# Initial-boundary value problems to semilinear
multi-term fractional differential equations

Sergii V. Siryk and Nataliya Vasylyeva  CONCEPT Lab, Istituto Italiano di
Tecnologia
Via Morego 30, 16163, Genova, Italy<EMAIL_ADDRESS>Institute of Applied
Mathematics and Mechanics of NASU
G.Batyuka st. 19, 84100 Sloviansk, Ukraine;
and Dipartimento di Matematica, Politecnico di Milano
Via Bonardi 9, 20133 Milano, Italy  nataliy<EMAIL_ADDRESS>
###### Abstract.

For $\nu,\nu_{i},\mu_{j}\in(0,1)$, we analyze the semilinear integro-
differential equation on the one-dimensional domain $\Omega=(a,b)$ in the
unknown $u=u(x,t)$

$\mathbf{D}_{t}^{\nu}(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{\nu_{i}}(\varrho_{i}u)-\sum_{j=1}^{N}\mathbf{D}_{t}^{\mu_{j}}(\gamma_{j}u)-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t),$

where $\mathbf{D}_{t}^{\nu},\mathbf{D}_{t}^{\nu_{i}},\mathbf{D}_{t}^{\mu_{j}}$
are Caputo fractional derivatives, $\varrho_{0}=\varrho_{0}(t)>0,$
$\varrho_{i}=\varrho_{i}(t)$, $\gamma_{j}=\gamma_{j}(t)$, $\mathcal{L}_{k}$
are uniform elliptic operators with time-dependent smooth coefficients,
$\mathcal{K}$ is a summable convolution kernel. Particular cases of this
equation are the recently proposed advanced models of oxygen transport through
capillaries. Under certain structural conditions on the nonlinearity $f$ and
orders $\nu,\nu_{i},\mu_{j}$, the global existence and uniqueness of classical
and strong solutions to the related initial-boundary value problems are
established via the so-called continuation arguments method. The crucial point
is searching suitable a priori estimates of the solution in the fractional
Hölder and Sobolev spaces. The problems are also studied from the numerical
point of view.

###### Key words and phrases:

a priori estimates, Caputo derivatives, nonlinear oxygen subdiffusion, global
solvability, numerical solutions

###### 2000 Mathematics Subject Classification:

Primary 35R11, 35B45, 35B65; Secondary 35Q92, 26A33, 65M22

##  1\. Introduction

Let $\Omega=(a,b)\subset\mathbb{R}$ be a segment, with a boundary
$\partial\Omega=\\{a\\}\cup\\{b\\}$. For an arbitrary fixed time $T>0$, we
denote

$\Omega_{T}=\Omega\times(0,T)\qquad\text{and}\qquad\partial\Omega_{T}=\partial\Omega\times[0,T].$

We consider the semilinear equation in the unknown function
$u=u(x,t):\Omega_{T}\to\mathbb{R}$,

$\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u)=g(x,t),$
(1.1)

subject either to the Dirichlet boundary condition (DBC)

$u=\psi(x,t)\quad\text{on}\quad\partial\Omega_{T},$ (1.2)

or to the Neumann boundary condition (NBC)

$\frac{\partial u}{\partial
x}=\psi_{1}(x,t)\qquad\text{on}\quad\partial\Omega_{T},$ (1.3)

where the functions $g,f,\psi,\psi_{1},\mathcal{K}$ are prescribed.

The equation is supplemented with the initial condition

$u(x,0)=u_{0}\quad\text{in}\quad\bar{\Omega}$ (1.4)

for some given initial datum $u_{0}$.

Here the $*$ denotes the usual time-convolution product on $(0,t)$, namely

$(\mathfrak{h}_{1}*\mathfrak{h}_{2})(t)=\int\limits_{0}^{t}\mathfrak{h}_{1}(t-s)\mathfrak{h}_{2}(s)ds,$

while the symbol $\mathbf{D}_{t}$ stands for the linear combinations of Caputo
fractional derivatives with respect to time, defined as

$\mathbf{D}_{t}u=\mathbf{D}_{t}^{\nu}(\varrho_{0}u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{\nu_{i}}(\varrho_{i}u)-\sum_{j=1}^{N}\mathbf{D}_{t}^{\mu_{j}}(\gamma_{j}u)$
(1.5)

for any fixed $\nu\in(0,1)$ $\nu_{i},\mu_{i}\in(0,\nu)$, and given positive
functions $\varrho_{0}=\varrho_{0}(t),$ $\varrho_{i}=\varrho_{i}(t)$,
$\gamma_{j}=\gamma_{j}(t),$ $i=1,...,M,$ and $j=1,...,N$. We agreed that if
$N=0$ or $M=0$ then the corresponding sum is missing from the above
representation. Here $\mathbf{D}_{t}^{\theta}$ denotes the Caputo fractional
derivative of order $\theta$ with respect to time. Let us recall the
definition of the Caputo fractional derivative in the case of
$\theta\in(0,1]$,

$\mathbf{D}_{t}^{\theta}u(x,t)=\begin{cases}\frac{1}{\Gamma(1-\theta)}\frac{\partial}{\partial
t}\int\limits_{0}^{t}\frac{u(x,s)-u(x,0)}{(t-s)^{\theta}}ds\quad\text{if}\quad\theta\in(0,1),\\\
\frac{\partial u}{\partial t}(x,t)\qquad\qquad\qquad\qquad\,\text{
if}\quad\theta=1,\end{cases}$

with $\Gamma$ being the Euler Gamma-function. An equivalent definition of this
derivative in the case of absolutely continuous functions reads

$\mathbf{D}_{t}^{\theta}u(x,t)=\begin{cases}\frac{1}{\Gamma(1-\theta)}\int\limits_{0}^{t}(t-s)^{-\theta}\frac{\partial
u}{\partial s}(x,s)ds\quad\text{if}\quad\theta\in(0,1),\\\ \frac{\partial
u}{\partial t}(x,t)\qquad\qquad\qquad\qquad\,\text{
if}\quad\theta=1.\end{cases}$

Coming to the operators involved, $\mathcal{L}_{i}$ are linear elliptic
operators of the second order with time-dependent coefficients, namely,

$\displaystyle\mathcal{L}_{1}u$
$\displaystyle=a_{2}\frac{\partial^{2}u}{\partial x^{2}}+a_{1}\frac{\partial
u}{\partial x}+a_{0}u,$ $\displaystyle\mathcal{L}_{2}u$
$\displaystyle=b_{2}\frac{\partial^{2}u}{\partial x^{2}}+b_{1}\frac{\partial
u}{\partial x}+b_{0}u,$

where $a_{i}=a_{i}(x,t),$ $b_{i}=b_{i}(x,t).$

The evolution equations with fractional derivatives play an important role in
the modleing of the so-called anomalous phenomena arising in Biology,
Geophysics, Chemistry and Physics (see e.g. [3, 2, 4, 30, 35, 36] and
references therein). It occurs that for certain processes the order of the
time-fractional derivatives from the corresponding model equation does not
remain constant. A possible method to control these phenomena is to exploit
the multi-term time-fractional diffusion-wave equation, see e.g. [32]. In
particular, partial case of equation (1.1) ($M=0$, $N=1$, $\varrho_{0}=1$ and
$\gamma_{1}=\text{constant}$) describes oxygen delivery through capillaries
[34, 42].

Published works related to the multi-term fractional diffusion/wave equations,
i.e. equations similar to (1.1) with the operator

$\mathbf{D}_{t}u=\sum_{i=1}^{N}q_{i}\mathbf{D}_{t}^{\nu_{i}}u,$ (1.6)

with $q_{i}$ being positive, and $0\leq\nu_{1}<\nu_{2}<...<\nu_{M}$, are quite
limited in spite of rich literature on their single-term version. Exact
solution of linear multi-term fractional diffusion equations with $q_{i}$
being positive constants on bounded domains are constructed employing
eigenfunction expansion in [7, 6, 14, 34, 45]. We quote [42, 34, 41, 45] where
certain numerical solutions are built to the corresponding initial-boundary
value problems to evolution equations with $\mathbf{D}_{t}$ given via (1.6).
Abstract multi-term time-fractional equations in Banach spaces are discussed
in [27]. Well-posedness along with a maximum principle and the long-time
asymptotic behavior of the solution for the initial-boundary value problems to
the these equation are studied in [19, 29, 31, 33] (see also references
therein). In fine, we refer to [28], where initial-boundary value problems to
this equation with the $x$-dependent coefficients $q_{i}$ are analyzed.

The principal distinction of equation (1.1) from the equations in the
aforementioned works is related to the representation of the operator
$\mathbf{D}_{t}$ (see (1.5)) as a linear combination of the multi-term
fractional derivatives. Therefore, for certain $\rho_{0}$ and $\gamma_{i}$,
$\mathbf{D}_{t}u$ can be rewritten in the form

$\frac{\partial}{\partial
t}\int_{0}^{t}\mathcal{N}(t-\tau)[u(x,\tau)-u(x,0)]d\tau$

with the kernel $\mathcal{N}$ being either a negative function or a function
alternating in sign. Indeed, choosing $M=0$, $N=1$, and

$\gamma_{1}=1+\varrho_{0},\quad\varrho_{0}\equiv C_{\varrho}>0,$

and appealing to [13, Lemma 4], we end up with the equality

$\mathbf{D}_{t}^{\nu}(\varrho_{0}u)-\mathbf{D}_{t}^{\nu_{1}}(\gamma_{1}u)=\frac{\partial}{\partial
t}[\mathcal{N}*(u-u(x,0))],$

where the kernel

$\mathcal{N}=C_{\varrho}\frac{t^{-\nu}}{\Gamma(1-\nu)}-(1+C_{\varrho})\frac{t^{-\mu_{1}}}{\Gamma(1-\mu_{1})},$

is negative  for $t>e^{-\gamma}$, $\gamma$ being the Euler-Mascheroni
constant. It is worth noting that the nonnegativity of the kernel
$\mathcal{N}$ plays a crucial role in the previous investigations of
fractional partial differential equations and related initial/initial-boundary
value problems. This assumption is removed in our research. Moreover, equation
(1.1) contains fractional derivatives calculated from the product of two
functions: the desired solution $u$ and the prescribed coefficients
$\varrho_{i}$, $\varrho_{0}$, $\gamma_{i}$. This peculiarity provides
additional difficulties since the typical Leibniz rule does not work in the
case of fractional derivatives.

The first result on the global classical solvability to the linear version
(1.1) with the operator $\mathbf{D}_{t}$ given by (1.5), where the
coefficients $\varrho_{i}=\varrho_{i}(x,t),$ $\gamma_{i}=\gamma_{i}(x,t)$ are
alternating sign, is discussed in [37]. To the authors’ best knowledge, there
are no works in the literature addressing to one-valued global solvability to
the quasilinear equation (1.1) in the case of $N\geq 1$ and positive
$\gamma_{i}$. The aim of the present paper is to fill this gap, providing a
well-posedness result along with the regularity of solutions in fractional
Hölder and Sobolev classes for any fixed time $T$, in the case of “power law”
memory kernel [38], i.e. satisfying for every $t\in[0,T]$ the bound

$|\mathcal{K}(t)|\leq Ct^{-\beta}$

for some positive constant $C$ and $\beta\in[0,1)$. Indeed, boundary problems
with kernels of this kind do have a practical interest: many viscoelastic
materials have rapidly decreasing memory in small time values, and are
therefore better described by kernels with singularities at the origin.

The technique of this paper heavily rely on the fact that we work in a one-
dimensional domain. Bedsides, the main tools in our analysis are a priori
estimates in fractional Sobolev and Hölder spaces. Our analysis is
complemented by numerical simulations.

### Outline of the paper

The paper is organized as follows: in Section 2, we introduce the notation and
the functional settings. The main assumptions are discussed in Section 3. The
principal results (Theorems 4.1 and 4.4) are stated in Section 4. Theorem 4.1
concerns to the classical global solvability of (1.1)-(1.5), while Theorem 4.4
touches the existence and uniqueness of strong solutions to these problems. In
Section 5, we recall some definitions together with some auxiliary technical
results from fractional calculus, playing a key role in the course of this
study. Section 6 is devoted to obtain a priori estimates in the fractional
Sobolev and Hölder spaces. The proof of Theorem 4.1 is carried out in Section
7. To this end, we exploit the continuation method, treating the family of
problems depending on the parameter $\lambda\in[0,1]$,

$\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u=\lambda[f(u_{0})-f(u)]+g(x,t)-f(u_{0}),$

subject to the conditions (1.2)-(1.4). At last, in Section 8, we prove Theorem
4.4 via construction of a strong solution as a limit of approximate smooth
solutions. In the final Section 9, we study (1.1)-(1.5) from the numerical
side.

##  2\. Functional Spaces and Notation

Throughout this work, the symbol $C$ will denote a generic positive constant,
depending only on the structural quantities of the problem. We will carry out
our analysis in the framework of the fractional Hölder spaces. To this end, in
what follows we take two arbitrary (but fixed) parameters

$\alpha\in(0,1)\quad\text{and}\quad\theta\in(0,1).$

For any nonnegative integer $l$, any Banach space
$(\mathbf{X},\|\cdot\|_{\mathbf{X}}),$ and any $p\geq 1,$ $s\geq 0$, we
consider the usual spaces

$\mathcal{C}^{s}([0,T],\mathbf{X}),\quad\mathcal{C}^{l+\alpha}(\bar{\Omega}),\quad
W^{s,p}(\Omega),\quad L_{p}(\Omega),\quad W^{s,p}((0,T),\mathbf{X}).$

Recall that for noninteger $s$, $W^{s,p}$ is called the Sobolev-Slobodeckii
space (for its definition and properties see, e.g., in [1, Chapter 1], [12,
Chapter 1]).

Denoting for $\beta\in(0,1)$

$\displaystyle\langle v\rangle^{(\beta)}_{x,\Omega_{T}}$
$\displaystyle=\sup\Big{\\{}\frac{|v(x_{1},t)-v(x_{2},t)|}{|x_{1}-x_{2}|^{\beta}}:\quad
x_{2}\neq x_{1},\quad x_{1},x_{2}\in\bar{\Omega},\quad t\in[0,T]\Big{\\}},$
$\displaystyle\langle v\rangle^{(\beta)}_{t,\Omega_{T}}$
$\displaystyle=\sup\Big{\\{}\frac{|v(x,t_{1})-v(x,t_{2})|}{|t_{1}-t_{2}|^{\beta}}:\quad
t_{2}\neq t_{1},\quad x\in\bar{\Omega},\quad t_{1},t_{2}\in[0,T]\Big{\\}},$

we assert the following definition.

######  Definition 2.1.

A function $v=v(x,t)$ belongs to the class
$\mathcal{C}^{l+\alpha,\frac{l+\alpha}{2}\theta}(\bar{\Omega}_{T})$, for
$l=0,1,2,$ if the function $v$ and its corresponding derivatives are
continuous and the norms here below are finite:

$\|v\|_{\mathcal{C}^{l+\alpha,\frac{l+\alpha}{2}\theta}(\bar{\Omega}_{T})}=\begin{cases}\|v\|_{\mathcal{C}([0,T],\mathcal{C}^{l+\alpha}(\bar{\Omega}))}+\sum_{|j|=0}^{l}\langle
D_{x}^{j}v\rangle^{(\frac{l+\alpha-|j|}{2}\theta)}_{t,{\Omega}_{T}},\qquad\qquad\qquad\qquad\quad\,l=0,1,\\\
\\\
\|v\|_{\mathcal{C}([0,T],\mathcal{C}^{2+\alpha}(\bar{\Omega}))}+\|\mathbf{D}_{t}^{\theta}v\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\theta}(\bar{\Omega}_{T})}+\sum_{|j|=1}^{2}\langle
D_{x}^{j}v\rangle^{(\frac{2+\alpha-|j|}{2}\theta)}_{t,{\Omega}_{T}},\quad
l=2.\end{cases}$

In a similar way, for $l=0,1,2,$ we introduce the space
$\mathcal{C}^{l+\alpha,\frac{l+\alpha}{2}\theta}(\partial\Omega_{T})$.

The properties of these spaces have been discussed in [21, Section 2]. It is
worth noting that, in the limiting case $\theta=1$, the class
$\mathcal{C}^{l+\alpha,\frac{l+\alpha}{2}\theta}$ coincides with the usual
parabolic Hölder space $H^{l+\alpha,\frac{l+\alpha}{2}}$ (see e.g.[25,
(1.10)-(1.12)]).

Finally, exploiting [44, Proposition 1], we introduce the space
$\mathcal{H}^{s}((0,T),\mathbf{X})$ for $s\in(0,1)$.

######  Definition 2.2.

For $s\in(0,1)$ we define the space

$\mathcal{H}^{s}((0,T),\mathbf{X})=\begin{cases}W^{s,2}((0,T),\mathbf{X}),\qquad\qquad\qquad\qquad
s\in(0,1/2),\\\ \\{v\in
W^{1/2,2}((0,T),\mathbf{X}),\quad\int_{0}^{T}\|v\|^{2}_{\mathbf{X}}\frac{dt}{t}<+\infty\\},\quad
s=1/2,\\\ \\{v\in W^{s,2}((0,T),\mathbf{X}),\quad v|_{t=0}=0\\},\qquad\qquad
s\in(1/2,1),\\\ \end{cases}$

subject toh the norms

$\|v\|_{\mathcal{H}^{s}((0,T),\mathbf{X})}=\begin{cases}\|v\|_{W^{s,2}((0,T),\mathbf{X})},\qquad\qquad\qquad
s\in(0,1),\ \ s\neq 1/2,\\\
\bigg{(}\|v\|^{2}_{W^{1/2,2}((0,T),\mathbf{X})}+\int_{0}^{T}\|v\|^{2}_{\mathbf{X}}\frac{dt}{t}\bigg{)}^{1/2},\quad
s=1/2.\end{cases}$

Setting $v(x,0)=v_{0}$ and taking into account [44, Propositions 3 and 7], we
arrive to the following norm equivalence

$C^{-1}\|v-v_{0}\|_{\mathcal{H}^{s}((0,T),\mathbf{X})}\leq\|\mathbf{D}_{t}^{s}v\|_{L_{2}((0,T),\mathbf{X})}\leq
C\|v-v_{0}\|_{\mathcal{H}^{s}((0,T),\mathbf{X})},$ (2.1)

for all $(v-v_{0})\in\mathcal{H}^{s}((0,T),\mathbf{X})$ and $s\in(0,1)$.

##  3\. General Assumptions

First, we state our general hypothesis on the structural terms of the model.
To this end, introducing

$\omega_{1-\nu}(t)=\frac{t^{-\nu}}{\Gamma(1-\nu)},$

we define the positive values $\nu^{*}$ and $T^{*}$ such that the kernels

$\mathcal{N}(t;\nu,\mu_{j})=\omega_{1-\nu}(t)-\omega_{1-\mu_{j}}(t),\quad
j=1,2,...,N,$

are nonnegative for all $t\in[0,T^{*}]$ and $0<\mu_{j}<\nu\leq\nu^{*}<1$.

h1 (Conditions on the fractional order of the derivatives):

We assume that

$\nu\in\begin{cases}(0,\nu^{*})\quad\text{if}\quad N\geq 1,\\\
(0,1)\quad\text{if}\quad
N=0,\end{cases}\quad\text{and}\quad\nu_{i},\mu_{j}\in\Big{(}0,\frac{\nu(2-\alpha)}{2}\Big{)},\quad
i=1,2,...,M,\,j=1,2,...,N,$ $0<\mu_{1}<...<\mu_{N}<\nu,\quad
0<\nu_{1}<...<\nu_{M}<\nu,\quad\nu_{i}\neq\mu_{j}\quad\text{for all}\quad
i=1,2,...,M,\,j=1,2,...,N,$

h2 (Ellipticity conditions):

There are positive constants $\delta_{i},$ $i=0,1,2,3,$ such that

$\displaystyle a_{2}(x,t)$
$\displaystyle\geq\delta_{0}>0,\quad\varrho_{0}(t)\geq\delta_{1}>0,$
$\displaystyle\varrho_{i}(t)$
$\displaystyle\geq\delta_{2}>0,\quad\gamma_{j}(t)\geq\delta_{3}>0,\quad
i=1,2,...,M,\,j=1,2,...,N,$

for any $(x,t)\in\bar{\Omega}_{T}$ and $t\in[0,T]$.

h3 (Regularity of the coefficients):

We require

$\displaystyle a_{k},b_{k}$
$\displaystyle\in\mathcal{C}^{\alpha,\frac{\alpha\nu}{2}}(\bar{\Omega}_{T}),\quad
k=0,1,2,\quad\frac{\partial a_{2}}{\partial x},\frac{\partial b_{2}}{\partial
x}\in\mathcal{C}(\bar{\Omega}_{T}),$
$\displaystyle\varrho_{0},\varrho_{i},\gamma_{j}$
$\displaystyle\in\mathcal{C}^{1}([0,T]),\quad i=1,2,...,M,\,j=1,2,...,N,$

and

$\frac{\partial\varrho_{i}}{\partial t},\frac{\partial\varrho_{0}}{\partial
t},\frac{\partial\gamma_{j}}{\partial t}\geq 0,$

for all $t\in[0,T]$ and $i=1,...,M,$ $j=1,2,...,N.$

Besides, in the case of $N\geq 1$ the representation holds

$\varrho_{0}=\varrho+\sum_{i=1}^{N}\gamma_{i}$

with the positive function $\varrho$ having the properties of the function
$\varrho_{0}$.

h4 (Condition on the kernel):

The summable kernel $\mathcal{K}$ fulfills the estimate

$|\mathcal{K}(t)|\leq\frac{C}{t^{\beta}},\quad\beta\in(0,1-\nu]$

for any $t\in[0,T]$.

h5 (Conditions on the given functions):

We require that the given functions possess the regularity:

(i) either

$\displaystyle u_{0}(x)$ $\displaystyle\in
C^{2+\alpha}(\bar{\Omega}),\qquad\qquad
g\in\mathcal{C}^{\alpha,\frac{\alpha\nu}{2}}(\bar{\Omega}_{T}),$
$\displaystyle\psi(x,t)$
$\displaystyle\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\partial\Omega_{T}),\quad\psi_{1}(x,t)\in\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}\nu}(\partial\Omega_{T}),$

(ii) or

$\psi,\psi_{1},g\equiv 0,\qquad
u_{0}\in\begin{cases}W^{2,2}(\Omega)\cap\overset{0}{W}\,^{1,2}(\Omega)\quad\text{in
the case of \bf{DBC}},\\\ W^{2,2}(\Omega)\qquad\qquad\quad\text{in the case of
\bf{NBC}}.\end{cases}$

h6 (Conditions on the nonlinearity):

The function $f(u)$ satisfies the one of two conditions:

(i) either $f(u)$ is the local Lipshits, i.e. for every $\rho>0$ there exists
a positive constant $C_{\rho}$ such that

$|f(u_{1})-f(u_{2})|\leq C_{\rho}|u_{1}-u_{2}|$

for any $u_{1},u_{2}\in[-\rho,\rho]$; and

there is a positive constant $L$ such that

$|f(u)|\leq L(1+|u|)\quad\text{for any}\quad u\in\mathbb{R};$ (3.1)

(ii) or

$\begin{cases}f\in\mathcal{C}^{1}(\mathbb{R}),\\\ |f(u)|\leq
L_{1}(1+|u|^{r}),\\\ uf(u)\geq-L_{2}+L_{3}|u|^{r+1},\\\ f^{\prime}(u)\geq-
L_{4},\end{cases}$ (3.2)

for some nonnegative constants $r$ and $L_{i}$, $i=1,2,3,4$.

h7 (Compatibility conditions):

The following compatibility conditions hold for every $x\in\partial\Omega$ at
the initial time $t=0$,

$\psi(x,0)=u_{0}(x)\quad\text{and}\quad\mathbf{D}_{t}\psi|_{t=0}=\mathcal{L}_{1}u_{0}(x)|_{t=0}-f(u_{0})+g(x,0)$

if the DBC (1.2) holds, and

$\frac{\partial u_{0}}{\partial x}(x)=\psi_{1}(x,0),$

if the NBC (1.3) holds.

