# Well-posedness for some third-order evolution differential equations: A
semigroup approach

Flank D. M. Bezerra  Departamento de Matemática, Universidade Federal da
Paraíba, 58051-900 João Pessoa PB, Brazil<EMAIL_ADDRESS>,
Alexandre N. Carvalho‡ Departamento de Matemática, Instituto de Ciências
Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos,
Caixa Postal 668, 13560-970 São Carlos SP, Brazil<EMAIL_ADDRESS>and
Lucas A. Santos  Instituto Federal da Paraíba, 58780-000 Itaporanga PB,
Brazil<EMAIL_ADDRESS>
###### Abstract.

In this paper, we discuss the well-posedness of the Cauchy problem associated
with the third-order evolution equation in time

$u_{ttt}+Au+\eta A^{\frac{1}{3}}u_{tt}+\eta A^{\frac{2}{3}}u_{t}=f(u)$

where $\eta>0$, $X$ is a separable Hilbert space, $A:D(A)\subset X\to X$ is an
unbounded sectorial operator with compact resolvent, and for some
$\lambda_{0}>0$ we have $\mbox{Re}\sigma(A)>\lambda_{0}$ and
$f:D(A^{\frac{1}{3}})\subset X\to X$ is a nonlinear function with suitable
conditions of growth and regularity.

Mathematics Subject Classification 2010: 34A08, 47D06, 47D03, 35K10.
Key words and phrases: Approximations; fractional powers; sectorial operator;
semigroups.

‡Research partially supported by CNPq # 303929/2015-4 and by FAPESP #
2003/10042-0, Brazil

###### Contents

  1. 1 Introduction
  2. 2 Functional Framework
    1. 2.1 Natural setting
    2. 2.2 New approach
  3. 3 Ill-posed problems
  4. 4 Parabolic differential equations
    1. 4.1 Sectoriality
    2. 4.2 Analysis by reducing the order
  5. 5 Remaks on the case $\eta=1$
  6. 6 Moore-Gibson-Thompson-type equations

##  1\. Introduction

In this paper, we discuss the well-posedness of the Cauchy problem associated
with the following third-order evolution equation in time

(1.1) $u_{ttt}+Au+\eta A^{\frac{1}{3}}u_{tt}+\eta A^{\frac{2}{3}}u_{t}=f(u)$

where $\eta>0$, $X$ is a separable Hilbert space and $A:D(A)\subset X\to X$ is
an unbounded sectorial operator with compact resolvent, and for some
$\lambda_{0}>0$ we have $\mbox{Re}\sigma(A)>\lambda_{0}$, that is,
$\mbox{Re}\lambda>\lambda_{0}$ for all $\lambda\in\sigma(A)$, where
$\sigma(A)$ is the spectrum of $A$. This allows us to define the fractional
power $A^{-\alpha}$ of order $\alpha\in(0,1)$ according to [2, Formula 4.6.9]
and [15, Theorem 1.4.2], as a closed linear operator on its domain with
inverse $A^{\alpha}$.

Denote by $X^{\alpha}=D(A^{\alpha})$ for $\alpha\in[0,1)$, taking $A^{0}:=I$
on $X^{0}:=X$ when $\alpha=0$. Recall that $X^{\alpha}$ is dense in $X$ for
all $\alpha\in(0,1]$, for details see [2, Theorem 4.6.5]. The fractional power
space $X^{\alpha}$ endowed with the norm

$\|\cdot\|_{X^{\alpha}}:=\|A^{\alpha}\cdot\|_{X}$

is a Banach space. It is not difficult to show that $A^{\alpha}$ is the
generator of a strongly continuous analytic semigroup on $X$, that we will
denote by $\\{e^{-tA^{\alpha}}:t\geqslant 0\\}$, see [15] for any
$\alpha\in[0,1]$. With this notation, we have
$X^{-\alpha}=(X^{\alpha})^{\prime}$ for all $\alpha>0$, see [2] for the
characterization of the negative scale.

Let $X_{-1}$ denote the extrapolation space of $X$ generated by $A$, and let
$\\{X_{-1}^{\alpha}:\alpha\geqslant 0\\}$ the fractional power scale generated
by operator $A$ in $X_{-1}$, see [2] and [3] for more details.

Here $f:D(A^{\frac{1}{3}})\subset X\to X$ is a nonlinear function with
suitable growth conditions and regularity in (1.1) for different cases of
$\eta>0$; namely, we consider:

  * •

If $0<\eta<1$, then we prove that the Cauchy problem defined by the linear
equation associated with (1.1) is ill-posed, consequently, the Cauchy problem
defined by (1.1) is ill-posed for any nonlinear function $f$, under the point
of view of the theory of strongly continuous semigroups of bounded linear
operators;

  * •

If $\eta=1$, then we assume that $f$ is twice continuously Fréchet
differentiable and Lipschitz continuous on bounded sets;

  * •

If $\eta>1$, then we assume that $f$ is an $\epsilon$-regular map relative to
the pair $(X^{\frac{1}{3}},X)$ for $\epsilon\geqslant 0$; that is, there exist
constants $c>0$, $\rho>1$, $\gamma(\epsilon)$ with
$\rho\epsilon\leqslant\gamma(\epsilon)<\frac{1}{3}$ such that
$f:X^{\frac{1}{3}+\epsilon}\to X^{\gamma(\epsilon)}$ and

(1.2) $\|f(\phi_{1})-f(\phi_{2})\|_{X^{\gamma(\epsilon)}}\leqslant
c\|\phi_{1}-\phi_{2}\|_{X^{\frac{1}{3}+\epsilon}}(1+\|\phi_{1}\|_{X^{\frac{1}{3}+\epsilon}}^{\rho-1}+\|\phi_{2}\|_{X^{\frac{1}{3}+\epsilon}}^{\rho-1}),$

for any $\phi_{1},\phi_{2}\in X^{\frac{1}{3}+\epsilon}$, see [3, Definition
2], [7] and [8] for more details.

For a better understanding of the $\epsilon$-regular map relative to the pair
$(X^{\frac{1}{3}},X)$ for $\epsilon\geqslant 0$, we construct the following
diagram.

$\nshortmid$$X$$\nshortmid$$X^{\frac{1}{3}}$$f$$\nshortmid$$X^{\gamma(\epsilon)}$$\nshortmid$$X^{\frac{1}{3}+\epsilon}$$f$$\cdots$$\cdots$$\cdots$$\cdots$
Figure 1. $X^{\frac{1}{3}+\epsilon}\subset X^{\frac{1}{3}}$ and
$X^{\gamma(\epsilon)}\subset X$

The evolution equations of third order in time have been studied extensively
in the Hilbert setting and much progress has been achieved. In [1, 6, 12, 16,
17, 18, 19, 21] and [22], and references therein, the Moore-Gibson-Thompson
(MGT) equation is studied in different contexts and results of existence,
stability and regularity of solutions are obtained by the spectral theory of
the MGT operator. In [14] the abstract linear equations of third order in time
is analyzed and results on (non)existence of solution are obtained. In [4] the
abstract linear equations of third order in time is analyzed and results on
(non) existence, stability and regularity of solution are obtained via theory
of fractional powers of closed and densely defined operators.

The article is organized in the following way. In Section 2 we present general
facts on spectral behavior of the our problem. In Section 3 we consider the
case $0\leqslant\eta<1$ and we obtain the result that shows that the problem
(1.1) is ill-posed under the point of view of the theory of strongly
continuous semigroups of bounded linear operators. In Section 4 we consider
the case $\eta>1$ and we obtain a result of existence, stability and
regularity of solutions for (1.1). In Section 5 we consider the case $\eta=1$
via theory of strongly continuous groups of bounded linear operators. Finally,
in Section 6 we explore our results on the Moore-Gibson-Thompson-type
equations, according to the references [1, 6, 12, 16, 17, 18, 19, 21] and
[22].

##  2\. Functional Framework

We first introduce some notations, we consider $Z=X^{\frac{2}{3}}\times
X^{\frac{1}{3}}\times X$ endowed with the norm given by

$\Big{\|}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{\|}^{2}_{Z}=\|u\|^{2}_{X^{\frac{2}{3}}}+\|v\|^{2}_{X^{\frac{1}{3}}}+\|w\|^{2}_{X},\
\forall\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}\in Z.$

###  2.1. Natural setting

We can rewrite the initial value problem associated with equation (1.1) as the
Cauchy problem in $Z$

(2.1) $\dfrac{d}{dt}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}+\mathbb{A}_{(\eta)}\Big{[}\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\Big{]}=F(\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}),\ t>0,$

and

(2.2) $\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}(0)=\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]},$

where $v=u_{t}$ and $w=v_{t}$ and the unbounded linear operator
$\mathbb{A}_{(\eta)}:D(\mathbb{A}_{(\eta)})\subset Z\to Z$ is defined by

(2.3) $D(\mathbb{A}_{(\eta)})=X^{1}\times X^{\frac{2}{3}}\times
X^{\frac{1}{3}}$

and

(2.4) $\mathbb{A}_{(\eta)}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}:=\Big{[}\begin{smallmatrix}0&-I&0\\\ 0&0&-I\\\
A&\eta A^{\frac{2}{3}}&\eta
A^{\frac{1}{3}}\end{smallmatrix}\Big{]}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}-v\\\ -w\\\ Au+\eta
A^{\frac{2}{3}}v+\eta A^{\frac{1}{3}}w\end{smallmatrix}\Big{]},\
\forall\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}\in
X^{1}\times X^{\frac{2}{3}}\times X^{\frac{1}{3}}.$

The nonlinearity $F$ given by

(2.5) $F(\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]})=\Big{[}\begin{smallmatrix}0\\\ 0\\\
f(u)\end{smallmatrix}\Big{]},$

where $f:D(A^{\frac{1}{3}})\subset X\to X$ is a Lipschitz continuous function
on bounded sets.

From now on, we denote

$Z^{1}=D(\mathbb{A}_{(\eta)})=X^{1}\times X^{\frac{2}{3}}\times
X^{\frac{1}{3}}.$

We also consider the following notion of mild solution for (2.1)-(2.2). Given
$\left[\begin{smallmatrix}u_{0}\\\ v_{0}\\\ w_{0}\end{smallmatrix}\right]\in
Z$ we say that $\left[\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\right]$
is a local mild solution of (2.1)-(2.2) provided
$\left[\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\right]\in
C([0,\tau_{u_{0},v_{0},w_{0}}),Z)$, $f(u)\in
C([0,\tau_{u_{0},v_{0},w_{0}}),X)$ and, for
$t\in(0,\tau_{u_{0},v_{0},w_{0}})$, $\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]$ satisfies the integral equation

(2.6) $\left[\begin{smallmatrix}u(t)\\\ v(t)\\\
w(t)\end{smallmatrix}\right]=e^{-\mathbb{A}_{(\eta)}t}\left[\begin{smallmatrix}u_{0}\\\
v_{0}\\\
w_{0}\end{smallmatrix}\right]+\int_{0}^{t}e^{-\mathbb{A}_{(\eta)}(t-s)}\left[\begin{smallmatrix}0\\\
0\\\ f(u(s))\end{smallmatrix}\right]ds.$

for some $\tau_{u_{0},v_{0},w_{0}}>0$.

######  Lemma 2.1.

Let $\mathbb{A}_{(\eta)}$ be the unbounded linear operator defined in
(2.3)-(2.4). The following conditions hold.

