# Global well-posedness for fractional Sobolev-Galpern type equations

Huy Tuan Nguyen a,b, Nguyen Anh Tuan c, Chao Yangd,111Corresponding author:
<EMAIL_ADDRESS>(Chao Yang). Other authors<EMAIL_ADDRESS>(Huy
Tuan Nguyen<EMAIL_ADDRESS>(Nguyen Anh Tuan).

aDepartment of Mathematics and Computer Science, University of Science, Ho Chi
Minh City, Vietnam
b Vietnam National University, Ho Chi Minh City, Vietnam
c Division of Applied Mathematics, Thu Dau Mot University, Binh Duong
Province, Vietnam
dCollege of Mathematical Sciences, Harbin Engineering University, 150001,
People’s Republic of China

###### Abstract

This article is a comparative study on an initial-boundary value problem for a
class of semilinear pseudo-parabolic equations with the fractional Caputo
derivative, also called the fractional Sobolev-Galpern type equations. The
purpose of this work is to reveal the influence of the degree of the source
nonlinearity on the well-posedness of the solution. By considering four
different types of nonlinearities, we derive the global well-posedness of mild
solutions to the problem corresponding to the four cases of the nonlinear
source terms. For the advection source function case, we apply a nontrivial
limit technique for singular integral and some appropriate choices of weighted
Banach space to prove the global existence result. For the gradient
nonlinearity as a local Lipschitzian, we use the Cauchy sequence technique to
show that the solution either exists globally in time or blows up at finite
time. For the polynomial form nonlinearity, by assuming the smallness of the
initial data we derive the global well-posed results. And for the case of
exponential nonlinearity in two-dimensional space, we derive the global well-
posedness by additionally use of Orlicz space.

Keywords: fractional pseudo-parabolic, globally Lipschitz source, exponential
nonlinearity, global well-posedness.
2020 MSC Classification: 35K20, 35K58.

##  1 Introduction

In the current work, we especially concern about the following time-fractional
pseudo-parabolic equation

$\displaystyle\partial_{t}^{\alpha}u-\partial_{t}^{\alpha}\Delta u-\Delta
u=H\left(u\right),$ (1)

wherein $\Omega$ is a bounded domain of $\mathbb{R}^{d}~{}(d\in\mathbb{N}),u$
is an unknown function from $\Omega\times(0,\infty)$ to $\mathbb{R}$, $H$ is
the source function which is going to be defined in more details later. The
notation $\partial_{t}^{\alpha}$ is abbreviated for the time-fractional
derivative of order $\alpha\in(0,1)$ in the Caputo sense, defined by

$\displaystyle\partial_{t}^{\alpha}u(t)=\int_{0}^{t}\frac{(t-\tau)^{\alpha-1}}{\Gamma(\alpha)}\partial_{\tau}u(\tau)\mathrm{d}\tau,$

provided that the right-hand side (RHS) of the above equation makes sense. In
addition, for Equation (1), we assume that the boundary $\partial\Omega$ of
$\Omega$ is sufficiently smooth. Also, for Equation (1) we consider the
homogeneous Dirichlet boundary condition

$\displaystyle u(x,t)=0,\quad(x,t)\in\partial\Omega\times(0,\infty),$ (2)

and the following initial value condition

$\displaystyle u(x,t)=u_{0}(x),\quad(x,t)\in\Omega\times\\{0\\},$ (3)

here, $u_{0}$ is an initial function which satisfies some specific assumptions
in different theorems claimed in the present paper. In Equation (1), the
operator $(-\Delta)=\mathcal{A}:\text{dom}(\mathcal{A})\subset
L^{2}(\Omega)\rightarrow L^{2}(\Omega)$ is uniformly symmetric elliptic. As we
know that, $\mathcal{A}$ owns a set of positive eigenvalues
$\\{\theta_{k}\\}_{k\in\mathbb{N}}$ whose elements organize an increasing
sequence. Corresponding to this set of Dirichlet eigenvalues, there is a set
of eigenfunctions $\\{\phi_{k}\\}_{k\in\mathbb{N}}$ which forms a complete
orthonormal of $L^{2}(\Omega)$. Our main goal is to investigate the globally
well-posed results for a mild solution of the initial-boundary value problem
(1)-(3), and we shall introduce the background of our work in the following
subsection.

###  1.1 Background of the problem

A generalized form of Equation (1) reflecting the theory of isotropic
incompressible homogeneous fluid was first considered by Coleman and Walter in
[16]. To investigate some non-steady fluid flows as the second-order fluids
due to the effect of pressure, Ting in [34] considered a special type
semilinear pseudo-parabolic equation in the following form

$\displaystyle a\partial_{t}u=b\Delta u+c\partial_{t}\Delta u+H(u).$ (4)

This equation along with a specified type of initial-boundary conditions
describes a bounded flow whose external force affects its solid boundary. And
such types of equations like (4) also appear in the study about thermodynamic
temperature [13] and population recovery [32]. In one-dimensional space, by
taking $b=0$ and $H(u)=-u_{x}-uu_{x}$ in (4), we have the regularized long-
wave equation or the Benjamin–Bona–Mahony BBM equation, which was proposed by
Benjamin, Bona, and Mahony in [7] with applications in the study of long waves
propagation. The work [7] was extended by Amick, Bona, and Schonbek in [3] by
investigating the large time behavior of solution to the Cauchy problem
related to the one-dimensional case of (4) with $H(u)=-u_{x}-uu_{x}$. This
nonlinear pseudo-parabolic equation can be seen as an additional consideration
of dissipation mechanisms for the BBM equation, also Celebi et.al. in [14]
considered the generalized BBM equation (GBBM) in the form of

$\displaystyle\partial_{t}u-\partial_{t}\Delta u-b\Delta
u+(\eta\cdot\nabla)u+\nabla\cdot F(u)=0,$

where $\eta\in\mathbb{R}^{d}$ is a constant vector, $F(u)$ is a
$d-$dimentional vector field, which received a lot of attentions in the PDEs
forum [1, 39, 30]. In addition, from [43, 25, 40, 19] and the references given
there, the readers will find that Equation (4) can also be used in biological
sciences, ﬁltration theory or the study of semiconductors.

Given the representativeness and application value of this kind of
mathematical model, it has attracted many mathematicians’ attention with many
rich results. It is an impossible task to mention all of them, so we only make
an overview of the works closely related to the research of this paper. In
[33], Showalter and Ting considered an initial value problem for a generalized
form of (4) as follows

$\displaystyle M\partial_{t}u+Lu=H,$

here, $M,L$ are second-order differential operators independent of $t$,
containing variable coefficients with some specific properties, $M$ is
uniformly strongly elliptic on a bounded open set
$\Omega\subset\mathbb{R}^{d}$. In the homogeneous case, based on properties of
the Friedrichs extensions of $M$ and $L$ which can be obtained by the Lax-
Milgram theorem, the authors constructed a group
$\left\\{E(t):t\in\mathbb{R}\right\\}$ which helps to get the unique existence
of the weak solution. Also, thanks to the results that $H^{1}_{0}(\Omega)\cap
H^{p}$ is invariant under effects of the group
$\left\\{E(t):t\in\mathbb{R}\right\\}$, the regularity of the solution was
proved. The main results of this work also include the asymptotic behavior for
the solution and the extended theories for the nonhomogeneous case. We also
notice that the different forms of the nonlinearities in (4) are of special
interest, and attract a lot of attention. In [43], Xu and Su considered a
pseudo-parabolic equation with the well-known polynomial source

$\displaystyle H(u)=u^{p},\quad\text{where}\quad p\in(1,\infty)\text{ if
}d=1,2,~{}\text{or}~{}p\in\left(1,\frac{d+2}{d-2}\right)\text{ if }d\geq 3.$

In consideration of the subcritical (resp. critical) case that the initial
energy is less than (resp. equal to) the depth of potential well
$J(u_{0})<\mathbf{d}$ (resp. $J(u_{0})=\mathbf{d}$) and the positive (resp.
non-negative) value at $u_{0}$ of the Nehari functional, i.e, $I(u_{0})>0$
(resp. $I(u_{0})\geq 0$), by using the Galerkin and the potential well theory,
the authors proved the global existence, uniqueness and the asymptotic
behavior of the solution. Otherwise, when $I(u_{0})<0$, the solution is proved
to be blowing up at a finite time. Further for the arbitrarily initial energy,
that is $J(u_{0})>0$, the comparison principle and variational methods are
adopted to obtain the finite-time blow-up results. And the blowup time for
such high energy case was also estimated in [44]. In [9, 10], besides studying
(4) with a source term satisfying growth conditions of polynomial type,
Caraballo and his colleagues also considered the inﬂuence of external forces
with some kind of delay. Namely, $H$ was given by $H(t,u)=f(t,u_{t})+g(u)$,
where $f(t,u_{t})$ is the time-dependent delay term caused by memory or
hereditary characteristics and $g\in C^{1}(\mathbb{R})$ satisfying

$\displaystyle\limsup_{|v|\to+\infty}\frac{g(v)}{v}\leq\frac{\theta_{1}}{6}\quad\text{
or }\quad\limsup_{|v|\to+\infty}\frac{g(v)}{v}\leq 0$

and

$\displaystyle\big{|}g(v)-g(w)\big{|}\leq
C|v-w|\left(1+|v|^{p-1}+|w|^{p-1}\right),\quad p>1,C>0.$

The above series of work has aroused great interest among colleagues. On the
one hand, there are a large number of practical problems surrounding this type
of model; on the other hand, there is a huge gap between the existing research
and the many different variants that exist widely. Therefore, a large number
of subsequent researches are rapidly developed around different variants of
the model, including the couple form of a parabolic system [42], the pseudo-
parabolic model (4) with the singular potential [26], a nonlocal form of (4)
with the nonlinearity $H(u)=|u|^{p-1}u-\fint_{\Omega}|u|^{p-1}u{\mathrm{d}}x$
[40], the nonlocal version with the conical degeneration by considering the
Fuchsian type Laplace operator [19]. In [46], J. Zhou used the $|x|^{\sigma}$
weighted $L^{p+1}(\Omega)$ to study the global solutions and blow-up solutions
to (4) with $H(u)=|x|^{\sigma}|u|^{p-1}u$ ($\sigma<0$). It’s also necessary to
consider the case where the nonlinear source function grows much faster than
the polynomial level. In this case, the nonlinearity of exponential type is
considered as an optimal alternative to the polynomial source. In [48] Zhu
et.al. investigated Equation (4) with $H$ as an exponential nonlinearity. By
the elliptic theory, the authors showed the local existence and uniqueness of
the solution. And once again, the potential well theory was applied to prove
that when the initial energy is low, this solution exists globally. A
sufficient condition for a blowing-up solution without any limit of initial
energy was also provided. The logarithmic forms of $H$ were concerned in [15,
18, 24]. In [15], Chen and Tian studied the subcritical and critical energy
cases for (4) with $H(u)=u\log|u|$. When $I(u_{0})<0$, the global existence
and uniqueness of solutions were proved. In contrary, if $I(u_{0})<0$, unlike
the polynomial cases mentioned above, the authors showed that the solution
doesn’t blow up in ﬁnite time but at $+\infty$. The same topic was also
concerned for an initial-boundary value problem for inﬁnitely degenerate
semilinear pseudo-parabolic equations with logarithmic nonlinearity in [18].
The global existence and the asymptotic behavior of the solutions were
discussed for the cases of subcritical/critical initial energy, and the
infinite time blow up was also showed. A periodic logarithmic nonlinearity
given by $H(u)=m(t)u\log|u|$ for (4) was concerned in [24] by Ji et.al. The
existence of the solution was established to show the instability further.
When $c=0$, (4) becomes the usual parabolic equation, which has been
investigated a lot over the years, we refer the reader to [50, 47, 17] and the
references therein.

The above work is a part of the representative work on the pseudo-parabolic
equations, but in fact, there are many results on such mathematical models,
and we obviously cannot list them all. Taking a glimpse of the whole leopard,
we can still see that this type of model is widely and intensively concerned
not only because of its physical and practical application background, but
also because of its interesting mathematical phenomena. In particular, the
above work has given us such an inspiration: the nonlinearity of the model
dramatically affects the properties and behaviors of the solution. Roughly
speaking, a weaker nonlinearity will ensure the global-in-time existence of
the solution, while a stronger nonlinearity will cause the dynamic properties
of the solution to be differentiated due to the scale of the initial data. We
hope to systematically describe this phenomenon in a unified work, which is
the original intention of this work. To achieve this goal, we have adopted the
strategy of gradually strengthening the degree of the nonlinearity, that is,
discussing the linear advective form inhomogeneous terms, and the
inhomogeneous terms strengthened to the squared nonlinearity, the power-type
nonlinearity, as well as the exponential nonlinear terms.

At the same time, we further expand our understanding of this type of problem
from the perspective of the equation structure, that is, we consider the time-
fractional derivative in the sense of Caputo for Equation (1). In recent
decades, the fractional calculus has been proved that it is useful in
application to many fields of science such as the study of Brownian motion
[41], chemical stimuluses of an organism with memory effects [27, 2], and
waves in linear viscoelastic media [31]. The greatest motivation for
considering Problem (1)-(3) comes from the fact that due to the non-local
nature of the fractional differential-integral operators, the time-fractional
pseudo-parabolic equation has not only provided both new insights into
physical models but been also very mathematically interesting. In view of one
of the the original ideas for the application of Equation (4), that is, the
study of some fluid flows, it is natural to propose the fractional derivative
in the investigation of some certain viscous fluids. In fact, many modified
versions of (4) have been proposed such as [6, 21]. Because the fractional
derivative will help us to capture the viscoelastic properties of the flow,
the time-fractional pseudo-parabolic equations are useful for describing the
behavior of some non-Newtonian fluids. In mathematical aspect, it is a hot
stream to consider the time-fractional version of the classical mathematical
models including the parabolic type equations or diffusion models [38, 49, 20,
37], the time-space fractional Shrödinger equation with polynomial type
nonlinearity [22], the time-fractional Navier-Stokes equations (FNS) [12, 45],
and also the time-fractional pseudo-parabolic equations [36]. Surprisingly in
[12], it was shown that the order of time-fractional derivative influences the
regularity not only in time variable but also in the spatial variable. In
[36], a first attempt to explore the influence of the degree of nonlinearity
on the dynamic behavior of the solution was conducted by considering the
logarithmic nonlinearity and globally Lipschitz nonlinearity. All of the above
achievements in this direction pushed us to consider not only the influence of
the degree of the nonlinearities but also the order of the time-fractional
derivative on the behavior of the solution to the time-fractional pseudo-
parabolic equations with four distinguished types of nonlinearities, in which
the degree of nonlinearities increase gradually.