######  Remark 3.1.

Thanks to Lemma 4.1 in [19], the equality is true

$(\mathcal{K}*\mathcal{L}_{2}u)(x,0)=0$

for any $u\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\partial\Omega_{T})$
and any $x\in\partial\Omega$. That explains the absence of the memory term
$(\mathcal{K}*\mathcal{L}_{2}u)$ in the compatibility conditions h7.

######  Remark 3.2.

It is worth noting that the existence of $\nu^{*}$ and $T^{*}$ in assumption
h1 is provided by [13, Lemma 4]. Indeed, this lemma establishes the existence
of the pair $(\nu_{\gamma},T_{\gamma})$, $0<\nu_{\gamma}<1$ and
$T_{\gamma}=e^{-\gamma}$ ($\gamma\approx 0.577$ is the Euler-Mascheroni
constant), such that the function $\omega_{1-\nu}(t)$ is strictly increasing
for all $\nu\in(0,\nu_{\gamma})$ and each $t\in[0,T_{\gamma}]$. Thus, this
assertion tells us that the kernels $\mathcal{N}(t;\nu,\mu_{j})$ are positive
if $0<\mu_{j}<\nu<\nu_{\gamma}$ and $t\in[0,T_{\gamma}]$. Hence, we can select
$\nu^{*}=\nu_{\gamma}$ and $T^{*}=T_{\gamma}$ in h1.

Nevertheless, if $T^{*}<T_{\gamma}$, then the value $\nu^{*}$ can be chosen
greater than $\nu_{\gamma}$. Unfortunately, an analytical proof of such
conjecture as well as explicit values of $\nu^{*}$ and $T^{*}$ seem to be out
of reach. This is the point of the story where the Numerics steps in. Indeed,
let us examine case $\mu_{j}=\frac{\nu}{j+1},$ $j=1,2,3,$ and $T^{*}\leq
0.11$. We find numerically $\nu_{j}^{*}=\nu^{*}(T^{*},\mu_{j})$ and
$\hat{\nu}_{\gamma}^{*}=\hat{\nu}_{\gamma}(T^{*})$, which provide for all
$t\in[0,T^{*}]$:

$\mathcal{N}\bigg{(}t;\nu,\frac{\nu}{j+1}\bigg{)}\geq
0\quad\text{for}\quad\frac{\nu}{j+1}<\nu\leq\nu_{j}^{*},\quad\text{and}$
$\omega_{1-\nu}(t)\quad\text{is strictly increasing for
}\quad\nu<\hat{\nu}^{*}_{\gamma}.$

Then, setting $\nu^{*}=\underset{j\in\\{1,2,3,\\}}{\min}\,\nu_{j}^{*}$, we
ensure the fulfillment of assumption h1 in the considered case.

In particular, our numeric calculations (presented with Figure 1 and Table 1)
demonstrate that if $T^{*}=0.1$, then $\nu^{*}=0.7200$, while
$\hat{\nu}^{*}_{\gamma}=0.5614$.

(a) $T^{*}=0.01$

(b) $T^{*}=0.1$

Figure 1. The behavior of the function $\omega_{1-\nu}(t)$ and the kernels $\mathcal{N}(t;\nu,\frac{\nu}{j+1})$ for $j=1,2,3,$ with (a) $T^{*}=0.01$, (b) $T^{*}=0.1$.  Table 1. Values of $\hat{\nu}^{*}_{\gamma}$ and $\nu^{*}_{j}$, $j=1,2,3,$ for the corresponding value $T^{*}$ $T^{*}$ | $0.01$ | $0.02$ | $0.03$ | $0.04$ | $0.05$ | $0.06$ | $0.07$ | $0.08$ | $0.09$ | $0.1$ | $0.11$
---|---|---|---|---|---|---|---|---|---|---|---
$\hat{\nu}^{*}_{\gamma}$ | 0.7703 | 0.7303 | 0.7003 | 0.6751 | 0.6527 | 0.6322 | 0.6131 | 0.5950 | 0.5779 | 0.5614 | 0.5455
$\nu^{*}_{1}$ | 0.9321 | 0.8984 | 0.8703 | 0.8451 | 0.8218 | 0.7999 | 0.7789 | 0.7586 | 0.7390 | 0.7200 | 0.7013
$\nu^{*}_{2}$ | 0.9616 | 0.9361 | 0.9133 | 0.8919 | 0.8714 | 0.8515 | 0.8321 | 0.8131 | 0.7944 | 0.7759 | 0.7577
$\nu^{*}_{3}$ | 0.9716 | 0.9505 | 0.9306 | 0.9116 | 0.8929 | 0.8746 | 0.8564 | 0.8385 | 0.8206 | 0.8023 | 0.7851

######  Remark 3.3.

We remark that the examples of the nonlinearity $f(u)$ and the kernel
$\mathcal{K}$ satisfying assumptions h6 and h4 are given in [21, Example 3.3],
[20, Remark 3.2] and [10, (1.4)], respectively.

##  4\. Main Results

Now we are in the position to state our first main result related with a
classical solvability of (1.1)-(1.4).

######  Theorem 4.1.

Let $T>0$ be arbitrarily given, and let assumptions h1-h4, h5 (i) and h7 hold.
Moreover, we assume that $f(u)$ meets requirement h6(i) if $N\geq 1$, while in
the case of $N=0$, $f(u)$ satisfies h6. Then, equation (1.1) with the initial
condition (1.4), subjects either to the DBC (1.2) or to the NBC (1.3), admits
a unique classical solution $u=u(x,t)$ satisfying regularity

$u\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T}),\quad\mathbf{D}_{t}^{\nu_{i}}u,\,\mathbf{D}_{t}^{\mu_{j}}u\in\mathcal{C}^{\alpha,\frac{\alpha\nu}{2}}(\bar{\Omega}_{T}),\,i=1,2,...,M,\,j=1,2,...,N.$

######  Remark 4.2.

Actually, with inessential modifications in the proof, the very same result
holds for the boundary value problem (1.1), (1.4) subject to either the mixed
boundary conditions, e.g.

$u(a,t)=\psi(a,t),\quad\frac{\partial u}{\partial x}(b,t)=\psi_{1}(b,t),\qquad
t\in[0,T],$

or the boundary conditions of the third kind.

Before stating our next result, we specify the notation of a strong solution
to (1.1)-(1.4).

######  Definition 4.3.

A function $u$ is called a strong solution to problems (1.1)-(1.4) if

$\bullet$ $u\in\mathcal{C}(\bar{\Omega}_{T})\cap
L_{2}((0,T),W^{2,2}(\Omega)),$
$\mathbf{D}_{t}^{\nu}u,\mathbf{D}_{t}^{\nu_{i}}u,\,\mathbf{D}_{t}^{\mu_{j}}u\in
L_{2}(\Omega_{T})$;

$\bullet$ the boundary and initial conditions (1.2)-(1.4) hold;

$\bullet$ for any fixed $T>0$ and any $\phi\in L_{2}(\Omega_{T})$,

$\int_{\Omega}\int_{0}^{T}(\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u))\phi
dxdt=\int_{\Omega}\int_{0}^{T}g\phi dxdt.$

Now we are ready to assert next results concerning to the existence of a
strong solution to (1.1)-(1.4).

######  Theorem 4.4.

Let assumptions h1-h4, h5(ii) hold. Let the nonlinearity $f(u)$ satisfy h6(i)
if $N\geq 1$, while $f(u)$ meets requirement h6 if $N=0$. Besides, in the case
of NBC, we require that

$\frac{\partial u_{0}}{\partial x}(a)=\frac{\partial u_{0}}{\partial x}(b)=0.$

Then problem (1.1)-(1.4) admits a unique strong solution $u=u(x,t)$.

######  Remark 4.5.

In general, assumption h3 on the coefficient $\varrho_{0}$ can be relaxed
under additional requirements on the orders $\nu_{i}$ and $\mu_{j}$ and
coefficients $\varrho_{i}$. Indeed, the results of Theorems 4.1 and 4.4 hold
if instead of the condition

$\varrho_{0}(t)=\varrho(t)+\sum_{j=1}^{N}\gamma_{j}(t)$

we will require that

$\displaystyle N$ $\displaystyle\leq M,\quad
0<\mu_{1}<\nu_{1}<\mu_{2}<\nu_{2}<...<\nu_{N-1}<\mu_{N}<\nu_{N}<...<\nu_{M}<\nu,$
$\displaystyle\varrho_{i}(t)$
$\displaystyle=\hat{\varrho}_{i}(t)+\gamma_{i}(t),\quad i=1,2,...,N,$

with $\hat{\varrho}_{i}(t)$ having the properties of the function
$\varrho_{i}(t)$.

Moreover, if $M=N=0,$ then Theorems 4.1 and 4.4 hold if
$\varrho=\varrho(x,t)\in\mathcal{C}^{1}(\bar{\Omega}_{T})$.

######  Remark 4.6.

Theorems 4.1 and 4.4 hold if $\mathbf{D}_{t}u$ in (1.5) is replaced by

$\mathbf{D}_{t}u=\varrho(t)\mathbf{D}_{t}^{\nu}u+\sum_{i=1}^{M}\varrho_{i}(t)\mathbf{D}_{t}^{\nu_{i}}u-\sum_{j=1}^{N}\gamma_{j}(t)\mathbf{D}_{t}^{\mu_{j}}u.$

######  Remark 4.7.

It is worth noting that our assumptions on the kernel $\mathcal{K}$ include
the case $\mathcal{K}=0$, telling that the multi-term subdiffusion equation:

$\mathbf{D}_{t}u-\mathcal{L}_{1}u+f(u)=g(x,t)$

fits in our analysis and described by the theorems above.

Finally, we remark that in our work we do not consider equation (1.1) if
$\nu=1$, since (thanks to presence the first order derivative in time in
(1.5)) this case is examined with simpler approach.

The remaining part of the paper is devoted to the proof of Theorems 4.1 and
4.4.

##  5\. Technical Results

In this section we describe some properties of fractional derivatives and
integrals, along with several technical assertions which will be used in the
course of our analysis. First, we define fractional Riemann-Liouville
integrals and derivatives.

Throughout this work, for any $\theta>0$ we denote (as we did before)

$\omega_{\theta}(t)=\frac{t^{\theta-1}}{\Gamma(\theta)},$ (5.1)

and define the fractional Riemann-Liouville integral and the derivative of
order $\theta$, respectively, of a function $v=v(x,t)$ with respect to time
$t$ as

$I_{t}^{\theta}v(x,t)=(\omega_{\theta}*v)(x,t),\quad\partial_{t}^{\theta}v(x,t)=\frac{\partial^{\lceil\theta\rceil}}{\partial
t^{\lceil\theta\rceil}}(\omega_{\lceil\theta\rceil-\theta}*v)(x,t),$

where $\lceil\theta\rceil$ is the ceiling function of $\theta$ (i.e. the
smallest integer greater than or equal to $\theta$).

Clearly, for $\theta\in(0,1)$ we have

$\partial_{t}^{\theta}v(x,t)=\frac{\partial}{\partial
t}(\omega_{1-\theta}*v)(x,t).$

Accordingly, the Caputo fractional derivative of the order $\theta\in(0,1)$ to
the function $v(x,t)$ can be represented as

$\mathbf{D}_{t}^{\theta}v(x,t)=\partial_{t}^{\theta}v(x,t)-\omega_{1-\theta}(t)v(x,0),$
(5.2)

if the both derivatives exist (see [17, (2.4.8)]).

In the first claim, which subsumes Propositions 4.1 and 4.2 in [21], we recall
some important relations for the fractional derivatives and integrals.

######  Proposition 5.1.

The following holds.

  1. (i)

Let $\theta,\theta_{1}\in(0,1),$ $t\in[0,T].$ Given any function
$w=w(t)\in\mathcal{C}^{\theta_{1}}([0,T])$

$I_{t}^{\theta}\partial_{t}^{\theta}w(t)=w(t).$

If in addition $\theta<2\theta_{1}$ and $p\geq 2$ is any even integer, it is
also true that

$\displaystyle\partial_{t}^{\theta}w^{p}(t)$
$\displaystyle\leq\partial_{t}^{\theta}w^{p}(t)+(p-1)w^{p}(t)\omega_{1-\theta}(t)$
$\displaystyle\leq pw^{p-1}(t)\partial_{t}^{\theta}w(t).$

If $w$ is a nonnegative function then these bounds hold for any integer odd
$p$.

  2. (ii)

For any given positive numbers $\theta_{1}$ and $\theta_{2}$, and any function
$k\in L_{1}(0,T),$ the following relations are fulfilled

$\omega_{\theta_{1}}*\omega_{\theta_{2}}=\omega_{\theta_{1}+\theta_{2}}(t),\,1*\omega_{\theta_{1}}=\omega_{1+\theta_{1}}(t),\
\omega_{\theta_{1}}(t)\geq CT^{\theta_{1}-1},\,\omega_{\theta_{1}}*k\leq
C\omega_{\theta_{1}+1}\leq C\omega_{\theta_{1}},$

for any $t\in[0,T]$. The positive constant $C$ depends only on $T$,
$\theta_{1}$ and $\|k\|_{L_{1}(0,T)}$.

Our next assertion is similar to point (i) of Proposition 5.1 stated to the
convolution with the kernel

$\mathcal{N}_{\theta}(t)=\mathcal{N}(t;\theta_{1},\theta_{2})=\omega_{1-\theta_{1}}(t)-\omega_{1-\theta_{2}}(t).$
(5.3)

######  Corollary 5.2.

Let $\theta^{*}$ and $T^{*}\leq T$ be the numbers such that the kernel
$\mathcal{N}_{\theta}(t)$ is positive for
$0<\theta_{2}<\theta_{1}\leq\theta^{*}\leq 1$ and $t\in[0,T^{*}]$. Then for
any given function $w\in\mathcal{C}^{\theta_{3}}([0,T]),$
$\theta_{3}\in(\theta_{1},1)$ and any even integer $p\geq 2$ the inequalities
are fulfilled

$\displaystyle\frac{d}{dt}(\mathcal{N}_{\theta}*w^{p})(t)$
$\displaystyle\leq\frac{d}{dt}(\mathcal{N}_{\theta}*w^{p})(t)+(p-1)w^{p}(t)\mathcal{N}_{\theta}(t)$
$\displaystyle\leq
pw^{p-1}(t)\frac{d}{dt}(\mathcal{N}_{\theta}*w)(t),\quad\forall
t\in[0,T^{*}].$ (5.4)

If additionally $w$ is nonnegative then theses bounds hold for any integer odd
$p$.

###### Proof.

It is worth noting that this claim is a simple consequence of the following
inequalities:

$\displaystyle\frac{d}{dt}(\mathcal{N}_{\theta}*w\,w_{1})(t)$
$\displaystyle=w\frac{d}{dt}(\mathcal{N}_{\theta}*w_{1})(t)+w_{1}(t)\frac{d}{dt}(\mathcal{N}_{\theta}*w)(t)-w(t)w_{1}(t)\mathcal{N}_{\theta}(t)$
$\displaystyle+\int_{0}^{t}[w(t)-w(s)][w_{1}(t)-w_{1}(s)]\frac{d}{d(t-s)}\mathcal{N}_{\theta}(t-s)ds,$
$\displaystyle\frac{d\mathcal{N}_{\theta}}{dt}(t)$ $\displaystyle\leq 0,$
(5.5)

which hold for any $w_{1}\in\mathcal{C}^{\theta_{3}}([0,T])$ and any
$t\in[0,T^{*}]$.

Indeed, substituting

$w_{1}(t)=\begin{cases}w\quad\text{if}\quad p=2,\\\ w^{2}\quad\text{if}\quad
p=3,\end{cases}$

in the first equality in (5), we derive the relations:

$\displaystyle\frac{d}{dt}(\mathcal{N}_{\theta}*w^{2})(t)$
$\displaystyle=2w\frac{d}{dt}(\mathcal{N}_{\theta}*w)(t)-\mathcal{N}_{\theta}(t)w^{2}(t)+\int_{0}^{t}[w(s)-w(t)]^{2}\frac{d}{d(t-s)}\mathcal{N}_{\theta}(t-s)ds,$
$\displaystyle\frac{d}{dt}(\mathcal{N}_{\theta}*w^{3})(t)$
$\displaystyle=w\bigg{[}\frac{d}{dt}(\mathcal{N}_{\theta}*w^{2})(t)+w(t)\frac{d}{dt}(\mathcal{N}_{\theta}*w)(t)-\mathcal{N}_{\theta}(t)w^{2}(t)\bigg{]}$
$\displaystyle+\int_{0}^{t}[w(s)-w(t)]^{2}[w(s)+w(t)]\frac{d}{d(t-s)}\mathcal{N}_{\theta}(t-s)ds.$

Then, taking into account the positivity of the kernel
$\mathcal{N}_{\theta}(t)$ for $\theta_{i}$ and $t$ meeting requirements of
this corollary, and appealing to the second inequality in (5), we arrive at
the desired estimates if $p=2,3$ (in the last case we also used the positivity
of $w$).