  * $i)$

The linear operator $\mathbb{A}_{(\eta)}$ is closed and densely defined;

  * $ii)$

Zero belongs to the resolvent set $\rho(\mathbb{A}_{(\eta)})$; namely, the
resolvent operator of $\mathbb{A}_{(\eta)}$ is the bounded linear operator
$\mathbb{A}_{(\eta)}^{-1}:Z\to Z$ given by

$\mathbb{A}_{(\eta)}^{-1}=\Big{[}\begin{smallmatrix}\eta A^{-\frac{1}{3}}&\eta
A^{-\frac{2}{3}}&A^{-1}\\\ -I&0&0\\\ 0&-I&0\end{smallmatrix}\Big{]}.$

Moreover, $\mathbb{A}_{(\eta)}$ has compact resolvent;

  * $iii)$

The spectrum set of $-\mathbb{A}_{(\eta)}$, $\sigma(-\mathbb{A}_{(\eta)})$, is
given by

$\sigma(-\mathbb{A}_{(\eta)})=\\{\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{z_{\eta}\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{\overline{z_{\eta}}\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\},$

where $\sigma(-A^{\frac{1}{3}})$ denote the spectrum set of $-A^{\frac{1}{3}}$
and

$z_{\eta}=\dfrac{1}{2}\Big{[}(\eta-1)-i\sqrt{3+2\eta-\eta^{2}}\Big{]}$

and

$\overline{z_{\eta}}=\dfrac{1}{2}\Big{[}(\eta-1)+i\sqrt{3+2\eta-\eta^{2}}\Big{]}.$

Proof: To prove part $i)$ we take a sequence
$\Big{(}\Big{[}\begin{smallmatrix}u_{n}\\\ v_{n}\\\
w_{n}\end{smallmatrix}\Big{]},\mathbb{A}_{(\eta)}\Big{[}\begin{smallmatrix}u_{n}\\\
v_{n}\\\ w_{n}\end{smallmatrix}\Big{]}\Big{)}$ is the graph of
$\mathbb{A}_{(\eta)}$, which converges in $Z$ to
$\Big{(}\Big{[}\begin{smallmatrix}\phi\\\ \varphi\\\
\psi\end{smallmatrix}\Big{]},\Big{[}\begin{smallmatrix}\nu\\\ \mu\\\
\zeta\end{smallmatrix}\Big{]}\Big{)}$. From this we easily conclude that
$\nu=-\varphi$, $\mu=-\psi$, and

$u_{n}\to\phi\ \mbox{in}\ X^{\frac{2}{3}}\Longrightarrow
A^{\frac{2}{3}}u_{n}\to A^{\frac{2}{3}}\phi\ \mbox{in}\ X,$ $v_{n}\to\varphi\
\mbox{in}\ X^{\frac{1}{3}}\Longrightarrow A^{\frac{1}{3}}v_{n}\to
A^{\frac{1}{3}}\varphi\ \mbox{in}\ X,$

and

$Au_{n}+\eta A^{\frac{2}{3}}v_{n}+\eta A^{\frac{1}{3}}w_{n}\to\zeta\
\mbox{in}\ X.$

Since

$A^{\frac{2}{3}}u_{n}+\eta A^{\frac{1}{3}}v_{n}+\eta w_{n}\to
A^{\frac{2}{3}}\phi+\eta A^{\frac{1}{3}}\varphi+\eta\psi\ \mbox{in}\ X,$

the closedness of $A^{\frac{1}{3}}$ imples that $A^{\frac{2}{3}}\phi+\eta
A^{\frac{1}{3}}\varphi+\eta\psi\in X^{\frac{1}{3}}$ and $A\phi+\eta
A^{\frac{2}{3}}\varphi+\eta A^{\frac{1}{3}}\psi=\zeta$, and we obtain the
result; that is, $\Big{[}\begin{smallmatrix}\phi\\\ \varphi\\\
\psi\end{smallmatrix}\Big{]}\in D(\mathbb{A}_{(\eta)})$ and
$\mathbb{A}_{(\eta)}\Big{[}\begin{smallmatrix}\phi\\\ \varphi\\\
\psi\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}\nu\\\ \mu\\\
\zeta\end{smallmatrix}\Big{]}$.

For the proof of $ii)$ the result easily follows from
$\mathbb{A}_{(\eta)}^{-1}:Z\to Z$ to be given by

$\mathbb{A}_{(\eta)}^{-1}=\Big{[}\begin{smallmatrix}\eta A^{-\frac{1}{3}}&\eta
A^{-\frac{2}{3}}&A^{-1}\\\ -I&0&0\\\ 0&-I&0\end{smallmatrix}\Big{]}$

which takes bounded subsets of $X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times X$
into bounded subsets of $X^{1}\times X^{\frac{2}{3}}\times X^{\frac{1}{3}}$.
The latter space is compactly embedded in $Z$ because the inclusions
$X^{\beta}\subset X^{\gamma}$ are compact, for $\beta>\gamma\geq 0$, provided
$A$ has compact resolvent. Finally, after considering the eigenvalue problem
for the operator $-\mathbb{A}_{(\eta)}$,

$-\mathbb{A}_{(\eta)}{\bf u}=\lambda{\bf u},$

and after straightforward calculations, part $iii)$ follows immediately from
the fact that $\mathbb{A}_{(\eta)}$ has compact resolvent. ∎

###  2.2. New approach

Let (1.1) be the original equation of third-order. Note that using the change
variable

$v=u_{t}+A^{\frac{1}{3}}u,$

and the function

$w=v_{t},$

we can rewrite the (1.1) as follows, a first-order evolution equation in time
for $w$

(2.7) $w_{t}+(\eta-1)A^{\frac{1}{3}}w+A^{\frac{2}{3}}v=f(u).$

The initial value problem associated with equation (2.7) as the Cauchy problem
in $Z$

(2.8) $\dfrac{d}{dt}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}+\mathbb{B}_{(\eta)}\Big{[}\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\Big{]}=F(\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}),\ t>0,$

and

(2.9) $\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}(0)=\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]},$

where $v=u_{t}$ and $w=v_{t}$ and the unbounded linear operator
$\mathbb{B}_{(\eta)}:D(\mathbb{B}_{(\eta)})\subset Z\to Z$ is defined by

(2.10) $D(\mathbb{B}_{(\eta)})=X^{1}\times X^{\frac{2}{3}}\times
X^{\frac{1}{3}},$

and

(2.11) $\mathbb{B}_{(\eta)}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}:=\Big{[}\begin{smallmatrix}A^{\frac{1}{3}}&-I&0\\\
0&0&-I\\\
0&A^{\frac{2}{3}}&(\eta-1)A^{\frac{1}{3}}\end{smallmatrix}\Big{]}\Big{[}\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}A^{\frac{1}{3}}u-v\\\
-w\\\ A^{\frac{2}{3}}v+(\eta-1)A^{\frac{1}{3}}w\end{smallmatrix}\Big{]},\
\forall\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}\in
X^{1}\times X^{\frac{2}{3}}\times X^{\frac{1}{3}}.$

The nonlinearity $F$ given by (2.5).

######  Lemma 2.2.

The following conditions hold.

  * $i)$

The linear operator $\mathbb{B}_{(\eta)}$ is closed and densely defined;

  * $ii)$

Zero belongs to the resolvent set $\rho(\mathbb{B}_{(\eta)})$; namely, the
resolvent operator of $\mathbb{B}_{(\eta)}$ is the bounded linear operator
$\mathbb{B}_{(\eta)}^{-1}:Z\to Z$ given by

$\mathbb{B}_{(\eta)}^{-1}=\Big{[}\begin{smallmatrix}A^{-\frac{1}{3}}&(\eta-1)A^{-\frac{2}{3}}&A^{-1}\\\
0&(\eta-1)A^{-\frac{1}{3}}&A^{-\frac{2}{3}}\\\
0&-I&0\end{smallmatrix}\Big{]}.$

Moreover, $\mathbb{B}_{(\eta)}$ has compact resolvent;

  * $iii)$

The spectrum set of $-\mathbb{B}_{(\eta)}$, $\sigma(-\mathbb{B}_{(\eta)})$, is
given by

$\sigma(-\mathbb{B}_{(\eta)})=\\{\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{c_{\eta}\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{d_{\eta}\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\},$

where $\sigma(-A^{\frac{1}{3}})$ denote the spectrum set of $-A^{\frac{1}{3}}$
and

$c_{\eta}=\dfrac{1}{2}\Big{[}(\eta-1)+\sqrt{\eta^{2}-2\eta-3}\Big{]}$

and

$d_{\eta}=\dfrac{1}{2}\Big{[}(\eta-1)-\sqrt{\eta^{2}-2\eta-3}\Big{]},$

Proof: To prove part $i)$ we take a sequence
$\Big{(}\Big{[}\begin{smallmatrix}u_{n}\\\ v_{n}\\\
w_{n}\end{smallmatrix}\Big{]},\mathbb{B}_{(\eta)}\Big{[}\begin{smallmatrix}u_{n}\\\
v_{n}\\\ w_{n}\end{smallmatrix}\Big{]}\Big{)}$ is the graph of
$\mathbb{B}_{(\eta)}$, which converges in $Z$ to
$\Big{(}\Big{[}\begin{smallmatrix}\phi\\\ \varphi\\\
\psi\end{smallmatrix}\Big{]},\Big{[}\begin{smallmatrix}\nu\\\ \mu\\\
\zeta\end{smallmatrix}\Big{]}\Big{)}$. Then we have

$u_{n}\to\phi\ \mbox{in}\ X^{\frac{2}{3}}$ $v_{n}\to\varphi\ \mbox{in}\
X^{\frac{1}{3}}$ $w_{n}\to\psi\ \mbox{in}\ X$

and

$A^{\frac{1}{3}}u_{n}-v_{n}\to\nu\ \mbox{in}\ X^{\frac{2}{3}}$ $-w_{n}\to\mu\
\mbox{in}\ X^{\frac{1}{3}}$
$A^{\frac{2}{3}}v_{n}+(\eta-1)A^{\frac{1}{3}}w_{n}\to\zeta\ \mbox{in}\ X$

From this, we easily conclude that

$\psi=-\mu\in X^{\frac{1}{3}}.$

From the closedness of $A^{\frac{1}{3}}$, we obtain

$A^{\frac{1}{3}}\varphi+(\eta-1)\psi\in X^{\frac{1}{3}}\ \ \text{and}\ \
A^{\frac{1}{3}}(A^{\frac{1}{3}}\varphi+(\eta-1)\psi)=\zeta,$

that is,

$\varphi\in X^{\frac{2}{3}}\ \text{and}\ \
A^{\frac{2}{3}}\varphi+(\eta-1)A^{\frac{1}{3}}\psi=\zeta$

From the closedness of $A^{\frac{1}{3}}$, we also obtain

$A^{\frac{2}{3}}\phi-A^{\frac{1}{3}}\varphi\in X^{\frac{1}{3}}\ \ \text{and}\
\
A^{\frac{1}{3}}(A^{\frac{2}{3}}\phi-A^{\frac{1}{3}}\varphi)=A^{\frac{2}{3}}\nu$

that is,

$\phi\in X^{1}\ \ \text{and}\ \ A^{\frac{1}{3}}\phi-\varphi=\nu$

From this, we conclude that $\Big{[}\begin{smallmatrix}\phi\\\ \varphi\\\
\psi\end{smallmatrix}\Big{]}\in D(\mathbb{B}_{(\eta)})$ and
$\mathbb{B}_{(\eta)}\Big{[}\begin{smallmatrix}\phi\\\ \varphi\\\
\psi\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}\nu\\\ \mu\\\
\zeta\end{smallmatrix}\Big{]}$.