###  1.2 Structure of the work

To provide an overview of the present paper, we give the outline of the work,
including some summaries of the mathematical contributions.

$\bullet$ Section 2 provides some basic knowledge about function spaces,
special operators, the mild formula, and some linear estimates.

$\bullet$ In Section 3, we consider the source term $H$ as a gradient type. To
prove the well-posedness results for $H$ in the advective form
$H(u)=(\eta\cdot\nabla)u$, we apply the technique for singular integration
developed in [11] to overcome the difficulties arising in finding the proper
functional space and proving the convergence in such space for the constructed
Picard sequence, without restriction on the time interval and the smallness
assumptions on the initial data. Also, in this section, for
$H(u)=(\eta\cdot\nabla)u+\nabla\cdot F(u)$, where $F$ is a vector field, we
firstly prove the local-in-time existence, then this solution is extended to
the one in some larger time interval. As a consequence, the mild solution is
shown to be the global-in-time solution or finite time blow up solution.

$\bullet$ Section 4 states our investigation of Problem (1)-(3) with the
polynomial nonlinearity and the exponential nonlinearity. Since the nonlinear
estimates for $H$ in these cases require much more strict conditions for the
parameters and dimension $d$, some smallness assumptions for $u_{0}$ are
necessary for getting the global well-posedness. Also, for the power-type
source term $H(u)=|u|^{p-1}u,~{}p>2$, the use of fractional Hilbert spaces and
Sobolev embeddings is very beneficial. The nonlinearity of exponential type is
even more difficult for us to control because of its rapid growth.
Fortunately, by using the Orlicz space, we overcome this challenge and obtain
the desired results.

##  2 Preliminaries

Entire this work, we always use the letter $I$ and the notation $\mathscr{T}$
to abbreviate, respectively, an interval of time and a time point in
$(0,\infty)$.

###  2.1 Basic materials

We first establish some functional space concepts. Suppose that $X$ is a
Banach space associated with the norm $\left\|\cdot\right\|_{X}$. We use the
notation $C(I\to X)$ to denote the space of all continuous functions
$w:I\rightarrow X$. If $I$ is compact, $C(I\to X)$ is a Banach space with the
norm

$\displaystyle\left\|w\right\|_{C(I\rightarrow X)}:=\sup_{t\in
I}\big{\|}w(t)\big{\|}_{X}<\infty.$

Suppose that $\Xi:\mathbb{R}\to\mathbb{R}^{+}$ is a Young function, i.e, a
convex function which is even, continuous on $[0,\infty)$, and satisfies

$\displaystyle\lim\limits_{z\to\infty}\frac{\Xi(z)}{z}=\infty\qquad\text{and}\qquad\lim\limits_{z\to
0}\frac{\Xi(z)}{z}=0.$

Then, we define the Orlicz space $L^{\Xi}(\Omega)$ as the space of all
measurable functions $w(x)$ such that

$\displaystyle\int_{\Omega}\Xi\left(\frac{\left|w(x)\right|}{\kappa}\right){\mathrm{d}}x<\infty\quad\text{for
some }\kappa>0.$

The space $L^{\Xi}(\Omega)$ is a Banach space with respect to the Luxemburg
norm

$\displaystyle\left\|w\right\|_{L^{\Xi}(\Omega)}:=\inf\left\\{\kappa\in\mathbb{R}~{}\Big{|}~{}\kappa>0,\int_{\Omega}\Xi\left(\frac{\left|w(x)\right|}{\kappa}\right){\mathrm{d}}x\leq
1\right\\}.$

######  Remark 2.1.

Note that, we can use the framework of Orlicz space to cover the definitions
of some well-known Lebesgue spaces as below

  1. (1)

Assume that $\Xi(z)=z^{p}$ with $1<p<\infty$. Then, $L^{\Xi}(\Omega)$ is the
usual Lebesgue space $L^{p}(\Omega)$.

  2. (2)

Let $\Xi$ be a Young function whose value equals to 0 on $[-1,1]$ and is not
be bound outside $[-1,1]$. Then, $L^{\Xi}(\Omega)$ is the usual Lebesgue space
$L^{\infty}(\Omega)$.

######  Remark 2.2.

From now on, we always use the symbols $L^{\Xi}(\Omega)$ to indicate the
Orlicz space with the Young function $\Xi(z)=e^{z^{2}}-1.$

For the purpose of deriving main estimates for Problem (1)-(3) involving the
exponential nonlinearity, we introduce the following lemma which can be found
in [4, Theorem 8.12] or [23, Lemma 2.1] about the relationship between the
space $L^{\Xi}(\Omega)$ and some usual Lebesgue spaces. For readers’
convenience, we only briefly present its proof.

######  Lemma 2.1.

For any $p\in\left[2,\infty\right)$, we have the estimate

$\displaystyle\left\|w\right\|_{L^{p}(\Omega)}\leq\Big{(}\Gamma\left(\frac{p}{2}+1\right)\Big{)}^{\frac{1}{p}}\left\|w\right\|_{L^{\Xi}(\Omega)}.$

###### Proof.

For $q\geq 1$, using the properties of the exponential function and the Gamma
function (see Definition 5.5), we get

$\displaystyle\frac{z^{q}}{\Gamma(q+1)}+1<e^{z}.$

Then, from the fact that
$\Big{\\{}\kappa\in\mathbb{R}~{}\big{|}~{}\kappa>0,\int_{\Omega}\Xi\big{(}|w(x)|\kappa^{-1}\mathrm{d}x\big{)}\leq
1\Big{\\}}=\left[\left\|w\right\|_{L^{\Xi}(\Omega)},\infty\right)$ and the
monotone convergence theorem, we obtain

$\displaystyle\int_{\Omega}\frac{\left(|w(x)|\left\|w\right\|^{-1}_{L^{\Xi}(\Omega)}\right)^{2q}}{\Gamma(q+1)}{\mathrm{d}}x\leq\int_{\Omega}\Xi\left(\frac{|w(x)|}{\left\|w\right\|_{L^{\Xi}(\Omega)}}\right){\mathrm{d}}x\leq
1.$

Then, by choosing $q=\frac{p}{2}$, we obtain the desired result. ∎

We note that the scalar product on $L^{2}(\Omega)$ between $w,v\in
L^{2}(\Omega)$ is given by

$\displaystyle\int_{\Omega}w({z})v({z})\mathrm{d}{z}.$

Then, from the spectral decomposition of the operator $\mathcal{A}$, for any
$\nu$, we can define the following Hilbert scale space

$\displaystyle D^{\nu}(\Omega):=\left\\{w\in
L^{2}(\Omega)~{}\Big{|}~{}\left\|w\right\|^{2}_{D^{\nu}(\Omega)}=\sum_{k=1}^{\infty}\theta_{k}^{\nu}\left(\int_{\Omega}w({z})\phi_{k}(z)\mathrm{d}{z}\right)^{2}<\infty\right\\}.$

In view of this setting, we define the space $D^{-\nu}(\Omega)$ by the dual
space of $D^{\nu}(\Omega)$ with respect to the pairing
$\left\langle\cdot,\cdot\right\rangle_{*}$, which is a Banach space equipped
with the norm

$\displaystyle\left\|w\right\|_{D^{-\nu}(\Omega)}=\left(\sum_{k=1}^{\infty}\theta_{k}^{-\nu}\left\langle
w,\phi_{k}\right\rangle_{*}^{2}\right)^{\frac{1}{2}}.$

######  Remark 2.3.

([5, Chapter 5]) If $w\in L^{2}(\Omega)$ and $v\in D^{\nu}(\Omega)$, we have

$\displaystyle\left\langle w,v\right\rangle_{*}=\left\langle
w,v\right\rangle:=\int_{\Omega}w({z})v({z})\mathrm{d}{z}.$

######  Remark 2.4.

Based on [8, Section 3], we see that $D^{\nu}(\Omega)$ coincides with the
Sobolev-Slobodecki space $W_{0}^{\nu,2}(\Omega)$ when $\nu\in(1/2,1].$

We next introduce the definition of the Mittag-Leffler function with two
parameters, which is the generalization of the classical Mittag-Leffler
function, as follows

$\displaystyle
E_{\alpha,\zeta}(z):=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha k+\zeta)},$

where $\alpha$ is a positive real number and $\zeta$ is a complex constant.
When $\zeta=1$, we use the symbol $E_{\alpha}$ instead of $E_{\alpha,1}$, and
[31, Section 1] shows that

$\displaystyle
E_{\alpha}(-z):=\int_{0}^{\infty}e^{-zr}\mathcal{W}_{\alpha}(r)\mathrm{d}r,$

where $\mathcal{W}_{\alpha}$ is the M-Wright type function. Also, the
following lemma is very useful to control the values of the Mittag-Leffler
function.

######  Lemma 2.2.

([31, Theorem 1.6]) Let $\alpha<1,~{}\zeta$ be a real constant, and
$\lambda\in\left(\frac{\pi\alpha}{2},\pi\right)$. Then, there exists a
positive constant $\mathcal{M}$ such that

$\displaystyle\Big{|}E_{\alpha,\zeta}(\zeta)\Big{|}\leq\frac{\mathcal{M}}{1+|\zeta|},$

whenever $|\zeta|\geq 0$ and $\lambda\leq|\arg(\zeta)|\leq\pi$.

######  Lemma 2.3.

([28, Appendix E]) Let $\alpha,t$ be in $(0,1)$ and $(0,\infty)$,
respectively. Then, for any $a>0$, we have

$\displaystyle\partial_{t}\big{(}E_{\alpha}(-at^{\alpha})\big{)}=-at^{\alpha-1}E_{\alpha,\alpha}(-at^{\alpha})\quad\text{and}\quad\partial_{t}\big{(}t^{\alpha-1}E_{\alpha,\alpha}(-at^{\alpha})\big{)}=t^{\alpha-2}E_{\alpha,\alpha-1}(-at^{\alpha}).$

###  2.2 Mild solution

For purpose of formulating a mild solution of Problem (1)-(3), the following
lemma on the Laplace transform of the Caputo derivative operator plays an
important role.

######  Lemma 2.4.

Suppose that the Laplace transform and the derivative of order $\alpha$ in the
Caputo sense of a function $w$ exist. Then, the equation below holds

$\displaystyle\widetilde{\partial_{t}^{\alpha}w}(s)=\mathcal{L}\\{\partial_{t}^{\alpha}w\\}(s)=s^{\alpha}\widetilde{w}(s)-s^{\alpha-1}w(0),$

where $\mathcal{L}$ stands for the Laplace transform operator.

Integrating the both sides of Equation (1) with an arbitrary function
$\phi_{k}$ and then applying the Laplace transform, we have

$\displaystyle
s^{\alpha}\int_{\Omega}[I+\mathcal{A}]\widetilde{u}({z},s)\phi_{k}({z})\mathrm{d}{z}+\int_{\Omega}\mathcal{A}\widetilde{u}({z},s)\phi_{k}({z})\mathrm{d}{z}$
$\displaystyle=$ $\displaystyle
s^{\alpha-1}\int_{\Omega}[I+\mathcal{A}]u_{0}({z})\phi_{k}({z})\mathrm{d}{z}+\int_{\Omega}\widetilde{H(u)}({z},s)\phi_{k}({z})\mathrm{d}{z}.$

This equation is equivalent to

$\displaystyle\int_{\Omega}\widetilde{u}({z},s)\phi_{k}({z})\mathrm{d}{z}=$
$\displaystyle\frac{s^{\alpha-1}(1+\theta_{k})}{s^{\alpha}(1+\theta_{k})+\theta_{k}}\int_{\Omega}u_{0}({z})\phi_{k}({z})\mathrm{d}{z}$
$\displaystyle+\frac{1}{s^{\alpha}(1+\theta_{k})+\theta_{k}}\int_{\Omega}\widetilde{H(u)}({z},s)\phi_{k}({z})\mathrm{d}{z}.$

Then by the inverse Laplace transform, we obtain

$\displaystyle\int_{\Omega}u({z},t)\phi_{k}({z})\mathrm{d}{z}=$
$\displaystyle\int_{\Omega}E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)u_{0}({z})\phi_{k}({z})\mathrm{d}{z}$
$\displaystyle+\int_{\Omega}\frac{(t-\tau)^{\alpha-1}}{1+\theta_{k}}E_{\alpha,\alpha}\left(\frac{-\theta_{k}(t-\tau)^{\alpha}}{1+\theta_{k}}\right)H(u({z},\tau))\phi_{k}({z})\mathrm{d}{z}\mathrm{d}\tau.$

Since $\\{\phi_{k}\\}_{k\in\mathbb{N}}$ is an orthonormal basis of
$L^{2}(\Omega)$, we can formulate the mild solution to Problem (1)-(3) in the
following way

$\displaystyle u(x,t)$
$\displaystyle=\int_{\Omega}G_{1}(x,{z},t)u_{0}({z})\mathrm{d}{z}+\int_{0}^{t}\int_{\Omega}G_{2}(x,{z},t-\tau)H(u({z},\tau))\mathrm{d}{z}\mathrm{d}\tau,$

where, the Green kernels $G_{1},G_{2}$ are given by

$\displaystyle\qquad G_{1}(x,{z},t)$
$\displaystyle=\sum_{k=1}^{\infty}E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\phi_{k}(x)\phi_{k}({z})$

and

$\displaystyle\qquad G_{2}(x,{z},t)$
$\displaystyle=\sum_{k=1}^{\infty}\frac{t^{\alpha-1}}{1+\theta_{k}}E_{\alpha,\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\phi_{k}(x)\phi_{k}({z}).$

For purpose of simplifying notations, we set

$\displaystyle\mathscr{S}(t)w(x):=\int_{\Omega}G_{1}(x,{z},t)w({z})\mathrm{d}{z}\quad\text{and}\quad\mathscr{R}(t)w(x):=\int_{\Omega}G_{2}(x,{z},t)w(z)\mathrm{d}{z}.$

Then, we can rewrite the formula for the mild solution in the following way

$\displaystyle
u(x,t)=\mathscr{S}(t)u_{0}(x)+\int_{0}^{t}\mathscr{R}(t-\tau)H(u(x,\tau))\mathrm{d}\tau.$

From the standpoint of this formulation concept, we derive some fundamental
linear estimate through the following lemma

######  Lemma 2.5.

Let $\alpha,\mu\in(0,1),~{}\nu\in[0,1]$ and $\nu^{*}\in[0,2]$. Assume that
$w\in D^{1}(\Omega)$. Then, we can find positive constants $C_{1}$ and $C_{2}$
such that

  1. (i)

$\big{\|}\mathscr{S}(t)\big{\|}_{\mathscr{L}\left(D^{\nu}(\Omega)\right)}\leq
C_{1}t^{-\alpha\mu}$;

  2. (ii)

$\big{\|}\mathscr{R}(t)\big{\|}_{\mathscr{L}\left(D^{{\nu-\nu^{*}}}(\Omega),D^{\nu}(\Omega)\right)}\leq
C_{2}t^{\alpha-1}$.