Finally, keeping in mind the obtained inequalities and exploiting the
induction, we end up with (5.2) for $p>3$. Thus, in order to complete the
verification of Corollary 5.2, we are left to prove (5). As for the second
inequality in (5), the straightforward calculations provide

$\frac{d\mathcal{N}_{\theta}}{dt}(t)=-\theta_{1}t^{-1}\mathcal{N}_{\theta}(t)-(\theta_{1}-\theta_{2})t^{-1}\omega_{1-\theta_{2}}(t)\leq
0\quad\text{for}\quad t\in[0,T^{\star}].$

Coming to the verification of the first equality in (5), we take advantage of
the definition of a derivative and have

$\frac{d}{dt}(\mathcal{N}_{\theta}*ww_{1})(t)=\underset{\varepsilon\to
0}{\lim}\frac{1}{\varepsilon}\bigg{[}\int_{0}^{t+\varepsilon}\mathcal{N}_{\theta}(t+\varepsilon-s)w_{1}(s)w(s)ds-\int_{0}^{t}\mathcal{N}_{\theta}(t-s)w_{1}(s)w(s)ds\bigg{]}.$

Then, exploiting the easily verified equality

$w(s)w_{1}(s)=[w(s)-w(t)][w_{1}(s)-w_{1}(t)]-w(t)w_{1}(t)+w(t)w_{1}(s)+w(s)w_{1}(t),$

we end up with the equality

$\displaystyle\frac{d}{dt}(\mathcal{N}_{\theta}*w_{1}w)(t)=$
$\displaystyle\underset{\varepsilon\to
0}{\lim}\frac{1}{\varepsilon}\Big{[}\int_{0}^{t+\varepsilon}\mathcal{N}_{\theta}(t+\varepsilon-s)[w_{1}(s)-w_{1}(t)][w(s)-w(t)]ds$
$\displaystyle-\int_{0}^{t}\mathcal{N}_{\theta}(t-s)[w_{1}(s)-w_{1}(t)][w(s)-w(t)]ds\Big{]}$
$\displaystyle+w_{1}(t)\underset{\varepsilon\to
0}{\lim}\frac{1}{\varepsilon}\Big{[}\int_{0}^{t+\varepsilon}\mathcal{N}_{\theta}(t+\varepsilon-s)w(s)ds-\int_{0}^{t}\mathcal{N}_{\theta}(t-s)w(s)ds\Big{]}$
$\displaystyle+w(t)\underset{\varepsilon\to
0}{\lim}\frac{1}{\varepsilon}\Big{[}\int_{0}^{t+\varepsilon}\mathcal{N}_{\theta}(t+\varepsilon-s)w_{1}(s)ds-\int_{0}^{t}\mathcal{N}_{\theta}(t-s)w_{1}(s)ds\Big{]}$
$\displaystyle-w_{1}(t)w(t)\underset{\varepsilon\to
0}{\lim}\frac{1}{\varepsilon}\Big{[}\int_{0}^{t+\varepsilon}\mathcal{N}_{\theta}(t+\varepsilon-s)ds-\int_{0}^{t}\mathcal{N}_{\theta}(t-s)ds\Big{]}.$

Finally, taking into account the smoothness of the functions $w$ and $w_{1}$,
we obtain the desired equality. This finishes the proof of Corollary 5.2. ∎

######  Corollary 5.3.

Let $0<\theta_{2}<\theta_{1}<\theta\leq 1$ and $w=w(t)\in\mathcal{C}([0,T])$
be a positive function, and let for any fixed $T>0$

$\displaystyle 0$
$\displaystyle<T_{1}=T_{1}(\theta_{1},\theta_{2})<\min\Big{\\{}T,\Big{(}\frac{\theta_{1}\Gamma(1+\theta_{1}-\theta_{2})}{\theta_{2}}\Big{)}^{\frac{1}{\theta_{1}-\theta_{2}}}\Big{\\}},$
$\displaystyle 0$
$\displaystyle<T_{2}=T_{2}(\theta_{1},\theta_{2},\theta)<\min\Big{\\{}T,\Big{(}\frac{\theta_{1}\Gamma(1+\theta-\theta_{2})}{\theta_{2}\Gamma(1+\theta-\theta_{1})}\Big{)}^{\frac{1}{\theta_{1}-\theta_{2}}}\Big{\\}}.$
(5.6)

Then the inequalities hold

$\displaystyle\theta_{1}I_{t}^{1}w(t)-\theta_{2}I_{t}^{1+\theta_{1}-\theta_{2}}w(t)$
$\displaystyle\geq 0\quad\text{for each}\quad t\in[0,T_{1}],$
$\displaystyle\theta_{1}I_{t}^{1+\theta-\theta_{1}}w(t)-\theta_{2}I_{t}^{1+\theta-\theta_{2}}w(t)$
$\displaystyle\geq 0\quad\text{for each}\quad t\in[0,T_{2}].$

###### Proof.

It is apparent that the second estimate is proved with the similar arguments
as the first. Thus, we restrict ourselves with the verification of the first
inequality. It is worth noting that this bound follows from the definition of
the fractional Riemann-Liouville integral and performing straightforward
calculations. Namely, appealing to (5.1), (5.3) and assumptions on
$\theta_{i}$, we easily conclude that

$\theta_{1}-\theta_{2}\omega_{1+\theta_{1}-\theta_{2}}(t)=\theta_{1}-\frac{\theta_{2}t^{\theta_{1}-\theta_{2}}}{\Gamma(1+\theta_{1}-\theta_{2})}\geq\theta_{1}-\frac{\theta_{2}T_{1}^{\theta_{1}-\theta_{2}}}{\Gamma(1+\theta_{1}-\theta_{2})}\geq
0.$

Thus, collecting these inequalities with the positivity of $w$, we end up with
the desired estimate

$\theta_{1}I_{t}^{1}w(t)-\theta_{2}I_{t}^{1+\theta_{1}-\theta_{2}}w(t)=\int_{0}^{t}[\theta_{1}-\theta_{2}\omega_{1+\theta_{1}-\theta_{2}}(\tau)]w(t-\tau)d\tau\geq
0,\quad t\in[0,T_{1}],$

which completes the proof of this corollary. ∎

Our next result deals with the fractional differentiation of a product, i.e.
$\mathbf{D}_{t}^{\theta}(w_{1}w_{2})$. We remark that the similar result under
stronger assumption on the function $w_{2}$ is established in [22, Corollary
3.1].

######  Proposition 5.4.

Let $\theta\in(0,1),$ and $w_{1}\in\mathcal{C}^{1}([0,T]),$
$w_{2}\in\mathcal{C}([0,T]).$

  1. (i)

If $\mathbf{D}_{t}^{\theta}w_{2}$ belongs either to $\mathcal{C}([0,T])$ or to
$L_{2}(0,T)$, then there is the equality

$\mathbf{D}_{t}^{\theta}(w_{1}w_{2})=w_{1}(t)\mathbf{D}_{t}^{\theta}w_{2}(t)+w_{2}(0)\mathbf{D}_{t}^{\theta}w_{1}(t)+\frac{\theta}{\Gamma(1-\theta)}\mathfrak{I}_{\theta}(t)$

with

$\mathfrak{I}_{\theta}(t)=\mathfrak{I}_{\theta}(t;w_{1},w_{2})=\int\limits_{0}^{t}\frac{[w_{1}(t)-w_{1}(s)][w_{2}(s)-w_{2}(0)]}{(t-s)^{1+\theta}}ds.$

Besides, the estimates hold

$\displaystyle\|\mathbf{D}_{t}^{\theta}(w_{1}w_{2})\|_{\mathcal{C}([0,T])}$
$\displaystyle\leq
C[\|\mathbf{D}_{t}^{\theta}w_{2}\|_{\mathcal{C}([0,T])}+\|w_{2}\|_{\mathcal{C}([0,T])}],$
$\displaystyle\|\mathfrak{I}_{\theta}(t)\|_{L_{2}(0,T)}$ $\displaystyle\leq
C\|w_{2}-w_{2}(0)\|_{L_{2}(0,T)},$
$\displaystyle\|\mathbf{D}_{t}^{\theta}(w_{1}w_{2})\|_{L_{2}(0,T)}$
$\displaystyle\leq
C[\|\mathbf{D}_{t}^{\theta}w_{2}\|_{L_{2}(0,T)}+\|w_{2}-w_{2}(0)\|_{L_{2}(0,T)}],$

where the positive constant $C$ depends only on $T$, $\theta$ and the norm of
the function $w_{1}$.

  2. (ii)

If $\partial_{t}^{\theta}w_{2}\in\mathcal{C}([0,T])$, then for any
$\theta_{1}\geq\theta$ and all $t\in[0,T]$ the equality holds

$\displaystyle I_{t}^{\theta_{1}}(w_{1}\partial_{t}^{\theta}w_{2})(t)$
$\displaystyle=I_{t}^{\theta_{1}-\theta}(w_{1}w_{2})(t)-w_{2}(0)[I_{t}^{\theta_{1}-\theta}w_{1}(t)-I_{t}^{\theta_{1}}(w_{1}\omega_{1-\theta})(t)]$
$\displaystyle-\theta I_{t}^{1+\theta_{1}-\theta}(\mathcal{W}(w_{1})w_{2})(t)$

with

$\mathcal{W}(w_{1})=\mathcal{W}(w_{1};t,\tau)=\int\limits_{0}^{1}\frac{\partial
w_{1}}{\partial z}(z)ds,\quad z=st+(1-s)(\tau),\quad 0<\tau<t<T.$

###### Proof.

As for the representation of $\mathbf{D}_{t}^{\theta}(w_{1}w_{2})$ stated in
(i) of this claim, it follows from the definition of Caputo fractional
derivative and smoothness of $w_{1}$ and $w_{2}$. Namely, we have

$\displaystyle\mathbf{D}_{t}^{\theta}(w_{1}w_{2})$
$\displaystyle=w_{1}(t)\mathbf{D}_{t}^{\theta}w_{2}+w_{2}(0)\mathbf{D}_{t}^{\theta}w_{1}+w^{\prime}_{1}(t)I_{t}^{1-\theta}(w_{2}-w_{2}(0))(t)$
$\displaystyle-\frac{1}{\Gamma(1-\theta)}\int_{0}^{t}[w_{2}(s)-w_{2}(0)]\frac{\partial}{\partial
t}\frac{[w_{1}(t)-w_{1}(s)]}{(t-s)^{\theta}}ds.$

Performing differentiation in the last integral arrives at the desired
representation.

Concerning the regularity of $\mathbf{D}_{t}^{\theta}(w_{1}w_{2})$ and
$\mathfrak{I}_{\theta}(t)$, they are simple consequence of the representation
to $\mathbf{D}_{t}^{\theta}(w_{1}w_{2})$, and of the properties of
$w_{1},w_{2},$ and the Young inequality to convolution.

In fine, point (ii) of this proposition follows from (5.2) and point (i) of
this claim, if one takes into account [17, Proposition 2.2] and semigroup
property of the fractional Riemann-Liouville integral. ∎

We now state and prove some inequalities that will be needed to obtain a
priori estimates of solutions to (1.1)-(1.4) in Section 6.5.

First, for each positive fixed $T_{3}$ and $T$, $0<T_{3}<T$, we introduce the
function

$\xi=\xi(t)\in\mathcal{C}_{0}^{\infty}(\mathbb{R}_{+}),\quad\xi\in[0,1],\quad\xi=\begin{cases}1,\quad
t\in[0,T_{3}/2],\\\ 0,\quad t\geq 3T_{3}/4.\end{cases}$ (5.7)

######  Lemma 5.5.

Let $T,T_{3}$ be arbitrary fixed, $T>T_{3}>0$, $\theta\in(0,1)$ and
$\theta_{1}\in(0,\theta]$. Then for any
$w\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\Omega}_{T_{3}})$
and $w_{1}\in\mathcal{C}^{1}([0,T])$ the equalities hold:

  1. (i)

$\|\xi w\|_{\mathcal{C}([0,T],\mathcal{C}^{1}(\bar{\Omega}))}\leq
C\|w\|_{\mathcal{C}([0,T_{3}],\mathcal{C}^{1}(\bar{\Omega}))};$

  2. (ii)

$\sum_{j=1}^{2}\big{[}\big{\|}\xi\frac{\partial^{j}w}{\partial
x^{j}}\big{\|}_{\mathcal{C}^{\alpha,\frac{2+\alpha-j}{2}\theta}(\bar{\Omega}_{T})}+\big{\|}\mathcal{K}*\xi\frac{\partial^{j}w}{\partial
x^{j}}\big{\|}_{\mathcal{C}^{\alpha,\frac{\theta\alpha}{2}}(\bar{\Omega}_{T})}\big{]}\leq
C\|w\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\Omega}_{T_{3}})};$

  3. (iii)

$\langle\xi w\rangle^{(\theta/2)}_{t,\Omega_{T}}\leq C[\langle
w\rangle^{(\theta/2)}_{t,\Omega_{T_{3}}}+\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}];$

  4. (iv)

$\|\mathfrak{I}_{\theta_{1}}(t;\xi,w_{1}w)\|_{\mathcal{C}^{\alpha,\frac{\theta\alpha}{2}}(\bar{\Omega}_{T})}\leq
C\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}([0,T_{3}],\mathcal{C}^{\alpha}(\bar{\Omega}))}+\langle
w\rangle^{(\theta/2)}_{t,\Omega_{T_{3}}}];$

  5. (v)

$\|\mathbf{D}_{t}^{\theta_{1}}(\xi
ww_{1})\|_{\mathcal{C}^{\alpha,\frac{\theta\alpha}{2}}(\bar{\Omega}_{T})}\leq
C[1+\|w_{1}\|_{\mathcal{C}^{1}([0,T])}][\|\mathbf{D}_{t}^{\theta_{1}}w\|_{\mathcal{C}^{\alpha,\frac{\theta\alpha}{2}}(\bar{\Omega}_{T_{3}})}+\|w\|_{\mathcal{C}^{\alpha,\frac{\theta\alpha}{2}}(\bar{\Omega}_{T_{3}})}+\|w_{0}\|_{\mathcal{C}^{\alpha}(\bar{\Omega})}];$

$\|\mathbf{D}_{t}^{\theta_{1}}(\xi
ww_{1})-\xi\mathbf{D}_{t}^{\theta_{1}}(ww_{1})\|_{\mathcal{C}^{\alpha,\frac{\theta\alpha}{2}}(\bar{\Omega}_{T})}\leq
C[1+\|w_{1}\|_{\mathcal{C}^{1}([0,T])}][\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\frac{\theta}{2})}+\|w\|_{\mathcal{C}([0,T_{3}],\mathcal{C}^{\alpha}(\bar{\Omega}))}+\|w_{0}\|_{\mathcal{C}^{\alpha}(\bar{\Omega})}],$
where $w_{0}=w(x,0);$

  6. (vi)

$\|\mathbf{D}_{t}^{\theta_{1}}(\xi w)\|_{L_{2}(0,T)}+\|\xi
w\|_{L_{2}((0,T),W^{2,2}(\Omega))}\leq
C[\|\mathbf{D}_{t}^{\theta_{1}}(w)\|_{L_{2}(0,T_{3})}+\|w\|_{L_{2}((0,T_{3}),W^{2,2}(\Omega))}]$.

Here the positive constant $C$ depends only on $T,\theta,\theta_{1}$ and the
norm of the function $\xi$.

###### Proof.

It is worth noting that the points (v) and (vi) are simple consequences of
Proposition 5.4 and the inequality in (iv) of this claim. Thus, to complete
the proof of this lemma, we are left to verify estimates in (i)-(iv).

The definition of the function $\xi(t)$ and the regularity of $w(x,t)$ arrive
at the inequality

$\|\xi w\|_{\mathcal{C}([0,T],\mathcal{C}^{2+\alpha}(\bar{\Omega}))}\leq
C\|w\|_{\mathcal{C}([0,T_{3}],\mathcal{C}^{2+\alpha}(\bar{\Omega}))},$

which in turn proves the point (i) of this claim. Besides, this bound tells us
that the verification of the estimate in (ii) will immediately follow from the
inequality

$\sum_{j=1}^{2}\bigg{\langle}\xi\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\rangle}_{t,\Omega_{T}}^{(\frac{2+\alpha-j}{2}\theta)}\leq
C\sum_{j=1}^{2}\bigg{[}\bigg{\langle}\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\rangle}_{t,\Omega_{T_{3}}}^{(\frac{2+\alpha-j}{2}\theta)}+\bigg{\|}\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\|}_{\mathcal{C}(\bar{\Omega}_{T_{3}})}\bigg{]}.$ (5.8)

For simplicity consideration, we first assume $0<t_{1}<t_{2}<T$, and set
$\Delta t=t_{2}-t_{1}$. Then, we discuss three options to the arrangement of
$t_{1}$ and $t_{2}$:

$\displaystyle\text{either}\quad 0<t_{1}<t_{2}\leq 3T_{3}/4,$
$\displaystyle\text{or}\quad 0<t_{1}\leq 3T_{3}/4<t_{2}\leq T,$
$\displaystyle\text{or}\quad 0<3T_{3}/4<t_{1}<t_{2}\leq T.$ (5.9)

In the first case, we have

$\bigg{|}\xi(t_{2})\frac{\partial^{j}w}{\partial
x^{j}}(x,t_{2})-\xi(t_{1})\frac{\partial^{j}w}{\partial
x^{j}}(x,t_{1})\bigg{|}\leq|\xi(t_{2})-\xi(t_{1})|\bigg{\|}\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\|}_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\xi(t_{1})\bigg{|}\frac{\partial^{j}w}{\partial
x^{j}}(x,t_{2})-\frac{\partial^{j}w}{\partial x^{j}}(x,t_{1})\bigg{|}.$

Collecting this estimate with smoothness of the functions $w$ and $\xi$, we
immediately obtain (5.8).

Coming to the second case, i.e. $t_{1}\leq\frac{3T_{3}}{4}<t_{2}$, the easily
verified inequalities

$\displaystyle\Delta t$ $\displaystyle>\frac{3T_{3}}{4}-t_{1},$
$\displaystyle\xi(t_{2})\frac{\partial^{j}w}{\partial
x^{j}}(x,t_{2})-\xi(t_{1})\frac{\partial^{j}w}{\partial x^{j}}(x,t_{1})$
$\displaystyle=\xi\bigg{(}\frac{3T_{3}}{4}\bigg{)}\frac{\partial^{j}w}{\partial
x^{j}}\bigg{(}x,\frac{3T_{3}}{4}\bigg{)}-\xi(t_{1})\frac{\partial^{j}w}{\partial
x^{j}}(x,t_{1}),$

provide the desired bound (5.8).

In the last case in (5), thanks to $\xi(t)=0$ for $t>\frac{3T_{3}}{4}$, we
have

$\xi(t_{2})\frac{\partial^{j}w}{\partial
x^{j}}(x,t_{2})-\xi(t_{1})\frac{\partial^{j}w}{\partial x^{j}}(x,t_{1})=0,$

which means that (5.8) holds.

As a result, gathering all estimates, we complete the proof of (5.8) and,
besides,

$\sum_{j=1}^{2}\bigg{\|}\xi\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\|}_{\mathcal{C}^{\alpha,\frac{2+\alpha-j}{2}\theta}(\bar{\Omega}_{T})}\leq
C\|w\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\Omega}_{T_{3}})}.$

Therefore, in order to complete the verification of the estimate in (ii), we
first take advantage of the representation

$\mathcal{K}*\xi\frac{\partial^{j}w}{\partial
x^{j}}=\begin{cases}\int_{0}^{t}\mathcal{K}(t-s)\xi(s)\frac{\partial^{j}w}{\partial
x^{j}}(x,s)ds,\qquad 0<t<\frac{3T_{3}}{4},\\\ \,\\\
\int_{0}^{\frac{3T_{3}}{4}}\mathcal{K}(t-s)\xi(s)\frac{\partial^{j}w}{\partial
x^{j}}(x,s)ds,\qquad t\geq\frac{3T_{3}}{4},\end{cases}$

and [19, Lemma 4.1], then, performing standard calculations, we conclude that

$\bigg{\|}\mathcal{K}*\xi\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\|}_{\mathcal{C}^{\alpha,\frac{\alpha\theta}{2}}(\bar{\Omega}_{T})}\leq
C\|\mathcal{K}\|_{L_{1}(0,T)}\bigg{\|}\frac{\partial^{j}w}{\partial
x^{j}}\bigg{\|}_{\mathcal{C}^{\alpha,\frac{\alpha\theta}{2}}(\bar{\Omega}_{T_{3}})}.$

Thus, the proof of the estimate in (ii) is finished.

Concerning the inequalities in (iii), they are obtained with the arguments
leading to (5.8).

At this point we examine the bound in (iv). Here we restrict ourselves with
the consideration of the case $\theta_{1}=\theta$. Another case is verified
with the similar manner.

In light of the definition of $\xi(t)$ (see (5.7)) and
$\mathfrak{I}_{\theta}(t)$, we deduce

$\displaystyle\mathfrak{I}_{\theta}(t)$
$\displaystyle=\begin{cases}0,\qquad\qquad t\leq T_{3}/2,\\\ \,\\\
\int_{0}^{t}\frac{[\xi(t)-\xi(\tau)][w(x,\tau)w_{1}(\tau)-w(x,0)w_{1}(0)]d\tau}{(t-\tau)^{1+\theta}},\quad
t\in(T_{3}/2,3T_{3}/4),\\\ \,\\\
\int_{0}^{3T_{3}/4}\frac{[\xi(t)-\xi(\tau)][w(x,\tau)w_{1}(\tau)-w(x,0)w_{1}(0)]d\tau}{(t-\tau)^{1+\theta}},\quad
t\geq 3T_{3}/4,\end{cases}$ $\displaystyle=\begin{cases}0,\qquad\qquad t\leq
T_{3}/2,\\\ \,\\\
\int_{0}^{3T_{3}/4}\frac{\mathcal{W}(\xi;t,\tau)[w(x,\tau)-w(x,0)]w_{1}(\tau)d\tau}{(t-\tau)^{\theta}}+\int_{0}^{3T_{3}/4}\frac{\mathcal{W}(\xi;t,\tau)[w_{1}(\tau)-w_{1}(0)]w(x,0)d\tau}{(t-\tau)^{\theta}},\quad
t\in(T_{3}/2,3T_{3}/4),\\\ \,\\\
\int_{0}^{t}\frac{\mathcal{W}(\xi;t,\tau)[w(x,\tau)-w(x,0)]w_{1}(\tau)d\tau}{(t-\tau)^{\theta}}+\int_{0}^{t}\frac{\mathcal{W}(\xi;t,\tau)[w_{1}(\tau)-w_{1}(0)]w(x,0)d\tau}{(t-\tau)^{\theta}},\quad
t\geq 3T_{3}/4,\end{cases}$ (5.10)

where the function $\mathcal{W}(\cdot)$ is defined in point (ii) of
Proposition 5.4.