For the proof of $ii)$ the result easily follows from
$\mathbb{B}_{(\eta)}^{-1}:Z\to Z$ to be given by

$\mathbb{B}_{(\eta)}^{-1}=\Big{[}\begin{smallmatrix}\eta
A^{-\frac{1}{3}}&(1-\eta)A^{-\frac{2}{3}}&A^{-1}\\\
0&(1-\eta)A^{-\frac{1}{3}}&A^{-\frac{2}{3}}\\\
0&-I&0\end{smallmatrix}\Big{]}.$

Finally, after considering the eigenvalue problem for the operator
$-\mathbb{B}_{(\eta)}$,

$-\mathbb{B}_{(\eta)}{\bf u}=\lambda{\bf u},$

and after straightforward calculations, part $iii)$ follows immediately from
the fact that $\mathbb{B}_{(\eta)}$ has compact resolvent. ∎

##  3\. Ill-posed problems

In this section we consider the case $0\leq\eta<1$. We prove that the Cauchy
problem defined by the linear equation associated with (1.1) is ill-posed in
$Z$, consequently, the Cauchy problem defined by (1.1) is ill-posed for any
nonlinear function $f$ in $Z$, under the point of view of the theory of
strongly continuous semigroups of bounded linear operators.

######  Lemma 3.1.

Let $\mathbb{A}_{(\eta)}$ be the unbounded linear operator defined in
(2.3)-(2.4). If $0\leq\eta<1$, then the unbounded linear operator
$-\mathbb{A}_{(\eta)}$ with $\mathbb{A}_{(\eta)}:D(\mathbb{A}_{(\eta)})\subset
Z\to Z$ is not the infinitesimal generator of a strongly continuous semigroup
in $Z$.

Proof: If $-\mathbb{A}_{(\eta)}$ generates a strongly continuous semigroup
$\\{e^{-\mathbb{A}_{(\eta)}t}:t\geqslant 0\\}$ in $Z$, it follows from [2]
that there exist constants $\omega({\eta)}\geq 0$ and $M(\eta)\geq 1$ such
that

(3.1) $\|e^{-\mathbb{A}_{(\eta)}t}\|_{\mathcal{L}(Y)}\leq
M(\eta)e^{\omega(\eta)t}\ \ \text{\ for \ }t\geqslant 0.$

Moreover, from [2] we have

(3.2)
$\\{\lambda\in\mathbb{C}:Re\lambda>\omega(\eta)\\}\subset\rho(-\mathbb{A}_{(\eta)})$

where $\rho(-\mathbb{A}_{(\eta)})$ denotes the resolvent set of the operator
$-\mathbb{A}_{(\eta)}$. From Lemma 2.1(iii) we have

$\sigma(-\mathbb{A}_{(\eta)})=\\{\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{z_{\eta}\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{\overline{z_{\eta}}\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\},$

where

$z_{\eta}=\dfrac{1}{2}\Big{(}(\eta-1)+i\sqrt{3+2\eta-\eta^{2}}\Big{)}$

and

$\overline{z_{\eta}}=\dfrac{1}{2}\Big{(}(\eta-1)-i\sqrt{3+2\eta-\eta^{2}}\Big{)},$

where $3+2\eta-\eta^{2}>0$ for any $0\leq\eta<1$, see Figure 2.

$\scriptstyle{\rm Re}$$\scriptstyle{\rm
Im}$$\theta_{\eta}$$\sigma(-A^{\frac{1}{3}})$$z_{\eta}\sigma(-A^{\frac{1}{3}})$$\overline{z_{\eta}}\sigma(-A^{\frac{1}{3}})$
Figure 2. Semi-lines contained the eigenvalues of $-\mathbb{A}_{(\eta)}$ and
$0<\theta_{\eta}<\dfrac{\pi}{2}$, $0\leqslant\eta<1$.

Since $\sigma(-A^{\frac{1}{3}})=\\{-\mu_{n}:n\in\mathbb{N}\\}$ with
$\mu_{n}\in\sigma(A^{\frac{1}{3}})$ for each $n\in\mathbb{N}$ and
$\mu_{n}\to\infty$ as $n\to\infty$ and $0\leq\eta<1$, we conclude that

$\sigma(-\mathbb{A}_{(\eta)})\cap\\{\lambda\in\mathbb{C}:Re\lambda>\omega(\eta)\\}\neq\emptyset$

This contradicts (3.4) and therefore $-\mathbb{A}_{(\eta)}$ can not be the
infinitesimal generator of a strongly continuous semigroup in $Z$. ∎

######  Theorem 3.2.

If $0\leqslant\eta<1$, then the Cauchy problem

$\dfrac{d}{dt}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}+\mathbb{A}_{(\eta)}\Big{[}\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}0\\\ 0\\\
0\end{smallmatrix}\Big{]},\ t>0,$

and

$\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}(0)=\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]},$

where $v=u_{t}$ and $w=v_{t}$ is ill-posed on $Z$.

Proof: The result easily follows from Lemma 3.1. ∎

######  Theorem 3.3.

The Cauchy problem (2.1)-(2.2) is ill-posed on $Z$. More precisely, let
$\left[\begin{smallmatrix}u_{0}\\\ v_{0}\\\ w_{0}\end{smallmatrix}\right]\in
Z$ does not exist $\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\in C([0,\tau_{u_{0},v_{0},w_{0}}),Z)$ with $f(u)\in
C([0,\tau_{u_{0},v_{0},w_{0}}),X)$ such that (2.6) holds, for any
$\tau_{u_{0},v_{0},w_{0}}>0$.

Proof: The result easily follows from Theorem 3.2. ∎

######  Remark 3.4.

If $\eta=0$ then thanks to the results in [4] the Cauchy linear problem
associated with (2.1) is ill-posed in $Z$, and therefore, the Cauchy problem
(2.1) is ill-posed in $Z$.

On the new approach present in Subsection 2.2 we have

######  Theorem 3.5.

Let $0\leqslant\eta<1$ and let $\mathbb{B}_{(\eta)}$ be the unbounded linear
operator defined in (2.10)-(2.11). Then the problem (2.8)-(2.9) is ill-posed
in the sense that it does not generate a strongly continuous semigroup of
bounded linear operators on the state space $Z$.

Proof: If $-\mathbb{B}_{(\eta)}$ generates a strongly continuous semigroup
$\\{e^{-\mathbb{B}_{(\eta)}t}:t\geqslant 0\\}$ on $Z$, it follows from Pazy
[20, Theorem 1.2.2] that there exist constants $\omega\geq 0$ and $M\geq 1$
such that

(3.3) $\|e^{-\mathbb{B}_{(\eta)}t}\|_{\mathcal{L}(Y)}\leq Me^{\omega t}\ \
\text{\ for \ }0\leq t<\infty.$

Moreover, from Pazy [20, Remark 1.5.4] we have

(3.4)
$\\{\lambda\in\mathbb{C}:Re\lambda>\omega\\}\subset\rho(-\mathbb{B}_{(\eta)}).$

where $\rho(-\mathbb{B}_{(\eta)})$ denotes the resolvent set of the operator
$-\mathbb{B}_{(\eta)}$.

From Lemma 2.2 we can consider a sequence
$(\lambda_{k}z_{\eta})_{k}\in\sigma(-\mathbb{B}_{(\eta)})$, for
$k=1,2,3,\dots$, with $\lambda_{k}\in\sigma(-A^{\frac{1}{3}})$ and
$|\lambda_{k}|\to\infty$ as $k\to\infty$. Note that

$arg(\lambda_{k}z_{\eta})=\arctan\left(\frac{2\sqrt{-\eta^{2}+2\eta+3}}{1-\eta}\right)$

and since $0\leq\eta<1$, we have

$0<arg(\lambda_{k}z_{\eta})<\pi/2$

for every $k\geq 1$ and $|\lambda_{k}z_{\eta}|\to\infty$ as $k\to\infty$. Then
we conclude that

$\sigma(-\mathbb{B}_{(\eta)})\cap\\{\lambda\in\mathbb{C}:Re\lambda>\omega\\}\neq\emptyset.$

This contradicts (3.4) and therefore $-\mathbb{B}_{(\eta)}$ can not be the
infinitesimal generator of a strongly continuous semigroup on $Z$. ∎

##  4\. Parabolic differential equations

In this section we consider the case $\eta>1$. Namely, thanks to the Lemma 2.2
we have the following illustration of the eigenvalues of
$-\mathbb{B}_{(\eta)}$.

$\scriptstyle{\rm Re}$$\scriptstyle{\rm
Im}$$\theta_{\eta}$$\sigma(-A^{\frac{1}{3}})$$z_{\eta}\sigma(-A^{\frac{1}{3}})$$\overline{z_{\eta}}\sigma(-A^{\frac{1}{3}})$
Figure 3. Semi-lines contained the eigenvalues of $-\mathbb{B}_{(\eta)}$ and
$0<\theta_{\eta}<\dfrac{\pi}{2}$, $\eta>1$.

###  4.1. Sectoriality

Initially, we prove the following theorem on the sectoriality of the operator
$\mathbb{B}_{(\eta)}$ for $\eta>1$.

######  Theorem 4.1.

Let $\eta>1$. The unbounded linear operator $\mathbb{B}_{(\eta)}$ defined in
(2.10)-(2.11) is a sectorial operator.

Proof: In this proof, $M$ will denote a positive constant, not necessarily the
same one. Let $\lambda\in\mathbb{C}$, then

$\lambda I-\mathbb{B}_{(\eta)}=\Big{[}\begin{smallmatrix}\lambda
I-A^{\frac{1}{3}}&I&0\\\ 0&\lambda I&I\\\ 0&-A^{\frac{2}{3}}&\lambda
I-(\eta-1)A^{\frac{1}{3}}\end{smallmatrix}\Big{]}.$

From Lemma2.2 it follows that

(4.1)
$\rho(\mathbb{B}_{(\eta)})=\rho(A^{\frac{1}{3}})\cap\rho(c_{\eta}A^{\frac{1}{3}})\cap\rho(d_{\eta}A^{\frac{1}{3}}).$

Note that for $\lambda\in\rho(\mathbb{B}_{(\eta)})$ we have

(4.2) $(\lambda
I-\mathbb{B}_{(\eta)})^{-1}=D_{\eta}(\lambda)^{-1}\left[\begin{smallmatrix}(\lambda
I-c_{\eta}A^{\frac{1}{3}})(\lambda I-d_{\eta}A^{\frac{1}{3}})&-(\lambda
I-(\eta-1)A^{\frac{1}{3}})&I\\\ 0&(\lambda I-A^{\frac{1}{3}})(\lambda
I-(\eta-1)A^{\frac{1}{3}})&-(\lambda I-A^{\frac{1}{3}})\\\ 0&(\lambda
I-A^{\frac{1}{3}})A^{\frac{2}{3}}&\lambda(\lambda
I-A^{\frac{1}{3}})\end{smallmatrix}\right]$

where

$D_{\eta}(\lambda)=(\lambda I-A^{\frac{1}{3}})(\lambda
I-c_{\eta}A^{\frac{1}{3}})(\lambda I-d_{\eta}A^{\frac{1}{3}})$

with

(4.3) $\displaystyle c_{\eta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\eta-1+\sqrt{\eta^{2}-2\eta-3}\right)$ (4.4)
$\displaystyle d_{\eta}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\eta-1-\sqrt{\eta^{2}-2\eta-3}\right).$

Since $Re(c_{\eta})>0$ and $Re(d_{\eta})>0$, for $\eta>1$,
$c_{\eta}A^{\frac{1}{3}}$ and $d_{\eta}A^{\frac{1}{3}}$ are sectorial
operators. Let
$S_{A^{\frac{1}{3}}},S_{c_{\eta}A^{\frac{1}{3}}},S_{d_{\eta}A^{\frac{1}{3}}}$
be sectors such that $S_{A^{\frac{1}{3}}}\subset\rho(A^{\frac{1}{3}})$,
$S_{c_{\eta}A^{\frac{1}{3}}}\subset\rho(c_{\eta}A^{\frac{1}{3}})$,
$S_{d_{\eta}A^{\frac{1}{3}}}\subset\rho(d_{\eta}A^{\frac{1}{3}})$ and

