$ from Lax-Milgram’s Lemma
the variational form has unique solution
$(u,\theta)\in[\mathfrak{D}(A^{\frac{1}{2}})\times\mathfrak{D}(A^{\frac{1}{2}})]$
and it satisfies (12)-(13), such that (14) is verify for
$(\psi_{1},\psi_{2})\in[\mathfrak{D}(A^{\frac{1}{2}})]^{2}$. From (15) and for
all $(\psi_{1},\psi_{2})\in[\mathfrak{D}(A^{\frac{1}{2}})]^{2}$, we have

$a^{2}\langle{A^{\frac{1}{2}}u},{A^{\frac{1}{2}}\psi_{1}}\rangle+\langle{A^{\frac{1}{2}}\theta},{A^{\frac{1}{2}}\psi_{2}}\rangle=\langle{-f^{4}-\eta
A^{\phi}f^{1}-\alpha\eta A^{\phi}f^{2}},{\psi_{2}}\rangle\\\ +\langle{-\eta
A^{\phi-1}f_{4}-\eta^{2}A^{2\phi-1}f^{1}-\alpha\eta^{2}A^{2\phi-1}f^{2}-\alpha
f^{3}-a^{2}\beta Af^{1}-f^{2}},{\psi_{1}}\rangle,$ (16)

choosing $\psi_{2}=0\in\mathfrak{D}(A^{\frac{1}{2}})$ from (16) and
$\forall\psi_{1}\in\mathfrak{D}(A^{\frac{1}{2}})$, we have

$a^{2}\langle{A^{\frac{1}{2}}u},{A^{\frac{1}{2}}\psi_{1}}\rangle=\langle{-\eta
A^{\phi-1}f_{4}-\eta^{2}A^{2\phi-1}f^{1}-\alpha\eta^{2}A^{2\phi-1}f^{2}-\alpha
f^{3}-a^{2}\beta Af^{1}-f^{2}},{\psi_{1}}\rangle,$

then

$a^{2}Au=-\eta
A^{\phi-1}f_{4}-\eta^{2}A^{2\phi-1}f^{1}-\alpha\eta^{2}A^{2\phi-1}f^{2}-\alpha
f^{3}\\\ -a^{2}\beta Af^{1}-f^{2}\quad{\rm
in}\quad[\mathfrak{D}(A^{\frac{1}{2}})]^{\prime}=\mathfrak{D}(A^{-\frac{1}{2}}).$
(17)

On the other hand, chossing $\psi_{1}=0\in\mathfrak{D}(A^{\frac{1}{2}})$ from
(17) and $\forall\psi_{2}\in\mathfrak{D}(A^{\frac{1}{2}})$, we have

$\langle{A\theta},{\psi_{2}}\rangle=\langle{-f^{4}-\eta
A^{\phi}f^{1}-\alpha\eta A^{\phi}f^{2}},{\psi_{2}}\rangle,$

then

$A\theta=-f^{4}-\eta A^{\phi}f^{1}-\alpha\eta A^{\phi}f^{2}\quad{\rm
in}\quad[\mathfrak{D}(A^{\frac{1}{2}})]^{\prime}=\mathfrak{D}(A^{-\frac{1}{2}}).$
(18)

(17) and (18) are solutions in the weak sense. From
$F=(f^{1},f^{2},f^{3},f^{4})^{T}\in\mathbb{H}$ and $0\leq\phi\leq 1$ implies
that $-\frac{1}{2}\leq\frac{2\phi-1}{2}\leq\frac{1}{2}$, we have
$\mathfrak{D}(A^{\frac{1}{2}})\hookrightarrow\mathfrak{D}(A^{\frac{2\phi-1}{2}})\hookrightarrow\mathfrak{D}(A^{-\frac{1}{2}})\hookrightarrow\mathfrak{D}(A^{\frac{1-2\phi}{2}})\hookrightarrow\mathfrak{D}(A^{\frac{1}{2}})$,
from (17), we have

$u=\dfrac{1}{a^{2}}\bigg{[}-\eta
A^{\phi-2}f_{4}-\eta^{2}A^{2\phi-2}f^{1}-\alpha\eta^{2}A^{2\phi-2}f^{2}-\alpha
A^{-1}f^{3}\\\ -a^{2}\beta f^{1}-A^{-1}f^{2}\bigg{]}\quad{\rm
in}\quad\mathfrak{D}(A^{\frac{1}{2}}).$ (19)

Now from $0\leq\phi\leq 1$ implies that $2\geq 2-\phi\geq
1\geq\frac{2-\phi}{2}\geq\frac{1}{2}$, we have
$\mathfrak{D}(A^{\frac{1}{2}})\hookrightarrow\mathfrak{D}(A^{\frac{2-\phi}{2}})\hookrightarrow\mathfrak{D}(A)\hookrightarrow\mathfrak{D}(A^{0})\hookrightarrow\mathfrak{D}(A^{\phi-1})\hookrightarrow\mathfrak{D}(A^{-1})$,
from (18), we have

$\theta=-A^{-1}f^{4}-\eta A^{\phi-1}f^{1}-\alpha\eta A^{\phi-1}f^{2}\quad{\rm
in}\quad\mathfrak{D}(A^{\frac{1}{2}}).$ (20)

then $a^{2}u+a^{2}\beta v-\eta A^{\phi-1}\theta\in\mathfrak{D}(A)$ and $\eta
A^{\psi-1}v+\alpha\eta A^{\psi-1}w+\theta\in\mathfrak{D}(A)$. The injectivity
of $\mathbb{B}$ follows from the uniqueness given by the Lemma of Lax-
Milgram’s. It remains to show that $\mathbb{B}^{-1}$ is a bounded operator.
From $v=f^{1}$, $w=f^{2}$ equations (19) and (20) and (8), we get

$\|U\|_{\mathbb{H}}^{2}=\alpha
a^{2}(\beta-\alpha)\|A^{\frac{1}{2}}f^{1}\|^{2}+a^{2}\|\alpha
A^{\frac{1}{2}}f^{1}+\dfrac{1}{a^{2}}\big{[}-\eta
A^{\phi-\frac{3}{2}}f_{4}-\eta^{2}A^{2\phi-\frac{3}{2}}f^{1}-\alpha\eta^{2}A^{2\phi-\frac{3}{2}}f^{2}\\\
-\alpha A^{-\frac{1}{2}}f^{3}-a^{2}\beta
A^{\frac{1}{2}}f^{1}-A^{-\frac{1}{2}}f^{2}\big{]}\|^{2}+\|\alpha
f^{2}+f^{1}\|^{2}\\\ +\|-A^{-1}f^{4}-\eta A^{\phi-1}f^{1}-\alpha\eta
A^{\phi-1}f^{2}\|^{2}$ (21)

Using norm $\|F\|_{\mathbb{H}}$ and applying inequalities Cauchy-Schwarz,
Young and Hölder and applying continuous embedding
$\mathfrak{D}(A^{\tau_{2}})\hookrightarrow\mathfrak{D}(A^{\tau_{1}}),\;\tau_{2}>\tau_{1}$,
after making various estimates, we obtain

$\|U\|^{2}_{\mathbb{H}}=\|\mathbb{B}^{-1}F\|_{\mathbb{H}}^{2}\leq
C\|F\|^{2}_{\mathbb{H}}.$

Therefore $\mathbb{B}^{-1}$ is bounded. So we come to the end of the proof of
this theorem. $\Box$
As a consequence of the previous Theorem(2), we obtain

###### Theorem 3

Given $U_{0}\in\mathbb{H}$ there exists a unique weak solution $U$ to the
problem (6) satisfying

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

Futhermore, if $U_{0}\in\mathfrak{D}(\mathbb{B}^{k}),\;k\in\mathbb{N}$, then
the solution $U$ of (6) satisfies

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

##  3 Stability

###  3.1 Exponential decay of the semigroup $S(t)=e^{t\mathbb{B}}$

In this section, we will study the asymptotic behavior of the semigroup of the
system (1)-(3). We will use the following spectral characterization of
exponential stability of semigroups due to Gearhart[10](Theorem 1.3.2 book of
Liu-Zheng [20]).

######  Theorem 4 (see [20])

Let $S(t)=e^{t\mathbb{B}}$ be a $C_{0}$-semigroup of contractions on a Hilbert
space $\mathbb{H}$. Then $S(t)$ is exponentially stable if and only if

$\rho(\mathbb{B})\supseteq\\{i\lambda/\lambda\in{\mathbb{R}}\\}\equiv
i{\mathbb{R}}$ (22)

and

$\limsup\limits_{|\lambda|\to\infty}\|(i\lambda
I-\mathbb{B})^{-1}\|_{\mathcal{L}(\mathbb{H})}<\infty$ (23)

holds.