Taking into account these equalities and performing standard technical
calculations, we end up with the estimate

$\|\mathfrak{I}_{\theta}(t)\|_{\mathcal{C}([0,T],\mathcal{C}^{\alpha}(\bar{\Omega}))}\leq
C\|\xi\|_{\mathcal{C}^{1}([0,T])}\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}([0,T_{3}],\mathcal{C}^{\alpha}(\bar{\Omega}))}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}].$ (5.11)

As for the Hölder regularity of $\mathfrak{I}_{\theta}(t)$ with respect to
time, we first assume that

$\Delta t=t_{2}-t_{1}<T_{3}/8,$ (5.12)

otherwise the bound of
$\langle\mathfrak{I}_{\theta}(t)\rangle_{t,\Omega_{T}}^{(\alpha\theta/2)}$
follows from (5.11).

Next we consider again (5) to the location of $t_{1},t_{2}$. If
$t_{1},t_{2}\in[0,3T_{3}/4]$, then using (5), we can write

$\mathfrak{I}_{\theta}(t_{2})-\mathfrak{I}_{\theta}(t_{1})\equiv\sum_{i=1}^{3}J_{i},$

where we set

$\displaystyle J_{1}$
$\displaystyle=\int_{0}^{t_{1}}\frac{\mathcal{W}(\xi;t_{2},t_{2}-\tau)[w(x,t_{2}-\tau)w_{1}(t_{2}-\tau)-w(x,t_{1}-\tau)w_{1}(t_{1}-\tau)]d\tau}{\tau^{\theta}},$
$\displaystyle J_{2}$
$\displaystyle=\int_{0}^{t_{1}}\frac{[\mathcal{W}(\xi;t_{2},t_{2}-\tau)-\mathcal{W}(\xi;t_{1},t_{1}-\tau)][w(x,t_{1}-\tau)w_{1}(t_{1}-\tau)-w(x,0)w_{1}(0)]d\tau}{\tau^{\theta}},$
$\displaystyle J_{3}$
$\displaystyle=\int_{t_{1}}^{t_{2}}\frac{\mathcal{W}(\xi;t_{2},t_{2}-\tau)[w(x,t_{2}-\tau)w_{1}(t_{2}-\tau)-w(x,0)w_{1}(0)]d\tau}{\tau^{\theta}}.$

According to the properties of $\xi,w$ and $w_{1}$, we immediately deduce

$|J_{1}|+|J_{2}|\leq C\Delta
t^{\theta/2}\|\xi\|_{\mathcal{C}^{2}([0,T])}\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}].$

As for $J_{3}$, we have

$\displaystyle|J_{3}|$ $\displaystyle\leq
C\|\xi\|_{\mathcal{C}^{1}([0,T])}\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}]\int_{t_{1}}^{t_{2}}\tau^{-\theta}(t_{2}-\tau)^{\theta/2}[1+(t_{2}-\tau)^{1-\theta/2}]d\tau$
$\displaystyle\leq C\Delta
t^{\theta/2}\|\xi\|_{\mathcal{C}^{2}([0,T])}\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}].$

It is worth noting that the positive constant $C$ depends only on $T$ and
$\theta$. Collecting these estimates arrives at the bound

$\langle\mathfrak{I}_{\theta}(t)\rangle_{t,\Omega_{T}}^{(\theta/2)}\leq
C\|\xi\|_{\mathcal{C}^{2}([0,T])}\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}].$ (5.13)

Concerning the case of $t_{1}\leq\frac{3T_{3}}{4}<t_{2}<T$, assumption (5.12)
tells us that

$t_{2}<\frac{7T_{3}}{8}<T_{3}.$

Hence, recasting the arguments leading to (5.13), we obtain the desired
estimate in the case of the second option in (5). Finally, analyzing the case
$\frac{3T_{3}}{4}\leq t_{1}<t_{2}\leq T$, two possibilities occur:

  1. (i)

either $t_{1}-\frac{3T_{3}}{4}\leq\frac{T_{3}}{8};$

  2. (ii)

or $t_{1}-\frac{3T_{3}}{4}>\frac{T_{3}}{8}.$

It is apparent that the option (i) is studied with the similar arguments
leading to (5.13). As for the case (ii), exploiting (5), we have

$\mathfrak{I}_{\theta}(t_{2})-\mathfrak{I}_{\theta}(t_{1})\equiv i_{1}+i_{2},$

where

$\displaystyle i_{1}$
$\displaystyle=\int_{0}^{3T_{3}/4}[w(x,\tau)w_{1}(\tau)-w(x,0)w_{1}(0)]\frac{\mathcal{W}(\xi;t_{2},\tau)-\mathcal{W}(\xi;t_{1},\tau)}{(t_{2}-\tau)^{\theta}}d\tau,$
$\displaystyle i_{2}$
$\displaystyle=\int_{0}^{3T_{3}/4}[w(x,\tau)w_{1}(\tau)-w(x,0)w_{1}(0)]\mathcal{W}(\xi;t_{2},\tau)[(t_{2}-\tau)^{-\theta}-(t_{1}-\tau)^{-\theta}]d\tau.$

According to the regularity of $\xi(t)$, we arrive at

$|i_{1}|\leq C\Delta
t\|w_{1}\|_{\mathcal{C}^{1}([0,T])}\|\xi\|_{\mathcal{C}^{2}(\bar{\Omega}_{T})}[\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}]\int_{0}^{3T_{3}/4}[\tau+\tau^{\theta/2}](t_{2}-\tau)^{-\theta}d\tau.$

To estimate the term $i_{2}$, we use the mean value theorem and the fact that
$t_{1}-\frac{3T_{3}}{4}>\frac{T_{3}}{8}$. Thus, we achieve

$|i_{2}|\leq C\frac{\Delta
t}{T_{3}^{2\theta}}\|\xi\|_{\mathcal{C}^{1}([0,T])}\|w_{1}\|_{\mathcal{C}^{1}([0,T])}[\|w\|_{\mathcal{C}(\bar{\Omega}_{T_{3}})}+\langle
w\rangle_{t,\Omega_{T_{3}}}^{(\theta/2)}].$

As a result, gathering this inequality with the estimate of $i_{1}$, we end up
with bound (5.13) in the third case in (5). Finally, (5.11) and (5.13)
completes the proof of point (iv) in this lemma. ∎

Next, we state and prove inequalities which are generalized the bounds in [20,
Lemma 4.2] and will be used later in this art.

######  Lemma 5.6.

Let
$w_{2}=w_{2}(x,t)\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\Omega}_{T}),$
$\theta\in(0,1)$, and let $w_{1}=w_{1}(t)\in\mathcal{C}^{1}([0,T])$ be a
positive function. We assume that

$\text{either}\quad
w_{2}|_{\partial\Omega_{T}}=0\quad\text{or}\quad\frac{\partial w_{2}}{\partial
x}|_{\partial\Omega_{T}}=0.$

Then for any $0<\theta_{2}<\theta_{1}\leq\theta$ and any integer even $p\geq
2$ the inequalities hold:

  1. (i)

$\displaystyle-
pI_{t}^{\theta}\bigg{(}\int_{\Omega}\partial_{t}^{\theta_{1}}(w_{1}w_{2})\frac{\partial}{\partial
x}\bigg{(}\frac{\partial w_{2}}{\partial x_{2}}\bigg{)}^{p-1}dx\bigg{)}(t)$
$\displaystyle\geq(p-1)I_{t}^{\theta}\Big{(}w_{1}\omega_{1-\theta_{1}}\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial
x_{2}}\bigg{)}^{p}dx\Big{)}(t)+I_{t}^{\theta-\theta_{1}}\bigg{(}w_{1}\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial x_{2}}\bigg{)}^{p}dx\bigg{)}(t)$ $\displaystyle-
w_{1}^{p}(0)\int_{\Omega}\bigg{(}\frac{\partial w_{2}}{\partial
x_{2}}(x,0)\bigg{)}^{p}dx[I_{t}^{\theta-\theta_{1}}(w_{1}^{1-p})(t)-I_{t}^{\theta}(w_{1}^{1-p}\omega_{1-\theta_{1}})(t)]$
$\displaystyle-\theta_{1}I_{t}^{1-\theta_{1}+\theta}\bigg{(}w_{1}^{p}\mathcal{W}(w_{1}^{1-p})\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial x_{2}}\bigg{)}^{p}dx\bigg{)}(t)\quad\text{for}\quad\forall
t\in[0,T],$

  2. (ii)

$\displaystyle-pI_{t}^{\theta}\bigg{(}\int_{\Omega}\frac{\partial}{\partial
t}(\mathcal{N}_{\theta}*w_{1}w_{2})\frac{\partial}{\partial
x}\bigg{(}\frac{\partial w_{2}}{\partial x_{2}}\bigg{)}^{p-1}dx\bigg{)}(t)$
$\displaystyle\geq(p-1)I_{t}^{\theta}\Big{(}w_{1}\mathcal{N}_{\theta}\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial
x_{2}}\bigg{)}^{p}dx\Big{)}(t)+I_{t}^{\theta-\theta_{1}}\bigg{(}w_{1}\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial x_{2}}\bigg{)}^{p}dx\bigg{)}(t)$ $\displaystyle-
w_{1}^{p}(0)\int_{\Omega}\bigg{(}\frac{\partial w_{2}}{\partial
x_{2}}(x,0)\bigg{)}^{p}dxI_{t}^{\theta-\theta_{1}}\bigg{(}w_{1}^{1-p}+I_{t}^{\theta_{1}}(\omega_{1-\theta_{1}}w_{1}^{1-p})\bigg{)}(t)$
$\displaystyle-
I_{t}^{\theta-\theta_{2}}\bigg{(}w_{1}\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial
x_{2}}\bigg{)}^{p}dx\bigg{)}(t)-\theta_{1}I_{t}^{1-\theta_{1}+\theta}\bigg{(}w_{1}^{p}\mathcal{W}(w_{1}^{1-p})\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial x_{2}}\bigg{)}^{p}dx\bigg{)}(t)$
$\displaystyle+\theta_{2}I_{t}^{1-\theta_{2}+\theta}\bigg{(}w_{1}^{p}\mathcal{W}(w_{1}^{1-p})\int_{\Omega}\bigg{(}\frac{\partial
w_{2}}{\partial x_{2}}\bigg{)}^{p}dx\bigg{)}(t)\quad\text{for}\quad\forall
t\in[0,T^{*}],$

where $\mathcal{N}_{\theta}$ is given with (5.3) and $\mathcal{W}(\cdot)$ is
defined in (ii) of Proposition 5.4.

###### Proof.

First we consider the case of homogeneous Dirichlet boundary condition. We
preliminarily observe that the estimate at the point (ii) is the same as the
one for the bound at (i) (where instead of Proposition 5.1, one should use
Corollary 5.2) of this claim. For these reason, we are left to tackle the
inequality in (i).

To this end, similar to the proof of Lemma 4.2 [20], we first construct the
mollification of the function $w_{2}$. For any positive $\mathfrak{d}<|b-a|$,
we introduce a cut function $\eta\in\mathcal{C}_{0}^{\infty}(\mathbb{R})$
taking values in $[0,1]$:

$\begin{cases}\eta=1,\quad\text{if}\quad
x\in(a-\frac{\mathfrak{d}}{4},b+\frac{\mathfrak{d}}{4}),\\\
\eta=0,\quad\text{if}\quad
x\in\mathbb{R}\backslash(a-\frac{\mathfrak{d}}{2},b+\frac{\mathfrak{d}}{2}),\end{cases}$

and then define the even extension $W_{\mathfrak{d}}(x,t)$ of the function
$w_{2}(x,t)$ in the segment $(a-\mathfrak{d},b+\mathfrak{d})$ as

$W_{\mathfrak{d}}(x,t)=\begin{cases}-w_{2}(2a-x,t),\quad\text{if}\quad
x\in(a-\mathfrak{d},a),\\\ w_{2}(x,t),\qquad\quad\quad\text{if}\quad
x\in[a,b],\\\ -w_{2}(2b-x,t),\quad\text{if}\quad
x\in(b,b+\mathfrak{d}).\end{cases}$

Then we build the zero extension of $W_{\mathfrak{d}}$ outside the segment
$(a-\mathfrak{d},b+\mathfrak{d})$ as

$W_{\eta}=W_{\eta}(x,t)=\eta(x)W_{\mathfrak{d}}(x,t).$

It is easily verify that
$W_{\eta}\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\mathbb{R}}_{T}),$
and

$\|W_{\eta}\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\mathbb{R}}_{T})}\leq
C\|w_{2}\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\theta}(\bar{\Omega}_{T})}.$

Setting the mollifier $J_{\varepsilon}(x)$ satisfying the properties:

$J_{\varepsilon}(x)\in\mathcal{C}_{0}^{\infty}(\mathbb{R}),\quad
J_{\varepsilon}(x)=0\quad\text{if}\quad|x|\geq\varepsilon,\quad\int_{-\infty}^{+\infty}J_{\varepsilon}(x)dx=1,$

we define the molification of the function $W_{\eta}$ as

$W_{\eta,\varepsilon}=\int_{\mathbb{R}}J_{\varepsilon}(x-y)W_{\eta}(y,t)dy.$

Clearly,

$W_{\eta,\varepsilon}\in\mathcal{C}([0,T],\mathcal{C}^{\infty}(\mathbb{R})),\quad\mathbf{D}_{t}^{\theta_{i}}W_{\eta,\varepsilon}\in\mathcal{C}([0,T],\mathcal{C}^{\infty}(\mathbb{R})),\,0<\theta_{i}\leq\theta,\,i=1,2,$

and

$W_{\eta,\varepsilon}=0\qquad\text{on}\quad\partial\Omega_{T}.$

The last equality tells us that

$\partial_{t}^{\theta_{2}}W_{\eta,\varepsilon}=\partial_{t}^{\theta_{1}}W_{\eta,\varepsilon}=\partial_{t}^{\theta}W_{\eta,\varepsilon}=0\qquad\text{for}\quad(x,t)\in\partial\Omega_{T}.$

Following standard approximation arguments (see e.g. [1, Chapter 1]), we have
the uniform convergences on $\bar{\Omega}_{T}$ as $\varepsilon\to 0$

$\displaystyle\frac{\partial^{i}W_{\eta,\varepsilon}}{\partial x^{i}}$
$\displaystyle\to\frac{\partial^{i}w_{2}}{\partial x^{i}},\quad
i=0,1,2,\quad\text{and}$
$\displaystyle\mathbf{D}_{t}^{\theta_{i}}W_{\eta,\varepsilon}$
$\displaystyle\to\mathbf{D}_{t}^{\theta_{i}}w_{2},\quad
0<\theta_{2}<\theta_{1}\leq\theta<1,$ $\displaystyle\frac{\partial}{\partial
x}\bigg{(}\frac{\partial W_{\eta,\varepsilon}}{\partial x}\bigg{)}^{p-1}$
$\displaystyle\to\frac{\partial}{\partial x}\bigg{(}\frac{\partial
w_{2}}{\partial x}\bigg{)}^{p-1},\quad p\geq 2.$

Moreover, exploiting [22, Corollary 3.1] and standard technical calculations,
we arrive at

$\displaystyle\partial_{t}^{\theta_{i}}(w_{1}W_{\eta,\varepsilon})$
$\displaystyle\to\partial_{t}^{\theta_{i}}(w_{1}w_{2}),\quad i=1,2,$
$\displaystyle\mathbf{D}_{t}^{\theta_{i}}(w_{1}W_{\eta,\varepsilon})$
$\displaystyle\to\mathbf{D}_{t}^{\theta_{i}}(w_{1}w_{2}).$

Now, we begin to prove the first inequality of this lemma for
$W_{\eta,\varepsilon}$. Namely, integration by parts together with
Propositions 5.1 and 5.4 yield

$\displaystyle-
pI_{t}^{\theta}\bigg{(}\int_{\Omega}\partial_{t}^{\theta_{1}}(w_{1}W_{\eta,\varepsilon})\frac{\partial}{\partial
x}\bigg{(}\frac{\partial W_{\eta,\varepsilon}}{\partial
x}\bigg{)}^{p-1}dx\bigg{)}(t)=pI_{t}^{\theta}\bigg{(}w_{1}^{1-p}\int_{\Omega}\partial_{t}^{\theta_{1}}\frac{\partial}{\partial
x}(w_{1}W_{\eta,\varepsilon})\bigg{(}\frac{\partial
w_{1}W_{\eta,\varepsilon}}{\partial x}\bigg{)}^{p-1}dx\bigg{)}(t)$
$\displaystyle\geq
I_{t}^{\theta}\bigg{(}w_{1}^{1-p}\int_{\Omega}\partial_{t}^{\theta_{1}}\bigg{(}\frac{\partial}{\partial
x}(w_{1}W_{\eta,\varepsilon})\bigg{)}^{p}dx\bigg{)}(t)+(p-1)I_{t}^{\theta}\bigg{(}w_{1}\omega_{1-\theta_{1}}\int_{\Omega}\bigg{(}\frac{\partial
W_{\eta,\varepsilon}}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)$
$\displaystyle=(p-1)I_{t}^{\theta}\bigg{(}w_{1}\omega_{1-\theta_{1}}\int_{\Omega}\bigg{(}\frac{\partial
W_{\eta,\varepsilon}}{\partial
x}\bigg{)}^{p}dx\bigg{)}(t)+I_{t}^{\theta-\theta_{1}}\bigg{(}w_{1}\int_{\Omega}\bigg{(}\frac{\partial
W_{\eta,\varepsilon}}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)$ $\displaystyle-
w_{1}^{p}(0)\int_{\Omega}\bigg{(}\frac{\partial W_{\eta,\varepsilon}}{\partial
x}(x,0)\bigg{)}^{p}dxI_{t}^{\theta-\theta_{1}}(w_{1}^{1-p})(t)-\theta_{1}I_{t}^{1-\theta_{1}+\theta}\bigg{(}\mathcal{W}(w_{1}^{1-p})w_{1}^{p}\int_{\Omega}\bigg{(}\frac{\partial
W_{\eta,\varepsilon}}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)$
$\displaystyle+w_{1}^{p}(0)\int_{\Omega}\bigg{(}\frac{\partial
W_{\eta,\varepsilon}}{\partial
x}(x,0)\bigg{)}^{p}dxI_{t}^{\theta}(w_{1}^{1-p}\omega_{1-\theta_{1}})(t)\equiv\mathcal{I}\bigg{(}t,w_{1},\frac{\partial
W_{\eta,\varepsilon}}{\partial x}\bigg{)}.$

Finally, taking into account of the positivity of $w_{1}$ and even $p$ to
control the last term in the right-hand side of the last inequality, we arrive
at the desired bound to the function $W_{\eta,\varepsilon}$. The conclusion
can be easily drawn from the uniform convergences

$I_{t}^{\theta}\bigg{(}\int_{\Omega}\partial_{t}^{\theta_{1}}(w_{1}W_{\eta,\varepsilon})\frac{\partial}{\partial
x}\bigg{(}\frac{\partial W_{\eta,\varepsilon}}{\partial
x}\bigg{)}^{p-1}dx\bigg{)}(t)\to
I_{t}^{\theta}\bigg{(}\int_{\Omega}\partial_{t}^{\theta_{1}}(w_{1}w_{2})\frac{\partial}{\partial
x}\bigg{(}\frac{\partial w_{2}}{\partial x}\bigg{)}^{p-1}dx\bigg{)}(t),$

and

$\mathcal{I}\bigg{(}t,w_{1},\frac{\partial W_{\eta,\varepsilon}}{\partial
x}\bigg{)}\to\mathcal{I}\bigg{(}t,w_{1},\frac{\partial w_{2}}{\partial
x}\bigg{)}.$

The case of $\frac{\partial u}{\partial x}=0$ on $\partial\Omega_{T}$ is
similar and left to the reader. This completes the proof of this lemma. ∎

For the reader’s convenience, we now recall the global classical solvability
of the linear version of problem (1.1)-(1.4). This result, stated as a lemma,
is proved in our previous work [37, Theorem 4.1, Remark 4.4], and will be a
key point in our analysis in Sections 6-7.

######  Lemma 5.7.

Let $\partial\Omega\in\mathcal{C}^{2+\alpha}$, $f(u)\equiv 0$, and let
$\nu\in(0,1],$ while $\nu_{i},\mu_{j}$ meet requirement h1. For any fixed
$T>0$, under assumptions h2-h4, and h5(i), h7, then the conclusions of Theorem
4.1 hold. Besides, this solution fulfills the estimate

$\displaystyle\|u\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}$
$\displaystyle\leq
C[\|u_{0}\|_{\mathcal{C}^{2+\alpha}(\bar{\Omega})}+\|g\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}+\|\psi\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\partial\Omega_{T})}]$

within the DBC (1.2), or

$\displaystyle\|u\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}$
$\displaystyle\leq
C[\|u_{0}\|_{\mathcal{C}^{2+\alpha}(\bar{\Omega})}+\|g\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}+\|\psi_{1}\|_{\mathcal{C}^{1+\alpha,\frac{1+\alpha}{2}\nu}(\partial\Omega_{T})}]$

within NBC (1.3). The generic constant $C$ is independent of the right-hand
sides of (1.1)-(1.4).

We conclude this preliminary section with the inequalities for every $v\in
W^{1,2}(\Omega)$ that will play a key role in the proof of Theorem 4.1 in
Subsection 6.2:

$\|v\|_{L_{\infty}(\Omega)}\leq
C\|v\|^{1/3}_{W^{1,2}(\Omega)}\|v\|^{2/3}_{L_{1}(\Omega)},$

and

$\|v\|_{L_{\infty}(\Omega)}\leq\varepsilon\|v\|_{W^{1,2}(\Omega)}+C\varepsilon^{-1/2}\|v\|_{L_{1}(\Omega)}.$
(5.14)

It is worth noting that the first inequality is the bound (2.19) in [26],
while estimate (5.14) is a simple consequence of the first and Young
inequality.