$\displaystyle\|\lambda I-A^{\frac{1}{3}}\|_{\mathcal{L}(X)}$ $\displaystyle<$
$\displaystyle\frac{M}{|\lambda|},\ \ \text{for each}\ \ \lambda\in
S_{A^{\frac{1}{3}}}$ $\displaystyle\|\lambda
I-c_{\eta}A^{\frac{1}{3}}\|_{\mathcal{L}(X)}$ $\displaystyle<$
$\displaystyle\frac{M}{|\lambda|},\ \ \text{for each}\ \ \lambda\in
S_{A^{\frac{1}{3}}}$ $\displaystyle\|\lambda
I-d_{\eta}A^{\frac{1}{3}}\|_{\mathcal{L}(X)}$ $\displaystyle<$
$\displaystyle\frac{M}{|\lambda|},\ \ \text{for each}\ \ \lambda\in
S_{A^{\frac{1}{3}}}$

for some $M>0$. We will prove that $\mathbb{B}_{\eta}$ is a sectorial operator
using the sector

$S_{\mathbb{B}_{(\eta)}}:=S_{A^{\frac{1}{3}}}\cap
S_{c_{\eta}A^{\frac{1}{3}}}\cap S_{d_{\eta}A^{\frac{1}{3}}}.$

It is immediate that
$S_{\mathbb{B}_{(\eta)}}\subset\rho(\mathbb{B}_{(\eta)})$. If $\lambda\in
S_{\mathbb{B}_{(\eta)}}$ and ${\bf u}=\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\in Z$ with $\|{\bf u}\|_{Z}\leqslant 1$, then
writing

$(\lambda I-\mathbb{B}_{(\eta)})^{-1}{\bf
u}=\left[\begin{smallmatrix}\varphi_{1}\\\ \varphi_{2}\\\
\varphi_{3}\end{smallmatrix}\right]$

where

(4.5) $\begin{split}\varphi_{1}&=(\lambda^{2}I-(\eta-1)\lambda
A^{\frac{1}{3}}-A^{\frac{2}{3}})D_{\eta}(\lambda)^{-1}u+(-\lambda
I+(\eta-1)A^{\frac{1}{3}})D_{\eta}(\lambda)^{-1}v+D_{\eta}(\lambda)^{-1}w,\\\
\varphi_{2}&=(\lambda^{2}I-\eta\lambda
A^{\frac{1}{3}}+(\eta-1)A^{\frac{2}{3}})D_{\eta}(\lambda)^{-1}v+(-\lambda
I+A^{\frac{1}{3}})D_{\eta}(\lambda)^{-1}w,\\\ \varphi_{3}&=(\lambda
A^{\frac{2}{3}}-A)D_{\eta}(\lambda)^{-1}v+(\lambda^{2}I-\lambda
A^{\frac{1}{3}})D_{\eta}(\lambda)^{-1}w.\end{split}$

In order to conclude that

$\|\varphi_{1}\|_{X^{\frac{2}{3}}}+\|\varphi_{2}\|_{X^{\frac{1}{3}}}+\|\varphi_{3}\|_{X}<\frac{M}{|\lambda|}$

We only need to show that $\lambda A^{\frac{1}{3}}D_{\eta}(\lambda)^{-1}$,
$A^{\frac{2}{3}}D_{\eta}(\lambda)^{-1}$, $\lambda^{2}D_{\eta}(\lambda)^{-1}$
$\in\mathcal{L}(X)$ and are bounded by $M/|\lambda|$, which is clear because
$D_{\eta}(\lambda)^{-1}\in\mathcal{L}(X)$ and

$\|D_{\eta}(\lambda)^{-1}\|_{\mathcal{L}(X)}<\frac{M}{|\lambda|^{3}}$

∎

######  Lemma 4.2.

Let $Z_{-1}$ denote the extrapolation space of $Z$ generated by
$\mathbb{B}_{(\eta)}$. Then

$Z_{-1}=X^{\frac{1}{3}}\times X\times X^{-\frac{1}{3}}.$

Proof: The extrapolation space of $Z$ is the completion of the normed space
$(Z,\|\mathbb{B}_{(\eta)}^{-1}\|_{Z})$. Since for
$\left[\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\right]\in Z$ we have

$\mathbb{B}_{(\eta)}^{-1}\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]=\Big{[}\begin{smallmatrix}\eta
A^{-\frac{1}{3}}u+(1-\eta)A^{-\frac{2}{3}}v+A^{-1}w\\\
(1-\eta)A^{-\frac{1}{3}}v+A^{-\frac{2}{3}}w\\\ v&\end{smallmatrix}\Big{]},$

and consequently

$\begin{split}&\Big{\|}\mathbb{B}_{(\eta)}^{-1}\left[\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\right]\Big{\|}_{X^{\frac{2}{3}}\times
X^{\frac{1}{3}}\times X}\\\ &=\|\eta
A^{-\frac{1}{3}}u+(1-\eta)A^{-\frac{2}{3}}v+A^{-1}w\|_{X^{\frac{2}{3}}}+\|(1-\eta)A^{-\frac{1}{3}}v+A^{-\frac{2}{3}}w\|_{X^{\frac{1}{3}}}+\|v\|_{X}\\\
&\leqslant\eta\|A^{\frac{1}{3}}u\|_{X}+\max\\{1,\eta-1\\}\|v\|_{X}+2\|A^{-\frac{1}{3}}w\|_{X}\\\
&\leqslant C_{1\eta}\Big{\|}\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\Big{\|}_{X\times X^{-\frac{1}{3}}\times
X^{-\frac{2}{3}}},\end{split}$

where $C_{1\eta}=\max\\{2,\eta\\}>0$.

By other hand

$\begin{split}&\Big{\|}\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\Big{\|}_{X^{\frac{1}{3}}\times X\times
X^{-\frac{1}{3}}}\\\
&=\|u\|_{X^{\frac{1}{3}}}+\|v\|_{X}+\|w\|_{X^{-\frac{1}{3}}}\\\
&=\dfrac{1}{\eta}\|\eta
A^{-\frac{1}{3}}u\|_{X^{\frac{2}{3}}}+\|v\|_{X}+\|w\|_{X^{-\frac{1}{3}}}\\\
&\leqslant\dfrac{1}{\eta}\|\eta
A^{-\frac{1}{3}}u+(1-\eta)A^{-\frac{2}{3}}v+A^{-1}w\|_{X^{\frac{2}{3}}}+\Big{(}2-\dfrac{1}{\eta}\Big{)}\|v\|_{X}+\Big{(}\dfrac{1}{\eta}+1\Big{)}\|w\|_{X^{-\frac{1}{3}}}\\\
&\leqslant\dfrac{1}{\eta}\|\eta
A^{-\frac{1}{3}}u+(1-\eta)A^{-\frac{2}{3}}v+A^{-1}w\|_{X^{\frac{2}{3}}}+\Big{(}2+\eta-\dfrac{2}{\eta}\Big{)}\|v\|_{X}+\Big{(}\dfrac{1}{\eta}+1\Big{)}\|(1-\eta)A^{-\frac{1}{3}}v+A^{-\frac{2}{3}}w\|_{X^{\frac{1}{3}}},\\\
&\leqslant
C_{2\eta}\Big{\|}\mathbb{B}_{(\eta)}^{-1}\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\Big{\|}_{X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X},\end{split}$

for some $C_{2\eta}=2+\eta-\dfrac{2}{\eta}>0$.

Hence, there exist $C_{1\eta}>0$ and $C_{2\eta}>0$ such that

$\Big{\|}\mathbb{B}_{(\eta)}^{-1}\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\Big{\|}_{X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X}\leqslant C_{1\eta}\Big{\|}\left[\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\right]\Big{\|}_{X\times X^{-\frac{1}{3}}\times
X^{-\frac{2}{3}}}\leqslant
C_{1\eta}C_{2\eta}\Big{\|}\mathbb{B}_{(\eta)}^{-1}\left[\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\right]\Big{\|}_{X^{\frac{2}{3}}\times
X^{\frac{1}{3}}\times X},$

completions of $(X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X,\|\mathbb{B}_{(\eta)}^{-1}\cdot\|_{X^{\frac{2}{3}}\times
X^{\frac{1}{3}}\times X})$ and $(X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X,\|\mathbb{B}_{(\eta)}^{-1}\cdot\|_{X^{\frac{1}{3}}\times X\times
X^{-\frac{1}{3}}})$ coincide. ∎

Consider the closed extension of $\mathbb{B}_{(\eta)}$ to $Z_{-1}$ (see [2,
Page 262]) and still denote it by $\mathbb{B}_{(\eta)}$. Then
$\mathbb{B}_{(\eta)}$ is a sectorial and positive operator in $Z_{-1}$ with
the domain $Z_{-1}^{1}=X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times X$; the
imaginary powers of $\mathbb{B}_{(\eta)}$ are bounded. Our next concern will
be to obtain embeddings of the spaces from the fractional power scale
$Z_{-1}^{\alpha}$, $\alpha\geqslant 0$, generated by
$(\mathbb{B}_{(\eta)},Z_{-1})$.

Before we can proceed we need the following general interpolation result.

######  Proposition 4.3.

Let $\mathcal{X}_{i}$, $\mathcal{Y}_{i}$, $\mathcal{M}_{i}$, $i=1,2,3$ be the
Banach spaces such that $\mathcal{X}_{1}\subset\mathcal{X}_{0}$,
$\mathcal{Y}_{1}\subset\mathcal{Y}_{0}$,
$\mathcal{M}_{1}\subset\mathcal{M}_{0}$ topologically and algebraically. Then

$[\mathcal{X}_{0}\times\mathcal{Y}_{0}\times\mathcal{M}_{0},\mathcal{X}_{1}\times\mathcal{Y}_{1}\times\mathcal{M}_{1}]_{\theta}=[\mathcal{X}_{0},\mathcal{X}_{1}]_{\theta}\times[\mathcal{Y}_{0},\mathcal{Y}_{1}]_{\theta}\times[\mathcal{M}_{0},\mathcal{M}_{1}]_{\theta},$

for any $\theta\in(0,1)$.

Proof: The proof is an immediate consequence of the definition of complex
interpolation spaces in [23, Section 1.9.2]. ∎

The following result also may be established by a straightforward extension of
[7, Theorem 2] so we omit its proof.

######  Lemma 4.4.

Denote by $\\{Z_{-1}^{\alpha}:\alpha\in[0,1]\\}$ the partial fractional power
scale generated by operator $\mathbb{B}_{(\eta)}$ in $Z_{-1}$. Then

$Z_{-1}^{k+\alpha}=X^{\frac{k+1+\alpha}{3}}\times X^{\frac{k+\alpha}{3}}\times
X^{\frac{k-1+\alpha}{3}},\quad\alpha\in[0,1],\ k\in\mathbb{N}.$

For better understanding the relation of the fractional power scale-spaces of
the operator $\mathbb{B}_{(\eta)}$, we construct the following diagram.

$Z$$Z^{1}$$\cdots$$(Z^{1})^{\prime}=Z^{-1}$$\cdots$$\mathbb{B}_{(\eta)}$$Z^{1}_{-1}$$Z^{\alpha}_{-1}$$\cdots$$Z_{-1}$$\cdots$$\mathbb{B}_{(\eta)}$
Figure 4. Fractional power scale generated by operator $\mathbb{B}_{(\eta)}$
in $Z_{-1}$ and $\alpha>1$.

In this section, we consider the case $\eta>1$. Now we prove some of the main
results of this paper.