###### Remark 5

Note that to show the condition (23) it is enough to show that: Let
$\delta>0$. There exists a constant $C_{\delta}>0$ such that the solutions of
the system (1)-(3) for $|\lambda|>\delta$, satisfy the inequality

$\dfrac{\|U\|_{\mathbb{H}}}{\|F\|_{\mathbb{H}}}\leq
C_{\delta}\quad\Longleftrightarrow\quad\|U\|^{2}_{\mathbb{H}}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}<\infty.$ (24)

In view this theorem(4), we will try to obtain some estimates for the solution
$U=(u,v,w,\theta)^{T}$ of the system $(i\lambda I-\mathbb{B})U=F$, where
$\lambda\in{\mathbb{R}}$ and $F=(f^{1},f^{2},f^{3},f^{4})^{T}\in\mathbb{H}$.
This system, written in components, reads

$\displaystyle i\lambda u-v$ $\displaystyle=$ $\displaystyle f^{1}\quad{\rm
in}\quad\mathfrak{D}(A^{\frac{1}{2}})$ (25) $\displaystyle i\lambda v-w$
$\displaystyle=$ $\displaystyle f^{2}\quad{\rm
in}\quad\mathfrak{D}(A^{\frac{1}{2}})$ (26) $\displaystyle i\alpha\lambda
w+a^{2}Au+a^{2}\beta Av-\eta A^{\phi}\theta+w$ $\displaystyle=$
$\displaystyle\alpha f^{3}\quad{\rm in}\quad\mathfrak{D}(A^{0})$ (27)
$\displaystyle i\lambda\theta+\eta A^{\phi}v+\alpha\eta A^{\phi}w+A\theta$
$\displaystyle=$ $\displaystyle f^{4}\quad{\rm in}\quad\mathfrak{D}(A^{0})$
(28)

From (10), we have the first estimate

$\displaystyle
a^{2}(\beta-\alpha)\|A^{\frac{1}{2}}v\|^{2}+\|A^{\frac{1}{2}}\theta\|^{2}$
$\displaystyle=$
$\displaystyle|a^{2}(\beta-\alpha)\|A^{\frac{1}{2}}v\|^{2}+\|A^{\frac{1}{2}}\theta\|^{2}|=|-{\rm
Re}\langle{\mathbb{B}U},{U}\rangle|$ $\displaystyle=$ $\displaystyle|{\rm
Re}\\{i\lambda\|U\|^{2}-\langle{\mathbb{B}U},{U}\rangle\\}|=|{\rm
Re}\\{\langle{(i\lambda I-\mathbb{B})U},{U}\rangle\\}|$ $\displaystyle=$
$\displaystyle|\langle{F},{U}\rangle|\leq\|F\|_{\mathbb{H}}\|\|U\|_{\mathbb{H}}.$

Therefore

$a^{2}(\beta-\alpha)\|A^{\frac{1}{2}}v\|^{2}+\|A^{\frac{1}{2}}\theta\|^{2}\leq\|F\|_{\mathbb{H}}\|\|U\|_{\mathbb{H}}.$
(29)

Next we show some lemmas that will lead us to the proof of the main theorem of
this section.

###### Lemma 6

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$|\lambda|\|A^{\frac{1}{2}}u\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad
0\leq\phi\leq 1.$ (30)

Proof: Taking the duality product between equation(25) and $Au$, taking
advantage of the self-adjointness of the powers of the operator $A$, we obtain

$i\lambda\|A^{\frac{1}{2}}u\|^{2}=\langle{A^{\frac{1}{2}}v},{A^{\frac{1}{2}}u}\rangle+\langle{A^{\frac{1}{2}}f^{1}},{A^{\frac{1}{2}}u}\rangle$

Applying Cauchy-Schwarz and Young inequalities and norms $\|F\|_{\mathbb{H}}$
and $\|U\|_{\mathbb{H}}$, for $\varepsilon>0$, exists $C_{\varepsilon}>0$ such
that

$|\lambda|\|A^{\frac{1}{2}}u\|^{2}\leq
C_{\varepsilon}\|A^{\frac{1}{2}}v\|^{2}+\varepsilon\|A^{\frac{1}{2}}u\|^{2}+C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}.$
(31)

Therefore for $|\lambda|>1$, we have
$\varepsilon\|A^{\frac{1}{2}}u\|^{2}\leq\varepsilon|\lambda|\|A^{\frac{1}{2}}u\|^{2}$.
Finish proof of this lemma.

$\Box$

###### Theorem 7

The semigroup $S(t)=e^{t\mathbb{B}}$ is exponentially stable when the
parameter $\phi$ assumes values in the interval $[0,1]$.

Proof: Let’s first check the condition (24), we will start by proving in the
following lemma the estimate of the term
$a^{2}\|A^{\frac{1}{2}}u+A^{\frac{1}{2}}v\|^{2}$ of $\|U\|_{\mathbb{H}}^{2}$:

###### Lemma 8

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$\|A^{\frac{1}{2}}u+A^{\frac{1}{2}}v\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad
0\leq\phi\leq 1.$ (32)

Proof: As $\|A^{\frac{1}{2}}u+A^{\frac{1}{2}}v\|^{2}\leq
C(\|A^{\frac{1}{2}}u\|^{2}+\|A^{\frac{1}{2}}v\|^{2})^{2}$ and for
$|\lambda|>1$ from Lemma(6) we have $\|A^{\frac{1}{2}}u\|\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}$, therefore additionally using
the estimate (29), we finish to proof this is lemma.

$\Box$

###### Lemma 9

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$\|w\|^{2}\leq C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm
for}\qquad 0\leq\phi\leq 1.$ (33)

Proof: Taking the duality product between equation (26) and $w$ and using
(27), taking advantage of the self-adjointness of the powers of the operator
$A$, we obtain

$\displaystyle\|w\|^{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{\alpha}\langle{v},{i\alpha\lambda
w}\rangle-\langle{f^{2}},{w}\rangle$ (34) $\displaystyle=$
$\displaystyle-\frac{1}{\alpha}\langle{v},{-a^{2}Au-a^{2}\beta Av+\eta
A^{\phi}\theta-w+\alpha f^{3}}\rangle-\langle{f^{2}},{w}\rangle$
$\displaystyle=$
$\displaystyle\dfrac{a^{2}}{\alpha}\langle{A^{\frac{1}{2}}v},{A^{\frac{1}{2}}u}\rangle+\dfrac{a^{2}\beta}{\alpha}\|A^{\frac{1}{2}}v\|^{2}-\dfrac{\eta}{\alpha}\langle{A^{\frac{1}{2}}v},{A^{\phi-\frac{1}{2}}\theta}\rangle+\dfrac{1}{\alpha}\langle{v},{w}\rangle-\dfrac{1}{\alpha}\langle{f^{2}},{w}\rangle.$

Applying estimates (29), (30) of Lemma(6) for $|\lambda|>1$, we finish to
proof this is lemma.

$\Box$
The following lemma estimates the term $\|v+\alpha w\|^{2}$ of
$\|U\|_{\mathbb{H}}^{2}$:

###### Lemma 10

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$\|v+\alpha w\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad
0\leq\phi\leq 1.$ (35)

Proof: As $\|v+\alpha w\|^{2}\leq C(\|v\|^{2}+\alpha^{2}\|w\|^{2})^{2}$ and as
$0\leq\frac{1}{2}$ applying continuous embedding, estimative (29) and
Lemma(9), we finish to proof this is lemma. $\Box$

Finally, using the Lemmas (8)-(10) and estimates (29), we get

$\|U\|_{\mathbb{H}}^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm for}\quad
0\leq\phi\leq 1.$ (36)

Therefore the condition (23) for $\phi\in[0,1]$ is verified. Next we show the
condition (22) of Theorem(4).

###### Lemma 11

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

$i\hskip 0.5pt\mathbb{R}\subset\varrho(\mathbb{B}).$ (37)

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 $U_{n}=(u_{n},v_{n},w_{n},\theta_{n})\in D(\mathbb{B})$ 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 (24), we have

$\|U_{n}\|^{2}_{\mathbb{H}}=a^{2}\alpha(\beta-\alpha)\|A^{\frac{1}{2}}u_{n}\|^{2}+a^{2}\|A^{\frac{1}{2}}u_{n}+\alpha
A^{\frac{1}{2}}v_{n}\|^{2}\\\ +\|v_{n}+\alpha
w_{n}\|^{2}+\|\theta_{n}\|^{2}\leq
C_{\delta}\|F_{n}\|_{\mathbb{H}}\|U_{n}\|_{\mathbb{H}}=C_{\delta}\|F_{n}\|_{\mathbb{H}}\rightarrow
0.$ (38)

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.

$\Box$
Therefore the semigroup $S(t)=e^{t\mathbb{B}}$ is exponentially stable for
$\phi\in[0,1]$, thus we finish the proof of this theorem. $\Box$

##  4 Regularity

###  4.1 Study of Analyticity and Lack Analyticity of the semigroup
$S(t)=e^{t\mathbb{B}}$

In this subsection we will show that the semigroup $S(t)$, is analytic for the
parameter $\phi=1$ and is not analytic when the parameter assumes values in
the interval $[0,1)$.

The following theorem characterizes the analyticity the semigroups $S(t)$.

######  Theorem 12 (see [20])

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

$\rho(\mathbb{B})\supseteq\\{i\lambda/\lambda\in{\mathbb{R}}\\}\equiv
i{\mathbb{R}}.$ (39)

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

$\limsup\limits_{|\lambda|\to\infty}\|\lambda(i\lambda
I-\mathbb{B})^{-1}\|_{\mathcal{L}(\mathbb{H})}<\infty,$ (40)

holds.

####  4.1.1 Analyticity

###### Theorem 13

The semigroup $S(t)=e^{t\mathbb{B}}$ is analytic for $\phi=1$.

Proof: From Lemma(11), (39) is verified. Therefore, it remains to prove (40),
for that it is enough to show that, let $\delta>0$. There exists a constant
$C_{\delta}>0$ such that the solutions of the system (1)-(3) for
$|\lambda|>\delta$, satisfy the inequality

$|\lambda|\dfrac{\|U\|_{\mathbb{H}}}{\|F\|_{\mathbb{H}}}\leq
C_{\delta}\quad\Longleftrightarrow\quad|\lambda|\|U\|^{2}_{\mathbb{H}}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}<\infty.$ (41)

Next we will prove some important lemmas to prove the analyticity of $S(t)$
when the parameter $\phi=1$.