##  6\. A Priori Estimates

In this section, we provide a priori estimates for the classical solutions to
the following family of problems for $\lambda\in[0,1],$

$\begin{cases}\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+\lambda
f(u)=g(x,t)\quad\text{in}\quad\Omega_{T},\\\
u(x,0)=u_{0}(x)\qquad\text{in}\qquad\bar{\Omega},\\\
u(x,t)=0,\qquad\text{on}\qquad\partial\Omega_{T}.\end{cases}$ (6.1)

These estimates are the crucial point in the proof of Theorems 4.1 and 4.4.

######  Lemma 6.1.

Let the assumptions of Theorem 4.1 hold with $\psi=\psi_{1}\equiv 0$.

If $u\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})$ is a
classical solution to (6.1). Then for any $\lambda\in[0,1],$ the following
inequalities are fulfilled

$\|u\|_{\mathcal{C}([0,T],\mathcal{C}^{1}(\bar{\Omega}))}+\langle
u\rangle_{t,\Omega_{T}}^{(\nu/2)}\leq
C(1+\|u_{0}\|_{W^{2,2}(\Omega)}+[\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}]^{\frac{1}{2}}),$
(6.2)
$\|u\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}\leq
C[1+\|u_{0}\|_{\mathcal{C}^{2+\alpha}(\bar{\Omega})}+\|g\|_{\mathcal{C}^{\alpha,\frac{\alpha}{2}\nu}(\bar{\Omega}_{T})}],$
(6.3)
$\displaystyle\|u\|_{L_{2}((0,T),W^{2,2}(\Omega))}+\|\mathbf{D}_{t}^{\nu}u\|_{L_{2}(\Omega_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T})}$
$\displaystyle\leq
C[1+\|u_{0}\|_{W^{2,2}(\Omega)}+[\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}]^{\frac{1}{2}}].$
(6.4)

Here the positive constant $C$ is independent of $\lambda$ and depends only on
$\nu,\nu_{i},\mu_{j}$, $T,L,L_{i}$, $r$ and the corresponding norms of the
coefficients $a_{i},b_{i},$ $\varrho_{0},$ $\varrho_{j}$, $\gamma_{j}$ and of
the kernel $\mathcal{K}$.

######  Remark 6.2.

Actually, our arguments in Section 6.1 tells us that the term
$\|u\|_{\mathcal{C}(\bar{\Omega}_{T})}$ is evaluated via minor norms

$\|u\|_{\mathcal{C}(\bar{\Omega}_{T})}\leq
C[1+\|u_{0}\|_{W^{1,2}(\Omega)}+[\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}]^{\frac{1}{2}}].$

First, we notice that estimate (6.3) is verified with the standard Schauder
approach and by means of Lemma 5.7 and bound (6.2). Thus, to prove Lemma 6.1,
we are left to produce inequalities (6.2) and (6.1). We preliminarily observe
that verification of these estimates in the case of absence of
$\mathbf{D}_{t}^{\mu_{i}}(\gamma_{i}u),$ $i=1,2,...,N,$ (i.e. $N=0$) is the
simpler and repeat the main steps (with minor changes) in arguments related
with the analysis of the case $N\geq 1$. Hence, here we focus on the case of
the presence of at least one fractional derivative
$\mathbf{D}_{t}^{\mu_{i}}(\gamma_{i}u)$ in the operator $\mathbf{D}_{t}u$. To
this end, appealing to assumption h3, we rewrite $\mathbf{D}_{t}u$ in the more
suitable form to our analysis

$\displaystyle\mathbf{D}_{t}u$
$\displaystyle=\,_{1}\mathbf{D}_{t}u+\,_{2}\mathbf{D}_{t}u,\quad\,_{1}\mathbf{D}_{t}u=\mathbf{D}_{t}^{\nu}(\varrho
u)+\sum_{i=1}^{M}\mathbf{D}_{t}^{\nu_{i}}(\varrho_{i}u),$
$\,{}_{2}\mathbf{D}_{t}u$
$\displaystyle=\sum_{j=1}^{N}[\mathbf{D}_{t}^{\nu}(\gamma_{j}u)-\mathbf{D}_{t}^{\mu_{j}}(\gamma_{j}u)]=\sum_{j=1}^{N}\frac{\partial}{\partial
t}\bigg{(}\mathcal{N}(t;\nu,\mu_{j})*(\gamma_{j}u-\gamma_{j}(0)u_{0})\bigg{)}$

(see (5.3) for the definition of the kernel $\mathcal{N}(t;\nu,\mu_{j})$).

Thanks to the positivity of the kernel $\mathcal{N}(t;\nu,\mu_{j})$ for
$t\in[0,T^{*}]$ (see h1), we first prove estimates (6.2) and (6.1) for
$t\in[0,T_{0}]$ where

$T_{0}<\min\bigg{\\{}T^{*},\underset{j\in\\{1,2,...,N\\}}{\min}\bigg{(}\frac{\nu\Gamma(1+\nu-\mu_{j})}{\mu_{j}}\bigg{)}^{\frac{1}{\nu-\mu_{j}}}\bigg{\\}}.$
(6.5)

After that, if $T\geq T_{0}$, we discuss the extension of these bounds to the
interval $(T_{0},T]$. It is worth noting that this step is absent in the case
of $N=0$, due to the proof of estimates (6.2) and (6.1) is carried out
immediately on entire time interval.

To verify (6.2) and (6.1) for $t\in[0,T_{0}]$, we will follow the strategy
containing fourth main steps. In the first, we evaluate the function $u(x,t)$
in the class $\mathcal{C}([0,T_{0}],W^{1,2}(\Omega))\cap
L_{2}((0,T_{0}),W^{2,2}(\Omega))$. Then, Sobolev embedding theorem (see, e.g.,
[1, Subsection 5.4]) allows us to readily obtain the bound of
$\|u\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}$ exploiting only the estimate on
$\|u\|_{\mathcal{C}([0,T_{0}],W^{1,2}(\Omega))}$. On the second stage, we
evaluate the term $\|\frac{\partial u}{\partial
x}\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}$ via integral iteration technique
adapted to the case of multi-term fractional derivatives. After that, to
complete the proof of (6.2), appealing to estimate of
$\|u\|_{\mathcal{C}([0,T_{0}],\mathcal{C}^{1}(\bar{\Omega}))}$, we arrive at
the bound of the Hölder seminorm of $u$ with respect to time. Finally, taking
into account (6.2), we achieve estimate (6.1) via evaluation of
$\|\mathbf{D}_{t}^{\nu}u\|_{L_{2}(\Omega_{T_{0}})}$,
$\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T_{0}})}$,
$\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T_{0}})}$.

Note that assumption h3 provides the existence of a constant $C_{0}$ such that

$\sum_{i=0}^{2}[\underset{\bar{\Omega}_{T}}{\sup}|a_{i}(x,t)|+\underset{\bar{\Omega}_{T}}{\sup}|b_{i}(x,t)|]+\underset{\bar{\Omega}_{T}}{\sup}\bigg{|}\frac{\partial
a_{2}}{\partial
x}(x,t)\bigg{|}+\underset{\bar{\Omega}_{T}}{\sup}\bigg{|}\frac{\partial
b_{2}}{\partial x}(x,t)\bigg{|}\leq C_{0}.$ (6.6)

###  6.1. Estimate of $\|u\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}$

We begin to evaluate the norm of $u$ in the space
$\mathcal{C}([0,T_{0}],L_{2}(\Omega))$. To this end, we multiply the equation
in (6.1) by $u(x,\tau)$, and then we integrate over $\Omega$ and compute the
fractional integral $I_{t}^{\nu}$. Thus, we have

$\sum_{j=1}^{5}\mathcal{R}_{j}(t)=0,$ (6.7)

where we put

$\displaystyle\mathcal{R}_{1}(t)$
$\displaystyle=I_{t}^{\nu}\bigg{(}\int_{\Omega}\,{}_{1}\mathbf{D}_{\tau}uudx\bigg{)}(t),\quad\mathcal{R}_{2}(t)=I_{t}^{\nu}\bigg{(}\int_{\Omega}\,{}_{2}\mathbf{D}_{\tau}uudx\bigg{)}(t),\quad\mathcal{R}_{3}(t)=-I_{t}^{\nu}\bigg{(}\int_{\Omega}\mathcal{L}_{1}uudx\bigg{)}(t),$
$\displaystyle\mathcal{R}_{4}(t)$
$\displaystyle=-I_{t}^{\nu}\bigg{(}\int_{\Omega}(\mathcal{K}*\mathcal{L}_{2}u)udx\bigg{)}(t),\quad\mathcal{R}_{5}(t)=I_{t}^{\nu}\bigg{(}\int_{\Omega}[\lambda
f(u)-g]udx\bigg{)}(t).$

At this point, we estimate each term $\mathcal{R}_{j}$, separately.

$\bullet$ By Propositions 5.1 and 5.4 and assumptions h2 and h3,

$\displaystyle\mathcal{R}_{1}(t)$
$\displaystyle\geq\frac{1}{2}I_{t}^{\nu}\Big{(}\varrho^{-1}\int_{\Omega}\partial_{t}^{\nu}(\varrho
u)^{2}dx\Big{)}(t)+\frac{1}{2}\sum_{i=1}^{M}I_{t}^{\nu}\Big{(}\varrho_{i}^{-1}\int_{\Omega}\partial_{t}^{\nu_{i}}(\varrho_{i}u)^{2}dx\Big{)}(t)$
$\displaystyle-\frac{1}{2}\int_{\Omega}u_{0}^{2}dx\Big{[}\varrho^{2}(0)I_{t}^{\nu}(\varrho^{-1}\omega_{1-\nu})+\sum_{i=1}^{M}\varrho_{i}^{2}(0)I_{t}^{\nu}(\varrho_{i}^{-1}\omega_{1-\nu_{i}})\Big{]}$
$\displaystyle\geq\frac{\delta_{1}}{2}\int_{\Omega}u^{2}(x,t)dx-\int_{\Omega}u_{0}^{2}dx\Big{[}2\varrho(0)+\sum_{i=1}^{M}\varrho_{i}(0)\omega_{1+\nu-\nu_{i}}(t)\Big{]}.$

$\bullet$ As for the term $\mathcal{R}_{2}(t)$, we recast the arguments
leading to the bound of $\mathcal{R}_{1}(t)$. Thus, exploiting Corollary 5.2,
Proposition 5.4, and conditions h2, h3, we immediately arrive at

$\displaystyle\mathcal{R}_{2}(t)$
$\displaystyle\geq\frac{1}{2}\sum_{j=1}^{N}I_{t}^{\nu}\bigg{(}\gamma^{-1}_{j}\int_{\Omega}\frac{\partial}{\partial\tau}(\mathcal{N}(\tau;\nu,\mu_{j})*(\gamma_{j}u)^{2})dx\bigg{)}(t)-\frac{1}{2}\int_{\Omega}u_{0}^{2}dx\sum_{j=1}^{N}I_{t}^{\nu}\bigg{(}\mathcal{N}(\tau;\nu,\mu_{j})\frac{\gamma^{2}_{j}(0)}{\gamma_{j}(\tau)}\bigg{)}(t)$
$\displaystyle\geq\frac{1}{2}\sum_{j=1}^{N}\gamma_{j}(t)\int_{\Omega}u^{2}(x,t)dx-\frac{1}{2}\sum_{j=1}^{N}I_{t}^{\nu-\mu_{j}}\bigg{(}\gamma_{j}\int_{\Omega}u^{2}dx\bigg{)}(t)-\frac{1}{2}\sum_{j=1}^{N}\gamma_{j}^{2}(0)\gamma_{j}^{-1}(t)\int_{\Omega}u_{0}^{2}dx$
$\displaystyle+\frac{1}{2}\sum_{j=1}^{N}\bigg{[}I_{t}^{1}\bigg{(}-\nu\mathcal{W}(\gamma_{j}^{-1})\int_{\Omega}(\gamma_{j}u)^{2}dx\bigg{)}(t)-I_{t}^{1+\nu-\mu_{j}}\bigg{(}-\mu_{j}\mathcal{W}(\gamma_{j}^{-1})\int_{\Omega}(\gamma_{j}u)^{2}dx\bigg{)}(t)\bigg{]}.$

After that, Corollary 5.3 with $T_{1}=T_{0}$ (see restriction (6.5)) and
assumption h3 tell us about the positivity of the last sum in the right-hand
side of this inequality. Thus, we have

$\mathcal{R}_{2}(t)\geq\frac{N}{2}\delta_{3}\int_{\Omega}u^{2}(x,t)dx-\frac{1}{2}\sum_{j=1}^{N}\gamma_{j}(T)I_{t}^{\nu-\mu_{j}}\bigg{(}\int_{\Omega}u^{2}dx\bigg{)}(t)-\frac{1}{2}\sum_{j=1}^{N}\gamma_{j}(0)\|u_{0}\|^{2}_{L_{2}(\Omega)}.$

$\bullet$ Integrating by parts and keeping in mind the homogeneous Dirichlet
boundary condition and according to assumptions h2-h3 and (6.6), we deduce

$\mathcal{R}_{3}(t)\geq\frac{3\delta_{0}}{4}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial
x}\bigg{)}^{2}dx\bigg{)}(t)-C_{1}I_{t}^{\nu}\bigg{(}\int_{\Omega}u^{2}dx\bigg{)},$

where

$C_{1}=C_{0}(1+4C_{0}/\delta_{0}).$

It is worth noting that, we used the Cauchy inequality to evaluate the term
$u\frac{\partial u}{\partial x}$.

$\bullet$ Coming to the term $\mathcal{R}_{4}(t)$, we integrate by parts and
take advantage of the Cauchy and the Poincare inequalities and requirements
h2-h3, (6.6). In summary, we obtain

$\mathcal{R}_{4}(t)\geq-\frac{\delta_{0}}{4}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial
x}\bigg{)}^{2}dx\bigg{)}(t)-CI_{t}^{\nu}\bigg{(}\mathcal{K}*\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial
x}\bigg{)}^{2}dx\bigg{)}(t)-CI_{t}^{\nu}\bigg{(}\int_{\Omega}u^{2}dx\bigg{)}(t),$

where the positive constant $C$ is independent of $\lambda$ and $T_{0}$.

$\bullet$ Exploiting assumption h6 (i) with the Cauchy inequality and point
(ii) of Proposition 5.1, we have

$\mathcal{R}_{5}(t)\geq-L|\Omega|\omega_{1+\nu}(t)-(L+2)I_{t}^{\nu}\bigg{(}\int_{\Omega}u^{2}dx\bigg{)}(t)-I_{t}^{\nu}\bigg{(}\int_{\Omega}g^{2}dx\bigg{)}(t).$

Finally, collecting all estimates of $\mathcal{R}_{j}(t)$, we end up with

$\displaystyle\int_{\Omega}u^{2}(x,t)dx+I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{2}dx\bigg{)}(t)$ $\displaystyle\leq
C\Big{\\{}\big{(}I_{t}^{\nu}+\sum_{i=1}^{N}I_{t}^{\nu-\mu_{i}}\big{)}\Big{(}\int_{\Omega}u^{2}dx\Big{)}(t)+I_{t}^{\nu}\Big{(}\mathcal{K}*\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{2}dx\Big{)}(t)$
$\displaystyle+1+\|u_{0}\|_{L_{2}(\Omega)}+I_{t}^{\nu}\bigg{(}\int_{\Omega}g^{2}dx\bigg{)}(t)\Big{\\}}.$

Appealing to associative properties of a convolution to handle the second term
in the right-hand side of this estimate, we use the Gronwall-type inequality
(4.3) in [21] and arrive at the desired bound

$\|u\|^{2}_{\mathcal{C}([0,T_{0}],L_{2}(\Omega))}+\underset{t\in[0,T_{0}]}{\sup}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{2}dx\bigg{)}(t)\leq
C[1+\|u_{0}\|_{L_{2}(\Omega)}+\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}]$
(6.8)

with the constant $C$ being independent of $\lambda$ and $T_{0}$. Moreover,
inequality (6.8) provides the estimate

$\bigg{\|}\frac{\partial u}{\partial x}\bigg{\|}_{L_{2}(\Omega_{T_{0}})}\leq
C[1+\|u_{0}\|^{2}_{L_{2}(\Omega)}+\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}].$
(6.9)

In order to complete the estimate of
$\|u\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}$, we need a similar bound for the
derivative $\frac{\partial u}{\partial x}$. To this end, we multiply the
equation in (6.1) by $\frac{\partial^{2}u}{\partial x^{2}}(x,\tau)$ and then
we integrate over $\Omega$ and compute the fractional integral $I_{t}^{\nu}$.
Taking into account h2-h4 and h6 (i) and applying Lemma 5.6 with $p=2$ and
(6.9), we obtain

$\displaystyle\frac{1}{2}[\delta_{1}+N\delta_{3}]\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{2}dx+\sum_{j=1}^{N}(\nu
I_{t}^{1}-\mu_{j}I_{t}^{1+\nu-\mu_{j}})\bigg{(}\gamma_{j}^{2}(-\mathcal{W}(\gamma_{j}^{-1}))\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{2}dx\bigg{)}(t)$
$\displaystyle+\frac{\delta_{0}}{2}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}(t)\leq\frac{1}{2}\sum_{j=1}^{N}\gamma_{j}(T)I_{t}^{\nu-\mu_{j}}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{2}dx\bigg{)}(t)$
$\displaystyle+\frac{3\|\mathcal{K}\|_{L_{1}(0,T)}C_{0}^{2}}{\delta_{0}}|\mathcal{K}|*I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}+C[1+\|u_{0}\|^{2}_{W^{1,2}(\Omega)}+\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}].$
(6.10)

At last, using Corollary 5.3 with $T_{1}=T_{0}$ (see (6.6)) to handle the
second term in the left-hand side of this inequality, and then applying
Gronwall-type inequality (4.3) in [21], we conclude that

$\underset{t\in[0,T_{0}]}{\sup}\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{2}dx+\underset{t\in[0,T_{0}]}{\sup}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}\leq
C[1+\|u_{0}\|^{2}_{W^{1,2}(\Omega)}+\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}]$
(6.11)

where the positive constant $C$ is independent of $\lambda$ and $T_{0}$.

Collecting this estimate with (6.8) and (6.9) and applying Sobolev embedding
theorem (as we wrote before) arrive at the desired estimate

$\displaystyle\|u\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}+\|u\|_{L_{2}((0,T_{0}),W^{2,2}(\Omega))}+\|u\|_{\mathcal{C}([0,T_{0}],W^{1,2}(\Omega))}$
$\displaystyle\leq
C[1+\|u_{0}\|_{W^{1,2}(\Omega)}+\sqrt{\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}}]\equiv
C\mathcal{F}(u_{0},g)$ (6.12)

with the positive constant $C$ being independent of $\lambda,T_{0}$.

Obviously, the last estimate allows us to evaluate the term
$\|\mathcal{K}*\mathcal{L}_{2}u\|_{L_{2}(\Omega_{T_{0}})}$. Indeed, the Young
inequality of a convolution (see, e.g. [9]) provides

$\|\mathcal{K}*\mathcal{L}_{2}u\|_{L_{2}(\Omega_{T_{0}})}\leq
C\|\mathcal{K}\|_{L_{1}(0,T)}\|\mathcal{L}_{2}u\|_{L_{2}(\Omega_{T_{0}})}.$

To manage the term $\|\mathcal{L}_{2}u\|_{L_{2}(\Omega_{T_{0}})}$, we apply
(6.1) and exploit the smoothness of the coefficients $b_{j},$ $j=0,1,2.$
Hence, we have

$\|\mathcal{K}*\mathcal{L}_{2}u\|^{2}_{L_{2}(\Omega_{T_{0}})}\leq
C\|u\|^{2}_{L_{2}((0,T_{0}),W^{2,2}(\Omega))}\leq
C[1+\|u_{0}\|^{2}_{W^{1,2}(\Omega)}+\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}],$
(6.13)

where the positive constant $C$ is independent of $\lambda$ and $T_{0}$. ∎

######  Remark 6.3.

The treatment of the term $I_{t}^{\nu}(\int_{\Omega}f(u)u_{xx}dx)(t)$ in (6.1)
in the case of (3.1) differs from the case of $f(u)$ satisfying (3.2). Indeed,
if (3.2) holds, we first rewrite the term
$I_{t}^{\nu}(\int_{\Omega}f(u)u_{xx}dx)(t)$ in the form

$I_{t}^{\nu}\bigg{(}\int_{\Omega}f(u)u_{xx}dx\bigg{)}(t)=I_{t}^{\nu}\bigg{(}\int_{\Omega}\bar{f}(u)u_{xx}dx\bigg{)}(t)+I_{t}^{\nu}\bigg{(}\int_{\Omega}[f(0)-L_{4}u]u_{xx}dx\bigg{)}(t)$

with

$\bar{f}(u)=f(u)-f(0)+L_{4}u.$

It is apparent that the second term in this representation is controlled with
the arguments leading to the estimate of $\mathcal{R}_{5}$. Coming the first
term, we note that the function $\bar{f}(u)$ meets the first three
requirements in (3.2) and, besides,

$\bar{f}(u)=0\quad\text{on}\quad\partial\Omega_{T},\quad\bar{f}^{\prime}(u)\geq
0.$

Thus, taking into account these relations and integrating by parts arrive at

$I_{t}^{\nu}\bigg{(}\int_{\Omega}\bar{f}(u)u_{xx}dx\bigg{)}(t)=I_{t}^{\nu}\bigg{(}\int_{\Omega}\bar{f}^{\prime}(u)(u_{x})^{2}dx\bigg{)}(t)\geq
0,$

which in turn provides (6.1).