######  Proposition 4.5.

Let $Z_{-1}$ denote the extrapolation space of $Z$ generated by
$\mathbb{A}_{\eta}$. Then

$Z_{-1}=X^{\frac{1}{3}}\times X\times X^{-\frac{1}{3}}.$

Proof: Again, following [2] and [8] we recall that the extrapolation space of
$Z$ is the completion of the normed space
$(Z,\|\mathbb{A}_{\eta}^{-1}\|_{Z})$. Since for
$\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}\in Z$

$\Big{\|}\mathbb{A}_{\eta}^{-1}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{\|}_{X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X}=\Big{\|}\Big{[}\begin{smallmatrix}\eta A^{-\frac{1}{3}}u+\eta
A^{-\frac{2}{3}}v+A^{-1}w\\\ -u\\\
-v\end{smallmatrix}\Big{]}\Big{\|}_{X^{\frac{2}{3}}\times
X^{\frac{1}{3}}\times X}\leqslant
3(1+\eta)\Big{\|}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{\|}_{X^{\frac{1}{3}}\times X\times
X^{-\frac{1}{3}}},$

and

$\Big{\|}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{\|}_{X^{\frac{1}{3}}\times X\times
X^{-\frac{1}{3}}}\leqslant
3(1+\eta)\Big{\|}\mathbb{A}_{\eta}^{-1}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{\|}_{X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X},$

completions of $(X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X,\|\mathbb{A}_{\eta}^{-1}\cdot\|_{X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X})$ and $(X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times
X,\|\cdot\|_{X^{\frac{1}{3}}\times X\times X^{-\frac{1}{3}}})$ coincide, see
the Figure 5. ∎

Consider the closed extension of $\mathbb{A}_{\eta}$ to $Z_{-1}$ (see [2, Page
262]) and still denote it by $\mathbb{A}_{\eta}$ with the domain
$Z_{-1}^{1}=X^{\frac{2}{3}}\times X^{\frac{1}{3}}\times X$.

For better understanding the relation of the fractional power scale-spaces of
the operator $\mathbb{A}_{\eta}$, we construct the following diagram.

$Z$$Z^{1}$$(Z^{1})^{\prime}=Z^{-1}$$\mathbb{A}_{\eta}$$Z^{1}_{-1}$$Z^{\alpha}_{-1}$$Z_{-1}$$\mathbb{A}_{\eta}$$\cdots$$\cdots$$\cdots$$\cdots$
Figure 5. Fractional power scale generated by operator $\mathbb{A}_{\eta}$ in
$Z_{-1}$ and $\alpha>1$.

###  4.2. Analysis by reducing the order

Namely, we consider the reduction of order

(4.6) $v=u_{t}+A^{\frac{1}{3}}u$

for positive time, where $u$ is an unknown function to be determined, and we
obtain the following equation of second order

(4.7) $v_{tt}+A^{\frac{2}{3}}v+(\eta-1)A^{\frac{1}{3}}v_{t}=g(v)$

for positive time, where $\eta>0$ and

(4.8) $g(v)=f(u).$

Note that we can reviews (4.7) as follows

(4.9) $\left[\begin{matrix}v\\\
v_{t}\end{matrix}\right]_{t}+\left[\begin{matrix}0&-I\\\
A^{\frac{2}{3}}&(\eta-1)A^{\frac{1}{3}}\end{matrix}\right]\left[\begin{matrix}v\\\
v_{t}\end{matrix}\right]=\left[\begin{matrix}0\\\
g(v)\end{matrix}\right],\quad\left[\begin{matrix}v(0)\\\
v_{t}(0)\end{matrix}\right]=\left[\begin{matrix}v_{0}\\\
w_{0}\end{matrix}\right]\in X^{\frac{1}{3}}\times X.$

Solving (4.9) we find $e^{-\Lambda t}\left[\begin{matrix}v_{0}\\\
w_{0}\end{matrix}\right]=\left[\begin{matrix}v(t,v_{0},w_{0})\\\
v_{t}(t,v_{0},w_{0})\end{matrix}\right]\in X^{\frac{1}{3}}\times X$ where
$\Lambda:D(\Lambda)\subset X^{\frac{1}{3}}\times X\to X^{\frac{1}{3}}\times X$
is defined by, $D(\Lambda)=X^{\frac{2}{3}}\times X^{\frac{1}{3}}$

$\Lambda\left[\begin{matrix}v_{0}\\\
w_{0}\end{matrix}\right]=\left[\begin{matrix}0&-I\\\
A^{\frac{2}{3}}&(\eta-1)A^{\frac{1}{3}}\end{matrix}\right]\left[\begin{matrix}v_{0}\\\
w_{0}\end{matrix}\right]=\left[\begin{matrix}-w_{0}\\\
A^{\frac{2}{3}}v_{0}\end{matrix}\right]$

It follows that $\mathbb{R}^{+}\ni t\mapsto v(t,v_{0},w_{0})\in
X^{\frac{1}{3}}$ is a continuous function. We can then try to solve

(4.10) $u_{t}+A^{\frac{1}{3}}u=v(t,v_{0},w_{0}),\quad u(0)=u_{0}$

through the variation of constants formula

(4.11)
$u(t)=e^{-A^{\frac{1}{3}}t}u(0)+\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}v(s)ds.$

for positive time.

We next solve (4.7) following the same ideas in [8, 9, 10] and [11]. Next, we
use (4.11) to obtain a local solution to our original differential equation. A
similar argument has been used in [5] and [16] for the classical MGT equation.

To better present our results we introduce some notations. We consider the
space $Y=X^{\frac{1}{3}}\times X$ equipped the norm given by

$\Big{\|}\begin{bmatrix}v\\\
w\end{bmatrix}\Big{\|}^{2}_{Y}=\|v\|^{2}_{X^{\frac{1}{3}}}+\|w\|^{2}_{X},\
\forall\begin{bmatrix}v\\\ w\end{bmatrix}\in Y.$

We can rewrite the initial value problem associated with equation (4.7) as the
Cauchy problem in $Y$

(4.12) $\dfrac{d}{dt}\begin{bmatrix}v\\\
w\end{bmatrix}+\varLambda\begin{bmatrix}v\\\
w\end{bmatrix}=G(\begin{bmatrix}v\\\ w\end{bmatrix}),\ t>0,$

and

(4.13) $\begin{bmatrix}v\\\ w\end{bmatrix}(0)=\begin{bmatrix}v_{0}\\\
w_{0}\end{bmatrix}$

where $w=v_{t}$ and the unbounded linear operator
$\varLambda:D(\varLambda)\subset Y\to Y$ is defined by

$D(\varLambda)=X^{\frac{2}{3}}\times X^{\frac{1}{3}}$

and

$\varLambda\begin{bmatrix}v\\\ w\end{bmatrix}:=\begin{bmatrix}0&-I\\\
A^{\frac{2}{3}}&(\eta-1)A^{\frac{1}{3}}\end{bmatrix}\begin{bmatrix}v\\\
w\end{bmatrix}=\begin{bmatrix}-w\\\
A^{\frac{2}{3}}v+(\eta-1)A^{\frac{1}{3}}w\end{bmatrix},\
\forall\begin{bmatrix}v\\\ w\end{bmatrix}\in X^{\frac{2}{3}}\times
X^{\frac{1}{3}}.$

From now on, we denote

$Y^{1}=D(\varLambda)=X^{\frac{2}{3}}\times X^{\frac{1}{3}}.$

The nonlinearity $G$ is given by

(4.14) $G(\begin{bmatrix}v\\\ w\end{bmatrix})=\begin{bmatrix}0\\\
g(v)\end{bmatrix}$

where $g$ is given by (4.8).

The following result may be established by a straightforward extension of [8,
Lemma 1] so we omit its proof.

######  Lemma 4.6.

The following conditions hold.

  * $i)$

The linear operator $\varLambda$ is closed and densely defined;

  * $ii)$

$0\in\rho(\varLambda)$ with
$\varLambda^{-1}=\Big{[}\begin{smallmatrix}(\eta-1)A^{-\frac{1}{3}}&A^{-\frac{2}{3}}\\\
-I&0\end{smallmatrix}\Big{]}$. Moreover, $\varLambda$ has compact resolvent;

  * $iii)$

The spectrum of $\varLambda$ consists of isolated eigenvalues
$\lambda^{\pm}_{n}$ given by

$\lambda^{\pm}_{n}=(\eta\pm\sqrt{\eta^{2}-1})\sqrt[3]{\mu_{n}},\
\mu_{n}\in\sigma(A);$

  * $iv)$

The linear operator $-\varLambda$ generates in $Y$ a $C^{0}$ analytic
semigroup $\\{e^{-\varLambda t}:t\geqslant 0\\}$;

  * $v)$

The semigroup $\\{e^{-\varLambda t}:t\geqslant 0\\}$ in $Y$ are compact and
asymptotically decaying;

  * $vi)$

For each $\begin{bmatrix}v\\\ w\end{bmatrix}\in Y^{1}$, we have

$\dfrac{1}{\eta}\Big{\|}\begin{bmatrix}v\\\
w\end{bmatrix}\Big{\|}_{Y^{1}}\leqslant\Big{\|}\varLambda\begin{bmatrix}v\\\
w\end{bmatrix}\Big{\|}_{Y}\leqslant\eta\Big{\|}\begin{bmatrix}v\\\
w\end{bmatrix}\Big{\|}_{Y^{1}}.$

The Lemma 4.6 allows us to define the fractional power $\varLambda^{-\alpha}$
of order $\alpha\in(0,1)$. Denote by $Y^{\alpha}=D(\varLambda^{\alpha})$ for
$\alpha\in[0,1)$, taking $\varLambda^{0}:=I$ on $Y^{0}:=Y$ when $\alpha=0$.
Recall that $Y^{\alpha}$ is dense in $Y$ for all $\alpha\in(0,1]$, for details
see [2, Theorem 4.6.5]. The fractional power space $X^{\alpha}$ endowed with
the norm

$\|\cdot\|_{Y^{\alpha}}:=\|\varLambda^{\alpha}\cdot\|_{Y}$

is a Banach space.

######  Remark 4.7.

Let $X_{-1}$ denote the extrapolation space of $X$ generated by
$A^{\frac{1}{3}}$. Consider the closed extension of $A^{\frac{1}{3}}$ to
$X_{-1}$ (see [2, Page 262]) and still denote it by $A^{\frac{1}{3}}$. Then
$A^{\frac{1}{3}}$ is a sectorial and positive operator in $X_{-1}$ with the
domain $X_{-1}^{\frac{1}{3}}$; the imaginary powers of $A^{\frac{1}{3}}$ are
bounded, see e.g. [7, Proposition 5]. Our next concern will be to obtain
embeddings of the spaces from the fractional power scale
$X_{-1}^{\frac{\alpha}{3}}$, $\alpha\geqslant 0$, generated by
$(A^{\frac{1}{3}},X_{-1})$.

For better understanding the relation of the fractional power scale-spaces of
the operator $A^{\frac{1}{3}}$, we construct the following diagram.

$X$$X^{\frac{1}{3}}$$(X^{\frac{1}{3}})^{\prime}=X^{-\frac{1}{3}}$$A^{\frac{1}{3}}$$X^{\frac{1}{3}}_{-1}$$X^{\frac{\alpha}{3}}_{-1}$$X_{-1}$$A^{\frac{1}{3}}$$\cdots$$\cdots$$\cdots$$\cdots$
Figure 6. Fractional power scale generated by operator $A^{\frac{1}{3}}$ in
$X_{-1}$ and $\alpha>1$.

The following result also may be established by a straightforward extension of
[7, Proposition 3] or [9, Theorem 2.3] so we omit its proof.

######  Theorem 4.8.

For each $\alpha\in[0,1]$, the fractional power spaces $Y^{\alpha}$ associated
to the operator $\varLambda$ coincide with $X^{\frac{1+\alpha}{3}}\times
X^{\frac{\alpha}{3}}$ with equivalent norms.