###### Proof.

(i) We consider three different cases of $\mu$ as follows to prove the
conclusion (i)

Case 1: $\mu=0$. The following inequality follows immediately from Lemma 2.2

$\displaystyle\big{\|}\mathscr{S}(t)w\big{\|}_{D^{\nu}(\Omega)}^{2}$
$\displaystyle=\sum_{k=1}^{\infty}\theta^{\nu}_{k}\left(E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq\sum_{k=1}^{\infty}\theta^{\nu}_{k}\left(\frac{\mathcal{M}}{1+\frac{\theta_{k}t^{\alpha}}{1+\theta_{k}}}\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq\mathcal{M}^{2}\left\|w\right\|_{D^{\nu}(\Omega)}.$

Case 2: $\mu=1$. For a given $w$ in $D^{\nu}(\Omega)$, Lemma 2.2 and
Parseval’s identity show us

$\displaystyle\big{\|}\mathscr{S}(t)w\big{\|}_{D^{\nu}(\Omega)}^{2}$
$\displaystyle=\sum_{k=1}^{\infty}\theta^{\nu}_{k}\left(E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq\sum_{k=1}^{\infty}\frac{2\mathcal{M}^{2}\theta^{\nu}_{k}(1+\theta_{k}^{2})}{\theta_{k}^{2}t^{2\alpha}}\big{\langle}w,\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq
2\mathcal{M}^{2}\left(1+\frac{1}{\theta^{2}_{1}}\right)t^{-2\alpha}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}.$

Case 3: $0<\mu<1$. Given $w\in D^{\nu}(\Omega)$, by analogous arguments as in
above, we get

$\displaystyle\big{\|}\mathscr{S}(t)w\big{\|}_{D^{\nu}(\Omega)}^{2}=\sum_{k=1}^{\infty}\theta_{k}^{\nu}\left(E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2}.$
(5)

Thanks to the relationship between the Mittag-Leffler function and the
M-Wright type function, the equality (5) becomes

$\displaystyle\big{\|}\mathscr{S}(t)w\big{\|}_{D^{\nu}(\Omega)}^{2}$
$\displaystyle=\sum_{k=1}^{\infty}\theta_{k}^{\nu}\left(\int_{0}^{\infty}\exp\left(\frac{-r\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\mathcal{W}_{\alpha}(r)\mathrm{d}r\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq\sum_{k=1}^{\infty}C_{\mu}^{2}t^{-2\alpha\mu}\left(\frac{\theta_{k}^{\nu}(1+\theta_{k})^{2\mu}}{\theta_{k}^{2\mu}}\right)\left(\int_{0}^{\infty}r^{-\mu}\mathcal{W}_{\alpha}(r)\mathrm{d}r\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2},$

wherein, we have used the fundamental inequality $e^{-z}\leq
C_{\mu}z^{-\mu},\mu\in(0,1),C_{\mu}>0$. Then, Lemma 5.2 implies

$\displaystyle\big{\|}\mathscr{S}(t)w\big{\|}_{D^{\nu}(\Omega)}\leq
C_{\mu}t^{-\alpha\mu}(\theta_{1}^{-1}+1)^{\mu}\frac{\Gamma(1-\mu)}{\Gamma(1-\alpha\mu)}\left\|w\right\|_{D^{\nu}(\Omega)}.$

(ii) We consider only the case $\nu^{*}=1$, the other cases are similar. For
given $w\in D^{\nu-1}(\Omega)$, we observe that

$\displaystyle\big{\|}\mathscr{R}(t)w\big{\|}_{D^{\nu}(\Omega)}^{2}\leq\sum_{k=1}^{\infty}\frac{\theta^{\nu}_{k}t^{2\alpha-2}}{(1+\theta_{k})^{2}}\left(E_{\alpha,\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\right)^{2}\big{\langle}w,\phi_{k}\big{\rangle}^{2}.$

From this, we can easily obtain the positive constant $C_{2}$ via the
inequality below

$\displaystyle\big{\|}\mathscr{R}(t)w\big{\|}_{D^{\nu}(\Omega)}\leq\frac{\mathcal{M}t^{\alpha-1}}{\theta_{1}^{\frac{1}{2}}}\left(\sum_{k=1}^{\infty}\theta_{k}^{\nu-1}\big{\langle}w,\phi_{k}\big{\rangle}^{2}\right)^{\frac{1}{2}},$

noting that we have applied Lemma 2.2 to get

$\displaystyle
E_{\alpha,\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\leq\frac{\mathcal{M}}{1+\left|\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right|}.$

The proof is completed. ∎

##  3 The generalized BBM equation

In this section, we investigate the initial value problem involving the
generalized BBM equation. More precisely, Equation (1) can be given in the
exact form

$\displaystyle\partial^{\alpha}_{t}u-\partial^{\alpha}_{t}\Delta u-b\Delta
u+(\eta\cdot\nabla)u+\nabla\cdot F(u)=0,$ (6)

whereby, $\eta$ is a $d-$dimensional constant vector and $F$ is a
$d-$dimensional vector field.

###  3.1 The time-fractional pseudo-parabolic equation with advection term

Throughout the current subsection, we assume that $F$ is the vector $0$ in
$H(u)=(\eta\cdot\nabla)u+\nabla\cdot F(u)$. Then, $H$ becomes a globally
Lipschitz source function. Indeed, by Cauchy–Schwarz inequality, we have

$\displaystyle\int_{\Omega}\left(\sum_{j=1}^{d}\eta_{j}u_{x_{j}}(x)\right)^{2}{\mathrm{d}}x\leq|\eta|^{2}\int_{\Omega}\left(\sum_{j=1}^{d}u^{2}_{x_{j}}(x)\right){\mathrm{d}}x=|\eta|^{2}\int_{\Omega}|\nabla
u(x)|^{2}{\mathrm{d}}x.$

It means that we can find a positive constant $\mathscr{C}_{1}$ independent of
$w,v\in D^{1}(\Omega)$ such that

$\displaystyle\big{\|}H(w)-H(v)\big{\|}_{L^{2}(\Omega)}\leq\mathscr{C}_{1}\left\|w-v\right\|_{D^{1}(\Omega)}.$
(7)

Our main principle is the successive approximation in some reasonable Banach
spaces. To this end, for any $a,\sigma>0$ we denote by
$\mathbb{Y}=\mathbb{Y}(a,\sigma)$, the space of all functions $w\in
C\left(I\to D^{1}(\Omega)\right)$ satisfying

$\displaystyle\bullet~{}\sup_{t\in I\setminus\\{0\\}}t^{\kappa}e^{-\sigma
t}\left\|w(t)\right\|_{D^{1}(\Omega)}<\infty,$
$\displaystyle\bullet~{}w(x,0)=u_{0}(x),$

and construct a sequence $\\{w_{n}\\}_{n=1}^{\infty}$ inductively in the
following way

$\displaystyle w_{1}(x,t)$ $\displaystyle:=\mathscr{S}(t)u_{0}(x),$
$\displaystyle w_{n+1}(x,t)$
$\displaystyle:=\mathscr{S}(t)u_{0}(x)+\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))\mathrm{d}\tau.$

Besides, we introduce some necessary lemmas to help us present the main points
of the proof more clearly.

######  Lemma 3.1.

([11, Lemma 8]) Let $h,\mu>0$ and $m,n>-1$ such that $m+n>-1$. Then,

$\displaystyle\lim_{\mu\rightarrow\infty}\left(\sup_{t\in
I\setminus\\{0\\}}t^{h}\int_{0}^{1}s^{m}(1-s)^{n}e^{-\mu
t(1-s)}\mathrm{d}s\right)=0.$

######  Lemma 3.2.

Presume that $u_{0}$ belongs to $D^{1}(\Omega)$ and $I=[0,\mathscr{T}]$. Then,
$\\{w_{n}\\}_{n=1}^{\infty}$ forms a subset of $C\left(I\rightarrow
D^{1}(\Omega)\right)$.

###### Proof.

Firstly, for any $t\geq 0$ and $\varepsilon>0$, Lemma 2.3 makes the following
formula hold

$\displaystyle\left|E_{\alpha}\left(\frac{-\theta_{k}(t+\varepsilon)^{\alpha}}{1+\theta_{k}}\right)-E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\right|=\int_{t}^{t+\epsilon}\frac{\theta_{k}r^{\alpha-1}}{1+\theta_{k}}E_{\alpha,\alpha}\left(\frac{-\theta_{k}r^{\alpha}}{1+\theta_{k}}\right)\mathrm{d}r.$

Then, for $u_{0}\in D^{1}(\Omega)$, one deduces that

$\displaystyle\big{\|}w_{1}(t+\varepsilon)-w_{1}(t)\big{\|}_{D^{1}(\Omega)}^{2}$
$\displaystyle=\sum_{k=1}^{\infty}\theta_{k}\left|E_{\alpha}\left(\frac{-\theta_{k}(t+\varepsilon)^{\alpha}}{1+\theta_{k}}\right)-E_{\alpha}\left(\frac{-\theta_{k}t^{\alpha}}{1+\theta_{k}}\right)\right|^{2}\big{\langle}u_{0},\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq\sum_{k=1}^{\infty}\theta_{k}\left(\int_{t}^{t+\epsilon}\frac{r^{\alpha-1}\mathcal{M}}{1+\left|\frac{-\theta_{k}r^{\alpha}}{1+\theta_{k}}\right|}\mathrm{d}r\right)^{2}\big{\langle}u_{0},\phi_{k}\big{\rangle}^{2}$
$\displaystyle\leq\frac{\mathcal{M}^{2}\big{(}(t-\varepsilon)^{\alpha}-t^{\alpha}\big{)}^{2}}{\alpha^{2}}\left\|u_{0}\right\|^{2}_{D^{1}(\Omega)}.$

One may obtain immediately for any $t\geq 0$ that

$\displaystyle\lim_{\varepsilon\to
0}\left\|w_{1}(t+\varepsilon)-w_{1}(t)\right\|_{D^{1}(\Omega)}=0,$ (8)

which means that $w_{1}$ is continuous on $I$ with respect to the
$D^{1}(\Omega)$ norm. On the other hand, for every $\varepsilon>0$, we get

$\displaystyle\big{\|}w_{2}(\varepsilon)-w_{1}(\varepsilon)\big{\|}_{D^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\varepsilon}\Big{\|}\mathscr{R}(\varepsilon-\tau)H(w_{n}(x,\tau))\Big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{\varepsilon}\Bigg{(}\sum_{k=1}^{\infty}\frac{\theta_{k}(\varepsilon-\tau)^{2(\alpha-1)}}{(1+\theta_{k})^{2}}E^{2}_{\alpha,\alpha}\left(\frac{\theta_{k}(\tau-\varepsilon)^{\alpha}}{1+\theta_{k}}\right)\Big{\langle}H(w_{1}(\tau)),\phi_{k}\Big{\rangle}^{2}\Bigg{)}^{\frac{1}{2}}\mathrm{d}\tau.$

Using Lemma 2.5(ii) and the fact that $H(0)=0$, we deduce

$\displaystyle\big{\|}w_{2}(\varepsilon)-w_{1}(\varepsilon)\big{\|}_{D^{1}(\Omega)}$
$\displaystyle\leq
C_{2}\int_{0}^{\varepsilon}(\varepsilon-\tau)^{\alpha-1}\Big{\|}H(w_{1}(\tau))\Big{\|}_{L^{2}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq\mathscr{C}_{1}C_{2}\left(\int_{0}^{\varepsilon}(\varepsilon-\tau)^{\alpha-1}\mathrm{d}\tau\right)\left(\sup_{t\in
I}\big{\|}w_{1}(t)\big{\|}_{D^{1}(\Omega)}\right).$

This result along with (8) help us conclude that $w_{2}$ is continuous at
$t=0$ with respect to the $D^{1}(\Omega)$ norm. We now turn to consider the
case when $t$ is positive. For this purpose, we proceed with two following
claims, which help us make the proof more clear.