###  6.2. The bound of $\|\frac{\partial u}{\partial
x}\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}$

In this subsection we aim to prove the bound

$\bigg{\|}\frac{\partial u}{\partial
x}\bigg{\|}_{\mathcal{C}(\bar{\Omega}_{T_{0}})}\leq\mathfrak{C}[1+\|u_{0}\|_{W^{2,2}(\Omega)}+(\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)})^{1/2}]$
(6.14)

with the positive constant $\mathfrak{C}$ is independent of $\lambda$ and
$T_{0}$ and depends only on $T,\nu,\nu_{i},\mu_{j},L,$
$\|\mathcal{K}\|_{L_{1}(0,T)},$ and the corresponding norms of the
coefficients.

For simplicity of consideration, we assume that

$\bigg{\|}\frac{\partial u}{\partial
x}\bigg{\|}_{\mathcal{C}(\bar{\Omega}_{T_{0}})}\geq\frac{\mathfrak{C}}{2}\mathcal{F}(u_{0},g),$
(6.15)

where $\mathcal{F}(u_{0},g)$ is defined in (6.1), otherwise we immediately
arrive at (6.14).

To verify (6.14) in the case of (6.15), we reason similarly to the case
analyzed in Subsection 6.1. Thus, we first multiply the equation in (6.1) by
$p\frac{\partial}{\partial x}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p-1}$ and then integrate over $\Omega$ and compute the fractional
integral $I_{t}^{\nu}$

$\displaystyle\varrho(0)\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p}dx+\frac{3\delta_{0}}{4}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}(t)$ $\displaystyle\leq
C\bigg{\\{}p(p-1)[1+\|u\|_{\mathcal{C}(\bar{\Omega}_{T_{0}})}]^{2}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p-2}dx\bigg{)}(t)$
$\displaystyle+p(p-1)I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)+\int_{\Omega}\bigg{(}\frac{\partial
u_{0}}{\partial x}\bigg{)}^{p}dx$
$\displaystyle+\bigg{(}\sum_{j=1}^{N}\gamma_{j}(T)T^{\mu_{N}-\mu_{j}}\bigg{)}I_{t}^{\nu-\mu_{N}}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)$
$\displaystyle+(p-1)N\Gamma(1+\nu)\underset{j\in(1,2,...,N)}{\sup}\|\gamma_{j}\|_{\mathcal{C}^{1}([0,T])}T^{1-\nu}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)\bigg{\\}}$
$\displaystyle+I_{t}^{\nu}\bigg{(}\int_{\Omega}(\mathcal{K}*\mathcal{L}_{2}u)p\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx\bigg{)}(t)$

with $C$ being independent of $\lambda$ and $T_{0}$. Here we used Corollary
5.3, Lemma 5.6 and assumptions h1-h3 to manage the term
$I_{t}^{\nu}\bigg{(}\int_{\Omega}\mathbf{D}_{t}up\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx\bigg{)}$, while to
handle the term
$I_{t}^{\nu}\bigg{(}\int_{\Omega}\mathcal{L}_{1}up\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx\bigg{)}$, we appealed
to h2-h3 and the Young inequality. Finally, we take advantage of (3.1) and
again the Young inequality to treat $I_{t}^{\nu}\bigg{(}\int_{\Omega}[\lambda
f(u)-g]p\frac{\partial}{\partial x}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p-1}dx\bigg{)}$. After that, keeping in mind restriction (6.15)
and inequality (6.1), we conclude that

$\displaystyle\varrho(0)\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p}dx+\frac{3\delta_{0}}{4}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}(t)$ $\displaystyle\leq
C\bigg{\\{}p(p-1)I_{t}^{\nu}\bigg{(}\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial
x}\bigg{|}^{p}\bigg{)}(t)+I_{t}^{\nu-\mu_{N}}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)$
$\displaystyle+\int_{\Omega}\bigg{(}\frac{\partial u_{0}}{\partial
x}\bigg{)}^{p}dx\bigg{\\}}+I_{t}^{\nu}\bigg{(}\int_{\Omega}(\mathcal{K}*\mathcal{L}_{2}u)p\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx\bigg{)}(t),$ (6.16)

where the positive constant $C$ is independent of $T_{0},\lambda,p$ and the
corresponding norms of $u_{0},g$.

Now we are left to evaluate the last term in the right-hand side of (6.2). To
this end, applying the Young inequality and restriction (6.6) provides the
estimate

$\displaystyle\int_{\Omega}(\mathcal{K}*\mathcal{L}_{2}u)p\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx$ $\displaystyle\leq
C_{0}\frac{2}{\varepsilon_{1}}p(p-1)\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial
x}(x,t)\bigg{|}^{p-2}\bigg{(}|\mathcal{K}|*\|u(\cdot,t)\|^{2}_{W^{2,2}(\Omega)}\bigg{)}(t)$
$\displaystyle+C_{0}p(p-1)\varepsilon_{1}(|\mathcal{K}|*1)(t)\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}(x,t)\bigg{)}^{2}dx$

where the small quantity $\varepsilon_{1}$ being specified later.

Exploiting assumption h4 and estimate (6.11) to manage the first term in the
right-hand side of this inequality, we easily conclude that

$\displaystyle
I_{t}^{\nu}\bigg{(}\int_{\Omega}(\mathcal{K}*\mathcal{L}_{2}u)p\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx\bigg{)}$
$\displaystyle\leq\frac{Cp(p-1)}{\varepsilon_{1}}I_{t}^{\nu}\bigg{(}\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial x}(x,t)\bigg{|}^{p-2}\bigg{)}(t)\mathcal{F}^{2}(u_{0},g)$
$\displaystyle+C_{0}p(p-1)\varepsilon_{1}\|\mathcal{K}\|_{L_{1}(0,T)}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}(x,t)\bigg{)}^{2}dx\bigg{)}(t).$

Applying restriction (6.15) to control the first term in the right-hand side
leads to the estimate

$\displaystyle
I_{t}^{\nu}\bigg{(}\int_{\Omega}(\mathcal{K}*\mathcal{L}_{2}u)p\frac{\partial}{\partial
x}\bigg{(}\frac{\partial u}{\partial x}\bigg{)}^{p-1}dx\bigg{)}$
$\displaystyle\leq\frac{2Cp(p-1)}{\mathfrak{C}^{2}\varepsilon_{1}}I_{t}^{\nu}\bigg{(}\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial x}(x,t)\bigg{|}^{p}\bigg{)}(t)$
$\displaystyle+C_{0}p(p-1)\varepsilon_{1}\|\mathcal{K}\|_{L_{1}(0,T)}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}(x,t)\bigg{)}^{2}dx\bigg{)}(t).$

Choosing

$\varepsilon_{1}=\frac{\delta_{0}}{4C_{0}\|\mathcal{K}\|_{L_{1}(0,T)}}$

and coming to estimate (6.2), we easily draw

$\displaystyle\varrho(0)\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p}dx+\frac{\delta_{0}}{2}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}(t)$ $\displaystyle\leq
C_{2}\bigg{\\{}p(p-1)I_{t}^{\nu}\bigg{(}\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial
x}\bigg{|}^{p}\bigg{)}(t)+I_{t}^{\nu-\mu_{N}}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)$
$\displaystyle+\int_{\Omega}\bigg{(}\frac{\partial u_{0}}{\partial
x}\bigg{)}^{p}dx\bigg{\\}},$ (6.17)

where the positive constant $C_{2}$ is independent of $T_{0},$ $\lambda,p$ and
the norms of $u_{0},g$.

Finally, to handle the term
$I_{t}^{\nu}\bigg{(}\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial x}(x,t)\bigg{|}^{p}\bigg{)}$, we exploit bound (5.14) with
$v=\bigg{(}\frac{\partial u}{\partial x}(x,t)\bigg{)}^{p/2}$ and have

$\displaystyle\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial u}{\partial
x}(x,t)\bigg{|}^{p}$
$\displaystyle\leq\frac{\varepsilon^{2}p^{2}}{4}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}(x,t)\bigg{)}^{2}dx+\varepsilon^{2}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p}dx$
$\displaystyle+\frac{C}{\varepsilon}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p/2}dx\bigg{)}^{2}$

with a small positive $\varepsilon$.

Then, setting here

$\varepsilon=\frac{\varepsilon_{0}}{\sqrt{|\Omega|}}$

with some positive $\varepsilon_{0}<1$ and computing the fractional integral,
we arrive at

$\displaystyle
I_{t}^{\nu}\bigg{(}\underset{\bar{\Omega}}{\sup}\bigg{|}\frac{\partial
u}{\partial x}(x,t)\bigg{|}^{p}\bigg{)}(t)$
$\displaystyle\leq\frac{\varepsilon_{0}^{2}p^{2}}{4[1-\varepsilon^{2}]|\Omega|}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}(x,t)\bigg{)}^{2}dx\bigg{)}(t)$
$\displaystyle+\frac{C\sqrt{|\Omega|}}{\varepsilon_{0}(1-\varepsilon_{0}^{2})}I_{t}^{\nu}\bigg{(}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}(x,t)\bigg{)}^{p/2}dx\bigg{)}^{2}\bigg{)}(t),$

Collecting this estimate with (6.2) and choosing

$\varepsilon_{0}^{2}=\frac{\delta_{0}|\Omega|}{\delta_{0}|\Omega|+p^{2}C_{2}},$

we achieve the bound

$\displaystyle\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}(x,t)\bigg{)}^{p}dx+p(p-1)I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p-2}\bigg{(}\frac{\partial^{2}u}{\partial
x^{2}}\bigg{)}^{2}dx\bigg{)}(t)$ $\displaystyle\leq
C\bigg{\\{}I_{t}^{\nu-\mu_{N}}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}(t)+\int_{\Omega}\bigg{(}\frac{\partial
u_{0}}{\partial x}\bigg{)}^{p}dx$
$\displaystyle+p^{3}I_{t}^{\nu}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p/2}dx\bigg{)}^{2}(t),$ (6.18)

where the positive constant $C$ is independent of $T_{0}$, $\lambda,$ $p$ and
the norms of $u_{0}$ and $g$.

In order to evaluate the term
$I_{t}^{\nu-\mu_{N}}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p}dx\bigg{)}(t)$ in the right-hand side of (6.2), we exploit the
Gronwal-type inequality [21, Proposition 4.3] and obtain

$\int_{\Omega}\bigg{(}\frac{\partial u}{\partial x}(x,t)\bigg{)}^{p}dx\leq
AE_{\nu-\mu_{N}}(Ct^{\nu-\mu_{N}})\quad\text{for any}\quad t\in[0,T_{0}],$

where we put

$A=C\bigg{[}\int_{\Omega}\bigg{(}\frac{\partial u_{0}}{\partial
x}\bigg{)}^{p}dx+p^{3}\bigg{\|}\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p/2}dx\bigg{\|}^{2}_{\mathcal{C}([0,T_{0}])}\bigg{]}$

and

$E_{\theta}(t)=\sum_{m=0}^{+\infty}\frac{z^{m}}{\Gamma(1+m\theta)}$

is the classical Mittag-Leffler function of the order $\theta$ (see, e.g., its
definition in [11, (2.2.4)]).

Applying this estimate to handle the first term in the right-hand side of
(6.2) and then taking into account formula (3.7.44) in [11] to compute the
fractional integral of Mittag-Leffler function produce the inequality

$\displaystyle\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p}dx$ $\displaystyle\leq
CE_{\nu-\mu_{N}}(CT^{\nu-\mu_{N}})\bigg{\\{}1+\int_{\Omega}\bigg{(}\frac{\partial
u_{0}}{\partial x}\bigg{)}^{p}dx$
$\displaystyle+p^{3}\bigg{\|}\int_{\Omega}\bigg{(}\frac{\partial u}{\partial
x}\bigg{)}^{p/2}dx\bigg{\|}^{2}_{\mathcal{C}([0,T_{0}])}\bigg{\\}}$

At last, denoting

$\mathcal{B}_{0}=CE_{\nu-\mu_{N}}(CT^{\nu-\mu_{N}})\qquad\mathcal{A}_{m}=\underset{t\in[0,T_{0}]}{\sup}\bigg{(}\int_{\Omega}\bigg{(}\frac{\partial
u}{\partial x}\bigg{)}^{p}dx\bigg{)}^{1/p}$

with $p=2^{m},$ $m\geq 1$, we derive the bound

$\displaystyle\mathcal{A}_{m}$
$\displaystyle\leq\mathcal{B}_{0}^{2^{-m}}[1+\|u_{0}\|_{W^{1,p}(\Omega)}+(2^{3m2^{-m}})\mathcal{A}_{m-1}]$
$\displaystyle\leq\mathcal{B}_{0}^{2^{-m}}[1+\|u_{0}\|_{\mathcal{C}^{1}(\bar{\Omega})}+(2^{3m2^{-m}})\mathcal{A}_{m-1}]$
$\displaystyle\leq\mathcal{B}_{0}^{2^{-m}}[1+\|u_{0}\|_{W^{2,2}(\Omega)}]+(2^{3m2^{-m}})\mathcal{A}_{m-1}].$
(6.19)

To handle the term $\|u_{0}\|_{\mathcal{C}^{1}(\bar{\Omega})}$ in these
inequalities, we used the Sobolev embedding theorem.

At this point, we analyze two possibilities:

(i) either
$\max\bigg{\\{}1+\|u_{0}\|_{W^{2,2}(\Omega)};\mathcal{A}_{m-1}\bigg{\\}}=1+\|u_{0}\|_{W^{2,2}(\Omega)}$,

(ii) or
$\max\bigg{\\{}1+\|u_{0}\|_{W^{2,2}(\Omega)};\mathcal{A}_{m-1}\bigg{\\}}=\mathcal{A}_{m-1}$.

Obviously, in the case of (i) passing to the limit as $m\to+\infty$ in (6.2),
we end up with the desired bound

$\underset{\bar{\Omega}_{T_{0}}}{\sup}\bigg{|}\frac{\partial u}{\partial
x}\bigg{|}\leq C[1+\|u_{0}\|_{W^{2,2}(\Omega)}].$

Conversely, if (ii) holds then

$\mathcal{A}_{m}\leq\mathcal{B}_{0}^{2^{-m}}\mathcal{A}_{m-1}\leq
C\exp\bigg{\\{}\sum_{m=1}^{+\infty}m2^{-m}\bigg{\\}}\mathcal{A}_{1}\leq
C\mathcal{A}_{1}$

and letting $m\to+\infty$, we deduce that

$\underset{\bar{\Omega}_{T_{0}}}{\sup}\bigg{|}\frac{\partial u}{\partial
x}\bigg{|}\leq C\mathcal{A}_{1}.$

Finally, applying (6.11) to control the term $\mathcal{A}_{1}$, we obtain
(6.14). ∎

###  6.3. The estimate of $\langle u\rangle_{t,{\Omega}_{T_{0}}}^{(\nu/2)}$

This subsection will be devoted to obtain Hölder regularity in time of the
solution $u$. Namely, we aim to achieve the bound

$\langle u\rangle_{t,{\Omega}_{T_{0}}}^{(\nu/2)}\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}.$

It is apparent that this estimate is a simple consequence of the inequality

$\displaystyle\Delta_{h}u$ $\displaystyle=|u(x,t+h)-u(y,t)|$
$\displaystyle\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}[h^{\nu/2}+|x-y|]$
(6.20)

for any $x,y\in\bar{\Omega},$ and $h\in(0,1)$ such that
$x+h^{\nu/2}\in\bar{\Omega}$ and $t\in[0,T_{0}]$. Indeed, substituting $x=y$
in (6.3), we immediately arrive at the desired bound.

First, appealing to the integral mean value theorem, we conclude that

$\int_{x}^{x+h^{\nu/2}}[u(z,t+h)-u(z,t)]dz=h^{\nu/2}[u(x^{*},t+h)-u(x^{*},t)]$
(6.21)

for some $x^{*}\in[x,x+h^{\nu/2}]$ and $t\in[0,T_{0}]$. Taking into account
this relation and (6.14), we deduce that

$\displaystyle\Delta_{h}u$
$\displaystyle\leq|u(x,t+h)-u(x^{*},t+h)|+|u(x^{*},t+h)-u(x^{*},t)|+|u(x^{*},t)-u(y,t)|$
$\displaystyle\leq
C\\{|x-x^{*}|+|x^{*}-y|\\}\underset{\bar{\Omega}_{T_{0}}}{\sup}\bigg{|}\frac{\partial
u}{\partial x}\bigg{|}+|u(x^{*},t+h)-u(x^{*},t)|$ $\displaystyle\leq
C[1+\|u_{0}\|_{W^{2,2}(\Omega)}+\|I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}\|^{1/2}_{\mathcal{C}(\bar{\Omega}_{T})}][h^{\nu/2}+|x^{*}-y|]$
$\displaystyle+|u(x^{*},t+h)-u(x^{*},t)|\equiv d_{1}+d_{2}.$ (6.22)

At this point, we evaluate each term $d_{i}$, separately.

$\bullet$ As for $d_{1}$, we should to evaluate $|x^{*}-y|$ in appropriate
way. To this end, we analyze three possibilities to the location of $y$:

i:

if $y\geq x^{*},$ then $|y-x^{*}|\leq|y-x|$;

ii:

if $x\leq y<x^{*},$ then $|y-x^{*}|\leq h^{\nu/2}$;

iii:

if $y<x,$ then $|y-x^{*}|\leq|x-y|+h^{\nu/2}$.

In summary, we derive the bound

$d_{1}\leq
C[|x-y|+h^{\nu/2}]\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}.$

$\bullet$ Concerning the estimate of $d_{2}$, we have to evaluate the left-
hand side of (6.21). To this end, exploiting (3.5.4) in [17] and [39, Chapter
1, Corollary 2] arrives at the easily verified relations

$u(z,t+h)-u(z,t)=I_{t+h}^{\nu}\mathbf{D}_{t+h}^{\nu}u(z,t+h)-I_{t}^{\nu}\mathbf{D}_{t}^{\nu}u(z,t),$

and, therefore,

$|u(x^{*},t+h)-u(x^{*},t)|\leq
Ch^{\nu/2}\underset{[0,T_{0}]}{\sup}\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{t}^{\nu}u(z,s)dz\bigg{|}.$
(6.23)

Thus, we are left to evaluate the term
$\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{t}^{\nu}u(z,s)dz$. Appealing to (i) in
Proposition 5.4 provides the representation

$\mathbf{D}_{t}u=\sum_{j=1}^{3}\mathfrak{D}_{j}u+\varrho_{0}\mathbf{D}_{t}^{\nu}u,$
(6.24)

where we set

$\displaystyle\mathfrak{D}_{1}u$
$\displaystyle=\sum_{i=1}^{M}\varrho_{i}\mathbf{D}_{t}^{\nu_{i}}u-\sum_{j=1}^{N}\gamma_{j}\mathbf{D}_{t}^{\mu_{j}}u,\quad\mathfrak{D}_{2}u=u_{0}\\{\mathbf{D}_{t}^{\nu}\varrho_{0}+\sum_{i=1}^{M}\mathbf{D}_{t}^{\nu_{i}}\varrho_{i}-\sum_{j=1}^{N}\mathbf{D}_{t}^{\mu_{j}}\gamma_{j}\\},$
$\displaystyle\mathfrak{D}_{3}u$
$\displaystyle=\frac{\nu}{\Gamma(1-\nu)}\int_{0}^{t}(t-s)^{-1-\nu}[\varrho_{0}(t)-\varrho_{0}(s)][u(x,s)-u_{0}(x)]ds$
$\displaystyle+\sum_{i=1}^{M}\frac{\nu_{i}}{\Gamma(1-\nu_{i})}\int_{0}^{t}(t-s)^{-1-\nu_{i}}[\varrho_{i}(t)-\varrho_{i}(s)][u(x,s)-u_{0}(x)]ds$
$\displaystyle-\sum_{j=1}^{N}\frac{\mu_{j}}{\Gamma(1-\mu_{j})}\int_{0}^{t}(t-s)^{-1-\mu_{j}}[\gamma_{j}(t)-\gamma_{j}(s)][u(x,s)-u_{0}(x)]ds.$

Using representation (6.24) and the equation in (6.1), we end up with

$\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{s}^{\nu}u(z,s)dz\bigg{|}\leq\sum_{j=1}^{6}|\mathcal{G}_{j}(u)|,$
(6.25)

where

$\displaystyle\mathcal{G}_{1}u$
$\displaystyle=\varrho_{0}^{-1}\int_{x}^{x+h^{\nu/2}}\mathcal{L}_{1}u(z,s)dz,\qquad\qquad\mathcal{G}_{2}u=\varrho_{0}^{-1}\int_{x}^{x+h^{\nu/2}}(\mathcal{K}*\mathcal{L}_{2}u)(z,s)dz,$
$\displaystyle\mathcal{G}_{3}u$
$\displaystyle=\varrho_{0}^{-1}\int_{x}^{x+h^{\nu/2}}[\lambda
f(u)-g(z,s)]dz,\quad\mathcal{G}_{4}u=\varrho_{0}^{-1}\int_{x}^{x+h^{\nu/2}}\mathfrak{D}_{3}u(z,s)dz,$
$\displaystyle\mathcal{G}_{5}u$
$\displaystyle=\varrho_{0}^{-1}\int_{x}^{x+h^{\nu/2}}\mathfrak{D}_{2}u(z,s)dz,\qquad\qquad\mathcal{G}_{6}u=\varrho_{0}^{-1}\int_{x}^{x+h^{\nu/2}}\mathfrak{D}_{1}u(z,s)dz.$

Hence, we are left to treat each $\mathcal{G}_{i}u$.