###### Lemma 14

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$|\lambda|\|\theta\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad
0\leq\phi\leq 1.$ (42)

Proof: Taking the duality product between equation (28) and $\lambda
A^{-1}\theta$, taking advantage of the self-adjointness of the powers of the
operator $A$, we obtain

$\displaystyle\lambda\|\theta\|^{2}$ $\displaystyle=$
$\displaystyle-i\lambda^{2}\|A^{-\frac{1}{2}}\theta\|^{2}-\eta\langle{-iw-
if^{2}},{A^{\phi-1}\theta}\rangle-\eta\langle{\alpha\lambda
w},{A^{\phi-1}\theta}\rangle+\langle{f^{4}},{\lambda A^{-1}\theta}\rangle$
(43) $\displaystyle=$
$\displaystyle-i\lambda^{2}\|A^{-\frac{1}{2}}\theta\|^{2}+i\eta\langle{w},{A^{\phi-1}\theta}\rangle+i\eta\langle{f^{2}},{A^{\phi-1}\theta}\rangle-i\eta
a^{2}\langle{A^{\frac{1}{2}}u},{A^{\phi-\frac{1}{2}}\theta}\rangle$
$\displaystyle-i\eta
a^{2}\beta\langle{A^{\frac{1}{2}}v},{A^{\phi-\frac{1}{2}}\theta}\rangle+i\eta^{2}\|A^{\frac{2\phi-1}{2}}\theta\|^{2}-i\eta\langle{w},{A^{\phi-1}\theta}\rangle+i\eta\alpha\langle{f^{3}},{A^{\phi-1}\theta}\rangle$
$\displaystyle-i\eta\langle{f^{4}},{A^{\phi-1}v}\rangle-i\alpha\eta\langle{f^{4}},{A^{\phi-1}w}\rangle-i\langle{f^{4}},{\theta}\rangle+i\|A^{-\frac{1}{2}}f^{4}\|^{2}.$

Taking the real part of (43) and simplifying, we get

$\displaystyle\lambda\|\theta\|^{2}$ $\displaystyle=$ $\displaystyle{\rm
Re}\\{i\eta\langle{f^{2}},{A^{\phi-1}\theta}\rangle-i\eta
a^{2}\langle{A^{\frac{1}{2}}u},{A^{\phi-\frac{1}{2}}\theta}\rangle-i\eta
a^{2}\beta\langle{A^{\frac{1}{2}}v},{A^{\phi-\frac{1}{2}}\theta}\rangle$ (44)
$\displaystyle+i\eta\alpha\langle{f^{3}},{A^{\phi-1}\theta}\rangle-i\eta\langle{f^{4}},{A^{\phi-1}v}\rangle-i\alpha\eta\langle{f^{4}},{A^{\phi-1}w}\rangle-i\langle{f^{4}},{\theta}\rangle\\}.$

As $|{\rm Re}\\{-i\eta
a^{2}\langle{A^{\frac{1}{2}}u},{A^{\phi-\frac{1}{2}}\theta}\rangle\\}|\leq
C\\{\|A^{\frac{1}{2}}u\|^{2}+\|A^{\phi-\frac{1}{2}}\theta\|^{2}\\}$. And for
$|\lambda|>1$ and $0\leq\phi\leq 1$ we have, $\phi-1\leq 0$ and
$\phi-\frac{1}{2}\leq\frac{1}{2}$ applying continuous embedding
$\mathfrak{D}(A^{\tau_{2}})\hookrightarrow\mathfrak{D}(A^{\tau_{1}}),\;\tau_{2}>\tau_{1}$,
estimates (29), (30) and applying Cauchy-Schwarz and Young inequalities, we
finish to proof this is lemma.

$\Box$

###### Lemma 15

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$\|A^{\frac{2\phi-1}{2}}w\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad
0\leq\phi\leq 1.$ (45)

Proof: Taking the duality product between equation (28) and $A^{\phi-1}w$,
taking advantage of the self-adjointness of the powers of the operator $A$, we
obtain

$\displaystyle\eta\alpha\|A^{\frac{2\phi-1}{2}}w\|^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\langle{A^{\phi-1}\theta},{i\alpha\lambda
w}\rangle-\eta\langle{A^{\frac{2\phi-1}{2}}v},{A^{\frac{2\phi-1}{2}}w}\rangle-\langle{A^{\frac{1}{2}}\theta},{A^{\frac{2\phi-1}{2}}w}\rangle+\langle{f^{4}},{A^{\phi-1}w}\rangle$
$\displaystyle=$
$\displaystyle-\dfrac{a^{2}}{\alpha}\langle{A^{\frac{2\phi-1}{2}}\theta},{A^{\frac{1}{2}}u}\rangle-\dfrac{a^{2}\beta}{\alpha}\langle{A^{\frac{2\phi-1}{2}}\theta},{A^{\frac{1}{2}}v}\rangle+\dfrac{\eta}{\alpha}\|A^{\frac{2\phi-1}{2}}\theta\|^{2}$
$\displaystyle-\dfrac{1}{\alpha}\langle{A^{-\frac{1}{2}}\theta},{A^{\frac{2\phi-1}{2}}w}\rangle+\langle{A^{\phi-1}\theta},{f^{3}}\rangle-\eta\langle{A^{\frac{2\phi-1}{2}}v},{A^{\frac{2\phi-1}{2}}w}\rangle$
$\displaystyle-\langle{A^{\frac{1}{2}}\theta},{A^{\frac{2\phi-1}{2}}w}\rangle+\langle{f^{4}},{A^{\phi-1}w}\rangle$

Applying Cauchy-Schwarz and Young inequalities and norms $\|F\|_{\mathbb{H}}$
and $\|U\|_{\mathbb{H}}$ and estimative (29), for $\varepsilon>0$, exists
$C_{\varepsilon}>0$ such that

$\|A^{\frac{2\phi-1}{2}}w\|^{2}\leq
C\\{\|A^{\frac{1}{2}}v\|^{2}+\|A^{\frac{1}{2}}u\|^{2}+\|A^{\frac{1}{2}}\theta\|^{2}\\}+C_{\varepsilon}\|A^{\frac{1}{2}}v\|^{2}\\\
+\varepsilon\|A^{\frac{2\phi-1}{2}}w\|^{2}+\|\theta\|\|f^{3}\|+\|f^{4}\|\|w\|.$
(47)

For $|\lambda|>1$, we have
$\|A^{\frac{1}{2}}u\|^{2}\leq|\lambda|\|A^{\frac{1}{2}}u\|^{2}$, from Lemma(6)
and using norms $\|F\|_{\mathbb{H}}$ and $\|U\|_{\mathbb{H}}$ finish proof of
this lemma.

$\Box$

###### Lemma 16

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$|\lambda|\|w\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad\phi=1.$
(48)

Proof: Let us decompose $w$ as $w=w_{1}+w_{2}$, such that

$i\alpha\lambda w_{1}+Aw_{1}=\alpha f^{3}\quad{\rm
in}\quad\mathfrak{D}(A^{0})\qquad{\rm and}\\\ i\alpha\lambda
w_{2}=-a^{2}Au-a^{2}\beta Av+\eta A\theta-w+Aw_{1}\quad{\rm
in}\quad\mathfrak{D}(A^{0}).$ (49)

Firstly, applying the product duality the first equation in (49)1 by $w_{1}$,
we have

$i\alpha\lambda\|w_{1}\|^{2}+\|A^{\frac{1}{2}}w_{1}\|^{2}=\langle{\alpha
f^{3}},{w_{1}}\rangle.$ (50)

Taking first the imaginary part of (50) and in the sequence the real part and
applying Cauchy-Schwarz inequality, we have

$\displaystyle\|w_{1}\|\leq
C\dfrac{[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}}{|\lambda|^{\frac{1}{2}}}\qquad{\rm
and}\qquad\|A^{\frac{1}{2}}w_{1}\|\leq
C[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}$ (51)

From $A^{\frac{1}{2}}w_{2}=A^{\frac{1}{2}}w-A^{\frac{1}{2}}w_{1}$, from
estimates (51)2 and Lemma(15) for $\phi=1$, we have

$\|A^{\frac{1}{2}}w_{2}\|\leq
C_{\delta}\\{\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\\}^{\frac{1}{2}}.$ (52)

Applying the operator $A^{-\frac{1}{2}}$ in the second equation of (49) and
using Lemmas (6) and (15), estimates (29) and (51)2, we have

$|\lambda|\|A^{-\frac{1}{2}}w_{2}||\leq
C\\{\|A^{\frac{1}{2}}u\|+\|A^{\frac{1}{2}}v\|+\|A^{\frac{1}{2}}\theta\|+\|A^{-\frac{1}{2}}w\|+\|A^{\frac{1}{2}}w_{1}\|\\}\\\
\leq C_{\delta}\\{\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\\}^{\frac{1}{2}}.$ (53)

Now, by Lion’s interpolations inequality ($-\frac{1}{2}<0<\frac{1}{2}$) and
estimatives (52) and (53), we derive

$\displaystyle\|w_{2}\|$ $\displaystyle\leq$ $\displaystyle
C\|A^{-\frac{1}{2}}w_{2}\|^{\frac{1}{2}}\|A^{\frac{1}{2}}w_{2}\|^{\frac{1}{2}}\leq
C_{\delta}|\lambda|^{-\frac{1}{2}}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\quad{\rm
for}\quad\phi=1.$

On the other hand, as $\|w\|\leq C\\{\|w_{1}\|+\|w_{2}\|\\}$ and
$-\frac{1}{2}\geq-\frac{1}{2}$ using estimative (51)1, we have

$|\lambda|\|w\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm for}\quad\phi=1.$ (54)

We finish the proof of this lemma.