Claim 1. For any $t,\varepsilon>0,k\in\mathbb{N}$, and $w_{1}\in
C\left(I\rightarrow D^{1}(\Omega)\right)$, we can apply Lemma 2.3 as follows

$\displaystyle\mathscr{E}(k,t,\tau,\varepsilon):$
$\displaystyle=\left|(t+\varepsilon-\tau)^{\alpha-1}E_{\alpha,\alpha}\left(\frac{-\theta_{k}(t+\varepsilon-\tau)^{\alpha}}{1+\theta_{k}}\right)-(t-\tau)^{\alpha-1}E_{\alpha,\alpha}\left(\frac{-\theta_{k}(t-\tau)^{\alpha}}{1+\theta_{k}}\right)\right|$
$\displaystyle=\left|\int_{t-\tau}^{t+\varepsilon-\tau}\frac{\theta_{k}r^{\alpha-2}}{1+\theta_{k}}E_{\alpha,\alpha-1}\left(\frac{-\theta_{k}r^{\alpha}}{1+\theta_{k}}\right)\mathrm{d}r\right|.$

Lemma 2.2 implies that, for any $\tau\in(0,t)$,

$\displaystyle\mathscr{E}(k,t,\tau,\varepsilon)\leq\mathcal{M}(1-\alpha)^{-1}\Big{|}(t+\varepsilon-\tau)^{\alpha-1}-(t-\tau)^{\alpha-1}\Big{|}.$
(9)

In view of this result, we infer the following estimate

$\displaystyle\int_{0}^{t}\Bigg{(}\sum_{k=1}^{\infty}\frac{\theta_{k}\mathscr{E}^{2}(k,t,\tau,\varepsilon)}{(1+\theta_{k})^{2}}\Big{\langle}H(w_{1}(\tau)),\phi_{k}\Big{\rangle}^{2}\Bigg{)}^{\frac{1}{2}}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{M}\mathscr{C}_{1}}{(1-\alpha)\theta_{1}^{\frac{1}{2}}}\left(\int_{0}^{t}\Big{|}(t+\varepsilon-\tau)^{\alpha-1}-(t-\tau)^{\alpha-1}\Big{|}\mathrm{d}\tau\right)\left(\sup_{t\in
I}\left\|w_{1}(t)\right\|_{D^{1}(\Omega)}\right).$

We notice that, for any $\tau\in(0,t)$, we have

$\displaystyle\begin{cases}\displaystyle\lim_{\varepsilon\rightarrow\infty}\Big{(}(t-\tau)^{\alpha-1}-(t+\varepsilon-\tau)^{\alpha-1}\Big{)}=0,\\\\[5.69046pt]
\displaystyle\Big{|}(t+\varepsilon-\tau)^{\alpha-1}-(t-\tau)^{\alpha-1}\Big{|}\leq
2(t-\tau)^{\alpha-1}.\end{cases}$

Then, applying the dominated convergence theorem, the following limit holds

$\displaystyle\lim_{\varepsilon\to
0}\left(\int_{0}^{t}\Bigg{(}\sum_{k=1}^{\infty}\frac{\theta_{k}\mathscr{E}^{2}(k,t,\tau,\varepsilon)}{(1+\theta_{k})^{2}}\Big{\langle}H(w_{1}(\tau)),\phi_{k}\Big{\rangle}^{2}\Bigg{)}^{\frac{1}{2}}\mathrm{d}\tau\right)=0.$

Claim 2. Using again Lemma 2.5(ii) and the fact that $w_{1}\in C\left(I\to
D^{1}(\Omega)\right)$, we obtain

$\displaystyle\int_{t}^{t+\epsilon}\Big{\|}\mathscr{R}(t+\varepsilon-\tau)H(w_{n}(x,\tau))\Big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$ $\displaystyle
C_{2}\int_{t}^{t+\epsilon}(t+\varepsilon-\tau)^{\alpha-1}\Big{\|}H(w_{1}(\tau))\Big{\|}_{L^{2}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\mathscr{C}_{1}C_{2}\alpha^{-1}\varepsilon^{\alpha}\left(\sup_{t\in
I}\left\|w_{1}(t)\right\|_{D^{1}(\Omega)}\right).$

Thus, one has

$\displaystyle\lim_{\varepsilon\to
0}\left(\int_{t}^{t+\epsilon}\Big{\|}\mathscr{R}(t+\varepsilon-\tau)H(w_{n}(x,\tau))\Big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau\right)=0.$

Now, the results of Claim 1 and Claim 2 along with the fact that $w_{1}\in
C\left(I\rightarrow D^{1}(\Omega)\right)$ yield that $w_{2}$ is continuous at
$t>0$. Note that, we have already proved the continuity at $0$ of $w_{2}$.
Therefore, we can conclude that $w_{2}\in C\left(I\rightarrow
D^{1}(\Omega)\right)$. Based on this result, the induction arguments show that
$w_{n+1}$ belongs to $C\left(I\rightarrow D^{1}(\Omega)\right)$ whenever
$w_{n}$ is in $C\left(I\to D^{1}(\Omega)\right)$, for every $n\in\mathbb{N}$.
The proof is completed. ∎

######  Theorem 3.3.

(Global existence) Assume that $u_{0}$ is an element of the space
$D^{1}(\Omega)$ and $I=[0,\mathscr{T}]$. Then, Problem $\eqref{Main
Equation}$-$\eqref{Initial condition}$ possesses at least one mild solution
$u$ in $C\left(I\rightarrow D^{1}(\Omega)\right)$.

###### Proof.

We first show that if $u_{0}$ belongs to $D^{1}(\Omega)$ and
$I=[0,\mathscr{T}]$, $\\{w_{n}\\}_{n=1}^{\infty}$ is a subset of the space
$\mathbb{Y}(\alpha,\sigma)$, for some big-value $\sigma$. If $u_{0}\in
D^{1}(\Omega)$, Lemma 2.5(i) shows that

$\displaystyle\left\|w_{1}(t)\right\|_{D^{1}(\Omega)}=\big{\|}\mathscr{S}(t)u_{0}\big{\|}_{D^{1}(\Omega)}\leq
C_{1}t^{-\alpha}\left\|u_{0}\right\|_{D^{1}(\Omega)}.$

It thus follows that

$\displaystyle\sup_{t\in I\setminus\\{0\\}}t^{\alpha}e^{-\sigma
t}\left\|w_{1}(t)\right\|_{D^{1}(\Omega)}<\infty.$ (10)

Next, we suppose that $w_{n}\in\mathbb{Y}(\alpha,\sigma)$. Then, by using
Lemma 2.5(ii), we have

$\displaystyle\left\|w_{n+1}(t)-w_{1}(t)\right\|_{D^{1}(\Omega)}$
$\displaystyle=$
$\displaystyle\Bigg{\|}\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\Bigg{\|}$
$\displaystyle\leq$ $\displaystyle
C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\big{\|}H(w_{n}(\tau))\big{\|}_{L^{2}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\mathscr{C}_{1}C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\left\|w_{n}(\tau)\right\|_{D^{1}(\Omega)}\mathrm{d}\tau.$

Multiplying the both sides of the above by $t^{\alpha}e^{-\sigma t},t>0$, we
obtain

$\displaystyle t^{\alpha}e^{-\sigma
t}\left\|w_{n+1}(t)-w_{1}(t)\right\|_{D^{1}(\Omega)}\leq\mathscr{C}_{1}C_{2}\mathcal{Q}\left(t,\alpha,\sigma\right)\left(\sup_{t\in
I}t^{\alpha}e^{-\sigma t}\|w_{n}(t)\|_{D^{1}(\Omega)}\right),$ (11)

where we set

$\displaystyle\mathcal{Q}\left(t,h,\sigma\right):=t^{h}\int_{0}^{t}(t-\tau)^{\alpha-1}\tau^{-h}e^{-\sigma(t-\tau)}\mathrm{d}\tau.$

By virtue of Lemma 3.1, (10), (11) and the triangle inequality, we deduce

$\displaystyle\sup_{t\in I\setminus\\{0\\}}\Big{(}t^{\alpha}e^{-\sigma
t}\left\|w_{n+1}(t)\right\|_{D^{1}(\Omega)}\Big{)}<\infty\quad\text{for some
large }\sigma.$

In addition, the use of Lemma 3.2 yields that $w_{n}$ is continuous on $I$,
for every $n\geq 1$. It follows that $w_{n+1}\in\mathbb{Y}(\alpha,\sigma)$.
Furthermore, $\\{w_{n}\\}_{n=1}^{\infty}$ can be proved to be a Cauchy
sequence in $\mathbb{Y}(\alpha,\sigma_{0})$. Let us verify this statement.
Firstly, Lemma 3.1 ensures the existence of a sufficiently large constant
$\sigma_{0}$ such that

$\displaystyle\sup_{t\in
I\setminus\\{0\\}}\mathcal{Q}\left(t,\alpha,\sigma_{0}\right)=\frac{3}{4\mathscr{C}_{1}C_{2}}.$

Secondly, assume that $w_{n-1},w_{n}\in\mathbb{Y}(\alpha,\sigma_{0}),n\geq 2$.
Then, by the same arguments as above, we obtain

$\displaystyle\left\|w_{n+1}(t)-w_{1}(t)\right\|_{D^{1}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\|H(w_{n}(\tau))-H(w_{n-1}(\tau))\|_{L^{2}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\mathscr{C}_{1}C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\|w_{n}(\tau)-w_{n-1}(\tau)\|_{D^{1}(\Omega)}\mathrm{d}\tau.$

It thus implies that

$\displaystyle
t^{\alpha}e^{-\sigma_{0}t}\left\|w_{n+1}(t)-w_{n}(t)\right\|_{D^{1}(\Omega)}\leq\frac{3}{4}\sup_{t\in
I\setminus\\{0\\}}\Big{(}t^{\alpha}e^{-\sigma_{0}t}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{D^{1}(\Omega)}\Big{)}.$

From the standpoint of this result, for any $n_{2}>n_{1}\geq 2$, we find

$\displaystyle
t^{\alpha}e^{-\sigma_{0}t}\left\|w_{n_{2}}(t)-w_{n_{1}}(t)\right\|_{D^{1}(\Omega)}\leq\sum_{n=n_{1}}^{n_{2}}\left(\frac{3}{4}\right)^{n-1}\sup_{t\in
I\setminus\\{0\\}}\Big{(}t^{\alpha}e^{-\sigma_{0}t}\|w_{2}(t)-w_{1}(t)\|_{D^{1}(\Omega)}\Big{)}.$
(12)

Then, the use of geometric series helps us to declare that
$\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence in
$\mathbb{Y}(\alpha,\sigma_{0})$. Now, the completeness of
$\mathbb{Y}(\alpha,\sigma_{0})$ provides a function $u$ which is the limit of
the sequence $\\{w_{n}\\}_{n=1}^{\infty}$. Applying the dominated Lebesgue
theorem, we have

$\displaystyle u$
$\displaystyle=\lim_{n\rightarrow\infty}\Bigg{(}\int_{\Omega}G_{1}(x,{z},t)u_{0}({z})\mathrm{d}{z}+\int_{0}^{t}\int_{\Omega}G_{2}(x,{z},t-\tau)H(w_{n}({z},\tau))\mathrm{d}{z}\mathrm{d}\tau\Bigg{)}$
$\displaystyle=\int_{\Omega}G_{1}(x,{z},t)u_{0}({z})\mathrm{d}{z}+\int_{0}^{t}\int_{\Omega}G_{2}(x,{z},t-\tau)H(u({z},\tau))\mathrm{d}{z}\mathrm{d}\tau.$

Then, we can conclude that $u$ is the mild solution to Problem (1)-(3). ∎

######  Theorem 3.4.

(Uniqueness and stability) Suppose that Problem $\eqref{Main
Equation}$-$\eqref{Initial condition}$ admits a mild solution $u$ in
$C\left(I\to D^{1}(\Omega)\right)$. Then, this solution is globally unique.
Furthermore, it depends continuously on the initial data.

###### Proof.

The uniqueness and the stability results of the mild solution can be obtained
by Gönwall’s inequality. Indeed, assume that $u,w$ are the milds solution of
Problem (1)-(3) in $\mathbb{Y}(\alpha,\sigma_{0})$ corresponding to the
initial data $u_{0},w_{0}\in D^{1}(\Omega)$. Then, we have

$\displaystyle\left\|u(t)-w(t)\right\|_{D^{1}(\Omega)}$ $\displaystyle\leq$
$\displaystyle C_{1}t^{-\alpha}\left\|u_{0}-w_{0}\right\|_{D^{1}(\Omega)}$
$\displaystyle+\mathscr{C}_{1}C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\|u(\tau)-w(\tau)\|_{D^{1}(\Omega)}\mathrm{d}\tau.$
(13)

Since $u,w$ are in the space $\mathbb{Y}(\alpha,\sigma_{0})$, then, Hölder’s
inequality shows that

$\displaystyle\left(t^{\alpha}e^{-\sigma_{0}t}\int_{0}^{t}(t-\tau)^{\alpha-1}\|u(\tau)-w(\tau)\|_{D^{1}(\Omega)}\mathrm{d}\tau\right)^{2}$
$\displaystyle\leq$
$\displaystyle\mathcal{Q}\left(t,2\alpha,2\sigma_{0}\right)\int_{0}^{t}(t-\tau)^{\alpha-1}\left(\tau^{\alpha}e^{-\sigma_{0}\tau}\left\|u(\tau)-v(\tau)\right\|_{D^{1}(\Omega)}\right)^{2}\mathrm{d}\tau.$

Accordingly, the inequality (3.1) becomes

$\displaystyle\left(t^{\alpha}e^{-\sigma_{0}t}\left\|u(t)-w(t)\right\|_{D^{1}(\Omega)}\right)^{2}$
$\displaystyle\leq$
$\displaystyle\left(C_{1}\left\|u_{0}-w_{0}\right\|_{D^{1}(\Omega)}\right)^{2}+\frac{3}{4}\int_{0}^{t}(t-\tau)^{\alpha-1}\left(\tau^{\alpha}e^{-\sigma_{0}\tau}\left\|u(\tau)-v(\tau)\right\|_{D^{1}(\Omega)}\right)^{2}\mathrm{d}\tau$

We now apply the fractional Grönwall’s inequality (see Lemm 5.4) to get the
following estimate

$\displaystyle
t^{\alpha}e^{-\sigma_{0}t}\left\|u(t)-w(t)\right\|_{D^{1}(\Omega)}\leq
C_{1}\left\|u_{0}-w_{0}\right\|_{D^{1}(\Omega)}E_{\alpha,1}\left(\frac{3\Gamma(\alpha)t^{\alpha}}{4}\right).$