$\bullet$ Taking into account (6.1) and (6.14), assumptions h2-h3, and easily
verified relation

$\int_{x}^{x+h^{\nu/2}}a_{2}(z,s)\frac{\partial^{2}u}{\partial
z^{2}}(z,s)dz=\int_{x}^{x+h^{\nu/2}}\frac{\partial}{\partial
z}\bigg{[}a_{2}(z,s)\frac{\partial u}{\partial
z}(z,s)\bigg{]}dz-\int_{x}^{x+h^{\nu/2}}\frac{\partial u}{\partial
z}(z,s)\frac{\partial a_{2}}{\partial z}(z,s)dz,$

we immediately deduce that

$|\mathcal{G}_{1}u|\leq\delta_{1}^{-1}[1+h^{\nu/2}]C_{0}\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}.$

$\bullet$ It is worth noting that, using the similar arguments and making
assumption h4 on the kernel $\mathcal{K}$, we get the inequality

$|\mathcal{G}_{2}u|\leq\frac{C_{0}(1+h^{\nu/4})\|\mathcal{K}\|_{L_{1}(0,T)}}{\delta_{1}}\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}.$

$\bullet$ Assumptions (3.1) on $f(u)$, h5 on $g$ and estimate (6.1) arrive at

$|\mathcal{G}_{3}u|\leq\delta_{1}^{-1}h^{\nu/2}[1+L]\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}.$

$\bullet$ In light of the regularity of the coefficients
$\varrho_{0},\varrho_{i},\gamma_{j}$ and the function $u_{0}$ (see h3 and h5),
we use (6.1) and obtain the bound

$|\mathcal{G}_{4}u|+|\mathcal{G}_{5}u|\leq
C\delta_{1}^{-1}h^{\nu/2}\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}$

with the constant $C$ depending only on $T,\nu,\mu_{j},\nu_{i},N,M$ and the
norms of the coefficients.

$\bullet$ Concerning $\mathcal{G}_{6}u,$ we exploit the representation (10.34)
in [23] and have

$|\mathcal{G}_{6}u|\leq\sum_{i=1}^{M}\varrho_{i}I_{s}^{\nu-\nu_{i}}\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{s}^{\nu}u(z,s)dz\bigg{|}+\sum_{j=1}^{N}\gamma_{j}I_{s}^{\nu-\mu_{j}}\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{s}^{\nu}u(z,s)dz\bigg{|}.$

Finally, gathering these estimates with (6.25), we obtain

$\displaystyle\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{s}^{\nu}u(z,s)dz\bigg{|}$
$\displaystyle\leq
C[1+h^{\frac{\nu}{2}}]\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}$
$\displaystyle+\bigg{[}\sum_{i=1}^{M}\varrho_{i}I_{s}^{\nu-\nu_{i}}+\sum_{j=1}^{N}\gamma_{j}I_{s}^{\nu-\mu_{j}}\bigg{]}\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{s}^{\nu}u(z,s)dz\bigg{|}.$

Then, Gronwall-type inequality (4.4) [21] tells us that

$\bigg{|}\int_{x}^{x+h^{\nu/2}}\mathbf{D}_{s}^{\nu}u(z,s)dz\bigg{|}\leq
C[1+h^{\frac{\nu}{2}}]\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]},$
(6.26)

where $C$ is independent of $T_{0},$ $\lambda,h$ and depends only on
$\nu,\mu_{i},\nu_{j},N,M,$ $\|\mathcal{K}\|_{L_{1}(0,T)}$, and the
corresponding norms of the coefficients.

As a result, (6.26) and (6.23) lead to

$\displaystyle d_{2}$ $\displaystyle=|u(x^{*},t+h)-u(x^{*},t)|$
$\displaystyle\leq
Ch^{\frac{\nu}{2}}[1+h^{\frac{\nu}{2}}]\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]},$

or coming to (6.3) and, taking into account the estimate of $d_{1}$, we
achieve

$\Delta_{h}u\leq
C\\{h^{\nu}+h^{\nu/2}+|x-y|\\}\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}\bigg{]}.$

It is apparent that (as we wrote above) this inequality provides the desired
estimate.

In summary, this bound completes the proof of (6.2), and therefore, the
verification (6.3) is finished. ∎

###  6.4. Verification of (6.1) if $t\in[0,T_{0}]$

In light of estimates (6.1) and (6.13), we conclude that (6.1) with $T=T_{0}$
will be proved if we obtain the inequality

$\displaystyle\|\mathbf{D}_{t}^{\nu}u\|_{L_{2}(\Omega_{T_{0}})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T_{0}})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T_{0}})}$
$\displaystyle\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]}.$
(6.27)

To this end, we use again representation (6.24) and the easily verified
estimates

$\displaystyle\|\mathbf{D}_{t}u\|_{L_{2}(\Omega_{T_{0}})}$ $\displaystyle\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]},$
$\displaystyle\|\mathbf{D}^{\nu}_{t}u\|_{L_{2}(\Omega_{T_{0}})}$
$\displaystyle\leq\frac{C}{\delta_{1}}\|\mathbf{D}_{t}u-\sum_{j=1}^{3}\mathfrak{D}_{j}u\|_{L_{2}(\Omega_{T_{0}})}\leq\frac{C}{\delta_{1}}[\|\mathbf{D}_{t}u\|_{L_{2}(\Omega_{T_{0}})}+\sum_{j=1}^{3}\|\mathfrak{D}_{j}u\|_{L_{2}(\Omega_{T_{0}})}],$
$\displaystyle\|\mathcal{D}_{2}u\|_{L_{2}(\Omega_{T_{0}})}+\|\mathfrak{D}_{3}u\|_{L_{2}(\Omega_{T_{0}})}$
$\displaystyle\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]}$
$\displaystyle\times[\|\varrho_{0}\|_{\mathcal{C}^{1}([0,T_{0}])}+\sum_{i=1}^{M}\|\varrho_{i}\|_{\mathcal{C}^{1}([0,T_{0}])}+\sum_{j=1}^{N}\|\gamma_{j}\|_{\mathcal{C}^{1}([0,T_{0}])}]$
(6.28)

with the constants being independent of $\lambda$ and $T_{0}$. We remark that
the last inequality in (6.4) is a simple consequence of assumption h3 and
estimates (6.1) and (6.14).

Besides, collecting all these estimates provides the bound

$\displaystyle\|\mathbf{D}^{\nu}_{t}u\|_{L_{2}(\Omega_{T_{0}})}$
$\displaystyle\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]}$
$\displaystyle+C_{3}[\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T_{0}})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T_{0}})}].$
(6.29)

As a result, in virtue of the last estimate, we are left to evaluate
$\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T_{0}})}$ and
$\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T_{0}})}$. To this end, we are
exploit [43, Theorem 2.2] (taking into account Definition 2.2 and estimate
(2.1)) and embedding Theorem 1.4.3.3 in [12]. Indeed, setting

$\mu_{0}=\frac{1}{4}\min\\{\mu_{1};\nu-\mu_{N}\\}\quad\text{and}\quad\nu_{0}=\frac{1}{4}\min\\{\nu_{1};\nu-\nu_{M}\\},$

and choosing $\bar{\nu}$ satisfying inequalities

$0<\bar{\nu}<\min\\{\nu-\mu_{N}-\mu_{0};\nu-\nu_{M}-\nu_{0}\\},$

we take advantage of [43, Theorem 2.2], [44, Propositions 3 and 7] and then
embedding Theorem 1.4.3.3 in [12] to conclude that

$\displaystyle
C_{\nu}\|u-u_{0}\|_{W^{\nu-\bar{\nu}}((0,T_{0}),L_{2}(\Omega))}\leq\|\mathbf{D}_{t}^{\nu}u\|_{L_{2}(\Omega_{T_{0}})},$
$\displaystyle\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T_{0}})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T_{0}})}\leq\sum_{i=1}^{M}C_{\nu_{i}}\|u-u_{0}\|_{W^{\nu_{i}+\nu_{0}}((0,T_{0}),L_{2}(\Omega))}$
$\displaystyle+\sum_{j=1}^{N}C_{\mu_{j}}\|u-u_{0}\|_{W^{\mu_{j}+\mu_{0}}((0,T_{0}),L_{2}(\Omega))}$
$\displaystyle\leq\varepsilon[\sum_{i=1}^{M}C_{\nu_{i}}+\sum_{j=1}^{N}C_{\mu_{j}}]\|u-u_{0}\|_{W^{\nu-\bar{\nu}}((0,T_{0}),L_{2}(\Omega))}$
$\displaystyle+C\bigg{\\{}\sum_{j=1}^{N}C_{\mu_{j}}\varepsilon^{-\frac{\mu_{j}+\mu_{0}}{\nu-\bar{\nu}-\mu_{j}-\mu_{0}}}+\sum_{i=1}^{M}C_{\nu_{i}}\varepsilon^{-\frac{\nu_{i}+\nu_{0}}{\nu-\bar{\nu}-\nu_{i}-\nu_{0}}}\bigg{\\}}\|u-u_{0}\|_{L_{2}(\Omega_{T_{0}})},$
(6.30)

where $C_{\nu}=C(\nu,\bar{\nu}),$ $C_{\nu_{i}}=C(\nu_{i},\nu_{0}),$
$C_{\mu_{j}}=C(\mu_{j},\mu_{0}),$ are positive constants defined in (i) of
Theorem 2.2 [43].

Gathering (6.4) and (6.4) and setting

$\varepsilon=\frac{C_{\nu}}{4C_{3}[\sum_{j=1}^{N}C_{\mu_{j}}+\sum_{i=1}^{M}C_{\nu_{i}}]},$

we end up with the bound

$\|u-u_{0}\|_{W^{\nu-\bar{\nu}}((0,T_{0}),L_{2}(\Omega))}\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]}$
(6.31)

with the positive quantity $C$ being independent of $\lambda$ and $T_{0}$. We
notice that, we exploited (6.1) to control the term
$\|u-u_{0}\|_{L_{2}(\Omega_{T_{0}})}$ in the right-hand side of the second
inequality in (6.4).

After that, (6.31) together with the second inequality in (6.4) arrive at the
estimate

$\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}u\|_{L_{2}(\Omega_{T_{0}})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}u\|_{L_{2}(\Omega_{T_{0}})}\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]},$

which in turn (see (6.4)) provides the desired bound

$\|D_{t}^{\nu}u\|_{L_{2}(\Omega_{T_{0}})}\leq
C\bigg{[}1+\|u_{0}\|_{W^{2,2}(\Omega)}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}\bigg{]}.$

This finishes the proof of (6.4) and, hence, (6.1) with $T=T_{0}$. ∎

###  6.5. Conclusion of the proof of Lemma 6.1

In light of the results described in Sections 6.1-6.4, we are left to extend
estimates (6.2)-(6.1) on the whole time interval $[0,T]$. To this end, we
first discuss the technique which allows us to extend these estimates to the
interval $[0,3T_{0}/2]$. Then, recasting this procedure a finite number of
times until the entire $[0,T]$ is exhausted.

First, we need in new function

$\mathcal{U}(x,t)=\xi(t)u(x,t),$ (6.32)

where $\xi(t)$ is defined in (5.7) with $T_{3}=T_{0}$, and $u(x,t)$ is a
solution of (6.1).

Next statement describes the main properties of this function.

######  Corollary 6.4.

The function $\mathcal{U}(x,t)$ solves the problem (6.1) in
$\bar{\Omega}_{T_{0}/2}$ and satisfies estimates:

$\displaystyle\|\mathcal{U}\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}\mathcal{U}\|_{\mathcal{C}^{\alpha,\frac{\alpha\nu}{2}}(\bar{\Omega}_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}\mathcal{U}\|_{\mathcal{C}^{\alpha,\frac{\alpha\nu}{2}}(\bar{\Omega}_{T})}\leq
C[1+\|u_{0}\|_{\mathcal{C}^{2+\alpha}(\bar{\Omega})}+\|g\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}],$
$\displaystyle\|\mathcal{U}\|_{\mathcal{C}([0,T],\mathcal{C}^{1}(\bar{\Omega}))}+\langle\mathcal{U}\rangle_{t,\Omega_{T}}^{(\nu/2)}\leq
C\bigg{[}1+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\underset{t\in[0,T]}{\sup}\|g\|_{L_{2}(\Omega)}+\|u_{0}\|_{W^{2,2}(\Omega)}\bigg{]},$
$\displaystyle\|\mathcal{U}\|_{L_{2}((0,T),W^{2,2}(\Omega))}+\|\mathbf{D}_{t}^{\nu}\mathcal{U}\|_{L_{2}(\Omega_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}\mathcal{U}\|_{L_{2}(\Omega_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}\mathcal{U}\|_{L_{2}(\Omega_{T})}+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}\|\mathcal{U}_{xx}\|^{2}_{L_{2}(\Omega)}\bigg{)}^{\frac{1}{2}}$
$\displaystyle\leq
C\bigg{[}1+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{1/2}+\|u_{0}\|_{W^{2,2}(\Omega)}\bigg{]},$
$\displaystyle\|\mathbf{D}_{t}\mathcal{U}-\mathcal{L}_{1}\mathcal{U}-\mathcal{K}*\mathcal{L}_{2}\mathcal{U}\|_{\mathcal{C}^{\alpha,\frac{\alpha\nu}{2}}(\bar{\Omega}_{T})}\leq
C\bigg{[}1+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{\frac{1}{2}}+\|u_{0}\|_{W^{2,2}(\Omega)}+\|g\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}\bigg{]},$
$\displaystyle\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|\mathbf{D}_{t}\mathcal{U}-\mathcal{L}_{1}\mathcal{U}-\mathcal{K}*\mathcal{L}_{2}\mathcal{U}\|^{2}_{L_{2}(\Omega)})\leq
C\bigg{[}1+\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)+\|u_{0}\|^{2}_{W^{2,2}(\Omega)}\bigg{]}$

Here the positive constant $C$ depends only on $T,\mu_{j},\nu_{i},\nu$ and the
corresponding norms of the coefficients of the operators $\mathcal{L}_{1},$
$\mathcal{L}_{2}$ and $\mathbf{D}_{t}$.

###### Proof.

Clearly, the first three inequalities are simple consequences of definition
(6.32), Lemma 5.5 (where $T_{3}=T_{0}$) and estimates (6.2)-(6.1) with
$T=T_{0}$, which are proved in Sections 6.1-6.4.

As for the fourth bound, taking into account that $u$ solves (6.1), we have
the representation

$\mathbf{D}_{t}\mathcal{U}-\mathcal{L}_{1}\mathcal{U}-\mathcal{K}*\mathcal{L}_{2}\mathcal{U}=\sum_{j=1}^{3}s_{j},$

where

$s_{1}=\xi[g(x,t)-\lambda f(u)],\quad s_{2}=\mathbf{D}_{t}(\xi
u)-\xi\mathbf{D}_{t}u,\quad
s_{3}=\xi(\mathcal{K}*\mathcal{L}_{2}u)(t)-(\mathcal{K}*\xi\mathcal{L}_{2}u)(t).$

At this point, we evaluate each $s_{j}$, separately.

$\bullet$ Appealing to h5 and h6 (i), we derive

$\|s_{1}\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}\leq
C[1+\|g\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}+\|u\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T_{0}})}],$

where the constant $C$ depends only on $\|\xi\|_{\mathcal{C}^{1}([0,T])}$,
$L,C_{\rho}$. Then exploiting inequality (6.2), we end up with

$\|s_{1}\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}\leq
C[1+\|g\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}+\|u_{0}\|_{W^{2,2}(\Omega)}].$

$\bullet$ Collecting the representation of the operator $\mathbf{D}_{t}$ with
statement (v) in Lemma 5.2 and the bound (6.2) with $T=T_{0}$ provides the
estimate

$\displaystyle\|s_{2}\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}$
$\displaystyle\leq
C[1+\|u_{0}\|_{\mathcal{C}^{\alpha}(\bar{\Omega})}+\|u\|_{\mathcal{C}^{\alpha,\nu/2}(\bar{\Omega}_{T_{0}})}]$
$\displaystyle\leq
C\bigg{[}\|u_{0}\|_{W^{2,2}(\Omega)}+1+\bigg{(}\underset{t\in[0,T]}{\sup}I_{t}^{\nu}(\|g\|^{2}_{L_{2}(\Omega)})(t)\bigg{)}^{\frac{1}{2}}\bigg{]}$

with the constant $C$ being independent of $\lambda$ and $T_{0}$.

$\bullet$ Standard calculations allow us to rewrite $s_{3}$ in the form

$s_{3}=\begin{cases}\xi(t)(\mathcal{K}*\mathcal{L}_{2}u)(t)-(\mathcal{K}*\xi\mathcal{L}_{2}u)(t),\qquad\qquad
t\in(0,3T_{0}/2],\\\ \,\\\
\int_{0}^{3T_{0}/2}\mathcal{K}(t-\tau)\xi(\tau)\mathcal{L}_{2}u(x,\tau)d\tau,\quad
t>3T_{0}/2.\end{cases}$

After that, [19, Lemma 4.1], Lemma 5.5 and assumptions of h2, h4 lead to the
estimate

$\|s_{3}\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}\leq
C[\|u_{0}\|_{W^{2,2}(\Omega)}+1+\|I_{t}^{\nu}\|g\|^{2}_{L_{2}(\Omega)}\|^{1/2}_{\mathcal{C}([0,T])}].$

As a result, gathering all estimates of $s_{i}$ leads to the searched
estimate.

Finally, we left remark that the verification of the last inequality in this
corollary is carried out with the similar arguments and with exploiting the
second and the third inequalities of this claim. This completes the proof of
Corollary 6.4. ∎

Now we introduce new unknown function

$\mathcal{V}=u-\mathcal{U}$ (6.33)

which solves the problem

$\begin{cases}\mathbf{D}_{t}\mathcal{V}-\mathcal{L}_{1}\mathcal{V}-\mathcal{K}*\mathcal{L}_{2}\mathcal{V}=g^{*}(x,t)-\lambda
f^{*}(\mathcal{V})\quad\text{in}\quad\Omega_{3T_{0}/2},\\\
\mathcal{V}(x,0)=0\quad\text{in}\quad\bar{\Omega},\\\
\mathcal{V}(x,t)=0\quad\text{on}\quad\partial\Omega_{3T_{0}/2},\end{cases}$

where we put

$g^{*}(x,t)=g(x,t)-\mathbf{D}_{t}\mathcal{U}+\mathcal{L}_{1}\mathcal{U}+\mathcal{K}*\mathcal{L}_{2}\mathcal{U},\quad
f^{*}(\mathcal{V})=f(\mathcal{V}+\mathcal{U}).$

The definition of the function $\mathcal{U}$ and Corollary 6.4 readily yield

$\displaystyle\|g^{*}\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{3T_{0}/2})}$
$\displaystyle\leq
C[1+\|u_{0}\|_{W^{2,2}(\Omega)}+\|g\|_{\mathcal{C}^{\alpha,\alpha\nu/2}(\bar{\Omega}_{T})}],$
$\displaystyle g^{*}(x,t)-\lambda f^{*}(\mathcal{V})$
$\displaystyle=0\quad\text{if}\quad t\in[0,T_{0}/2],\quad x\in\bar{\Omega},$

and $f^{*}(\mathcal{V})$ meets requirements h6 (i). Besides, the last equality
here means that compatibility conditions hold. Finally, we introduce the new
time-variable

$\sigma=t-\frac{T_{0}}{2},\quad\sigma\in[-T_{0}/2,T_{0}]$

and recast arguments of the end of Section 6.3 in [37]. Thus, we deduce

$\begin{cases}\bar{\mathbf{D}}_{\sigma}\bar{\mathcal{V}}-\bar{\mathcal{L}}_{1}\bar{\mathcal{V}}-\mathcal{K}*\bar{\mathcal{L}}_{2}\bar{\mathcal{V}}=\bar{g}^{*}(x,\sigma)-\lambda\bar{f}^{*}(\bar{\mathcal{V}})\quad\text{in}\quad\Omega_{T_{0}},\\\
\bar{\mathcal{V}}(x,0)=0\quad\text{in}\quad\bar{\Omega},\\\
\bar{\mathcal{V}}(x,\sigma)=0\quad\text{on}\quad\partial\Omega_{T_{0}},\end{cases}$
(6.34)

and $\bar{\mathcal{V}}(x,\sigma)=0$ if $\sigma\in[-T_{0}/2,0]$. Here we denote

$\bar{\mathcal{V}}(x,\sigma)=\mathcal{V}(x,\sigma+T_{0}/2),\quad\bar{g}^{*}(x,\sigma)=g^{*}(x,\sigma+T_{0}/2)\quad\bar{f}^{*}(\bar{\mathcal{V}})=f^{*}(\mathcal{V})|_{t=\sigma+T_{0}/2},$

and we call $\bar{\mathcal{L}}_{i}$, $\bar{\mathbf{D}}_{\sigma}$ the operators
$\mathcal{L}_{i}$ and $\mathbf{D}_{\sigma}$, respectively, with the bar
coefficients. It is easy to verify that the coefficients
$\bar{\mathcal{L}}_{i}$, $\bar{\mathbf{D}}_{\sigma}$ and the function
$\bar{g}^{*}$, $\bar{f}^{*}$ meet the requirements of Lemma 6.1.