$\Box$

###### Lemma 17

Let $\delta>0$. There exists $C_{\delta}>0$ such that the solutions of the
system (1)-(3) for $|\lambda|>\delta$, satisfy

$|\lambda|\|A^{\frac{1}{2}}v\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\qquad{\rm for}\qquad\phi=1.$
(55)

Proof: Taking the duality product between equation (27) and $w$ and using (25)
and (26), taking advantage of the self-adjointness of the powers of the
operator $A$, we obtain

$\displaystyle i\alpha\lambda\|w\|^{2}$ $\displaystyle=$
$\displaystyle-\|w\|^{2}-a^{2}\langle{Au},{i\lambda
v-f^{2}}\rangle-a^{2}\beta\langle{Av},{i\lambda v-f^{2}}\rangle$ (56)
$\displaystyle+\eta\langle{A^{\phi}\theta},{w}\rangle+\alpha\langle{f^{3}},{w}\rangle$
$\displaystyle=$ $\displaystyle-\|w\|^{2}+a^{2}\langle{A(i\lambda
u)},{v}\rangle+a^{2}\langle{A^{\frac{1}{2}}u},{A^{\frac{1}{2}}f^{2}}\rangle+i\lambda
a^{2}\beta\|A^{\frac{1}{2}}v\|^{2}$
$\displaystyle+a^{2}\beta\langle{A^{\frac{1}{2}}v},{A^{\frac{1}{2}}f^{2}}\rangle+\eta\langle{A^{\phi}\theta},{w}\rangle+\alpha\langle{f^{3}},{w}\rangle$
$\displaystyle=$
$\displaystyle-\|w\|^{2}+a^{2}\|A^{\frac{1}{2}}v\|^{2}+a^{2}\langle{A^{\frac{1}{2}}f^{1}},{A^{\frac{1}{2}}v}\rangle+a^{2}\langle{A^{\frac{1}{2}}u},{A^{\frac{1}{2}}f^{2}}\rangle$
$\displaystyle+i\lambda
a^{2}\beta\|A^{\frac{1}{2}}v\|^{2}+a^{2}\beta\langle{A^{\frac{1}{2}}v},{A^{\frac{1}{2}}f^{2}}\rangle+\eta\langle{A^{\phi}\theta},{w}\rangle+\alpha\langle{f^{3}},{w}\rangle.$

On the other hand, taking the duality product between $\frac{1}{\alpha}\theta$
and equation (28), and using (25) and (26), taking advantage of the self-
adjointness of the powers of the operator $A$, we obtain

$\displaystyle\eta\langle{A^{\phi}\theta},{w}\rangle=-\dfrac{i\lambda}{\alpha}\|\theta\|^{2}-\dfrac{\eta}{\alpha}\langle{A^{\phi-\frac{1}{2}}\theta},{A^{\frac{1}{2}}v}\rangle-\dfrac{1}{\alpha}\|A^{\frac{1}{2}}\theta\|^{2}+\dfrac{1}{\alpha}\langle{\theta},{f^{4}}\rangle.$
(57)

Using (57) in (56) and taking imaginary part, for $0\leq\phi\leq 1$, we get

$\displaystyle\lambda a^{2}\beta\|A^{\frac{1}{2}}v\|^{2}$ $\displaystyle=$
$\displaystyle\alpha\lambda\|w\|^{2}+\dfrac{\lambda}{\alpha}\|\theta\|^{2}+{\rm
Im}\\{-a^{2}\langle{A^{\frac{1}{2}}f^{1}},{A^{\frac{1}{2}}v}\rangle-a^{2}\langle{A^{\frac{1}{2}}u},{A^{\frac{1}{2}}f^{2}}\rangle$
$\displaystyle-a^{2}\beta\langle{A^{\frac{1}{2}}v},{A^{\frac{1}{2}}f^{2}}\rangle-\alpha\langle{f^{3}},{w}\rangle+\dfrac{\eta}{\alpha}\langle{A^{\phi-\frac{1}{2}}\theta},{A^{\frac{1}{2}}v}\rangle-\dfrac{1}{\alpha}\langle{\theta},{f^{4}}\rangle\\}.$

Applying Cauchy-Schwarz and Young inequalities, Lemmas(14) and (16), norms
$\|F\|_{\mathbb{H}}$ and $\|U\|_{\mathbb{H}}$ and estimative (29), we finish
the proof of this lemma.

$\Box$

Finally, from Lemmas(6), (14), (16) and (17), the proof of this theorem is
finished.

$\Box$

####  4.1.2 Lack Analyticity

###### Theorem 18

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

Proof: We apply Theorem (12) to show this result. Since the operator $A$ is
strictly positive, selfadjoint 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 $D(A^{0})$-norm eigenvector associated to
the eigenvalue $\sigma_{n}$, that is,

$Ae_{n}=\sigma_{n}e_{n},\quad\|e_{n}\|_{\mathfrak{D}(A^{0})}=1,\quad
n\in{\mathbb{N}}.$ (59)

Next we will show the lack of analyticity for
$\phi\in(\frac{1}{2},\frac{3}{4}]$:
Consider the eigenvalues and eigenvectors of the operator $A$ as in (59). Let
$F_{n}=(0,0,-\frac{e_{n}}{2\alpha},\frac{e_{n}}{2})\in\mathbb{H}$. The
solution $U_{n}=(u_{n},v_{n},w_{n},\theta_{n})$ of the system
$(i\lambda_{n}I-\mathbb{B})U_{n}=F_{n}$ satisfies $v_{n}=i\lambda_{n}u_{n}$,
$w_{n}=-\lambda_{n}^{2}u_{n}$, $A^{\psi}e_{n}=\sigma_{n}^{\psi}e_{n}$ and the
following equations

$\displaystyle-i\alpha\lambda^{3}_{n}u_{n}+a^{2}Au_{n}+ia^{2}\beta\lambda_{n}Au_{n}-\lambda^{2}_{n}u_{n}-\eta
A^{\phi}\theta_{n}$ $\displaystyle=$ $\displaystyle-e_{n},$ $\displaystyle
i\eta\lambda_{n}A^{\phi}u_{n}-\alpha\eta\lambda^{2}_{n}A^{\phi}u_{n}+i\lambda_{n}\theta_{n}+A\theta_{n}$
$\displaystyle=$ $\displaystyle e_{n}.$

Equivalently,

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}I-a^{2}\beta\lambda_{n}A\big{]}+\lambda^{2}_{n}I-a^{2}A\\}u_{n}+\eta
A^{\phi}\theta_{n}$ $\displaystyle=$ $\displaystyle e_{n},$
$\displaystyle\\{i\eta\lambda_{n}A^{\phi}-\alpha\eta\lambda^{2}_{n}A^{\phi}\\}u_{n}+\\{i\lambda_{n}I+A\\}\theta_{n}$
$\displaystyle=$ $\displaystyle e_{n}.$

Let us see whether this system admits solutions of the form

$u_{n}=\mu_{n}e_{n},\quad\theta_{n}=\nu_{n}e_{n},$

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

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}-a^{2}\beta\lambda_{n}\sigma_{n}]+\lambda_{n}^{2}-a^{2}\sigma_{n}\\}\mu_{n}+\eta\sigma_{n}^{\phi}\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (60)
$\displaystyle\\{i\eta\lambda_{n}\sigma_{n}^{\phi}-\alpha\eta\lambda^{2}_{n}\sigma_{n}^{\phi}\\}\mu_{n}+\\{i\lambda_{n}+\sigma_{n}\\}\nu_{n}$
$\displaystyle=$ $\displaystyle 1.$ (61)

In this point, we introduce the numbers

$\sigma_{n}:=\dfrac{\lambda_{n}^{2}}{a^{2}}.$ (62)

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

$|\sigma_{n}|\approx|\lambda_{n}|^{2},\quad\alpha\lambda_{n}^{3}-a^{2}\beta\lambda_{n}\sigma_{n}=(\alpha-\beta)\lambda_{n}^{3},\quad\eta\sigma_{n}^{\phi}=\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi}\\\
i\eta\lambda_{n}\sigma_{n}^{\phi}-\alpha\eta\lambda_{n}^{2}\sigma_{n}^{\phi}=i\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha\eta}{a^{2\phi}}\lambda_{n}^{2+2\phi}\quad{\rm
and}\quad
i\lambda_{n}+\sigma_{n}=i\lambda_{n}+\dfrac{1}{a^{2}}\lambda_{n}^{2}.$ (63)

Using identities of (62) and (63) in system (60)-(61), we obtain

$\displaystyle
i(\alpha-\beta)\lambda_{n}^{3}\mu_{n}+\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi}\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (64)
$\displaystyle\\{i\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi+1}-\dfrac{\alpha\eta}{a^{2\phi}}\lambda_{n}^{2+2\phi}\\}\mu_{n}+\\{i\lambda_{n}+\dfrac{1}{a^{2}}\lambda_{n}^{2}\\}\nu_{n}$
$\displaystyle=$ $\displaystyle 1.$ (65)

Then

$\Delta=\left|\begin{array}[]{cc}i(\alpha-\beta)\lambda_{n}^{3}&\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi}\\\
\\{i\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi+1}-\dfrac{\alpha\eta}{a^{2\phi}}\lambda_{n}^{2+2\phi}\\}&\\{i\lambda_{n}+\dfrac{1}{a^{2}}\lambda_{n}^{2}\\}\end{array}\right|=(\beta-\alpha)\lambda^{4}_{n}+\dfrac{\alpha\eta^{2}}{a^{4\phi}}\lambda_{n}^{2+4\phi}\\\
-i\bigg{[}\dfrac{\beta-\alpha}{a^{2}}\lambda_{n}^{5}+\dfrac{\eta^{2}}{a^{4\phi}}\lambda_{n}^{4\phi+1}\bigg{]}$
(66)

Therefore, for $\phi\in\big{[}\frac{1}{2},\frac{3}{4}\big{]}$ and
$\beta-\alpha>0$, we have

$|\Delta|\approx\left\\{\begin{array}[]{ccc}|\lambda_{n}|^{\max\\{4,5\\}}=|\lambda_{n}|^{5}\hskip
85.35826pt{\rm for}&0\leq\phi\leq\dfrac{1}{2}&\\\
|\lambda_{n}|^{\max\\{2+4\phi,5\\}}=\left\\{\begin{array}[]{ccc}|\lambda_{n}|^{5}&{\rm
for}&\dfrac{1}{2}\leq\phi\leq\dfrac{3}{4}\\\ \\\ |\lambda_{n}|^{2+4\phi}&{\rm
for}&\dfrac{3}{4}\leq\phi\leq 1.\end{array}\right.\end{array}\right.$ (67)