######  Lemma 4.9.

Let $Y_{-1}$ denote the extrapolation space of $Y$ generated by $\varLambda$.
Then

$Y_{-1}=X\times X^{-\frac{1}{3}}.$

Proof: Following [2] and [8] we recall that the extrapolation space of $Y$ is
the completion of the normed space $(Y,\|\varLambda^{-1}\|_{Y})$. Since for
$\begin{bmatrix}v\\\ w\end{bmatrix}\in Y$

$\Big{\|}\varLambda^{-1}\begin{bmatrix}v\\\
w\end{bmatrix}\Big{\|}_{X^{\frac{1}{3}}\times
X}\leqslant\eta\Big{\|}\begin{bmatrix}v\\\ w\end{bmatrix}\Big{\|}_{X\times
X^{-\frac{1}{3}}}\leqslant\eta^{2}\Big{\|}\varLambda^{-1}\begin{bmatrix}v\\\
w\end{bmatrix}\Big{\|}_{X^{\frac{1}{3}}\times X},$

completions of $(X^{\frac{1}{3}}\times
X,\|\varLambda^{-1}\cdot\|_{X^{\frac{1}{3}}\times X})$ and
$(X^{\frac{1}{3}}\times X,\|\varLambda^{-1}\cdot\|_{X\times
X^{-\frac{1}{3}}})$ coincide, see the Figure 7. ∎

Consider the closed extension of $\varLambda$ to $Y_{-1}$ (see [2, Page 262])
and still denote it by $\varLambda$. Then $\varLambda$ is a sectorial and
positive operator in $Y_{-1}$ with the domain
$Y_{-1}^{1}=X^{\frac{1}{3}}\times X$; the imaginary powers of $\varLambda$ are
bounded, see e.g. [7, Proposition 5]. Our next concern will be to obtain
embeddings of the spaces from the fractional power scale $Y_{-1}^{\alpha}$,
$\alpha\geqslant 0$, generated by $(\varLambda,Y_{-1})$.

The following result also may be established by a straightforward extension of
[7, Theorem 2] so we omit its proof.

######  Lemma 4.10.

Denote by $\\{Y_{-1}^{\alpha}:\alpha\in[0,1]\\}$ the partial fractional power
scale generated by operator $\varLambda$ in $Y_{-1}$. Then

$Y_{-1}^{k+\alpha}=X^{\frac{k+\alpha}{3}}\times
X^{\frac{k-1+\alpha}{3}},\quad\alpha\in[0,1],\ k\in\mathbb{N}.$

For better understanding the relation of the fractional power scale-spaces of
the operator $\varLambda$, we construct the following diagram.

$Y$$Y^{1}$$\cdots$$(Y^{1})^{\prime}=Y^{-1}$$\cdots$$\varLambda$$Y^{1}_{-1}$$Y^{\alpha}_{-1}$$\cdots$$Y_{-1}$$\cdots$$\varLambda$
Figure 7. Fractional power scale generated by operator $\varLambda$ in
$Y_{-1}$ and $\alpha>1$.

Since the semigroup $\\{e^{-\varLambda t}:t\geqslant 0\\}$ generated by
$-\varLambda$ in $Y_{-1}$ is analytic and the linear Cauchy problem 4.12-4.13
with $\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix}\in Y_{-1}$ has a unique
solution

(4.15) $\begin{bmatrix}v\\\ w\end{bmatrix}(t)=e^{-\varLambda
t}\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix},\ t\geqslant 0.$

In the following theorem we explain the smoothing action of the solution to
the linear Cauchy problem 4.12-4.13, it follows from [9, Theorem 2.4] and [11,
Theorem 2.1.1].

######  Theorem 4.11.

If $t>0$ and $\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix}\in Y_{-1}$, then

$\begin{bmatrix}v\\\ w\end{bmatrix}(t)=e^{-\varLambda
t}\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix}\in Y_{-1}^{\alpha},\ \mbox{for
each}\ \alpha\geqslant 0.$

In particular, if $t>0$ and $\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix}\in
Y^{1}_{-1}$, then

$\begin{bmatrix}v\\\ w\end{bmatrix}(t)=e^{-\varLambda
t}\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix}\in Y_{-1}^{1}.$

Moreover,

$\begin{bmatrix}v\\\ w\end{bmatrix}(\cdot)\in
C^{1}((0,\infty),Y^{\alpha}_{-1}),\ \mbox{for each}\ \alpha\in[0,1).$

In particular, if $\begin{bmatrix}v_{0}\\\ w_{0}\end{bmatrix}\in Y^{1}_{-1}$,
then

$\begin{bmatrix}v\\\ w\end{bmatrix}(\cdot)\in C^{1}([0,\infty),Y_{-1}).$

######  Theorem 4.12.

Assume that $f$ is $\epsilon-$regular map relative to the pair
$(X^{\frac{1}{3}},X)$ for $\epsilon\geqslant 0$, in the sense of (1.2), then
$g$ is $\epsilon-$regular map relative to the pair
$(X_{-1}^{\frac{1}{3}},X_{-1})$, where
$\\{X_{-1}^{\frac{\alpha}{3}}:\alpha\geqslant 0\\}$ denotes the fractional
power scale generated by operator $A^{\frac{1}{3}}$ in $X_{-1}$.

Proof: We will prove that there exist constants $c>0$, $\rho>1$,
$\gamma(\epsilon)>0$ with $\rho\epsilon\leqslant\gamma(\epsilon)<\frac{1}{3}$
such that $g:X_{-1}^{\frac{1}{3}+\epsilon}\to X_{-1}^{\gamma(\epsilon)}$ and

$\|g(\phi_{1})-g(\phi_{2})\|_{X_{-1}^{\gamma(\epsilon)}}\leqslant
c\|\phi_{1}-\phi_{2}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}(1+\|\phi_{1}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1}+\|\phi_{2}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1}),$

for any $\phi_{1},\phi_{2}\in X_{-1}^{\frac{1}{3}+\epsilon}$, see the figures
below.

For a better understanding of the $\epsilon$-regular map relative to the pair
$(X_{-1}^{\frac{1}{3}},X_{-1})$ for $\epsilon\geqslant 0$, we construct the
following diagram.

$\nshortmid$$X_{-1}$$\nshortmid$$X_{-1}^{\frac{1}{3}}$$g$$\nshortmid$$X_{-1}^{\gamma(\epsilon)}$$\nshortmid$$X_{-1}^{\frac{1}{3}+\epsilon}$$g$$\cdots$$\cdots$$\cdots$$\cdots$
Figure 8. $X_{-1}^{\frac{1}{3}+\epsilon}\subset X_{-1}^{\frac{1}{3}}$ and
$X_{-1}^{\gamma(\epsilon)}\subset X_{-1}$

Since $f$ is an $\epsilon$-regular map relative to the pair
$(X^{\frac{1}{3}},X)$ for $\epsilon\geqslant 0$, we have the following: if
$g(\phi_{i})=f(\psi_{i})$, where

$\psi_{i}=e^{-A^{\frac{1}{3}}t}u(0)+\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{i}(s)ds,\quad
i=1,2,$

and $\\{e^{-A^{\frac{1}{3}}t};t\geqslant 0\\}$ denotes the $C^{0}-$semigroup
corresponding to (4.6) with $v\equiv 0$ subject to initial condition
$u(0)=u_{0}\in X^{\frac{1}{3}+\epsilon}$ then for each $t\geqslant 0$,
$g=f\circ e^{-A^{\frac{1}{3}}t}$ is an $\epsilon$-regular map relative to the
pair $(X_{-1}^{\frac{1}{3}},X_{-1})$ for $\epsilon\geqslant 0$ because by
Remark 4.7 we have

$\begin{split}&\|g(\phi_{1})-g(\phi_{2})\|_{X_{-1}^{\gamma(\epsilon)}}\\\
&\leqslant c_{0}\|f(\psi_{1})-f(\psi_{2})\|_{X^{\gamma(\epsilon)}}\\\
&\leqslant
c_{0}\|\psi_{1}-\psi_{2}\|_{X^{\frac{1}{3}+\epsilon}}(1+\|\psi_{1}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1}+\|\psi_{2}\|_{X^{\frac{1}{3}+\epsilon}}^{\rho-1})\\\
&=c_{1}\Big{\|}\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{1}(s)ds-\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{2}(s)ds\Big{\|}_{X^{\frac{1}{3}+\epsilon}}\times\\\
&\times\Big{(}1+\Big{\|}e^{-A^{\frac{1}{3}}t}u(0)+\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{1}(s)ds\Big{\|}_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1}+\Big{\|}e^{-A^{\frac{1}{3}}t}u(0)+\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{2}(s)ds\Big{\|}_{X^{\frac{1}{3}+\epsilon}}^{\rho-1}\Big{)}\\\
&=c_{1}\Big{\|}\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}(\phi_{1}-\phi_{2})(s)ds\Big{\|}_{X^{\frac{1}{3}+\epsilon}}\times\\\
&\times\Big{(}1+\Big{\|}e^{-A^{\frac{1}{3}}t}u(0)+\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{1}(s)ds\Big{\|}_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1}+\Big{\|}e^{-A^{\frac{1}{3}}t}u(0)+\displaystyle\int_{0}^{t}e^{-A^{\frac{1}{3}}(t-s)}\phi_{2}(s)ds\Big{\|}_{X^{\frac{1}{3}+\epsilon}}^{\rho-1}\Big{)}\\\
&\leqslant
c_{2}\|\phi_{1}-\phi_{2}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}(1+\|\phi_{1}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1}+\|\phi_{2}\|_{X_{-1}^{\frac{1}{3}+\epsilon}}^{\rho-1})\end{split}$

for some positive constants $c_{0},c_{1}$ and $c_{2}$. ∎

Thanks to [7, Corollary 2] we have the following on the $\epsilon-$regular map
$G$ defined in (4.14).

######  Proposition 4.13.

For $\epsilon\geqslant 0$, $G$ is $\epsilon-$regular map relative to the pair
$(Y^{1}_{-1},Y_{-1})$; that is, there exist constants $C>0$, $\rho>1$,
$\gamma(\epsilon)$ with $\rho\epsilon\leqslant\gamma(\epsilon)<1$ such that
such that $G:Y_{-1}^{1+\epsilon}\to Y_{-1}^{\gamma(\epsilon)}$ and

(4.16) $\Big{\|}G(\begin{bmatrix}\phi\\\
\varphi\end{bmatrix})-G(\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix})\Big{\|}_{Y_{-1}^{\gamma(\epsilon)}}\leqslant
C\Big{\|}\begin{bmatrix}\phi\\\
\varphi\end{bmatrix}-\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix}\Big{\|}_{Y_{-1}^{1+\epsilon}}\Big{(}1+\Big{\|}\begin{bmatrix}\phi\\\
\varphi\end{bmatrix}\Big{\|}_{Y_{-1}^{1+\epsilon}}^{\rho-1}+\Big{\|}\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix}\Big{\|}_{Y_{-1}^{1+\epsilon}}^{\rho-1}\Big{)},$

for any $\begin{bmatrix}\phi\\\
\varphi\end{bmatrix},\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix}\in Y_{-1}^{1+\epsilon}.$

Proof: Let $\begin{bmatrix}\phi\\\
\varphi\end{bmatrix},\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix}\in Y_{-1}^{1+\epsilon}$. Thanks to Lemma 4.10 we
have

$\Big{\|}G(\begin{bmatrix}\phi\\\
\varphi\end{bmatrix})-G(\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix})\Big{\|}_{Y_{-1}^{\gamma(\epsilon)}}=\Big{\|}\begin{bmatrix}0\\\
g(\phi)-g(\phi^{\prime})\end{bmatrix}\Big{\|}_{Y_{-1}^{\gamma(\epsilon)}}=\|g(\phi)-g(\phi^{\prime})\|_{X^{\frac{-1+\gamma(\epsilon)}{3}}}$

and from fractional power scale generated by operator $A^{\frac{1}{3}}$ in
$X_{-1}$ we obtain