The proof is completed. ∎

###  3.2 The non-trivial case of the vector field

In this subsection, we assume that the vector field $F$ in
$H(u)=(\eta\cdot\nabla)u+\nabla\cdot F(u)$, is given by
$F(u)=(u^{2},u^{2},...,u^{2})$ and $d\in\\{2,3,4\\}$. Then, we can check that
$H$ is a local Lipschitzian. To verify this statement, we first recall the
embeddings $L^{\frac{2d}{d+2}}(\Omega)\hookrightarrow D^{-1}(\Omega)$ in Lemma
5.3, which helps us to find

$\displaystyle\Big{\|}\nabla\cdot F(u)\Big{\|}_{D^{-1}(\Omega)}\leq
2^{\frac{2d+2}{d+2}}\Big{\|}\big{|}\nabla
u\big{|}u\Big{\|}_{L^{\frac{2d}{d+2}}(\Omega)}\leq
2^{\frac{2d+2}{d+2}}\big{\|}u\big{\|}_{L^{d}(\Omega)}\big{\|}\nabla
u\big{\|}_{L^{2}(\Omega)},$

where the Hölder inequality with two conjugate indices $\frac{d+2}{2}$ and
$\frac{d+2}{d}$ has been applied to get the second inequality. We now can
easily check that

$\displaystyle\begin{cases}D^{1}(\Omega)\hookrightarrow
L^{d}(\Omega),\qquad&\text{if }~{}d=2,\\\ D^{1}(\Omega)\hookrightarrow
L^{\frac{2d}{d-2}}(\Omega)\hookrightarrow L^{d}(\Omega),&\text{if
}~{}d\in\\{3,4\\}.\end{cases}$

Therefore, there exists a positive constant $C_{\Omega}$ from the above
embeddings such that that

$\displaystyle\Big{\|}\big{|}\nabla u\big{|}u\Big{\|}_{D^{-1}(\Omega)}\leq
2^{\frac{2d+2}{d+2}}C_{\Omega}\big{\|}u\big{\|}^{2}_{D^{1}(\Omega)}.$

As a consequence of the above estimates, we can find a constant $C(\Omega,d)$
such that for any $u,v\in D^{1}(\Omega)$ there holds

$\displaystyle\Big{\|}\nabla\cdot F(u)-\nabla\cdot
F(v)\Big{\|}_{D^{-1}(\Omega)}\leq
C(\Omega,d)\left(\big{\|}u\big{\|}_{D^{1}(\Omega)}+\big{\|}v\big{\|}_{D^{1}(\Omega)}\right)\left\|u-v\right\|_{D^{1}(\Omega)}.$

In addition, from (7) and the fact that $L^{2}(\Omega)\hookrightarrow
D^{-1}(\Omega)$, for any $u,v\in D^{1}(\Omega)$, there exists a positive
constant $\overline{\mathscr{C}}_{1}$ independent of $u,v$ such that the
following estimate holds

$\displaystyle\big{\|}H\left(u\right)-H\left(v\right)\big{\|}_{D^{-1}(\Omega)}\leq\overline{\mathscr{C}}_{1}\left(\big{\|}u\big{\|}_{D^{1}(\Omega)}+\big{\|}v\big{\|}_{D^{1}(\Omega)}+1\right)\left\|u-v\right\|_{D^{1}(\Omega)}$
(14)

The main aim of this part is to investigate the well-posed results that either
the mild solution exists globally or blows up in finite maximal time.

######  Theorem 3.5.

(Local existence) Let $\alpha\in(0,1)$. If $u_{0}\in D^{1}(\Omega)$, Problem
(1)-(3) posesses a locally unique mild solution.

###### Proof.

Thanks to the assumption $u_{0}\in D^{1}(\Omega)$, Lemma 2.5(i) implies

$\displaystyle\big{\|}\mathscr{S}(t)u_{0}\big{\|}_{D^{1}(\Omega)}\leq
C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}.$ (15)

For any $w,v\in C(I\to D^{1}(\Omega))$, from Lemma 2.5(ii) and inequality (14)
we have

$\displaystyle\int_{0}^{t}\big{\|}\mathscr{R}(t-\tau)\Big{(}H(w(\tau))-H(v(\tau))\Big{)}\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$ $\displaystyle
C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\left\|H(w(\tau))-H(v(\tau))\right\|_{D^{-1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\overline{\mathscr{C}}_{1}C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\left(\left\|w(\tau)\right\|_{D^{1}(\Omega)}+\left\|v(\tau)\right\|_{D^{1}(\Omega)}+1\right)\left\|w(\tau)-v(\tau)\right\|_{D^{1}(\Omega)}\mathrm{d}\tau.$

This implies that

$\displaystyle\int_{0}^{t}\big{\|}\mathscr{R}(t-\tau)\Big{(}H(w(\tau))-H(v(\tau))\Big{)}\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\overline{\mathscr{C}}_{1}C_{2}\alpha^{-1}\mathscr{T}^{\alpha}\Bigg{(}\left\|w\right\|_{C(I\to
D^{1}(\Omega))}+\left\|v\right\|_{C(I\to
D^{1}(\Omega))}+1\Bigg{)}\left\|w-v\right\|_{C(I\to D^{1}(\Omega))}.$ (16)

Then, if we consider the Picard sequence

$\displaystyle w_{1}(x,t)$ $\displaystyle:=\mathscr{S}(t)u_{0}(x),$
$\displaystyle w_{n+1}(x,t)$
$\displaystyle:=\mathscr{S}(t)u_{0}(x)+\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))\mathrm{d}\tau,$

Inequality (15) immediately shows that $w_{1}\in C(I\to D^{1}(\Omega))$.
Besides, for $w_{n}\in C(I\to D^{1}(\Omega))$, Estimate (3.2) gives us the
following result

$\displaystyle\left\|w_{n+1}(\tau)-w_{1}(\tau)\right\|_{D^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{t}\big{\|}\mathscr{R}(t-\tau)H(w_{n}(\tau))\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\overline{\mathscr{C}}_{1}C_{2}\alpha^{-1}\mathscr{T}^{\alpha}\Big{(}\left\|w_{n}\right\|^{2}_{C(I\to
D^{1}(\Omega))}+\left\|w_{n}\right\|_{C(I\to D^{1}(\Omega))}\Big{)},$

i.e.,

$\displaystyle\left\|w_{n+1}-w_{1}\right\|_{C(I\to
D^{1}(\Omega))}\leq\overline{\mathscr{C}}_{1}C_{2}\alpha^{-1}\mathscr{T}^{\alpha}\Big{(}\left\|w_{n}\right\|^{2}_{C(I\to
D^{1}(\Omega))}+\left\|w_{n}\right\|_{C(I\to D^{1}(\Omega))}\Big{)}.$

Then, we can conclude that $\\{w_{n}\\}_{n=1}^{\infty}\subset C(I\to
D^{1}(\Omega)).$ Furthermore, if $\mathscr{T}$ is small enough, Estimate (3.2)
infers for $w_{n-1},w_{n},w_{n+1}\in C(I\to D^{1}(\Omega))$ that

$\displaystyle\left\|w_{n+1}-w_{n}\right\|_{C(I\to
D^{1}(\Omega))}\leq\frac{3}{4}\left\|w_{n}-w_{n-1}\right\|_{C(I\to
D^{1}(\Omega))}.$

It means $\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence in $C(I\to
D^{1}(\Omega))$. Consequently, there exists a locally unique mild solution $u$
of Problem (1)-(3) in $C(I\to D^{1}(\Omega))$. ∎

######  Lemma 3.6.

Let $u_{0}$ be in $D^{1}(\Omega)$. Suppose that $u$ is a unique mild solution
to Problem (1)-(3) on $[0,\mathscr{T}]$. Then this solution can be extended to
$[0,\mathscr{T}+T],$ for some $T>0.$

###### Proof.

Assume that $u$ is the unique mild solution of Problem (1)-(3) on
$[0,\mathscr{T}]$ and $I=[0,\mathscr{T}+T]$, where $T>0$ will be chosen later.
For a constant $R>0$, we consider the space below

$\displaystyle\mathbb{E}:=\left\\{w\in C(I\to
D^{1}(\Omega))\quad\Biggl{|}\begin{array}[]{*{11}{ll}}&w(t,\cdot)=u(t,\cdot),&\forall
t\in\left[0,\mathscr{T}\right],\\\
&\left\|w(t,\cdot)-u(T,\cdot)\right\|_{D^{1}(\Omega)}\leq R,&\forall
t\in\left[\mathscr{T},\mathscr{T}+T\right]\end{array}\right\\}.$

We aim to to show that Problem (1)-(3) has a unique mild solution that belongs
to $\mathbb{E}$, then it is the extension of $u$ to $I$. To this end, we take
an element $\overline{w}$ of $\mathbb{E}$ and define the sequence
$\\{w_{n}\\}_{n=1}^{\infty}$ as

$\displaystyle w_{1}(x,t)$
$\displaystyle:=\mathscr{S}(t)u_{0}(x)+\int_{0}^{t}\mathscr{R}(t-\tau)H(\overline{w}(x,\tau))\mathrm{d}\tau,$
$\displaystyle w_{n+1}(x,t)$
$\displaystyle:=\mathscr{S}(t)u_{0}(x)+\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))\mathrm{d}\tau.$

We can check easily for every $t\in[0,\mathscr{T}]$, that

$\displaystyle w_{1}(x,t)=w_{2}(x,t)=...=w_{n}(x,t)=u(x,t),\quad\forall
x\in\Omega.$

On the contrary, when $t\in[\mathscr{T},\mathscr{T}+T]$, a simple calculation
shows that

$\displaystyle\left\|u_{1}(t)-u(\mathscr{T})\right\|_{D^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\big{\|}\mathscr{S}(t)u_{0}-\mathscr{S}(\mathscr{T})u_{0}\big{\|}_{D^{1}(\Omega)}$
$\displaystyle+\int_{\mathscr{T}}^{t}\big{\|}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle+\int_{0}^{\mathscr{T}}\big{\|}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))-\mathscr{R}(\mathscr{T}-\tau)H(w_{n}(x,\tau))\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau.$

Similar to Lemma 3.2, we derive

$\displaystyle\big{\|}\mathscr{S}(t)u_{0}-\mathscr{S}(\mathscr{T})u_{0}\big{\|}_{D^{1}(\Omega)}\leq\alpha^{-1}\mathcal{M}(t-\mathscr{T})^{\alpha}\left\|u_{0}\right\|_{D^{1}(\Omega)}\leq\alpha^{-1}\mathcal{M}T^{\alpha}\left\|u_{0}\right\|_{D^{1}(\Omega)}$

and

$\displaystyle\int_{0}^{\mathscr{T}}\big{\|}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))-\mathscr{R}(\mathscr{T}-\tau)H(w_{n}(x,\tau))\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{M}\overline{\mathscr{C}}_{1}C_{2}\big{|}t^{\alpha}-(t-\mathscr{T})^{\alpha}-\mathscr{T}^{\alpha}\big{|}}{\alpha(1-\alpha)\theta^{\frac{1}{2}}_{1}}\Big{(}\left\|w_{n}\right\|^{2}_{C(I\to
D^{1}(\Omega))}+\left\|w_{n}\right\|_{C(I\to D^{1}(\Omega))}\Big{)}.$

In addition, the following estimate also holds

$\displaystyle\int_{\mathscr{T}}^{t}\left\|\mathscr{R}(t-\tau)H(w_{n}(x,\tau))\right\|_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\overline{\mathscr{C}}_{1}C_{2}\int_{\mathscr{T}}^{t}(t-\tau)^{\alpha-1}\Big{(}\left\|w_{n}\right\|^{2}_{D^{1}(\Omega)}+\left\|w_{n}\right\|_{D^{1}(\Omega)}\Big{)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{\overline{\mathscr{C}}_{1}C_{2}(t-\mathscr{T})^{\alpha}}{\alpha}\Big{(}\left\|w_{n}\right\|^{2}_{C(I\to
D^{1}(\Omega))}+\left\|w_{n}\right\|_{C(I\to D^{1}(\Omega))}\Big{)}$
$\displaystyle\leq$
$\displaystyle\frac{\overline{\mathscr{C}}_{1}C_{2}T^{\alpha}}{\alpha}\Big{(}\left\|w_{n}\right\|^{2}_{C(I\to
D^{1}(\Omega))}+\left\|w_{n}\right\|_{C(I\to D^{1}(\Omega))}\Big{)}.$

From the above estimates, by taking $T$ small enough, one has

$\displaystyle\left\|u_{1}(t)-u(\mathscr{T})\right\|_{D^{1}(\Omega)}\leq R.$

It follows that $w_{1}$ is in $\mathbb{E}$. The same arguments show that if
$w_{n}\in\mathbb{E}$ for $n\geq 1$, then $w_{n+1}$ also belongs to
$\mathbb{E}$. To complete the proof, we have to show that
$\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence in $\mathbb{E}$. Indeed, for
$w_{n-1},w_{n}\in\mathbb{E}$ and $t\geq\mathscr{T}$, we obtain

$\displaystyle\left\|w_{n+1}(t)-w_{n}(t)\right\|_{D^{1}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\int_{\mathscr{T}}^{t}\big{\|}\mathscr{R}(t-\tau)H(w_{n}(\tau))-w_{n-1}(x,\tau)\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\int_{\mathscr{T}}^{t}\overline{\mathscr{C}}_{1}C_{2}(t-\tau)^{\alpha-1}\Bigg{(}\left\|w_{n-1}(\tau)\right\|_{D^{1}(\Omega)}+\left\|w_{n}(\tau)\right\|_{D^{1}(\Omega)}+1\Bigg{)}\left\|w_{n}(\tau)-w_{n-1}(\tau)\right\|_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$ $\displaystyle
2\overline{\mathscr{C}}_{1}C_{2}T^{\alpha}\alpha^{-1}\Bigg{(}\left(\left\|u(\mathscr{T})\right\|_{D^{1}(\Omega)}+R\right)+1\Bigg{)}\left\|w_{n}-w_{n-1}\right\|_{C(I\to
D^{1}(\Omega))},$

provided that for any $w\in\mathbb{E}$, by the triangle inequality, the
following estimate holds for any $t\in[\mathscr{T},\mathscr{T}+T]$

$\displaystyle\left\|w(t)\right\|_{D^{1}(\Omega)}\leq\left\|u(\mathscr{T})\right\|_{D^{1}(\Omega)}+R.$

Hence, if $T$ is sufficiently small, we obtain

$\displaystyle\left\|w_{n+1}-w_{n}\right\|_{C(I\to
D^{1}(\Omega))}\leq\frac{3}{4}\left\|w_{n}-w_{n-1}\right\|_{C(I\to
D^{1}(\Omega))},$

the same arguments as (12) show that $\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy
sequence in $\mathbb{E}$. As we said before, $\mathbb{E}$ is a Banach space,
then we can find a limit function of $\\{w_{n}\\}_{n=1}^{\infty}$ which is the
unique mild solution to Problem (1)-(3) on $[0,\mathscr{T}+T].$ The proof is
completed. ∎

######  Theorem 3.7.

(Global and finite time blowup solution) Let $u_{0}$ be in $D^{1}(\Omega)$ and
$u$ be the unique mild solution to Problem (1)-(3). If we define

$\displaystyle\mathscr{T}_{\max}:=\sup\Big{\\{}\mathscr{T}>0~{}\big{|}~{}u\text{
exists uniquely on }[0,\mathscr{T}]\Big{\\}},$

then either $\mathscr{T}_{\max}=\infty$ or
$\lim\limits_{t\to\mathscr{T}_{\max}^{-}}\left\|u(t)\right\|_{D^{1}(\Omega)}=\infty$.