Then, we repeat the arguments of Sections 6.1-6.4 in the case of problem
(6.34) and obtain estimates (6.2)-(6.1) to the function $\bar{\mathcal{V}}$.
In fine, taking into account representation (6.33) and Corollary 6.4, we
extend these estimates to the interval $[0,3T_{0}/2]$. Therefore, after
repeating this procedure finite times, we get (6.2)-(6.1) for all $t\in[0,T]$.
∎

##  7\. Proof of Theorem 4.1

Here we proceed with a detailed proof of this theorem in the case of DBC (1.2)
and $f(u)$ satisfying h6 (i). Another cases are analyzed with the similar
arguments and left to the readers.

First of all, we reduce problem (1.1), (1.2), (1.4) to the problem with
homogeneous initial and boundary conditions. To this end, we apply [21, Remark
3.1] and Lemma 5.7 (in this art) to the linear problem for the unknown
function $\mathfrak{U}=\mathfrak{U}(x,t):\Omega_{T}\to\mathbb{R}:$

$\begin{cases}\mathbf{D}_{t}\mathfrak{U}-\mathcal{L}_{1}\mathfrak{U}-\mathcal{K}*\mathcal{L}_{2}\mathfrak{U}=g(x,t)-f(u_{0})\quad\text{in}\,\Omega_{T},\\\
\mathfrak{U}(x,0)=u_{0}(x)\quad\text{in}\quad\bar{\Omega},\\\
\mathfrak{U}(x,t)=\psi(x,t)\quad\text{on}\quad\partial\Omega_{T},\end{cases}$

and obtain the one-valued global classical solvability of this problem
$\mathfrak{U}\in\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})$
satisfying the bound

$\displaystyle\|\mathfrak{U}\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\bar{\Omega}_{T})}+\sum_{i=1}^{M}\|\mathbf{D}_{t}^{\nu_{i}}\mathfrak{U}\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}+\sum_{j=1}^{N}\|\mathbf{D}_{t}^{\mu_{j}}\mathfrak{U}\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}$
$\displaystyle\leq
C[1+\|u_{0}\|_{\mathcal{C}^{2+\alpha}(\bar{\Omega})}+\|g\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}+\|\psi\|_{\mathcal{C}^{2+\alpha,\frac{2+\alpha}{2}\nu}(\partial\Omega_{T})}]$
$\displaystyle\equiv C\mathfrak{G}(u_{0},f,\psi).$

Here we used assumption h6 (i) and [21, Remark 3.1] to control the term
$\|f(u_{0})\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}$.
Then we look for a solution of the original problem (1.1), (1.2), (1.4) in the
form

$u(x,t)=v(x,t)+\mathfrak{U}(x,t),$

where the unknown function $v$ solves the problem

$\begin{cases}\mathbf{D}_{t}v-\mathcal{L}_{1}v-\mathcal{K}*\mathcal{L}_{2}v=G(x,t)-F(v)\quad\text{in}\quad\Omega_{T},\\\
v(x,0)=0\quad\text{in}\quad\bar{\Omega},\\\
v(x,t)=0\quad\text{on}\quad\partial\Omega_{T}.\end{cases}$ (7.1)

Here we set

$G(x,t)=f(u_{0})-f(\mathfrak{U}),\qquad
F(v)=f(v+\mathfrak{U})-f(\mathfrak{U}).$

######  Remark 7.1.

Assumptions h6 (i) and the estimate of $\mathfrak{U}$ readily ensure the
following relations to the functions $F$ and $G$:

$\|G\|_{\mathcal{C}^{\alpha,\frac{\nu\alpha}{2}}(\bar{\Omega}_{T})}\leq
C\mathfrak{G}(u_{0},f,\psi),$

and for all $v_{i}\in[-\rho,\rho]$ and $v\in\mathbb{R}$ there holds

$|F(v_{1})-F(v_{2})|\leq C_{\rho}|v_{1}-v_{2}|,\quad|F(v)|\leq L^{*}(1+|v|),$

with

$L^{*}=L(1+2\underset{\bar{\Omega}_{T}}{\sup}|\mathfrak{U}|)\leq
L[1+2C\mathfrak{G}(u_{0},f,\psi)].$

Moreover, the direct calculations and the properties of the function
$\mathfrak{U}$ arrive at the equalities:

$G(x,0)=0\quad\text{for}\quad x\in\bar{\Omega},\quad F(0)=0\quad\text{for
each}\quad(x,t)\in\bar{\Omega}_{T}.$

Thus, we easily conclude that $F$ and $G$ meet requirements of Theorem 4.1
and, hence, we are left to prove this theorem in the case of problem (7.1).

To this end, we rely on the so-called continuation arguments, in analogy to
the case of semilinear problem to the subdiffusion equations with a single-
term fractional derivative described in [21, Section 5.2]. This approach deals
with the analysis of the family of problems for $\lambda\in[0,1]$,

$\begin{cases}\mathbf{D}_{t}v-\mathcal{L}_{1}v-\mathcal{K}*\mathcal{L}_{2}v=G(x,t)-\lambda
F(v)\quad\text{in}\,\Omega_{T},\\\ v(x,0)=0\quad\text{in}\quad\bar{\Omega},\\\
v(x,t)=0\quad\text{on}\quad\partial\Omega_{T}.\end{cases}$ (7.2)

Let $\Lambda$ be the set of those $\lambda$ for which (7.2) is solvable on
$[0,T]$. For $\lambda=0$, (7.2) is a linear problem studied in detailed in
[37] (see also Lemma 5.7 here). Hence, keeping in mind assumptions h1-h5, h7
and Remark 7.1, we can apply Lemma 5.7 to (7.2) with $\lambda=0$ and obtain
the global classical solvability in the corresponding classes. Therefore,
$0\in\Lambda$. The next step demonstrates that $\Lambda$ is open and closed at
the same time. To this end, we repeat step-by-step the arguments given in [21,
Section 5.2] and, exploiting Lemma 6.1 (where $f=F$, $g=G$, $u_{0}=0$), we
complete the proof of Theorem 4.1.∎

##  8\. Proof of Theorem 4.4

Here we focus on the proof of this theorem in the case of DBC (1.2). The case
of NBC (1.3) is treated with the similar arguments. We will follow the
strategy consisting in two steps. In the first, we assume the existence of
functions $U_{0,n}$ which approximate the initial data
$u_{0}\in\overset{0}{W}\,^{1,2}(\Omega)\cap W^{2,2}(\Omega)$ and satisfy
assumptions h5 (i) and h7. Other words, the function
$U_{0,n}\in\mathcal{C}_{0}^{\infty}(\bar{\Omega})$ has the properties:

$U_{0,n}\quad\text{converges to}\,\,u_{0}\quad\text{in}\quad
W^{2,2}(\Omega);\quad\begin{cases}U_{0,n}(a)=U_{0,n}(b)=0,\\\
a_{2}(a,0)\frac{\partial^{2}U_{0,n}}{\partial
x^{2}}(a)+a_{1}(a,0)\frac{\partial U_{0,n}}{\partial x}(a)+f(0)=0,\\\
a_{2}(b,0)\frac{\partial^{2}U_{0,n}}{\partial
x^{2}}(b)+a_{1}(b,0)\frac{\partial U_{0,n}}{\partial x}(b)+f(0)=0.\end{cases}$
(8.1)

We discuss subsequently the technique of the construction of $U_{0,n}$.

Appealing to Theorem 4.1, we build smooth solutions
$u_{n}=u_{n}(x,t):\Omega_{T}\to\mathbb{R}$ to the approximated problems

$\mathbf{D}_{t}u_{n}-\mathcal{L}_{1}u_{n}-\mathcal{K}*\mathcal{L}_{2}u_{n}+f(u_{n})=0\quad\text{in}\quad\Omega_{T},\quad
u_{n}(x,0)=U_{0,n}\quad\text{in}\quad\bar{\Omega},\quad
u_{n}(x,t)=0\quad\text{on}\quad\partial\Omega_{T}.$ (8.2)

Then we extract a convergent subsequence and pass to the limit in the
equation. Using the Banach-Alaoglu Theorem and estimates stated in Lemma 6.1
(with $\lambda=1,\,g\equiv 0$), we can apply a standard arguments to choose a
subsequence of $u_{n}$ (which we relabel) such that, for any fixed time $T>0$

$\displaystyle u_{n}$ $\displaystyle\rightharpoonup u\quad\text{weakly
in}\quad L_{2}((0,T),W^{2,2}(\Omega)),$ $\displaystyle K*u_{n}$
$\displaystyle\rightharpoonup K*u\quad\text{weakly in}\quad
L_{2}((0,T),W^{2,2}(\Omega)),$ $\displaystyle u_{n}$ $\displaystyle\rightarrow
u\quad\text{uniformly}\quad\mathcal{C}^{\alpha^{*},\frac{\alpha^{*}\nu}{2}}(\bar{\Omega}_{T})\quad\text{for
any}\quad\alpha^{*}\in(0,1),$ $\displaystyle f(u_{n})$
$\displaystyle\rightarrow
f(u)\quad\text{uniformly}\quad\mathcal{C}(\bar{\Omega}_{T}),$
$\displaystyle\mathbf{D}_{t}^{\nu}u_{n}$
$\displaystyle\rightharpoonup\mathbf{D}_{t}^{\nu}u_{n}\quad\text{weakly
in}\quad L_{2}(\Omega_{T}),$ $\displaystyle\mathbf{D}_{t}^{\nu_{i}}u_{n}$
$\displaystyle\rightharpoonup\mathbf{D}_{t}^{\nu_{i}}u_{n},\,i=1,2,...,M,\quad\text{weakly
in}\quad L_{2}(\Omega_{T}),$ $\displaystyle\mathbf{D}_{t}^{\mu_{j}}u_{n}$
$\displaystyle\rightharpoonup\mathbf{D}_{t}^{\mu_{j}}u_{n},\,j=1,2,...,N,\quad\text{weakly
in}\quad L_{2}(\Omega_{T}),$

for some $u$ belonging to all the spaces above.

Therefore, such $u$ satisfies estimate (6.1) (with $g\equiv 0$) and

$\|u\|_{\mathcal{C}^{\alpha*,\frac{\alpha*\nu}{2}}(\bar{\Omega}_{T})}\leq
C[1+\|u_{0}\|_{W^{2,2}(\Omega)}],$

besides, $u$ satisfies equation (1.1) a.e. in $\Omega_{T}$, along with (1.4)
and homogeneous boundary condition (1.2).

As for the uniqueness this solution, this fact is proved with the standard
arguments. Namely, assuming the existence of two solutions $u$ and $u^{*}$
satisfying the same data, we consider homogenous problem (1.1), (1.2) and
(1.4) for the difference $U^{*}=u-u^{*}$, and recasting the proof leading to
estimate (6.11) (see also Section 6.5 concerning this estimate to whole time
interval), we get

$\|U^{*}\|_{\mathcal{C}([0,T],L_{2}(\Omega))}\leq 0\quad\Rightarrow\quad
u\equiv u^{*}.$

In summary, in order to complete the proof of the existence of a unique strong
solution to (1.1), (1.2) and (1.4) satisfying the regularity required by
Theorem 4.4, we are left to build the function $U_{0,n}$ satisfying (8.1). To
this end, we introduce new function

$U_{0}(x)=u_{0}(x)-P_{u_{0}}(x),$

where $P_{u_{0}}(x)$ is the $5^{\text{th}}$ degrees polynomial satisfying
equalities

$\begin{cases}P_{u_{0}}(a)=P_{u_{0}}(b)=0,\\\
P^{\prime}_{u_{0}}(a)=u^{\prime}_{0}(a),\\\
P^{\prime}_{u_{0}}(b)=u^{\prime}_{0}(b),\\\
P^{\prime\prime}_{u_{0}}(a)=\frac{1}{a_{2}(a,0)}[-f(0)-a_{1}(a,0)u^{\prime}_{0}(a)],\\\
P^{\prime\prime}_{u_{0}}(b)=\frac{1}{a_{2}(b,0)}[-f(0)-a_{1}(b,0)u^{\prime}_{0}(b)].\end{cases}$

Here we used the smoothness of $u_{0}$ and embedding theorem which provides
existence (in the classical sense) of $u^{\prime}_{0}(a)$ and
$u^{\prime}_{0}(b)$.

We easily conclude that $U_{0}(x)\in\overset{0}{W}\,^{2,2}(\Omega)$ and,
hence, we can approximate $U_{0}(x)$ by the functions
$\bar{U}_{0,n}\in\mathcal{C}_{0}^{\infty}(\bar{\Omega})$ in the norm of
$W^{2,2}(\Omega)$. Then, setting

$U_{0,n}(x)=\bar{U}_{0,n}(x)+P_{u_{0}}(x),$

and performing standard calculations, we conclude that
$U_{0,n}(x)\in\mathcal{C}^{2+\alpha}(\bar{\Omega})$ and satisfies conditions
(8.1). This means that $U_{0,n}$ satisfies assumptions of Theorem 4.1 and
approximate the initial data $u_{0}$ in the corresponding classes. Finally,
coming to (8.2), we carry out the procedure described above and complete the
proof of Theorem 4.4. ∎.

##  9\. Numerical Simulations

We finally presents some numerical tests aimed to illustrate our theoretical
results (Theorems 4.1 and 4.4) and demonstrate some effects of fractional
derivatives which do not have yet analytical proofs. Namely, if all
derivatives in the operator $\mathbf{D}_{t}$ (see (1.5)) are fractional, then
we observe rapid changes in the solutions for small time, and the solutions
slow down as time goes up (see Example 9.2). It differs from the case $\nu=1$
in (1.5), where this effect is negligible (and is diminishing as fractional
orders grow, as we observe in Example 9.2).

For simplicity, we focus on the initial-boundary value problem in the one-
dimensional domain $\Omega=(0,1)$ (multi-dimensional generalization of the
proposed finite-difference scheme is straightforward and boils down to adding
new terms that approximate the occurring new spatial partial derivatives in a
completely similar manner):

$\begin{cases}\mathbf{D}_{t}^{\nu}(\varrho_{0}u)+\mathbf{D}_{t}^{\nu_{1}}(\varrho_{1}u)-\mathbf{D}_{t}^{\mu_{1}}(\gamma_{1}u)-\mathfrak{a}\frac{\partial^{2}u}{\partial
x^{2}}+\mathfrak{d}\frac{\partial u}{\partial x}\
-(\mathcal{K}*b\frac{\partial^{2}u}{\partial
x^{2}})=f(x,t,u)+g(x,t)\quad\text{in }\Omega_{T},\\\ \vskip 0.85358pt\cr
u(x,0)=u_{0}(x),\qquad\qquad\qquad\quad x\in[0,1],\\\ \vskip
5.69054pt\cr\mathfrak{c}_{1}\frac{\partial u}{\partial
x}(0,t)+\mathfrak{c}_{2}u(0,t)=\varphi_{1}(t),\quad t\in[0,T],\\\ \vskip
5.69054pt\cr\mathfrak{c}_{3}\frac{\partial u}{\partial
x}(1,t)+\mathfrak{c}_{4}u(1,t)=\varphi_{2}(t),\quad t\in[0,T].\end{cases}$
(9.1)

We introduce the space-time mesh with nodes

$x_{k}=kh,\quad\sigma_{j}=j\sigma,\quad k=0,1,\ldots,K,\quad
j=0,1,\ldots,J,\quad h=L/K,\quad\sigma=T/J.$

For these examples, we actually take $L=1$. Denoting the finite-difference
approximation of the solution $u$ at the point $(x_{k},\sigma_{j})$ by
$u^{j}_{k}$ and calling

$\displaystyle\mathfrak{a}^{j+1}_{k}=\mathfrak{a}(x_{k},\sigma_{j+1}),\qquad\mathfrak{d}^{j+1}_{k}=\mathfrak{d}(x_{k},\sigma_{j+1}),\qquad
b^{j}_{k}=b(x_{k},\sigma_{j}),$
$\displaystyle\mathcal{K}_{m,j}=\int_{\sigma_{m}}^{\sigma_{m+1}}{\mathcal{K}}(\sigma_{j+1}-s)ds,\qquad\rho_{m}=(-1)^{m}\binom{\nu}{m},\qquad\tilde{\rho}_{m}=(-1)^{m}\binom{\nu_{1}}{m},\qquad\bar{\rho}_{m}=(-1)^{m}\binom{\mu_{1}}{m},$
$\displaystyle\varrho_{0,k}^{j+1}=\varrho_{0}(x_{k},\sigma_{j+1}),\qquad\varrho^{j+1}_{1,k}=\varrho_{1}(x_{k},\sigma_{j+1}),\qquad\gamma^{j+1}_{1,k}=\gamma_{1}(x_{k},\sigma_{j+1}),$

we approximate the differential equation in (9.1) at each time level
$\sigma_{j+1}$ and spatial point $x_{k}$, so to obtain the finite-difference
scheme

$\displaystyle\sigma^{-\nu}\sum\limits_{m=0}^{j+1}(\varrho_{0,k}^{j+1-m}u^{j+1-m}_{k}-\varrho_{0,k}^{0}u_{0}(x_{k}))\rho_{m}+\sigma^{-\nu_{1}}\sum\limits_{m=0}^{j+1}(\varrho_{1,k}^{j+1-m}u^{j+1-m}_{k}-\varrho_{1,k}^{0}u_{0}(x_{k}))\tilde{\rho}_{m}$
$\displaystyle-\sigma^{-\mu_{1}}\sum\limits_{m=0}^{j+1}(\gamma_{1,k}^{j+1-m}u^{j+1-m}_{k}-\gamma_{1,k}^{0}u_{0}(x_{k}))\bar{\rho}_{m}-\frac{\mathfrak{a}^{j+1}_{k}}{h^{2}}(u^{j+1}_{k-1}-2u^{j+1}_{k}+u^{j+1}_{k+1})+\frac{\mathfrak{d}^{j+1}_{k}}{2h}(u^{j+1}_{k+1}-u^{j+1}_{k-1})$
$\displaystyle=\sum_{m=0}^{j}\left(b^{m}_{k}\frac{u^{m}_{k-1}-2u^{m}_{k}+u^{m}_{k+1}}{h^{2}}+b^{m+1}_{k}\frac{u^{m+1}_{k-1}-2u^{m+1}_{k}+u^{m+1}_{k+1}}{h^{2}}\right)\\!\frac{\mathcal{K}_{m,j}}{2}+f(x_{k},\sigma_{j},u_{k}^{j})+g(x_{k},\sigma_{j+1}),$

for

$k=1,\ldots,K-1\qquad\text{and}\qquad j=0,1,\ldots,J-1.$

Here, the derivatives $u_{x}$ and $u_{xx}$ are approximated by the standard
central finite-difference formulas, the trapezoid rule is employed to
approximate the integrals in the sum

$\sum_{m=0}^{j}\int^{\sigma_{m+1}}_{\sigma_{m}}{\mathcal{K}}(\sigma_{j+1}-s)b(x,s)u_{xx}(x,s)ds$

representing the convolution term in (9.1), and the Grünwald-Letnikov (GL)
formula [17, 8, 15] is applied to approximate the fractional derivatives
$\mathbf{D}_{t}^{\nu}(\varrho_{0}u)$, $\mathbf{D}_{t}^{\nu_{1}}(\varrho_{1}u)$
and $\mathbf{D}_{t}^{\mu_{1}}(\gamma_{1}u)$. Alternatively, here one might
also use the “Leibniz rule” for Caputo derivatives (see [22, Corollary 3.1])
to reduce the treating of fractional derivatives terms in (9.1) to the
approximation of fractional derivatives of the segregated function $u$ only
(but not its products with $\varrho_{i}$ or $\gamma_{i}$) by the cost of
adding integral extra terms (which can be approximated using the same approach
as it was done for the convolution term in (9.1)) and making slight changes in
the right part $g$. However we will not pursue this way further here. Also, in
order to achieve an improvement in the temporal discretization accuracy
(primarily, owing to the approximation of the fractional derivatives) we apply
the Richardson extrapolation (see [8]). Finally, two fictitious mesh points
outside the spatial domain to approximate the derivatives in the boundary
conditions with the second order of accuracy are exploited [37, 23]. Further
improvement in the accuracy of calculations may be reached by resorting to
finite element methods [15, 40], albeit we do not have the possibility to
pursue this direction further here.

######  Example 9.1.

Consider problem (9.1) with $T=1$ and

$\displaystyle\mathcal{K}(t)$
$\displaystyle=t^{-1/3},\quad\mathfrak{a}(x,t)=\cos(\pi
x/4)+t,\quad\mathfrak{d}(x,t)=x+t,\quad b(x,t)=t^{1/3}+\sin(\pi x),\quad
u_{0}(x)=\cos(\pi x),$ $\displaystyle\varrho_{0}(t)$
$\displaystyle=1+t,\quad\varrho_{1}=1/2,\quad\gamma_{1}(t)=(1+t^{2})/2,\quad\mathfrak{c}_{1}=\mathfrak{c}_{3}=1,\quad\mathfrak{c}_{2}=\mathfrak{c}_{4}=0,\quad\varphi_{1}(t)=\varphi_{2}(t)=0,$