$\Big{\|}G(\begin{bmatrix}\phi\\\
\varphi\end{bmatrix})-G(\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix})\Big{\|}_{Y_{-1}^{\gamma(\epsilon)}}\leqslant
c\|g(\phi)-g(\phi^{\prime})\|_{X_{-1}^{\frac{\gamma(\epsilon)}{3}}}$

for some $c>0$, and by Theorem 4.12 we get

$\begin{split}\Big{\|}G(\begin{bmatrix}\phi\\\
\varphi\end{bmatrix})-G(\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix})\Big{\|}_{Y_{-1}^{\gamma(\epsilon)}}&\leqslant
C\|\phi-\phi^{\prime}\|_{X_{-1}^{\frac{1}{3}+\tilde{\epsilon}}}(1+\|\phi\|_{X_{-1}^{\frac{1}{3}+\tilde{\epsilon}}}^{\rho-1}+\|\phi^{\prime}\|_{X_{-1}^{\frac{1}{3}+\tilde{\epsilon}}}^{\rho-1})\\\
&\leqslant
C\|\phi-\phi^{\prime}\|_{X^{\tilde{\epsilon}}}(1+\|\phi\|_{X^{\tilde{\epsilon}}}^{\rho-1}+\|\phi^{\prime}\|_{X^{\tilde{\epsilon}}}^{\rho-1})\\\
&\leqslant C\Big{\|}\begin{bmatrix}\phi\\\
\varphi\end{bmatrix}-\begin{bmatrix}\phi^{\prime}\\\
\varphi^{\prime}\end{bmatrix}\Big{\|}_{Y_{-1}^{1+\epsilon}}(1+\|\phi\|_{X^{\tilde{\epsilon}}}^{\rho-1}+\|\phi^{\prime}\|_{X^{\tilde{\epsilon}}}^{\rho-1})\end{split}$

for some $C>0$, $\rho>1$ and $\gamma(\epsilon)>0$ with
$\rho\tilde{\epsilon}\leqslant\frac{\gamma(\epsilon)}{3}<\frac{1}{3}$
$(\tilde{\epsilon}=\frac{\epsilon}{3})$.

Hence, by Lemma 4.10 we obtain (4.16) and the proof of the result is complete.
∎

######  Definition 4.14.

Let $\epsilon\geqslant 0$, $\tau>0$, $\begin{bmatrix}v_{0}\\\
w_{0}\end{bmatrix}\in Y_{-1}^{1}$. We say that $\begin{bmatrix}v\\\
w\end{bmatrix}:[t_{0},\tau]\to Y_{-1}^{1}$ is an $\epsilon-$regular mild
solution ($\epsilon-$solution for shor) to (4.12)-(4.13) if
$\begin{bmatrix}v\\\ w\end{bmatrix}\in C([t_{0},\tau],Y_{-1}^{1})\cap
C((t_{0},\tau],Y_{-1}^{1+\epsilon})$, and $\begin{bmatrix}v\\\
w\end{bmatrix}(t)$ satisfies

$\begin{bmatrix}v(t)\\\
w(t)\end{bmatrix}=e^{-\varLambda(t-t_{0})}\begin{bmatrix}v_{0}\\\
w_{0}\end{bmatrix}+\int_{t_{0}}^{t}e^{-\varLambda(t-s)}G(\begin{bmatrix}v(s)\\\
w(s)\end{bmatrix})ds.$

Thanks to [3, Theorem 1], see also [7, Theorem 3], we have the following on
existence of $\epsilon$-regular solution to (4.12)-(4.13) on certain interval
$[0,\tau]$.

######  Theorem 4.15.

Let $\epsilon\geqslant 0$, $\tau>0$, $\begin{bmatrix}v_{0}\\\
w_{0}\end{bmatrix}\in Y_{-1}^{1}$. Then exists a unique $\epsilon$-regular
solution to (4.12)-(4.13) on certain interval $[0,\tau]$. This solution
satisfies

$\begin{bmatrix}v\\\ w\end{bmatrix}\in C((0,\tau],Y_{-1}^{1})\cap
C((0,\tau],Y_{-1}^{1+\theta}),\quad 0\leqslant\theta<\gamma(\epsilon),$

and

$t^{\theta}\|\begin{bmatrix}v(t)\\\ w(t)\end{bmatrix}\|_{Y_{-1}^{1+\theta}}\to
0,\quad\mbox{as}\quad t\searrow 0,\quad 0<\theta<\gamma(\epsilon).$

From this, we establish local well posedness for the Cauchy problem
(2.1)-(2.2). We would like to study this problem in the phase space $Z$. To
pose the problem in the mentioned space we will need to consider the nonlinear
term $F$ as a map with values in the extrapolated space $Z_{-1}$ associated to
$\mathbb{A}_{\eta}$ in $Z$.

######  Definition 4.16.

Let $\epsilon=0$, $\tau>0$, $\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}\in Z^{1}_{-1}$ and
$\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}(\cdot):[0,\tau]\to Z_{-1}^{1}$. We say that
$\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}(\cdot)$ is an
$\epsilon-$regular solution to (2.1)-(2.2) on $[0,\tau]$ if and only if

  * i)

$\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}(\cdot)\in
C([0,\tau],Z_{-1}^{1})\cap C((0,\tau],Z_{-1}^{1+\epsilon})$;

  * ii)

$\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}(\cdot)$
satisfies the Cauchy integral formula:

(4.17) $\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}(t)=e^{-\mathbb{A}_{\eta}t}\Big{[}\begin{smallmatrix}u_{0}\\\
v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}+\displaystyle\int_{0}^{t}e^{-\mathbb{A}_{\eta}(t-s)}F\Big{(}\Big{[}\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\Big{]}(s)\Big{)}ds,\ t\in[0,\tau].$

######  Theorem 4.17.

Let $\epsilon=0$, $\tau>0$, $\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}\in Z^{1}_{-1}$. Then there exists a unique
$\epsilon-$regular solution to (2.1)-(2.2) on certain interval $[0,\tau]$. In
addition, we have

$\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}\in
C((0,\tau],Z_{-1}^{1})\cap C((0,\tau],Z_{-1}^{1+\theta}),\quad
0\leqslant\theta<\gamma(\epsilon),$

and

$t^{\theta}\|\begin{bmatrix}u(t)\\\ v(t)\\\
w(t)\end{bmatrix}\|_{Z_{-1}^{1+\theta}}\to 0,\quad\mbox{as}\quad t\searrow
0,\quad 0<\theta<\gamma(\epsilon).$

Proof: We also know that, for $\epsilon>0$, $t^{\epsilon}\left\|e^{-\Lambda
t}\left[\begin{matrix}v_{0}\\\
w_{0}\end{matrix}\right]\right\|_{Y^{\epsilon}}\stackrel{{\scriptstyle t\to
0}}{{\longrightarrow}}0$. Knowing that
$Y^{\epsilon}=X^{\frac{1}{3}+\epsilon}\times X^{\epsilon}$, we have that
$t^{\epsilon}\|v(t,v_{0},w_{0})\|_{X^{\frac{1}{3}+\epsilon}}\stackrel{{\scriptstyle
t\to 0}}{{\longrightarrow}}0$. Then the integral defining
$u(t,u_{0},v_{0},w_{0})$ is convergent and the resulting function is
continuous at $t=0$. In fact

$\begin{split}\|u-u_{0}\|_{X^{\frac{2}{3}}}&\leqslant\|e^{-A^{\frac{1}{3}}t}u_{0}-u_{0}\|_{X^{\frac{2}{3}}}+\displaystyle\int_{0}^{t}\|e^{-A^{\frac{1}{3}}(t-s)}\|_{\mathcal{L}(X,X^{1-\epsilon})}\|v(s)\|_{X^{\frac{1}{3}+\epsilon}}ds\\\
&\leqslant\|e^{-A^{\frac{1}{3}}t}u_{0}-u_{0}\|_{X^{\frac{2}{3}}}+M\displaystyle\int_{0}^{t}(t-s)^{1-\epsilon}s^{-\epsilon}s^{\epsilon}\|v(s)\|_{X^{\frac{1}{3}+\epsilon}}ds\\\
&\leqslant\|e^{-A^{\frac{1}{3}}t}u_{0}-u_{0}\|_{X^{\frac{2}{3}}}+M\displaystyle\int_{0}^{1}(1-s)^{-1+\epsilon}s^{-\epsilon}ds\
{\displaystyle\sup_{s\in[0,t]}s^{\epsilon}\|v(s)\|_{X^{\frac{1}{3}+\epsilon}}}\end{split}$

This ensures the continuity.

Finally, the result follows from (4.6) and Theorem 4.15. ∎

##  5\. Remaks on the case $\eta=1$

In this section we consider the case $\eta=1$. We note that the initial value
problem associated with equation (2.7) as the Cauchy problem in $Z$

(5.1) $\dfrac{d}{dt}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}+\mathbb{B}_{(1)}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}=F_{1}(\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}),\ t>0,$

and

(5.2) $\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}(0)=\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]},$

where $v=u_{t}$ and $w=v_{t}$ and the unbounded linear operator
$\mathbb{B}_{(1)}:D(\mathbb{B}_{(1)})\subset Z\to Z$ is defined by

(5.3) $D(\mathbb{B}_{(1)})=X^{1}\times X^{\frac{2}{3}}\times X^{\frac{1}{3}},$

and

(5.4) $\mathbb{B}_{(1)}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}:=\Big{[}\begin{smallmatrix}A^{\frac{1}{3}}&-I&0\\\
0&0&-I\\\
0&A^{\frac{2}{3}}&0\end{smallmatrix}\Big{]}\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}=\Big{[}\begin{smallmatrix}A^{\frac{1}{3}}u-v\\\
-w\\\ A^{\frac{2}{3}}v\end{smallmatrix}\Big{]},\
\forall\Big{[}\begin{smallmatrix}u\\\ v\\\ w\end{smallmatrix}\Big{]}\in
X^{1}\times X^{\frac{2}{3}}\times X^{\frac{1}{3}}.$

The nonlinearity $F_{1}$ given by (2.5), where $f:D(A^{\frac{2}{3}})\subset
X\to X$ is a Lipschitz continuous function on bounded sets.

Thanks to Lemma 2.2, in particular, we have

######  Lemma 5.1.

Let $\mathbb{B}_{(1)}$ be the unbounded linear operator defined in
(5.3)-(5.4). The following conditions hold.

  * $i)$

The unbounded linear operator $\mathbb{B}_{(1)}$ is closed and densely
defined;

  * $ii)$

Zero belongs to the resolvent set $\rho(\mathbb{B}_{(1)})$; namely, the
resolvent operator of $\mathbb{B}_{(1)}$ is the bounded linear operator
$\mathbb{B}_{(1)}^{-1}:Z\to Z$ given by

(5.5)
$\mathbb{B}_{(1)}^{-1}=\Big{[}\begin{smallmatrix}A^{-\frac{1}{3}}&0&A^{-1}\\\
0&0&A^{-\frac{2}{3}}\\\ 0&-I&0\end{smallmatrix}\Big{]}.$

Moreover, $\mathbb{B}_{(1)}$ has compact resolvent;

  * $iii)$

The spectrum set of $-\mathbb{B}_{1}$, $\sigma(-\mathbb{B}_{(1)})$, is given
by

$\sigma(-\mathbb{B}_{(1)})=\\{\lambda\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{\lambda
i\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\}\cup\\{-\lambda
i\in\mathbb{C}:\lambda\in\sigma(-A^{\frac{1}{3}})\\},$

where $\sigma(-A^{\frac{1}{3}})$ denote the spectrum set of
$-A^{\frac{1}{3}}$.