For other hand,

$\big{|}\Delta_{\nu_{n}}\big{|}=\bigg{|}\left|\begin{array}[]{cc}i(\alpha-\beta)\lambda_{n}^{3}&1\\\
\\\
\\{i\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi+1}-\dfrac{\alpha\eta}{a^{2\phi}}\lambda_{n}^{2+2\phi}\\}&1\end{array}\right|\bigg{|}=\big{|}-i\\{\beta-\alpha)\lambda_{n}^{3}+\dfrac{\eta}{a^{2\phi}}\lambda_{n}^{2\phi+1}\\}\\\
+\dfrac{\alpha\eta}{a^{2\phi}}\lambda_{n}^{2+2\phi}\big{|}\approx|\lambda_{n}|^{2+2\phi}\quad{\rm
for}\quad\dfrac{1}{2}\leq\phi\leq 1.$ (68)

then

$|\nu_{n}|=\bigg{|}\dfrac{\Delta_{\nu_{n}}}{\Delta}\bigg{|}\approx|\lambda_{n}|^{2\phi-3}\quad{\rm
for}\quad\dfrac{1}{2}\leq\phi\leq\dfrac{3}{4}.$ (69)

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

$\|U_{n}\|_{\mathbb{H}}\geq
K\|A^{\frac{1}{2}}v_{n}\|=K|\sigma_{n}|^{\frac{1}{2}}|\nu_{n}|\|e_{n}\|\approx
K|\lambda_{n}|^{2\phi-2}\quad{\rm
for}\quad\dfrac{1}{2}\leq\phi\leq\dfrac{3}{4}.$ (70)

On the other hand:

$|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\geq K|\lambda_{n}|^{2\phi-1}\quad{\rm
for}\quad\dfrac{1}{2}\leq\phi\leq\dfrac{3}{4}.$

Therefore: Let $\frac{1}{2}\leq\phi\leq\frac{3}{4}$: as
$2\phi-1>0\Leftrightarrow\phi>\frac{1}{2}$, then for $\phi$ in
$(\frac{1}{2},\frac{3}{4}]$ we have
$\|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\to\infty$ approaches infinity as
$|\lambda_{n}|\to\infty$. Therefore for $\phi$ in $(\frac{1}{2},\frac{3}{4}]$
the (40) condition fails. Consequently to semigroup $S(t)$ is not analytic for
$\phi\in(\frac{1}{2},\frac{3}{4}]$.

Next, we will show the lack of analyticity for
$\phi\in(0,\frac{1}{2}]\cup[\frac{3}{4},1)$:
Consider the eigenvalues and eigenvectors of the operator $A$ as in (59). Let
$F_{n}=(0,0,-\frac{e_{n}}{\alpha},0)\in\mathbb{H}$. The solution
$U_{n}=(u_{n},v_{n},w_{n},\theta_{n})$ of the system
$(i\lambda_{n}I-\mathbb{B})U_{n}=F_{n}$ satisfies $v_{n}=i\lambda_{n}u_{n}$,
$w_{n}=-\lambda_{n}^{2}u_{n}$, $A^{\psi}e_{n}=\sigma_{n}^{\psi}e_{n}$ and the
following equations

$\displaystyle-i\alpha\lambda^{3}_{n}u_{n}+a^{2}Au_{n}+ia^{2}\beta\lambda_{n}Au_{n}-\lambda^{2}_{n}u_{n}-\eta
A^{\phi}\theta_{n}$ $\displaystyle=$ $\displaystyle-e_{n},$ $\displaystyle
i\eta\lambda_{n}A^{\phi}u_{n}-\alpha\eta\lambda^{2}_{n}A^{\phi}u_{n}+i\lambda_{n}\theta_{n}+A\theta_{n}$
$\displaystyle=$ $\displaystyle 0.$

Equivalently,

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}I-a^{2}\beta\lambda_{n}A\big{]}+\lambda^{2}_{n}I-a^{2}A\\}u_{n}+\eta
A^{\phi}\theta_{n}$ $\displaystyle=$ $\displaystyle e_{n},$
$\displaystyle\\{i\eta\lambda_{n}A^{\phi}-\alpha\eta\lambda^{2}_{n}A^{\phi}\\}u_{n}+\\{i\lambda_{n}I+A\\}\theta_{n}$
$\displaystyle=$ $\displaystyle 0.$

Let us see whether this system admits solutions of the form

$u_{n}=\mu_{n}e_{n},\quad\theta_{n}=\nu_{n}e_{n},$

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

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}-a^{2}\beta\lambda_{n}\sigma_{n}]+\lambda_{n}^{2}-a^{2}\sigma_{n}\\}\mu_{n}+\eta\sigma_{n}^{\phi}\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (71)
$\displaystyle\\{i\eta\lambda_{n}\sigma_{n}^{\phi}-\alpha\eta\lambda^{2}_{n}\sigma_{n}^{\phi}\\}\mu_{n}+\\{i\lambda_{n}+\sigma_{n}\\}\nu_{n}$
$\displaystyle=$ $\displaystyle 0.$ (72)

In this point, we introduce the numbers

$\sigma_{n}:=\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}.$ (73)

Then

$|\sigma_{n}|\approx|\lambda_{n}|^{2},\quad\lambda_{n}^{2}-a^{2}\sigma_{n}=\frac{\beta-\alpha}{\beta}\lambda_{n}^{2},\quad\eta\sigma_{n}^{\phi}=\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\\\
i\eta\lambda_{n}\sigma_{n}^{\phi}-\alpha\eta\lambda_{n}^{2}\sigma_{n}^{\phi}=i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}\\\
{\rm and}\quad
i\lambda_{n}+\sigma_{n}=i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}.$
(74)

Using identities of (73) and (74) in system (71)-(72), we obtain

$\displaystyle\bigg{\\{}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}\bigg{\\}}\mu_{n}+\bigg{\\{}\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\bigg{\\}}\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (75)
$\displaystyle\bigg{\\{}i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}\bigg{\\}}\mu_{n}+\bigg{\\{}i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\bigg{\\}}\nu_{n}$
$\displaystyle=$ $\displaystyle 1.$ (76)

Then

$\Delta=\left|\begin{array}[]{cc}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}&\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\\\
i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}&i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\end{array}\right|\\\
=\dfrac{\alpha(\beta-\alpha)}{a^{2}\beta^{2}}\lambda^{4}_{n}+\dfrac{\alpha^{2\phi+1}\eta^{2}}{a^{4\phi}\beta^{2\phi}}\lambda_{n}^{2+4\phi}+i\bigg{[}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{3}-\dfrac{\eta^{2}\alpha^{2\phi}}{a^{4\phi}\beta^{2\phi}}\lambda_{n}^{4\phi+1}\bigg{]}$
(77)

Therefore, for $\phi\in(0,\frac{1}{2}]\cup[\frac{3}{4},1)$ and
$\beta-\alpha>0$, we have

$|\Delta|\approx\left\\{\begin{array}[]{ccc}|\lambda_{n}|^{\max\\{4,3\\}}=|\lambda_{n}|^{4}\hskip
42.67912pt{\rm for}&0\leq\phi\leq\dfrac{1}{2}&\\\
|\lambda_{n}|^{\max\\{2+4\phi,1+4\phi\\}}=|\lambda_{n}|^{2+4\phi}\quad\quad{\rm
for}&\dfrac{1}{2}\leq\phi\leq 1.&\end{array}\right.$ (78)

For other hand,

$\big{|}\Delta_{\nu_{n}}\big{|}=\bigg{|}\left|\begin{array}[]{cc}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}&1\\\
\\\
\\{i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}\\}&0\end{array}\right|\bigg{|}\\\
=\bigg{|}\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi+2}-i\dfrac{\alpha^{\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}\bigg{|}\approx|\lambda_{n}|^{2\phi+2}.$
(79)

then

$|\nu_{n}|=\bigg{|}\dfrac{\Delta_{\nu_{n}}}{\Delta}\bigg{|}\approx\left\\{\begin{array}[]{ccc}|\lambda_{n}|^{2\phi-2}&{\rm
for}&0\leq\phi\leq\dfrac{1}{2},\\\ \\\ |\lambda_{n}|^{-2\phi}&{\rm
for}&\dfrac{1}{2}\leq\phi\leq 1.\end{array}\right.$ (80)

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

$\|U_{n}\|_{\mathbb{H}}\geq
K\|A^{\frac{1}{2}}v_{n}\|=K|\sigma_{n}|^{\frac{1}{2}}|\nu_{n}|\|e_{n}\|\\\
\approx K\left\\{\begin{array}[]{ccc}(i)\;|\lambda_{n}|^{2\phi-1}&{\rm
for}&0\leq\phi\leq\dfrac{1}{2},\\\ \\\ (ii)\;|\lambda_{n}|^{1-2\phi}&{\rm
for}&\dfrac{1}{2}\leq\phi\leq\dfrac{3}{4},\\\ \\\
(iii)\;|\lambda_{n}|^{1-2\phi}&{\rm for}&\dfrac{3}{4}\leq\phi\leq
1.\end{array}\right.$ (81)

On the other hand:

$|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\geq
K\left\\{\begin{array}[]{ccc}(i)\;|\lambda_{n}|^{2\phi}&{\rm
for}&0\leq\phi\leq\dfrac{1}{2}\\\ \\\ (ii)\;|\lambda_{n}|^{2-2\phi}&{\rm
for}&\dfrac{1}{2}\leq\phi\leq 1.\end{array}\right.$

Therefore : First(i), let $0\leq\phi\leq\frac{1}{2}$: as
$2\phi>0\Leftrightarrow 0<\phi$, then for $\phi$ in $(0,\frac{1}{2}]$ we have
$\|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\to\infty$. And Second(ii), let
$\dfrac{1}{2}\leq\phi\leq 1$: as $2-2\phi>0\Leftrightarrow\phi<1$, then for
$\phi$ in $[\frac{1}{2},1)$ we have
$\|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\to\infty$ approaches infinity as
$|\lambda_{n}|\to\infty$. Therefore for $\phi$ in $(0,1)$ the (40) condition
fails. Consequently to semigroup $S(t)$ in particular is not analytic for
$\phi\in(0,\frac{1}{2}]\cup[\frac{3}{4},1)$.