###### Proof.

Assume that the maximal time point $\mathscr{T}_{\max}$ is finite and
$I=[0,\mathscr{T})$. Let us make a contradict assumption that there exists a
finite constant $\mathscr{M}$ such that

$\displaystyle\left\|u(t)\right\|_{D^{1}(\Omega)}\leq\mathscr{M},\quad\forall
t\in[0,\mathscr{T}_{\max}).$ (17)

Let $\\{t_{n}\\}_{n=1}^{\infty}$ be a sequence in $I$ satisfying
$t_{n}\xrightarrow{n\rightarrow\infty}\mathscr{T}_{\max}$. We aim to show that
$\\{u_{n}\\}_{n=1}^{\infty}=\\{u(t_{n})\\}_{n=1}^{\infty}$ is a Cauchy
sequence in $D^{1}(\Omega)$. In fact, for $t_{n}<t_{m}$, we find that

$\displaystyle\left\|u_{m}-u_{n}\right\|_{D^{1}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\big{\|}\mathscr{S}(t_{m})u_{0}-\mathscr{S}(t_{n})u_{0}\big{\|}_{D^{1}(\Omega)}$
$\displaystyle+\int_{t_{n}}^{t_{m}}\big{\|}\mathscr{R}(t_{m}-\tau)H(u(\tau))\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle+\int_{0}^{t_{n}}\big{\|}\mathscr{R}(t_{m}-\tau)H(u(\tau))-\mathscr{R}(t_{n}-\tau)H(u(\tau))\big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau=:\sum_{j\in\\{1,2,3\\}}\mathcal{I}_{j}.$

On one hand, since $u_{0}$ is in $D^{1}(\Omega)$, we have

$\displaystyle\mathcal{I}_{1}\leq\alpha^{-1}\mathcal{M}(t_{m}-t_{n})^{\alpha}\left\|u_{0}\right\|_{D^{1}(\Omega)}\leq\alpha^{-1}\mathcal{M}\Big{(}(t_{m}-\mathscr{T}_{\max})^{\alpha}+(\mathscr{T}_{\max}-t_{n})^{\alpha}\Big{)}\left\|u_{0}\right\|_{D^{1}(\Omega)}.$

Then, for any $\varepsilon>0$, we can find a number $n_{1}$ such that

$\displaystyle\mathcal{I}_{1}\leq\frac{\varepsilon}{3},\quad\forall m,n\geq
n_{1}.$

On the other hand, the use of Lemma 2.5(ii) shows that

$\displaystyle\mathcal{I}_{2}$ $\displaystyle\leq
C_{2}\int_{t_{n}}^{t_{m}}(t_{m}-\tau)^{\alpha-1}\big{\|}H(u(\tau))\big{\|}_{D^{-1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq\overline{\mathscr{C}}_{1}C_{2}\int_{t_{n}}^{t_{m}}(t_{m}-\tau)^{\alpha-1}\Big{(}\left\|u(\tau)\right\|^{2}_{D^{1}(\Omega)}+\left\|u(\tau)\right\|_{D^{1}(\Omega)}\Big{)}\mathrm{d}\tau$
$\displaystyle\leq\alpha^{-1}\big{(}\mathscr{M}^{2}+\mathscr{M}\big{)}\overline{\mathscr{C}}_{1}C_{2}\Big{(}(t_{m}-\mathscr{T}_{\max})^{\alpha}+(\mathscr{T}_{\max}-t_{n})^{\alpha}\Big{)}.$

Then, we can also find a number $n_{2}$ such that

$\displaystyle\mathcal{I}_{2}\leq\frac{\varepsilon}{3},\quad\forall m,n\geq
n_{2}.$

Also, it’s obviously that

$\displaystyle\Big{\|}\mathscr{R}(t_{m}-\tau)H(u(\tau))-\mathscr{R}(t_{n}-\tau)H(u(\tau))\Big{\|}^{2}_{D^{1}(\Omega)}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}\frac{\theta_{k}\mathscr{E}^{2}(k,t_{n},\tau,t_{m}-t_{n})}{(1+\theta_{k})^{2}}\Big{\langle}H(u(\tau)),\phi_{k}\Big{\rangle}^{2}$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{\infty}\frac{\theta_{k}\mathscr{E}^{2}(k,t_{n},\tau,t_{m}-t_{n})}{(1+\theta_{k})^{2}}\Big{\langle}H(u(\tau)),\phi_{k}\Big{\rangle}^{2}.$

Then, from (9), the following inequality holds

$\displaystyle\int_{0}^{t_{n}}\Big{\|}\mathscr{R}(t_{m}-\tau)H(u(\tau))-\mathscr{R}(t_{n}-\tau)H(u(\tau))\Big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{M}\overline{\mathscr{C}}_{1}C_{2}}{(1-\alpha)}\int_{0}^{t_{n}}\Big{(}(t_{n}-\tau)^{\alpha-1}-(t_{m}-\tau)^{\alpha-1}\Big{)}\big{\|}H(u(\tau))\big{\|}_{D^{-1}(\Omega)}\mathrm{d}\tau$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{M}\big{(}\mathscr{M}^{2}+\mathscr{M}\big{)}\overline{\mathscr{C}}_{1}C_{2}}{\alpha(1-\alpha)}\big{|}t_{m}^{\alpha}-(t_{m}-t_{n})^{\alpha}-t_{n}^{\alpha}\big{|}$
$\displaystyle\leq$
$\displaystyle\frac{\mathcal{M}\big{(}\mathscr{M}^{2}+\mathscr{M}\big{)}\overline{\mathscr{C}}_{1}C_{2}}{\alpha(1-\alpha)}\Big{(}\big{|}t_{n}^{\alpha}-\mathscr{T}_{\max}^{\alpha}\big{|}+\big{|}t_{n}-\mathscr{T}_{\max}\big{|}^{\alpha}+\big{|}t_{m}-\mathscr{T}_{\max}\big{|}^{\alpha}+\big{|}t_{m}^{\alpha}-\mathscr{T}_{\max}^{\alpha}\big{|}\Big{)}.$

The convergence of $\\{t_{n}\\}_{n=1}^{\infty}$ gives us a large number
$n_{3}$ such that

$\displaystyle\mathcal{I}_{3}\leq\frac{\varepsilon}{3},\quad\forall m,n\geq
n_{3}.$

Hence, for every $\varepsilon>0$, we can always find a number
$n_{0}=\max\left\\{n_{1},n_{2},n_{3}\right\\}$ that whenever $m,n\geq n_{0}$,
we have

$\displaystyle\left\|u_{m}-u_{n}\right\|_{D^{1}(\Omega)}\leq\varepsilon.$

Accordingly, we can conclude that $\\{u_{n}\\}_{n=1}^{\infty}$ is a Cauchy
sequence in $D^{1}(\Omega)$. Thanks to the completeness of $D^{1}(\Omega)$,
there exists a unique function $\overline{u}$ which is the limit of $u_{n}$ as
$n$ to infinity, i.e.

$\displaystyle\overline{u}=\lim\limits_{n\to\infty}u(x,t_{n})=u(x,\mathscr{T}_{\max}).$

Thus we have extended $u$ over the interval $[0,\mathscr{T}_{\max}]$. Now,
Lemma 3.6 is available for extending $u$ to some large time interval. This
fact leads us to a contradiction. Then, we can finish our proof by making the
conclusion that

$\displaystyle\lim\limits_{t\to\mathscr{T}_{\max}^{-}}\left\|u(t)\right\|_{D^{1}(\Omega)}=\infty,$

provided $\mathscr{T}_{\max}<\infty.$ ∎

##  4 The other Lipschitz cases of the source functions

Throughout this section, we concern in the following form of the initial value
problem (1)-(3)

$\displaystyle\begin{cases}\displaystyle\partial_{t}^{\alpha}u-\partial_{t}^{\alpha}\Delta
u-\Delta
u=H\left(u\right),\quad&\text{in}\quad\Omega\times(0,\infty),\\\\[4.26773pt]
\displaystyle
u=0,&\text{on}\quad\partial\Omega\times(0,\infty),\\\\[4.26773pt]
\displaystyle u=u_{0},&\text{in}\quad\Omega\times\\{0\\}.\end{cases}$ (18)

In particular, our main results for a mild solution $u:[0,\infty)\to\Omega$
are going to revolve around the following two hypotheses for the source
functions $H$.

$(H_{1})$ The nonlinearity of polynomial type

$\displaystyle\quad~{}H(u)=|u|^{p-1}u,\quad p>2.$

$(H_{2})$ The nonlinearity of exponential type

$\displaystyle\quad~{}H(u)=u^{3}e^{u^{2}}.$

###  4.1 The polynomial nonlinearity

For the hypothesis $(H_{1})$ for the source function, we aim to establish the
global-in-time well-posedness result for the mild solution to Problem (1)-(3).
To this end we first provide some nonlinear control on $H$.

######  Lemma 4.1.

Let $d\geq 2$ and $p>2$. Then, if

$\displaystyle 0<\nu<\min\left\\{1,\frac{d+2-2(p-1)d}{2-4(p-1)}\right\\},$

we can find a independent constant $\mathscr{C}_{2}>0$ such that

$\displaystyle\big{\|}H(u)-H(v)\big{\|}_{D^{\nu-1}(\Omega)}\leq\mathscr{C}_{2}\left(\left\|u\right\|_{D^{\nu}(\Omega)}^{p-1}+\left\|v\right\|_{D^{\nu}(\Omega)}^{p-1}\right)\left\|u-v\right\|_{D^{\nu}(\Omega)},$

for every $u,v\in D^{\nu}(\Omega)$.

###### Proof.

Firstly, for any $q\geq 1$, Hölder’s inequality gives us the following
estimate

$\displaystyle\big{\|}|u|^{p-1}(u-v)\big{\|}_{L^{q}(\Omega)}\leq\left\|u-v\right\|_{L^{2q}(\Omega)}\left\|u\right\|_{L^{2(p-1)q}(\Omega)}^{p-1}.$

Then, if we choose $q=\frac{2d}{d-2(\nu-1)}>1$, Lemma 5.3 shows that the
Lebesgue space $L^{q}(\Omega)$ embeds into the Hilbert space
$D^{\nu-1}(\Omega)$. Also, by another use of Lemma 5.3 and the assumptions for
$d,p,\nu$ and $\Omega$, we have

$\displaystyle D^{\nu}(\Omega)\hookrightarrow
L^{2(p-1)q}(\Omega)\hookrightarrow L^{2q}(\Omega)\hookrightarrow
L^{q}(\Omega).$

Combining the above result with the fact $(a+b)^{m}\leq
2^{m}(a^{m}+b^{m}),~{}a,b,m>0$ and the triangle inequality, for any $u,v\in
D^{\nu}(\Omega)$ we have

$\displaystyle\big{\|}H(u)-H(v)\big{\|}_{D^{\nu-1}(\Omega)}\leq
2^{q}C_{\Omega}\left(\left\|u\right\|_{D^{\nu}(\Omega)}^{p-1}+\left\|v\right\|_{D^{\nu}(\Omega)}^{p-1}\right)\left\|u-v\right\|_{D^{\nu}(\Omega)},$

where $C_{\Omega}$ comes from the embeddings. The proof is completed. ∎

######  Definition 4.2.

Let $I=[0,\mathscr{T}]$ and $\sigma,M>0$. Then, we define a subset of
$C\left(I\to D^{\nu}(\Omega)\right)$ as follows

$\displaystyle\mathbb{X}=\mathbb{X}(\sigma,M,\nu):=\left\\{w\in C(I\rightarrow
D^{\nu}(\Omega))~{}\Big{|}~{}w(0)=u_{0}~{}\text{ and }\sup_{t\in
I\setminus\\{0\\}}t^{\sigma}\left\|w(t)\right\|_{D^{\nu}(\Omega)}<M\right\\}.$

######  Remark 4.1.

Observe that $\mathbb{X}$ is a Banach space involving the norm

$\displaystyle\left\|w\right\|_{\mathbb{X}}=\sup_{t\in
I\setminus\\{0\\}}t^{\sigma}\left\|w(t)\right\|_{D^{\nu}(\Omega)}.$

######  Theorem 4.3.

Let $\alpha\in(0,1)$ and $p>2$ such that $\frac{p}{p-1}<\alpha^{-1}$. For a
given datum $u_{0}\in D^{\nu}(\Omega)$ sufficiently small and
$I=[0,\mathscr{T}]$. Problem $\eqref{Main Equation}$-$\eqref{Initial
condition}$ admits a unique mild solution that belongs to the space
$\mathbb{X}=\mathbb{X}\left(\frac{\alpha}{p-1},2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)},\nu\right)$.