$\scriptstyle{\rm Re}$$\scriptstyle{\rm
Im}$$\sigma(-A^{\frac{1}{3}})$$i\sigma(-A^{\frac{1}{3}})$$-i\sigma(-A^{\frac{1}{3}})$
Figure 9. Semi-lines contained the eigenvalues of $-\mathbb{B}_{1}$.

Proof: The proof of $i)$ easily follows from (5.4). For the proof of $ii)$ the
result easily follows from (5.5). Finally, for the proof of $iii)$ the result
easily follows from (5.4). ∎

######  Theorem 5.2.

Let $\mathbb{B}_{(1)}$ be the unbounded linear operator defined in
(5.3)-(5.4). The unbounded linear operator $-\mathbb{B}_{(1)}$ is not a
dissipative operator on the state space $Z$.

Proof: Indeed, let $u\in X^{1}$ be such that $u\neq 0$ and let
$\Big{[}\begin{smallmatrix}u\\\ 2A^{\frac{1}{3}}u\\\
0\end{smallmatrix}\Big{]}\in Z^{1}$. Note that

$\Big{\langle}\mathbb{B}_{(1)}\Big{[}\begin{smallmatrix}u\\\
2A^{\frac{1}{3}}u\\\ 0\end{smallmatrix}\Big{]},\Big{[}\begin{smallmatrix}u\\\
2A^{\frac{1}{3}}u\\\ 0\end{smallmatrix}\Big{]}\Big{\rangle}=-\langle
A^{\frac{1}{3}}u,u\rangle_{X^{\frac{2}{3}}}$

and consequently

$Re\Big{\langle}-\mathbb{B}_{(1)}\Big{[}\begin{smallmatrix}u\\\
2A^{\frac{1}{3}}u\\\ 0\end{smallmatrix}\Big{]},\Big{[}\begin{smallmatrix}u\\\
2A^{\frac{1}{3}}u\\\ 0\end{smallmatrix}\Big{]}\Big{\rangle}>0$

and the prove is complete. ∎

Theorem5.2 ensures that the linear operator $-\mathbb{B}_{(1)}$ is not an
infinitesimal generator of a specific type of strongly continuous semigroup in
$Z$; that is, the strongly continuous semigroup of contractions in $Z$.

##  6\. Moore-Gibson-Thompson-type equations

In this subsection we present boundary-initial value problems associated with
a Moore-Gibson-Thompson equation with fractional damped and strongly damped
linear wave equation, where our results from previous section can be applied.
Namely, let $\Omega\subset\mathbb{R}^{N}$, $N\geqslant 3$, be a bounded smooth
domain, and the initial-boundary value problems

(6.1) $\begin{cases}u_{ttt}-\Delta
u+\eta(-\Delta)^{\frac{1}{3}}u_{tt}+\eta(-\Delta)^{\frac{2}{3}}u_{t}=f(u),&\
t>0,\ x\in\Omega,\\\ u(0,x)=\varphi(x),\ u_{t}(0,x)=\xi(x),\
u_{tt}(0,x)=\psi(x),&\ x\in\Omega,\\\ u(t,x)=0,&\ t\geqslant 0,\
x\in\partial\Omega,\end{cases}$

where $\eta>1$.

The nonlinearity $f:\mathbb{R}\to\mathbb{R}$ in (6.1) is a continuously
differentiable function satisfying for some $1<\rho<\frac{3N+4}{3N-8}$ the
growth condition

(6.2) $|f^{\prime}(s)|\leqslant C(1+|s|^{\rho-1}).$

Here we consider $X=L^{2}(\Omega)$ and the negative Laplacian operator

$Au=-\Delta u,$

with domain

$D(A)=H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$

which is a sectorial operator and it bounded imaginary powers, and
consequently the spaces $X^{\alpha}$, $\alpha\in[0,1]$, are characterized with
the aid of complex interpolation as

$X^{\alpha}=[H^{2}(\Omega)\cap
H^{1}_{0}(\Omega),L^{2}(\Omega)]_{\alpha}=H^{2\alpha}_{\\{I\\}}(\Omega)$

and

$X^{-\alpha}=(H^{2\alpha}_{\\{I\\}}(\Omega))^{\prime}$

where $[\cdot,\cdot]_{\alpha}$ denotes the complex interpolation function (see
[2] and [13]). In particular $X=X^{0}=L^{2}(\Omega)$,
$X^{\frac{1}{2}}=H_{0}^{1}(\Omega)$,
$X^{-\frac{1}{2}}=(H_{0}^{1}(\Omega))^{\prime}$ and $X^{1}=H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$.

With this set-up we will consider problem (6.1) in the form (2.8)-(2.9) with
$u_{0}=\varphi,v_{0}=A^{\frac{1}{3}}\varphi+\xi$, and
$w_{0}=A^{\frac{1}{3}}\xi+\psi$.

Let $F:Z_{-1}^{1}\to Z_{-1}^{\alpha}$, $\alpha\geqslant 0$, be a locally
Lipschitz continuous map, as well as in (2.5). Recall that a mild solution of
(2.8)-(2.9) on $[0,\tau]$ is a function $\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{(}\cdot,\Big{[}\begin{smallmatrix}u_{0}\\\
v_{0}\\\ w_{0}\end{smallmatrix}\Big{]}\Big{)}\in C([0,\tau],Z_{-1}^{1})$ which
satisfies

$\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{(}t,\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}\Big{)}=e^{-\mathbb{B}_{\eta}t}\Big{[}\begin{smallmatrix}u_{0}\\\
v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}+\displaystyle\int_{0}^{t}e^{-\mathbb{B}_{\eta}(t-s)}F\Big{(}\Big{[}\begin{smallmatrix}u\\\
v\\\ w\end{smallmatrix}\Big{]}\Big{(}s,\Big{[}\begin{smallmatrix}u_{0}\\\
v_{0}\\\ w_{0}\end{smallmatrix}\Big{]}\Big{)}\Big{)}ds,$

for $t\in[0,\tau]$. We say that (2.8)-(2.9) is locally well posed in
$Z_{-1}^{1}$ is for any $\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}\in Z_{-1}^{1}$ there is a unique mild solution

$t\mapsto\Big{[}\begin{smallmatrix}u\\\ v\\\
w\end{smallmatrix}\Big{]}\Big{(}t,\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\
w_{0}\end{smallmatrix}\Big{]}\Big{)}$

of (2.8)-(2.9) defined on a maximal interval of existence
$[0,t_{u_{,}v_{0},w_{0}})$ and depending continuously on the initial data
$\Big{[}\begin{smallmatrix}u_{0}\\\ v_{0}\\\ w_{0}\end{smallmatrix}\Big{]}$.

As a consequence of the Sobolev embeddings we obtain the following result cf.
[11, Proposition 1.3.8].

######  Proposition 6.1.

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded domain of class $C^{m}$ and
$(A,D(A))$ be a sectorial operator in $L^{p}(\Omega)$, $1<p<\infty$, with
$D(A)\subset W^{2m,p}(\Omega)$ for some $m\geqslant 1$. Then for
$\alpha\in[0,1]$ the following inclusion holds.

$X^{\alpha}\subset W^{s,q}(\Omega)$

if $2m\alpha-\frac{N}{p}\geqslant s-\frac{N}{q}$, $1<p\leqslant q<\infty$,
$s\geqslant 0$.

From we have

######  Theorem 6.2.

The problem (2.8)-(2.9) with $\eta>1$ is locally posed in $Z_{-1}^{1}$
whenever $f$ satisfies (6.2) for some $1<\rho<\frac{3N+4}{3N-8}$

Proof: The map $F$ defined as in (2.5) is Lipschitz continuous on bounded sets
from $Z_{-1}^{1}$ into $Z_{-1}^{\sigma}=X^{\frac{1+\sigma}{3}}\times
X^{\frac{\sigma}{3}}\times X^{\frac{-1+\sigma}{3}}$ whenever
$0<\sigma\leqslant\tilde{\sigma}$, and
$\tilde{\sigma}=\min\\{1,(\rho-1)(2-\frac{3}{4}N)+1\\}$
($X^{\frac{2}{3}}\hookrightarrow
L^{\frac{6N(\rho-1)}{8+4(2\sigma)}}(\Omega)$). Indeed, if $B$ is a bounded
subset of $Z_{-1}^{1}$ and $\Big{[}\begin{smallmatrix}u_{1}\\\ v_{1}\\\
w_{1}\end{smallmatrix}\Big{]},\Big{[}\begin{smallmatrix}u_{2}\\\ v_{2}\\\
w_{2}\end{smallmatrix}\Big{]}\in B$, we have

$\Big{\|}F\Big{(}\Big{[}\begin{smallmatrix}u_{1}\\\ v_{1}\\\
w_{1}\end{smallmatrix}\Big{]}\Big{)}-F\Big{(}\Big{[}\begin{smallmatrix}u_{2}\\\
v_{2}\\\
w_{2}\end{smallmatrix}\Big{]}\Big{)}\Big{\|}_{Z_{-1}^{\sigma}}\leqslant
c_{1}\|f(u_{1})-f(u_{2})\|_{X^{\frac{-1+\sigma}{3}}}.$

Since $L^{\frac{6N}{3N+4(1-\sigma)}}(\Omega)\hookrightarrow
H^{\frac{2(1-\sigma)}{3}}_{\\{I\\}}(\Omega)=X^{\frac{-1+\sigma}{3}}$ we obtain

$\Big{\|}F\Big{(}\Big{[}\begin{smallmatrix}u_{1}\\\ v_{1}\\\
w_{1}\end{smallmatrix}\Big{]}\Big{)}-F\Big{(}\Big{[}\begin{smallmatrix}u_{2}\\\
v_{2}\\\
w_{2}\end{smallmatrix}\Big{]}\Big{)}\Big{\|}_{Z_{-1}^{\sigma}}\leqslant
c_{2}\|f(u_{1})-f(u_{2})\|_{L^{\frac{6N}{3N+4(1-\sigma)}}(\Omega)}$

and thanks to (6.2) there exists $C>0$ such that

$\forall s_{1},s_{2}\in\mathbb{R},\quad|f(s_{1})-f(s_{2})|\leqslant
C|s_{1}-s_{2}|(1+|s_{1}|^{\rho-1}+|s_{2}|^{\rho-1})$

and consequently

$\begin{split}\Big{\|}F\Big{(}\Big{[}\begin{smallmatrix}u_{1}\\\ v_{1}\\\
w_{1}\end{smallmatrix}\Big{]}\Big{)}-F\Big{(}\Big{[}\begin{smallmatrix}u_{2}\\\
v_{2}\\\
w_{2}\end{smallmatrix}\Big{]}\Big{)}\Big{\|}_{Z_{-1}^{\sigma}}&\leqslant
c_{3}\|u_{1}-u_{2}\|_{X^{\frac{2}{3}}}\Big{(}1+\|u_{1}\|^{\rho-1}_{L^{\frac{6N(\rho-1)}{8+4(1-\sigma)}}(\Omega)}+\|u_{2}\|^{\rho-1}_{L^{\frac{6N(\rho-1)}{8+4(1-\sigma)}}(\Omega)}\Big{)}\\\
&\leqslant c_{4}\Big{\|}\Big{[}\begin{smallmatrix}u_{1}\\\ v_{1}\\\
w_{1}\end{smallmatrix}\Big{]}-\Big{[}\begin{smallmatrix}u_{2}\\\ v_{2}\\\
w_{2}\end{smallmatrix}\Big{]}\Big{\|}_{Z_{-1}^{1}}.\end{split}$

The proof now follows from [15]. ∎

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