Next we will show the lack of analyticity for $\phi=0$:

Consider the eigenvalues and eigenvectors of the operator $A$ as in (59). Let
$F_{n}=(0,0,-\frac{e_{n}}{\alpha},0)\in\mathbb{H}$. The solution
$U_{n}=(u_{n},v_{n},w_{n},\theta_{n})$ of the system
$(i\lambda_{n}I-\mathbb{B})U_{n}=F_{n}$ satisfies $v_{n}=i\lambda_{n}u_{n}$,
$w_{n}=-\lambda_{n}^{2}u_{n}$, $A^{\psi}e_{n}=\sigma_{n}^{\psi}e_{n}$ and the
following equations

$\displaystyle-i\alpha\lambda^{3}_{n}u_{n}+a^{2}Au_{n}+ia^{2}\beta\lambda_{n}Au_{n}-\lambda^{2}_{n}u_{n}-\eta\theta_{n}$
$\displaystyle=$ $\displaystyle-e_{n},$ $\displaystyle
i\eta\lambda_{n}u_{n}-\alpha\eta\lambda^{2}_{n}u_{n}+i\lambda_{n}\theta_{n}+A\theta_{n}$
$\displaystyle=$ $\displaystyle 0.$

Equivalently,

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}I-a^{2}\beta\lambda_{n}A\big{]}+\lambda^{2}_{n}I-a^{2}A\\}u_{n}+\eta\theta_{n}$
$\displaystyle=$ $\displaystyle e_{n},$
$\displaystyle\\{i\eta\lambda_{n}I-\alpha\eta\lambda^{2}_{n}I\\}u_{n}+\\{i\lambda_{n}I+A\\}\theta_{n}$
$\displaystyle=$ $\displaystyle 0.$

Let us see whether this system admits solutions of the form

$u_{n}=\mu_{n}e_{n},\quad\theta_{n}=\nu_{n}e_{n},$

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

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}-a^{2}\beta\lambda_{n}\sigma_{n}]+\lambda_{n}^{2}-a^{2}\sigma_{n}\\}\mu_{n}+\eta\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (82)
$\displaystyle\\{i\eta\lambda_{n}-\alpha\eta\lambda^{2}_{n}\\}\mu_{n}+\\{i\lambda_{n}+\sigma_{n}\\}\nu_{n}$
$\displaystyle=$ $\displaystyle 0.$ (83)

In this point, we introduce the numbers

$\sigma_{n}:=\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}.$ (84)

Then

$|\sigma_{n}|\approx|\lambda_{n}|^{2},\qquad\lambda_{n}^{2}-a^{2}\sigma_{n}=\frac{\beta-\alpha}{\beta}\lambda_{n}^{2},\qquad\eta=\eta\\\
i\eta\lambda_{n}-\alpha\eta\lambda_{n}^{2}=i\eta\lambda_{n}-\alpha\eta\lambda_{n}^{2}\quad{\rm
and}\quad
i\lambda_{n}+\sigma_{n}=i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}.$
(85)

Using identities of (84) and (85) in system (82)-(83), we obtain

$\displaystyle\bigg{\\{}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}\bigg{\\}}\mu_{n}+\eta\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (86)
$\displaystyle\bigg{\\{}i\eta\lambda_{n}-\alpha\eta\lambda_{n}^{2}\bigg{\\}}\mu_{n}+\bigg{\\{}i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\bigg{\\}}\nu_{n}$
$\displaystyle=$ $\displaystyle 1.$ (87)

Then

$\Delta=\left|\begin{array}[]{cc}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}&\eta\\\
i\eta\lambda_{n}-\alpha\eta\lambda_{n}^{2}&i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\end{array}\right|\\\
=\dfrac{\alpha(\beta-\alpha)}{a^{2}\beta^{2}}\lambda^{4}_{n}+\alpha\eta^{2}\lambda_{n}^{2}+i\bigg{[}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{3}-\eta^{2}\lambda_{n}\bigg{]}$
(88)

Therefore, as $\beta-\alpha>0$, we have

$|\Delta|\approx|\lambda_{n}|^{\max\\{4,3\\}}=|\lambda_{n}|^{4}.$ (89)

For other hand,

$\big{|}\Delta_{\mu_{n}}\big{|}=\bigg{|}\left|\begin{array}[]{cc}1&\eta\\\ \\\
0&i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\end{array}\right|\bigg{|}=\bigg{|}i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\bigg{|}\approx|\lambda_{n}|^{2}.$
(90)

then

$|\mu_{n}|=\bigg{|}\dfrac{\Delta_{\mu_{n}}}{\Delta}\bigg{|}\approx|\lambda_{n}|^{-2}.$
(91)

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

$\|U_{n}\|_{\mathbb{H}}\geq
K\|A^{\frac{1}{2}}v_{n}\|=K|\sigma_{n}|^{\frac{1}{2}}|\lambda_{n}||\mu_{n}|\approx
K|\lambda_{n}|^{0}.$ (92)

On the other hand:

$|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\geq K|\lambda_{n}|^{1}.$

Therefore: for $\phi=0$, we have
$\|\lambda_{n}|\|U_{n}\|_{\mathbb{H}}\to\infty$ approaches infinity as
$|\lambda_{n}|\to\infty$. Therefore for $\phi=0$ the (40) condition fails.
Consequently to semigroup $S(t)$ is not analytic for $\phi=0$. This completes
the proof of this theorem.

$\Box$

###  4.2 Gevrey Sharp-class and non-existence of Classe Gevrey

Before exposing our results, it is useful to recall the next definition and
result presented in [13] (adapted from [25], Theorem 4, p. 153]).

###### Definition 19

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

$||S^{(n)}(t)||_{\mathcal{L}(\mathbb{H})}\leq C\mu^{n}(n!)^{s},\text{ for all
}\quad t\in K,n=0,1,2...$ (93)

######  Theorem 20 ([25])

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 $0<\Psi<1$:

$\lim\limits_{|\lambda|\to\infty}\sup|\lambda|^{\Psi}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathcal{L}(\mathbb{H})}<\infty.$ (94)

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

Our main result in this subsection is as follows:

###### Theorem 21

Let $S(t)=e^{t\mathbb{B}}$ strongly continuos-semigroups of contractions on
the Hilbert space $\mathbb{H}$, the semigroups $S(t)$ is of Grevrey class
$s_{i}$ $i=1,2$, for every $s_{1}>\frac{1}{\Psi_{1}(\phi)}=\frac{1}{2-2\phi}$
for $\phi\in(\frac{1}{2},\frac{3}{4}]$ and
$s_{2}>\frac{1}{\Psi_{2}(\phi)}=\frac{1}{2\phi-1}$ for
$\phi\in[\frac{3}{4},1)$, such that we have the resolvent estimative:

$(i)\;\limsup_{|\lambda|\to\infty}|\lambda|^{2-2\phi}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathcal{L}(\mathbb{H})}<\infty,\qquad{\rm
for}\quad\frac{1}{2}<\phi\leq\frac{3}{4}\quad{\rm
and}\quad\lambda\in{\mathbb{R}},$ (95)
$(ii)\;\limsup_{|\lambda|\to\infty}|\lambda|^{2\phi-1}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathcal{L}(\mathbb{H})}<\infty,\quad{\rm
for}\quad\frac{3}{4}\leq\phi<1\quad{\rm and}\quad\lambda\in{\mathbb{R}},$ (96)

(iii) The semigroup $S(t)=e^{t\mathbb{B}}$ does not admit Gevrey class when
the parameter $\phi$ belongs to the interval $[0,\frac{1}{2}]$.

Proof: (i): Note that the estimate

$|\lambda|^{2-2\phi}||(i\lambda
I-\mathbb{B})^{-1}F||_{\mathbb{H}}=|\lambda|^{2-2\phi}\|U\|_{\mathbb{H}}\leq
C_{\delta}\|F\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{1}{2}<\phi\leq\dfrac{3}{4}$ (97)

implies the inequality (95). Therefore from now on we will show (97), for this
purpose let us estimate the term $|\lambda|^{2-2\phi}\|w\|^{2}$, we assume
$\lambda\in\mathbb{R}$ with $|\lambda|>1$, we shall borrow some ideias from
[19]. Let us decompose $w$ as $w=w_{1}+w_{2}$, such that

$i\alpha\lambda w_{1}+Aw_{1}=\alpha f^{3}\quad{\rm
in}\quad\mathfrak{D}(A^{0})\qquad{\rm and}\\\ i\alpha\lambda
w_{2}=-a^{2}Au-a^{2}\beta Av+\eta A^{\phi}\theta-w+Aw_{1}\quad{\rm
in}\quad\mathfrak{D}(A^{0}).$ (98)

Firstly, applying the product duality the first equation in (98)1 by $w_{1}$,
we have

$i\alpha\lambda\|w_{1}\|^{2}+\|A^{\frac{1}{2}}w_{1}\|^{2}=\langle{\alpha
f^{3}},{w_{1}}\rangle.$ (99)

Taking first the imaginary part of (99) and in the sequence the real part and
applying Cauchy-Schwarz inequality, we have

$\displaystyle\|w_{1}\|\leq
C\dfrac{[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}}{|\lambda|^{\frac{1}{2}}}\qquad{\rm
and}\qquad\|A^{\frac{1}{2}}w_{1}\|\leq
C[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}$ (100)

From
$A^{\frac{2\phi-1}{2}}w_{2}=A^{\frac{2\phi-1}{2}}w-A^{\frac{2\phi-1}{2}}w_{1}$,
from estimates (108)2 and Lemma(15), we have

$\|A^{\frac{2\phi-1}{2}}w_{2}\|\leq
C_{\delta}\\{\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\\}^{\frac{1}{2}}\quad{\rm
for}\quad 0\leq\phi\leq 1.$ (101)

Applying the operator $A^{-\frac{1}{2}}$ in the second equation of (107) and
using Lemmas (6) and (15), estimates (29) and (108)2, we have