###### Proof.

In the same spirit with Section 3, we consider again the sequence
$\\{w_{n}\\}_{n=1}^{\infty}$ established by

$\displaystyle w_{1}(x,t)$ $\displaystyle:=\mathscr{S}(t)u_{0}(x),$
$\displaystyle w_{n+1}(x,t)$
$\displaystyle:=\mathscr{S}(t)u_{0}(x)+\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(x,\tau))\mathrm{d}\tau.$

By analogous argument as Lemma 3.2, we can prove that
$\\{w_{n}\\}_{n=1}^{\infty}$ is a subset of $C(I\rightarrow D^{\nu}(\Omega))$.
So, we need only to focus on deriving that $\\{w_{n}\\}_{n=1}^{\infty}$ is a
convergent sequence in
$\mathbb{X}=\mathbb{X}\left(\alpha\mu,2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)},\nu\right)$,
where $\mu=\frac{1}{p-1}$. The Picard iteration process for this proof
includes three main steps.

$\bullet$ Step 1. We show that $w_{1}$ is in $\mathbb{X}$.

Indeed, since $\mu\in(0,1)$ and $u_{0}\in D^{\nu}(\Omega)$, thanks to Lemma
2.5(i), we obtain

$\displaystyle\left\|w_{1}(t)\right\|_{D^{\nu}(\Omega)}\leq
C_{1}t^{-\alpha\mu}\left\|u_{0}\right\|_{D^{\nu}(\Omega)},$

which implies

$\displaystyle t^{\alpha\mu}\left\|w_{1}(t)\right\|_{D^{\nu}(\Omega)}\leq
C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}.$

$\bullet$ Step 2. Assume that $w_{n}$ belongs to $\mathbb{X}$ for any
$n\in\mathbb{N}$, then $w_{n+1}\in\mathbb{X}$.

Indeed, Lemma 2.5(ii) provides us the following estimate

$\displaystyle\left\|w_{n+1}(t)-w_{1}(t)\right\|_{D^{\nu}(\Omega)}\leq
C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\|H(w_{n}(\tau))\|_{D^{\nu-1}(\Omega)}\mathrm{d}\tau.$
(19)

Using Lemma 4.1 with $v=0$ and the fact that $w_{n}\in\mathbb{X}$, we have

$\displaystyle\left\|w_{n+1}(t)-w_{1}(t)\right\|_{D^{\nu}(\Omega)}$
$\displaystyle\leq\mathscr{C}_{2}C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\left\|w_{n}(\tau)\right\|_{D^{\nu}(\Omega)}^{p}\mathrm{d}\tau$
$\displaystyle\leq\mathscr{C}_{2}C_{2}\left(\int_{0}^{t}(t-\tau)^{\alpha-1}\tau^{-p\alpha\mu}\mathrm{d}\tau\right)\left\|w_{n}\right\|_{\mathbb{X}}^{p}$
$\displaystyle\leq\mathscr{C}_{2}C_{2}\left(\int_{0}^{t}(t-\tau)^{\alpha-1}\tau^{-p\alpha\mu}\mathrm{d}\tau\right)\left(2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}\right)^{p}.$

Also, from our assumptions, we find that

$\displaystyle p\alpha\mu<1\quad\text{and}\quad 1-p\mu+\mu=0,$

which tells that

$\displaystyle
t^{\alpha\mu}\int_{0}^{t}(t-\tau)^{\alpha-1}\tau^{-p\alpha\mu}\mathrm{d}\tau=\beta(\alpha,1-p\alpha\mu),$

where the Beta function is defined in Definition 5.5. From the standpoint of
this result, we infer

$\displaystyle t^{\alpha\mu}\left\|w_{n+1}(t)\right\|_{D^{\nu}(\Omega)}\leq
C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}+\mathscr{C}_{2}C_{2}\beta(\alpha,1-p\alpha\mu)\left(2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}\right)^{p}.$
(20)

Hence, the statement that $w_{n+1}\in\mathbb{X}$ follows from the small data
$u_{0}D^{\nu}(\Omega)$, then Step 2 is completed.

$\bullet$ Step 3. We show that $\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy
sequence.

Suppose that $w_{n}$ and $w_{n-1}$ are in $\mathbb{X}$. Our techniques are not
too different from Step 2. Indeed, we have

$\displaystyle\left\|w_{n+1}(t)-w_{n}(t)\right\|_{D^{\nu}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\mathscr{C}_{2}C_{2}\int_{0}^{t}(t-\tau)^{\alpha-1}\left(\left\|w_{n}(\tau)\right\|_{D^{\nu}(\Omega)}^{p-1}+\left\|w_{n-1}(\tau)\right\|_{D^{\nu}(\Omega)}^{p-1}\right)\left\|w_{n}(\tau)-v_{n-1}(\tau)\right\|_{D^{\nu}(\Omega)}\mathrm{d}\tau.$
(21)

Since $\\{w_{n}\\}_{n=1}^{\infty}\in\mathbb{X}$, multiplying the both sides of
(4.1) by $t^{\alpha\mu}$, we have

$\displaystyle
t^{\alpha\mu}\left\|w_{n+1}(t)-w_{n}(t)\right\|_{D^{\nu}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
2\mathscr{C}_{2}C_{2}\left(t^{\alpha\mu}\int_{0}^{t}(t-\tau)^{\alpha-1}\tau^{-p\alpha\mu}\mathrm{d}\tau\right)\left(2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}\right)^{p-1}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{X}}$
$\displaystyle\leq$ $\displaystyle
2\mathscr{C}_{2}C_{2}\beta(\alpha,1-p\alpha\mu)\left(2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}\right)^{p-1}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{X}}.$
(22)

Then, we can take the supremum over the interval $I\setminus\\{0\\}$ on both
sides of (4.1) to obtain

$\displaystyle\left\|w_{n+1}(t)-w_{n}(t)\right\|_{\mathbb{X}}\leq
2\mathscr{C}_{2}C_{2}\beta(\alpha,1-p\alpha\mu)\left(2C_{1}\left\|u_{0}\right\|_{D^{\nu}(\Omega)}\right)^{p-1}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{X}}.$

Therefore, if $\left\|u_{0}\right\|_{D^{\nu}(\Omega)}$ is sufficiently small,
$\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence in $\mathbb{X}$.

From the above three steps, we can use again the arguments performed as in the
proofs of Section 3 to find the limit function $u$ of
$\\{w_{n}\\}_{n=1}^{\infty}$ which is the unique mild solution of Problem
(1)-(3). The proof is completed. ∎

###  4.2 Exponential nonlinearity

This subsection concerns the global well-posedness of Problem (1)-(3) with the
source function $H(u)=u^{3}e^{u^{2}}$. Here for any $w,v\in\mathbb{R}$, we
have

$\displaystyle|H(w)-H(v)|\leq\mathscr{C}_{3}\left|w-v\right|\left(w^{2}e^{w^{2}}+v^{2}e^{v^{2}}\right),$

for some indepent $\mathscr{C}_{3}>0.$ We recall the following theorem in [35,
Theorem 2] or [29, Section 1] which is used to control the solution operator
in the framework of Orlicz space.

######  Theorem 4.4.

Let $\Omega\subset\mathbb{R}^{2}$ be a cone domain. Then,
$W_{0}^{1,2}(\Omega)$ embeds continuously into the Orlicz space
$L^{\Xi}(\Omega)$.

######  Lemma 4.5.

Let $\alpha\in\left(0,\frac{2}{3}\right),$ $\Omega\subset\mathbb{R}^{2}$ be a
bounded domain with sufficiently smooth boundary, and $M$ be a finite
constant. Suppose that $I=[0,\mathscr{T}]$ and $w,v$ are functions in
$C\left(I\to L^{\Xi}(\Omega)\right)$ such that

$\displaystyle\max\left(\sup_{t\in
I}\left\|w(t)\right\|_{L^{\Xi}(\Omega)},\sup_{t\in
I}\left\|t^{\frac{\alpha}{2}}w(t)\right\|_{L^{\Xi}(\Omega)},\sup_{t\in
I}\left\|w(t)\right\|_{L^{\Xi}(\Omega)},\sup_{t\in
I}\left\|t^{\frac{\alpha}{2}}v(t)\right\|_{L^{\Xi}(\Omega)}\right)<M.$

If $M$ is small enough, we can derive the following inequalities

$\displaystyle\left\|\int_{0}^{t}\mathscr{R}(t-\tau)\big{(}H(w(\tau))-H(v(\tau))\big{)}\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{2\mathscr{C}_{3}\mathscr{T}^{\alpha}}{\alpha}\left\|w-v\right\|_{L^{\Xi}(\Omega)}M^{2}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6M^{2}\right)^{\frac{1}{6}}\right)$
(23)

and

$\displaystyle
t^{\frac{\alpha}{2}}\left\|\int_{0}^{t}\mathscr{R}(t-\tau)\big{(}H(w(\tau))-H(v(\tau))\big{)}\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{2\mathscr{C}_{3}\left\|w-v\right\|_{L^{\Xi}(\Omega)}}{\big{(}\beta\left(\alpha,1-\frac{3\alpha}{2}\big{)}\right)^{-1}}M^{2}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6M^{2}\right)^{\frac{1}{6}}\right).$
(24)

###### Proof.

Assume that $w$ and $v$ are functions in $L^{\Xi}(\Omega)$. Then, we can use
Theorem 4.4 to obtain

$\displaystyle\left\|\int_{0}^{t}\mathscr{R}(t-\tau)\left(H(w(\tau))-H(v(\tau))\right)\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}\leq\int_{0}^{t}\Big{\|}\mathscr{R}(t-\tau)\left(H(w(\tau))-H(v(\tau))\right)\Big{\|}_{D^{1}(\Omega)}\mathrm{d}\tau.$

Using Lemma 2.5, we have

$\displaystyle\left\|\int_{0}^{t}\mathscr{R}(t-\tau)\left(H(w(\tau))-H(v(\tau))\right)\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}\leq\int_{0}^{t}(t-\tau)^{\alpha-1}\Big{\|}H(w(\tau))-H(v(\tau))\Big{\|}_{L^{2}(\Omega)}\mathrm{d}\tau.$
(25)

To continuous, we make a nonlinear estimate for the source function with
$w=w(t),v=v(t),~{}t\in I$, as follows

$\displaystyle\left\|H(w)-H(v)\right\|_{L^{2}(\Omega)}$ (26)
$\displaystyle\leq$
$\displaystyle\mathscr{C}_{3}\left\|(w-v)(u^{2}+v^{2})\right\|_{L^{2}(\Omega)}+\mathscr{C}_{3}\sum_{u\in\\{w,v\\}}\left\|\left|w-v\right||u|^{2}\left(e^{u^{2}}-1\right)\right\|_{L^{2}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\mathscr{C}_{3}\sum_{u\in\\{w,v\\}}\left(\left\|w-v\right\|_{L^{4}(\Omega)}\left\|u\right\|^{2}_{L^{4}(\Omega)}+\left\|w-v\right\|_{L^{6}(\Omega)}\left\|u\right\|^{2}_{L^{6}(\Omega)}\left\|e^{u^{2}}-1\right\|_{L^{6}(\Omega)}\right),$

here, we have used Hölder’s inequality. If
$\left\|u\right\|_{L^{\Xi}(\Omega)}<(\frac{1}{6})^{\frac{1}{2}}$, it follows
from the techniques used in [23, Lemma 3.2] that

$\displaystyle\left\|e^{u^{2}}-1\right\|^{6}_{L^{6}(\Omega)}$
$\displaystyle\leq\int_{\Omega}\left(e^{6u^{2}(x)}-1\right){\mathrm{d}}x\leq\int_{\Omega}\left(\exp\left({\frac{6\left\|u\right\|^{2}_{L^{\Xi}(\Omega)}u^{2}(x)}{\left\|u\right\|^{2}_{L^{\Xi}(\Omega)}}}\right)-1\right){\mathrm{d}}x\leq
6\left\|u\right\|^{2}_{L^{\Xi}(\Omega)},$ (27)

where we have used the fact that

$\displaystyle\left\\{\kappa\in\mathbb{R}~{}\Big{|}~{}\kappa>0,\int_{\Omega}\Xi\left(\frac{|w(x)|}{\kappa}\right)\mathrm{d}x\leq
1\right\\}=\left[\left\|w\right\|_{L^{\Xi}(\Omega)},\infty\right).$

According to (26), (27) and Lemma 2.1, we find that

$\displaystyle\left\|H(w)-H(v)\right\|_{L^{2}(\Omega)}$
$\displaystyle\leq\mathscr{C}_{3}\sum_{u\in\\{w,v\\}}\left\|w-v\right\|_{L^{\Xi}(\Omega)}\left\|u\right\|^{2}_{L^{\Xi}(\Omega)}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6\left\|u\right\|^{2}_{L^{\Xi}(\Omega)}\right)^{\frac{1}{6}}\right).$
(28)

Inclusion inequalities (25) and (28) allow us to deduce the inequality (4.5).
Furthermore, by using (4.5) and

$\displaystyle
t^{\frac{\alpha}{2}}\int_{0}^{t}(t-\tau)^{\alpha-1}\tau^{-\frac{3\alpha}{2}}\mathrm{d}\tau=\beta\left(\alpha,1-\frac{3\alpha}{2}\right),$

we get the result (4.5). ∎

######  Theorem 4.6.

Let $\Omega$ be a bounded domain of $\mathbb{R}^{2}$ with sufficiently smooth
boundary $\partial\Omega$ and $H(u)=u^{3}e^{u^{2}}$. Assume that
$I=[0,\mathscr{T}]$ and $u_{0}\in D^{1}(\Omega)$ is small enough. Then, there
exists a unique mild solution of Problem (1)-(3) in $C\left(I\to
L^{\Xi}(\Omega)\right).$

###### Proof.

We start by considering again the sequence $\\{w_{n}\\}_{n=1}^{\infty}$ as in
Theorem 4.3 and the following space

