∎

11institutetext: Yuri Luchko1,∗ 22institutetext: Department of Mathematics,
Physics, and Chemistry, Berlin University of Applied Sciences and Technology,
Luxemburger Str. 10, Berlin – 13353, Germany
22email<EMAIL_ADDRESS>∗ corresponding author 33institutetext: Masahiro
Yamamto2 44institutetext: Department of Mathematical Sciences, The University
of Tokyo, Komaba, Meguro, Tokyo – 153, Japan
Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti,
Romania
Correspondence member of Accademia Peloritana dei Pericolanti, Palazzo
Università, Piazza S. Pugliatti 1, Messina – 98122, Italy
44email<EMAIL_ADDRESS>
# Comparison principles for the time-fractional diffusion equations with the
Robin boundary conditions. Part I: Linear equations

Yuri Luchko1 Masahiro Yamamoto2

(Received: XX 2023 / Revised: …. / Accepted: ……)

###### Abstract

The main objective of this paper is analysis of the initial-boundary value
problems for the linear time-fractional diffusion equations with a uniformly
elliptic spatial differential operator of the second order and the Caputo type
time-fractional derivative acting in the fractional Sobolev spaces. The
boundary conditions are formulated in form of the homogeneous Neumann or Robin
conditions. First we discuss the uniqueness and existence of solutions to
these initial-boundary value problems. Under some suitable conditions on the
problem data, we then prove positivity of the solutions. Based on these
results, several comparison principles for the solutions to the initial-
boundary value problems for the linear time-fractional diffusion equations are
derived.

###### Keywords:

fractional calculus (primary), fractional diffusion equation, positivity of
solutions, comparison principle

###### MSC:

35B51 (primary) 35R11 26A33

††journal: Fract. Calc. Appl. Anal.

##  1 Introduction

In this paper, we deal with a linear time-fractional diffusion equation in the
form

$\partial_{t}^{\alpha}(u(x,t)-a(x))=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}u(x,t))$
$+\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}u(x,t)+c(x,t)u(x,t)+F(x,t),\quad
x\in\Omega,\,0<t<T,$ (1.1)

where $\partial_{t}^{\alpha}$ is the Caputo fractional derivative of order
$\alpha\in(0,1)$ defined on the fractional Sobolev spaces (see Section 2 for
the details) and $\Omega\subset\mathbb{R}^{d},\ d=1,2,3$ is a bounded domain
with a smooth boundary $\partial\Omega$. All the functions under consideration
are supposed to be real-valued.

In what follows, we always assume that the following conditions are satisfied:

$\left\\{\begin{array}[]{rl}&a_{ij}=a_{ji}\in C^{1}(\overline{\Omega}),\quad
1\leq i,j\leq d,\\\ &b_{j},\,c\in C^{1}([0,T];C^{1}(\overline{\Omega}))\cap
C([0,T];C^{2}(\overline{\Omega})),\quad 1\leq j\leq d,\\\ &\mbox{and there
exists a constant $\kappa>0$ such that}\\\
&\sum_{i,j=1}^{d}a_{ij}(x)\xi_{i}\xi_{j}\geq\kappa\sum_{j=1}^{d}\xi_{j}^{2},\quad
x\in\Omega,\,\xi_{1},...,\xi_{d}\in\mathbb{R}.\end{array}\right.$ (1.2)

Using the notations $\partial_{j}=\frac{\partial}{\partial x_{j}}$,
$j=1,2,...,d$, we define a conormal derivative $\partial_{\nu_{A}}w$ with
respect to the differential operator
$\sum_{i,j=1}^{d}\partial_{j}(a_{ij}\partial_{i})$ by

$\partial_{\nu_{A}}w(x)=\sum_{i,j=1}^{d}a_{ij}(x)\partial_{j}w(x)\nu_{i}(x),\quad
x\in\partial\Omega,$ (1.3)

where $\nu=\nu(x)=:(\nu_{1}(x),...,\nu_{d}(x))$ is the unit outward normal
vector to $\partial\Omega$ at the point
$x:=(x_{1},...,x_{d})\in\partial\Omega$.

For the equation (1.1), we consider the initial-boundary value problems with
the homogeneous Neumann boundary condition

$\partial_{\nu_{A}}u=0\quad\mbox{on $\partial\Omega\times(0,T)$}$ (1.4)

or the more general homogeneous Robin boundary condition

$\partial_{\nu_{A}}u+\sigma(x)u=0\quad\mbox{on $\partial\Omega\times(0,T)$},$
(1.5)

where $\sigma$ is a sufficiently smooth function on $\partial\Omega$ that
satisfies the condition $\sigma(x)\geq 0,\ x\in\partial\Omega$.

For partial differential equations of the parabolic type that correspond to
the case $\alpha=1$ in the equation (1.1), several important qualitative
properties of solutions to the corresponding initial-boundary value problems
are known. In particular, we mention a maximum principle and a comparison
principle for the solutions to these problems (PW , RR ).