$|\lambda|\|A^{-\frac{1}{2}}w_{2}||\leq
C\\{\|A^{\frac{1}{2}}u\|+\|A^{\frac{1}{2}}v\|+\|A^{\phi-\frac{1}{2}}\theta\|\\\
+\|A^{-\frac{1}{2}}w\|+\|A^{\frac{1}{2}}w_{1}\|\\}\leq
C_{\delta}\\{\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\\}^{\frac{1}{2}}.$ (102)

Now, by Lion’s interpolations inequality
($-\frac{1}{2}<0<\frac{\phi-1}{2(2-\phi)}$) and estimative (102), and from
$\frac{1}{2}<\phi<1$ we have $\frac{\phi-1}{2(2-\phi)}\leq\frac{2\phi-1}{2}$,
applying continuous embedding
$\mathfrak{D}(A^{\tau_{2}})\hookrightarrow\mathfrak{D}(A^{\tau_{1}}),\;\tau_{2}>\tau_{1}$
and (101), we derive

$\|w_{2}\|\leq
C\|A^{-\frac{1}{2}}w_{2}\|^{\phi-1}\|A^{\frac{\phi-1}{2(2-\phi)}}w_{2}\|^{2-\phi}\\\
\leq
C_{\delta}|\lambda|^{-(\phi-1)}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\quad{\rm
for}\quad\dfrac{1}{2}<\phi\leq\dfrac{3}{4}.$ (103)

On the other hand, as $\|w\|\leq C\\{\|w_{1}\|+\|w_{2}\|\\}$ and for
$\frac{1}{2}<\phi\leq\frac{3}{4}$ we have $-(\phi-1)\geq-\frac{1}{2}$, using
estimates (100)1 and (103), we get

$|\lambda|^{2-2\phi}\|w\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{1}{2}<\phi\leq\dfrac{3}{4}.$ (104)

Now applying the estimate (104) and Lemma(14) on (4.1.1) for
$\frac{1}{2}\leq\phi\leq\frac{3}{4}$, we get

$|\lambda|\|A^{\frac{1}{2}}v\|\leq|\lambda|^{2\phi-1}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{1}{2}<\phi\leq\dfrac{3}{4},$ (105)

for other hand, using Lemmas(6), (14) and estimates (104) and (105), we finish
the proof of inequality (97).
Proof: (ii): From other hand, note that the estimate

$|\lambda|^{2\phi-1}||(i\lambda
I-\mathbb{B})^{-1}F||_{\mathbb{H}}=|\lambda|^{2\phi-1}\|U\|_{\mathbb{H}}\leq
C_{\delta}\|F\|_{\mathbb{H}}\quad{\rm for}\quad\dfrac{3}{4}\leq\phi<1,$ (106)

implies the inequality (96). Therefore from now on we will show (106), for
this purpose let us estimate the term $|\lambda|^{2\phi-1}\|w\|^{2}$, we
assume $\lambda\in\mathbb{R}$ with $|\lambda|>1$. Now let us decompose $w$ as
$w=w_{1}+w_{2}$, such that

$i\alpha\lambda w_{1}+Aw_{1}=\alpha f^{3}\quad{\rm
in}\quad\mathfrak{D}(A^{0})\qquad{\rm and}\\\ i\alpha\lambda
w_{2}=-a^{2}Au-a^{2}\beta Av+\eta A^{\phi}\theta-w+Aw_{1}\quad{\rm
in}\quad\mathfrak{D}(A^{0}).$ (107)

Firstly, applying the product duality the first equation in (107)1 by $w_{1}$,
and taking first the imaginary part and in the sequence the real part and
applying Cauchy-Schwarz inequality, we have

$\displaystyle\|w_{1}\|\leq
C\dfrac{[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}}{|\lambda|^{\frac{1}{2}}}\qquad{\rm
and}\qquad\|A^{\frac{1}{2}}w_{1}\|\leq
C[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}.$ (108)

Applying the operator $A^{-\frac{1}{2}}$ in the second equation of (107) and
using Lemmas (6) and (15), estimates (29) and (108)2, we have

$|\lambda|\|A^{-\frac{1}{2}}w_{2}||\leq
C\\{\|A^{\frac{1}{2}}u\|+\|A^{\frac{1}{2}}v\|+\|A^{\phi-\frac{1}{2}}\theta\|\\\
+\|A^{-\frac{1}{2}}w\|+\|A^{\frac{1}{2}}w_{1}\|\\}\leq
C_{\delta}\\{\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\\}^{\frac{1}{2}}.$ (109)

Now, by Lion’s interpolations inequality
($-\frac{1}{2}<0<\frac{2\phi-1}{2(3-2\phi)}$) and estimative (109) and as
$\frac{2\phi-1}{2(3-2\phi)}<\frac{2\phi-1}{2}$, applying continuous embedding
$\mathfrak{D}(A^{\tau_{2}})\hookrightarrow\mathfrak{D}(A^{\tau_{1}}),\;\tau_{2}>\tau_{1}$
and (109), we derive

$\|w_{2}\|\leq
C\|A^{-\frac{1}{2}}w_{2}\|^{\frac{2\phi-1}{2}}\|A^{\frac{2\phi-1}{2(3-2\phi)}}w_{2}\|^{\frac{3-2\phi}{2}}\\\
\leq
C_{\delta}|\lambda|^{-\frac{2\phi-1}{2}}[\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}]^{\frac{1}{2}}\qquad{\rm
for}\qquad\dfrac{3}{4}\leq\phi<1.$ (110)

On the other hand, as $\|w\|\leq C\\{\|w_{1}\|+\|w_{2}\|\\}$ and for
$\frac{1}{2}<\phi<1$ we have $-\frac{2\phi-1}{2}\geq-\frac{1}{2}$, using
estimates (108)1 and (110), we get

$|\lambda|^{2\phi-1}\|w\|^{2}\leq
C_{\delta}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{3}{4}\leq\phi<1.$ (111)

Now applying the estimate (111) and Lemma(14) on (4.1.1) for
$\frac{3}{4}\leq\phi<1$, we get

$|\lambda|\|A^{\frac{1}{2}}v\|\leq|\lambda|^{2\phi-1}\|F\|_{\mathbb{H}}\|U\|_{\mathbb{H}}\quad{\rm
for}\quad\dfrac{3}{4}\leq\phi<1,$ (112)

for other hand, using Lemmas(6), (14) and estimates (111) and (112), we finish
the proof of (106).

Proof: (iii):.
Consider $0\leq\phi\leq\frac{1}{2}$ and the eigenvalues and eigenvectors of
the operator $A$ as in (59). Let
$F_{n}=(0,0,-\frac{e_{n}}{\alpha},0)\in\mathbb{H}$. The solution
$U_{n}=(u_{n},v_{n},w_{n},\theta_{n})$ of the system
$(i\lambda_{n}I-\mathbb{B})U_{n}=F_{n}$ satisfies $v_{n}=i\lambda_{n}u_{n}$,
$w_{n}=-\lambda_{n}^{2}u_{n}$, $A^{\psi}e_{n}=\sigma_{n}^{\psi}e_{n}$ and the
following equations

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}I-a^{2}\beta\lambda_{n}A\big{]}+\lambda^{2}_{n}I-a^{2}A\\}u_{n}+\eta
A^{\phi}\theta_{n}$ $\displaystyle=$ $\displaystyle e_{n},$
$\displaystyle\\{i\eta\lambda_{n}A^{\phi}-\alpha\eta\lambda^{2}_{n}A^{\phi}\\}u_{n}+\\{i\lambda_{n}I+A\\}\theta_{n}$
$\displaystyle=$ $\displaystyle 0.$

Let us see whether this system admits solutions of the form

$u_{n}=\mu_{n}e_{n},\quad\theta_{n}=\nu_{n}e_{n},$

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

$\displaystyle\\{i\big{[}\alpha\lambda^{3}_{n}-a^{2}\beta\lambda_{n}\sigma_{n}]+\lambda_{n}^{2}-a^{2}\sigma_{n}\\}\mu_{n}+\eta\sigma_{n}^{\phi}\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (113)
$\displaystyle\\{i\eta\lambda_{n}\sigma_{n}^{\phi}-\alpha\eta\lambda^{2}_{n}\sigma_{n}^{\phi}\\}\mu_{n}+\\{i\lambda_{n}+\sigma_{n}\\}\nu_{n}$
$\displaystyle=$ $\displaystyle 0.$ (114)

In this point, we introduce the numbers
$\sigma_{n}:=\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}$. We have that

$|\sigma_{n}|=|\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}|\approx|\lambda_{n}|^{2},\quad\lambda_{n}^{2}-a^{2}\sigma_{n}=\frac{\beta-\alpha}{\beta}\lambda_{n}^{2},\quad\eta\sigma_{n}^{\phi}=\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\\\
i\eta\lambda_{n}\sigma_{n}^{\phi}-\alpha\eta\lambda_{n}^{2}\sigma_{n}^{\phi}=i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}\\\
{\rm and}\quad
i\lambda_{n}+\sigma_{n}=i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}.$
(115)

Using identities of (115) in system (113)-(114), we obtain

$\displaystyle\bigg{\\{}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}\bigg{\\}}\mu_{n}+\bigg{\\{}\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\bigg{\\}}\nu_{n}$
$\displaystyle=$ $\displaystyle 1,$ (116)
$\displaystyle\bigg{\\{}i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}\bigg{\\}}\mu_{n}+\bigg{\\{}i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\bigg{\\}}\nu_{n}$
$\displaystyle=$ $\displaystyle 1.$ (117)

Then

$\Delta=\left|\begin{array}[]{cc}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{2}&\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\\\
i\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{1+2\phi}-\dfrac{\alpha^{1+\phi}\eta}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2+2\phi}&i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\end{array}\right|\\\
=\dfrac{\alpha(\beta-\alpha)}{a^{2}\beta^{2}}\lambda^{4}_{n}+\dfrac{\alpha^{2\phi+1}\eta^{2}}{a^{4\phi}\beta^{2\phi}}\lambda_{n}^{2+4\phi}+i\bigg{[}\dfrac{\beta-\alpha}{\beta}\lambda_{n}^{3}-\dfrac{\eta^{2}\alpha^{2\phi}}{a^{4\phi}\beta^{2\phi}}\lambda_{n}^{4\phi+1}\bigg{]}$
(118)