$\displaystyle\mathbb{U}:=\left\\{w\in C\left(I\to
L^{\Xi}(\Omega)\right)~{}\Big{|}~{}\left\|w\right\|_{\mathbb{U}}<2C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}\right\\},$

where, the $\mathbb{U}$-norm is given by

$\displaystyle\left\|w\right\|_{\mathbb{U}}:=\max\left\\{\sup_{t\in
I}\left\|w\right\|_{L^{\Xi}(\Omega)},\sup_{t\in
I}t^{\frac{\alpha}{2}}\left\|w\right\|_{L^{\Xi}(\Omega)}\right\\}.$

Using Theorem 4.4 and considering Lemma 2.5 in two cases: $\mu=0$ and
$\mu=\frac{1}{2}$, we have

$\displaystyle\begin{cases}\displaystyle\left\|w_{1}(t)\right\|_{L^{\Xi}(\Omega)}\leq\left\|\mathscr{S}(t)u_{0}\right\|_{D^{1}(\Omega)}\leq
C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)},\qquad&\mu=0,~{}t\in
I\\\\[5.69046pt] \displaystyle
t^{\frac{\alpha}{2}}\left\|w_{1}(t)\right\|_{L^{\Xi}(\Omega)}\leq
t^{\frac{\alpha}{2}}\left\|\mathscr{S}(t)u_{0}\right\|_{D^{1}(\Omega)}\leq
C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)},&\mu=\frac{1}{2},~{}t\in
I.\end{cases}$

This result yields that $w_{1}\in\mathbb{U}.$ Next we assume that $w_{n}$ is
in $\mathbb{U}$, then we can show that $w_{n+1}$ is also in $\mathbb{U}$ In
fact, for given small data $u_{0}$, set

$\displaystyle\mathscr{T}_{*}:=\left(\frac{32\mathscr{C}_{3}\left\|u_{0}\right\|_{D^{1}(\Omega)}^{2}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)}{\alpha
C_{1}}\right)^{\frac{-1}{\alpha}}.$

On the one hand, when $t\leq\mathscr{T}_{*}$, we apply Lemma 4.5 with $v\equiv
0$ to find that

$\displaystyle\left\|\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq\frac{2\mathscr{C}_{3}\mathscr{T}_{*}^{\alpha}}{\alpha(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{-3}}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)$
$\displaystyle=2^{-1}C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}.$

On the other hand, if $t$ is larger than $\mathscr{T}_{*}$ defined above, we
can deduce that

$\displaystyle\left\|\int_{\mathscr{T}_{*}}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\mathscr{T}_{*}^{-\frac{\alpha}{2}}t^{\frac{\alpha}{2}}\left\|\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{2\mathscr{C}_{3}\mathscr{T}_{*}^{-\frac{\alpha}{2}}(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{3}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)}{\left(\beta\left(\alpha,1-\frac{3\alpha}{2}\right)\right)^{-1}}$
$\displaystyle=$
$\displaystyle\frac{128\left\|u_{0}\right\|_{D^{1}(\Omega)}^{4}\left(\mathscr{C}_{3}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)\right)^{\frac{3}{2}}}{\left(\beta\left(\alpha,1-\frac{3\alpha}{2}\right)\right)^{-1}}.$

Hence, we can choose $\left\|u_{0}\right\|_{D^{1}(\Omega)}$ small enough to
obtain

$\displaystyle\left\|\int_{\mathscr{T}_{*}}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}\leq\frac{C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}}{2}.$
(29)

Combining all of the above arguments, whether $t\leq\mathscr{T}_{*}$ or
$t>\mathscr{T}_{*},$ we can see that

$\displaystyle\left\|\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}\leq
C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}.$ (30)

In addition, apply again Lemma 25, for any $w_{n}\in\mathbb{U}$, we obtain

$\displaystyle
t^{\frac{\alpha}{2}}\left\|\int_{\mathscr{T}_{*}}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{2\mathscr{C}_{3}(2C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)})^{3}}{\left(\beta\left(\alpha,1-\frac{3\alpha}{2}\right)\right)^{-1}}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right).$
(31)

Therefore, for any $w_{n}\in\mathbb{U}$, we deduce the following two claims.

Claim 1. Combining the conclusions that $w_{1}\in\mathbb{U}$ and the
inequality (30) gives us

$\displaystyle\left\|w_{n+1}(t)\right\|_{L^{\Xi}(\Omega)}\leq\left\|\mathscr{S}(t)u_{0}\right\|_{L^{\Xi}(\Omega)}+\left\|\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}\leq
2C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}.$

Claim 2. Similarly, since $w_{1}\in\mathbb{U}$ and the estimate (4.2) holds,
we have

$\displaystyle
t^{\frac{\alpha}{2}}\left\|w_{n+1}(t)\right\|_{L^{\Xi}(\Omega)}\leq
t^{\frac{\alpha}{2}}\left\|\mathscr{S}(t)u_{0}\right\|_{L^{\Xi}(\Omega)}+t^{\frac{\alpha}{2}}\left\|\int_{0}^{t}\mathscr{R}(t-\tau)H(w_{n}(\tau))\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}\leq
2C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)},$

as long as $\left\|u_{0}\right\|_{D^{1}(\Omega)}$ is sufficiently small.

These claims show that for any $t\in I$,
$\big{\|}w_{n+1}\big{\|}_{\mathbb{U}(t)}$ is less than or equals to
$2C_{1}\left\|u_{0}\right\|_{D^{1}(\Omega)}$. It means that $w_{n+1}$ belongs
to $\mathbb{U}$ whenever $w_{n}$ is in $\mathbb{U}$, provided that the
continuity of $w_{n+1}$ is inferred by Lemma 3.2. The remaining work is to
show that $\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence with respect to
the $\mathbb{U}$ norm. Based on this result, we can easily obtain the unique
mild solution $u$ of Problem (1)-(3) which is the limit function of the
sequence $\\{w_{n}\\}_{n=1}^{\infty}$. To this end, we take two elements of
the sequence, $w_{n-1},w_{n}\in\mathbb{U}$. Analogous to the way we find the
estimate (30), we set

$\displaystyle\mathscr{T}_{**}:=\left(\frac{64\mathscr{C}_{3}\left\|u_{0}\right\|^{2}_{D^{1}(\Omega)}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)}{\alpha
C_{1}}\right)^{\frac{-1}{\alpha}}.$

On the one hand, for any $t>\mathscr{T}_{**}$, we have

$\displaystyle\Bigg{\|}\int_{0}^{t}\mathscr{R}(t-\tau)\big{(}H(w_{n}(\tau))-H(w_{n-1}(\tau))\big{)}\mathrm{d}\tau\Bigg{\|}_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\underbrace{\int_{0}^{\mathscr{T}_{**}}\Big{\|}\mathscr{R}(t-\tau)\big{(}H(w_{n}(\tau))-H(w_{n-1}(\tau))\big{)}\Big{\|}_{L^{\Xi}(\Omega)}\mathrm{d}\tau}_{\mathrm{I}_{1}}$
$\displaystyle+\underbrace{\int_{\mathscr{T}_{**}}^{t}\Big{\|}\mathscr{R}(t-\tau)\big{(}H(w_{n}(\tau))-H(w_{n-1}(\tau))\big{)}\Big{\|}_{L^{\Xi}(\Omega)}\mathrm{d}\tau}_{\mathrm{I}_{2}}.$

In view of Lemma 4.5, one finds that

$\displaystyle\mathrm{I}_{1}$
$\displaystyle\leq\frac{2\mathscr{C}_{3}\mathscr{T}_{**}^{\alpha}}{\alpha(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{-2}}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}}$
$\displaystyle\leq\frac{1}{4}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}}.$

Applying Lemma 4.5 in the same way as in (29), we have

$\displaystyle\mathrm{I}_{2}$
$\displaystyle\leq\frac{2\mathscr{C}_{3}\mathscr{T}_{**}^{-\frac{\alpha}{2}}(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)}{\left(\beta\left(\alpha,1-\frac{3\alpha}{2}\right)\right)^{-1}}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}}$
$\displaystyle=\frac{64\left\|u_{0}\right\|_{D^{1}(\Omega)}^{3}\left(\mathscr{C}_{3}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)\right)^{\frac{3}{2}}}{\left(\beta\left(\alpha,1-\frac{3\alpha}{2}\right)\right)^{-1}}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}}$
$\displaystyle\leq\frac{1}{4}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}},$

provided that $\left\|u_{0}\right\|_{D^{1}(\Omega)}$ is sufficiently small. In
addition, for any $t<\mathscr{T}_{**}$, we note that

$\displaystyle\Bigg{\|}\int_{0}^{t}\mathscr{R}(t-\tau)\big{(}H(w_{n}(\tau))-H(w_{n-1}(\tau))\big{)}\mathrm{d}\tau\Bigg{\|}_{L^{\Xi}(\Omega)}\leq\mathrm{I}_{1}\leq\mathrm{I}_{1}+\mathrm{I}_{2}.$

Wherefore, for any $t\in I$ the following estimate holds

$\displaystyle\sup_{t\in I}\left\|w_{n+1}(t)-w_{n}(t)\right\|_{D^{1}(\Omega)}$
$\displaystyle=\sup_{t\in
I}\left\|\int_{0}^{t}\mathscr{R}(t-\tau)\big{(}H(w_{n}(\tau))-H(w_{n-1}(\tau))\big{)}\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq\mathrm{I}_{1}+\mathrm{I}_{2}\leq\frac{1}{2}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}}.$
(32)

On the other hand, by replacing respectively $w,v$ by
$w_{n},w_{n-1}\in\mathbb{U}$ in Lemma 4.5, the following inequality holds

$\displaystyle
t^{\frac{\alpha}{2}}\left\|\int_{0}^{t}\mathscr{R}(t-\tau)\big{(}H(w_{n}(\tau))-H(w_{n-1}(\tau))\big{)}\mathrm{d}\tau\right\|_{L^{\Xi}(\Omega)}$
$\displaystyle\leq$
$\displaystyle\frac{2\mathscr{C}_{3}(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}}{\left(\beta\left(\alpha,1-\frac{3\alpha}{2}\right)\right)^{-1}}\left(\left(\Gamma\left(3\right)\right)^{\frac{1}{2}}+\left(\Gamma\left(4\right)\right)^{\frac{1}{2}}\left(6(2\left\|u_{0}\right\|_{D^{1}(\Omega)})^{2}\right)^{\frac{1}{6}}\right)\left\|w_{n}(t)-w_{n-1}(t)\right\|_{\mathbb{U}}.$

As a result, if $\left\|u_{0}\right\|_{D^{1}(\Omega)}$ is small enough, we
deduce

$\displaystyle\sup_{t\in
I\setminus\\{0\\}}t^{\frac{\alpha}{2}}\left\|w_{n+1}(t)-w_{n}(t)\right\|_{L^{\Xi}(\Omega)}\leq\frac{1}{2}\left\|w-v\right\|_{\mathbb{U}}.$
(33)

On account of (4.2), (4.2) and (33), we can conclude that

$\displaystyle\left\|w_{n+1}(t)-w_{n}(t)\right\|_{\mathbb{U}}\leq\frac{1}{2}\left\|w_{n}(t)-w_{n-1}(t)\right\|_{L^{\Xi}(\Omega)},$

for every $w_{n},w_{n-1}\in\mathbb{U}$. It means that
$\\{w_{n}\\}_{n=1}^{\infty}$ is a Cauchy sequence in $\mathbb{U}$. The proof
is completed. ∎

######  Remark 4.2.

We note that in the above proof, we consider only the case
$\mathscr{T}>\max\left(\mathscr{T}_{*},\mathscr{T}_{**}\right)$. When,
$\mathscr{T}<\min\left(\mathscr{T}_{*},\mathscr{T}_{**}\right)$, the proof is
similar and easier, so we omit it here.

##  5 Appendix

######  Definition 5.1.

Let $\alpha\in(0,1)$. Then, the definition of the M-Wright type function
$\mathcal{W}_{\alpha}$ is given by

$\displaystyle\mathcal{W}_{\alpha}(r):=\sum_{k=0}^{\infty}\frac{r^{k}}{k!\Gamma(-\alpha
k+1-\alpha)}.$

######  Lemma 5.2.

([12, Section 2]) For $\alpha\in(0,1)$ and $\mu\in(-1,\infty)$, there holds

$\displaystyle\int_{0}^{\infty}r^{\mu}\mathcal{W}_{\alpha}(r)\mathrm{d}r=\frac{\Gamma(1+\mu)}{\Gamma(1+\alpha\mu)}.$

######  Lemma 5.3.

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth boundary and
$p\in[1,\infty)$. Then,

  1. (i)

if $~{}0\leq\nu<\frac{d}{2}$ and $1\leq p\leq\frac{2d}{d-2\nu}$, or
$\nu=\frac{d}{2}$ and $1\leq p<\infty$, we have

$\displaystyle D^{\nu}(\Omega)\hookrightarrow L^{p}(\Omega);$

  2. (ii)

if $\frac{-d}{2}<\nu\leq 0$ and $p\geq\frac{2d}{d-2\nu}$, we have

$\displaystyle D^{\nu}(\Omega)\hookleftarrow L^{p}(\Omega).$

######  Lemma 5.4.

(Fractional Grönwall inequality) Let $m,n$ be positive constants and
$\alpha\in(0,1).$ Suppose that function $w\in L^{\infty}(0,T]$ satisfies the
following inequality

$\displaystyle w(t)\leq
m+n\int_{0}^{t}(t-\tau)^{\alpha-1}w(\tau)\mathrm{d}\tau,\quad\text{for
all~{}}t\in(0,T],$

then, the result below is satisfied

$\displaystyle w(t)\leq mE_{\alpha}\big{(}n\Gamma(\alpha)t^{\alpha}\big{)}.$

######  Definition 5.5.

Let $p,q>0$. Then, the Beta function and the Gamma function can be defined
respectively by

$\displaystyle\beta(p,q)=\int_{0}^{1}(1-z)^{p-1}z^{q-1}\mathrm{d}z\quad\text{
and }\quad\Gamma(p)=\int_{0}^{\infty}z^{p-1}e^{-z}\mathrm{d}z.$

## Acknowledgement

The first author (Huy Tuan Nguyen) was supported by Vietnam National
Foundation for Science and Technology Development (NAFOSTED) under grant
number 101.02-2019.09.

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