Therefore, for $\phi\in[0,\frac{1}{2}]$ and $\beta-\alpha>0$, we have

$|\Delta|\approx|\lambda_{n}|^{\max\\{4,3\\}}=|\lambda_{n}|^{4}\qquad{\rm
for}\quad 0\leq\phi\leq\dfrac{1}{2}.$ (119)

For other hand,

$\big{|}\Delta_{\mu_{n}}\big{|}=\bigg{|}\left|\begin{array}[]{cc}1&\dfrac{\eta\alpha^{\phi}}{a^{2\phi}\beta^{\phi}}\lambda_{n}^{2\phi}\\\
\\\
0&i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\end{array}\right|\bigg{|}=\bigg{|}i\lambda_{n}+\dfrac{\alpha}{a^{2}\beta}\lambda_{n}^{2}\bigg{|}\approx|\lambda_{n}|^{2}.$
(120)

then

$|\mu_{n}|=\bigg{|}\dfrac{\Delta_{\mu_{n}}}{\Delta}\bigg{|}\approx|\lambda_{n}|^{-2}\qquad{\rm
for}\qquad 0\leq\phi\leq\dfrac{1}{2}.$ (121)

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

$\|U_{n}\|_{\mathbb{H}}\geq
K\|A^{\frac{1}{2}}v_{n}\|=K|\lambda_{n}|\|A^{\frac{1}{2}}u_{n}\|=K|\lambda_{n}||\sigma_{n}|^{\frac{1}{2}}|\nu_{n}|\|e_{n}\|\\\
\approx K|\lambda_{n}|^{0}\quad{\rm for}\quad 0\leq\phi\leq\dfrac{1}{2}.$
(122)

Finally, if there is a Gevrey class for $S(t)$ when $\phi\in[0,\frac{1}{2}]$,
there must exist a $\Psi\in(0,1)$ such that the identity

$\lim\limits_{|\lambda_{n}|\to\infty}\sup|\lambda_{n}|^{\Psi}||(i\lambda
I-\mathbb{B})^{-1}||_{\mathcal{L}(\mathbb{H})}<\infty.$ (123)

is verified. On the other hand, multiplying both sides of the inequality (122)
by $|\lambda_{n}|^{\Psi}$ for $\Psi\in(0,1)$ and considering
$0\leq\phi\leq\dfrac{1}{2}$, we have
$|\lambda_{n}|^{\Psi}\|U_{n}\|_{\mathbb{H}}\to\infty$ approaches infinity as
$|\lambda_{n}|\to\infty$. Therefore for $\phi$ in $[0,\frac{1}{2}]$ the (123)
condition fails. Consequently the semigroup $S(t)$ does not admit a Gevrey
class for $\phi\in[0,\frac{1}{2}]$. We finish the proof of this theorem.
$\Box$

######  Remark 22 (Gevrey Sharp-class)

The two Gevrey classes determined above are Sharp, meaning by “Gevrey Sharp”,
the statement of the following theorem:

###### Theorem 23

The two Gevrey classes determined by the functions $\Psi_{1}(\phi)=2-2\phi$
for $\phi\in(\frac{1}{2},\frac{3}{4}]$ and $\Psi_{2}(\phi)=2\phi-1$ for
$\phi\in[\frac{3}{4},1)$ are sharp, in the sense:

$(i):\;{\rm If}\;\Pi_{1}:=\Psi_{1}+\delta_{1}=2-2\phi+\delta_{1}\quad{\rm
for\;all}\quad\delta_{1}>0\quad{\rm
and}\quad\frac{1}{2}<\phi\leq\dfrac{3}{4},$ (124)

then

$s_{1}>\dfrac{1}{\Pi_{1}}\qquad{\rm
for}\qquad\dfrac{1}{2}<\phi\leq\dfrac{3}{4},$

is not a Gevrey class of the semigroup $S(t)=e^{t\mathbb{B}}$.

$(ii):\;{\rm If}\;\Pi_{2}:=\Psi_{2}+\delta_{2}=2\phi-1+\delta_{2}\quad{\rm
for\;all}\quad\delta_{2}>0\quad{\rm and}\quad\frac{3}{4}\leq\phi<1,$ (125)

then

$s_{2}>\dfrac{1}{\Pi_{2}}\qquad{\rm for}\qquad\dfrac{3}{4}\leq\phi<1,$

is not a Gevrey class of the semigroup $S(t)=e^{t\mathbb{B}}$.

Proof: We will use the results of items $(i)$ and $(ii)$ of Theorem(21) and
the estimates (95) and (96). In effect, from estimates (70) and item $(iii)$
of (81), we have

$|\lambda_{n}|^{\Pi_{1}}\|U_{n}\|_{\mathbb{H}}=K|\lambda_{n}|^{2-2\phi+\delta_{1}}\|U_{n}\|_{\mathbb{H}}\geq
K|\lambda_{n}|^{\delta_{1}}\to\infty,\quad{\rm
when}\quad|\lambda_{n}|\to\infty$

and

$|\lambda_{n}|^{\Pi_{2}}\|U_{n}\|_{\mathbb{H}}=K|\lambda_{n}|^{2\phi-1+\delta_{2}}\|U_{n}\|_{\mathbb{H}}\geq
K|\lambda_{n}|^{\delta_{2}}\to\infty,\quad{\rm
when}\quad|\lambda_{n}|\to\infty$

Therefore $\Pi_{1}$ and $\Pi_{2}$ not verify the conditions (95) and (96) of
the Theorem(21) concerning class Gevrey.

Then the Gevrey classe $s_{1}>\frac{1}{2-2\phi}$ for
$\phi\in(\frac{1}{2},\frac{3}{4}]$ and $s_{2}>\frac{1}{2\phi-1}$ for
$\phi\in[\frac{3}{4},1)$ the semigroup $S(t)$ are Sharp.

$\Box$

## References

  * [1] M. S. Alves, C. Buriol, M. V. Ferreira, J. E. M. Rivera, M. Sepúlveda and O. Vera, Asymptotic behaviour for the Vibrations modeled by the standard linear solid model with a thermal effect, Journal of Mathematical Analysis and Applications, 399 (2013), 472-479.
  * [2] K. Ammari, F. Shel and L. Tebou, Regularity of the semigroups associated with some damped coupled elastic systems II: A nondegenerate fractional damping case, Mathematical Methods in the Applied Sciences, (2022).
  * [3] M. Conti, V. Pata, M. Pellicer and R. Quintanilla, On the analyticity of the MGT-viscoelastic plate with heat conduction, Journal Differ Equ 269 (2020), 7862-7880.
  * [4] M. Conti, L. Liverani and V. Pata, The MGT-Fourier model in the supercritical case, Journal Differ Equ 301 (2021), 543-567.
  * [5] M. Conti, L. Liverani and V. Pata, Thermoelasticity with Antidissipation, Discrete and Continuous Dynamical Systems,- Series S, 15-8 (2022) 2173-2188.
  * [6] F. Dell’Oro, J. E. M. Rivera and V. Pata, Stability properties of an abstract system with applications to linear thermoelastic plates, Journal of Evolution Equations, Vol 13 N 4 (2013), 777-794.
  * [7] F. Dell’Oro and V. Pata, On the Moore-Thompson Equation and Its relation to Linear Viscoelasticity, Appl. Math. Optim. 76 (2017), 641-655.
  * [8] F. Dell’Oro and V. Pata, On the analyticity of the abstract MGT-Fourier system, Meccanica (2022).
  * [9] P. Gamboa, H. H. Nguyen, O. Vera and V. Poblete, Vibrations Modeled by the standard linear model of viscoelasticity with boundary dissipation, Electronic Journal of Differential Equations Vol. 2017- 163 (2017), 1-10.
  * [10] Gearhart. Spectral theory for contraction semigroups on Hilbert spaces. Trans. Amer. Math. Soc. 236 (1978), 385-394.
  * [11] 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), 70–88.
  * [12] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gilbson-Thompson equation arising in high intensity ultrasound, 40, Nro 4 (2011), 971-988.
  * [13] 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), 2875-2889.
  * [14] Z. Kuang, Z. Liu and H. D. F. Sare, Regularity analysis for an abstract thermoelastic system with inertial term, ESAIM: Control, Optimisation and Calculus of Variations, S24, 27 (2021).
  * [15] Z. Kuang, Z. Liu and L. Tebou, Optimal semigroup regularity for velocity coupled elastic systems: A degenerate fractional damping case, ESAIM: Control, Optimisation and Calculus of Variations 28, 46 (2022).
  * [16] I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free boundary conditions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 27 (1998), 457-482.
  * [17] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part II: General decay of energy, Journal Differential Equations, 259 (2015) 7610-7635.
  * [18] I. Lasiecka and X. Wang, Moore-Gibson-Thompson equation with memory, part I: Exponential decay of energy, ZAMP 67, 17 (2016) 1-23.
  * [19] Z. Liu and M. Renardy, A note on the equations of thermoelastic plate, Appl. Math. Lett., 8, (1995), 1-6.
  * [20] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Chapman & Hall CRC Research Notes in Mathematics, Boca Raton, FL, 398 (1999).
  * [21] R. Quintanilla, Moore-Gibson-Thompson thermoelasticity, Math. Mech. Solids 24 (2019), 4020-4031.
  * [22] R. Quintanilla, Moore-Gibson-Thompson thermoelasticity with two temperatures, Applications in Engineering Science 1 (2020).
  * [23] 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), 7085–7134.
  * [24] 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).
  * [25] S. W. Taylor, Gevrey Regularity of Solutions of Evolution Equations and Boundary Controllability, Thesis (Ph.D.) University of Minnesota, (1989), 182 pp.
  * [26] 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), 539-544.
  * [27] L. Tebou, Regularity and Stability for a plate model involving fractional rotational forces and damping, Zeitschrift für angewandte Mathematik and Physik ZAMP, 72:158 (2021).