The main purpose of this paper is the comparison principles for the linear
time-fractional diffusion equation (1.1) with the Neumann or the Robin
boundary conditions.

For the equations of type (1.1) with the Dirichlet boundary conditions, the
maximum principles in different formulations were derived and used in Bor ;
Lu1 ; luchko-1 ; luchko-2 ; Lu2 ; LY1 ; LY2 ; LY3 ; Za . For a maximum
principle for the time-fractional transport equations we refer to LSY . In Kir
, a maximum principle for the more general space- and time-space-fractional
partial differential equations has been derived.

Because any maximum principle involves the Dirichlet boundary values, its
formulation in the case of the Neumann or Robin boundary conditions requires
more cares. For this kind of the boundary conditions, both positivity of
solutions and the comparison principles can be derived under some suitable
restrictions on the problem data. One typical result of this sort says that
the solution $u$ to the equation (1.1) with the boundary condition (1.4) or
(1.5) and an appropriately formulated initial condition is non-negative in
$\Omega\times(0,T)$ if the initial value $a$ and the non-homogeneous term $F$
are non-negative in $\Omega$ and in $\Omega\times(0,T)$, respectively. Such
positivity properties and their applications have been intensively discussed
and used for the partial differential equations of parabolic type ($\alpha=1$
in the equation (1.1)), see, e.g., E , Fr , Pao2 , or RR .

However, to the best knowledge of the authors, no results of this kind have
been published for the time-fractional diffusion equations in the case of the
Neumann or Robin boundary conditions. The main subject of this paper is in
derivation of a positivity property and the comparison principles for the
linear equation (1.1) with the boundary condition (1.4) or (1.5) and an
appropriately formulated initial condition. In the subsequent paper, these
result will be extended to the case of the semilinear time-fractional
diffusion equations. The arguments employed in these papers rely on an
operator theoretical approach to the fractional integrals and derivatives in
the fractional Sobolev spaces that is an extension of the theory well-known in
the case $\alpha=1$, see, e.g., He , Pa , Ta . We also refer to the recent
publications Al-R  and Lu  devoted to the comparison principles for solutions
to the fractional differential inequalities with the general fractional
derivatives and for solutions to the ordinary fractional differential
equations, respectively.

The rest of this paper is organized as follows. In Section 2, some important
results regarding the unique existence of solutions to the initial-boundary
value problems for the linear time-fractional diffusion equations are
presented. Section 3 is devoted to a proof of a key lemma that is a basis for
the proofs of the comparison principles for the linear and semilinear time-
fractional diffusion equations. The lemma asserts that each solution to (1.1)
is non-negative in $\Omega\times(0,T)$ if $a\geq 0$ and $F\geq 0$, provided
that $u$ is assumed to satisfy some extra regularity. In Section 4, we prove a
comparison principle that is our main result for the problem (1.1) for the
linear time-fractional diffusion equation. Moreover, we establish the order-
preserving properties for other problem data (the zeroth-order coefficient $c$
of the equation and the coefficient $\sigma$ of the Robin condition). Finally,
a detailed proof of an important auxiliary statement is presented in an
Appendix.

##  2 Well-posedness results

For $x\in\Omega,\thinspace 0<t<T$, we define an operator

$-Av(x,t):=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v(x,t)+\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}v(x,t)+c(x,t)v(x,t)$
(2.1)

and assume that the conditions (1.2) for the coefficients $a_{ij},b_{j},c$ are
satisfied.

In this section, we deal with the following initial-boundary value problem for
the linear time-fractional diffusion equation (1.1) with the time-fractional
derivative of order $\alpha\in(0,1)$

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u(x,t)-a(x))+Au(x,t)=F(x,t),\quad
x\in\Omega,\thinspace 0<t<T,\\\ &\partial_{\nu_{A}}u+\sigma(x)u(x,t)=0,\quad
x\in\partial\Omega,\,0<t<T,\end{array}\right.$ (2.2)

along with the initial condition (2.3) formulated below.

To appropriately define the Caputo fractional derivative $d_{t}^{\alpha}w(t)$,
$0<\alpha<1$, we start with its definition on the space

${{}_{0}C^{1}[0,T]}:=\\{u\in C^{1}[0,T];\thinspace u(0)=0\\}$

that reads as follows:

$d_{t}^{\alpha}w(t)=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}\frac{dw}{ds}(s)ds,\
w\in{{}_{0}C^{1}[0,T]}.$

Then we extend this operator from the domain
$\mathcal{D}(d_{t}^{\alpha}):={{}_{0}C^{1}[0,T]}$ to $L^{2}(0,T)$ taking into
account its closability (Yo ). As have been shown in KRY , there exists a
unique minimum closed extension of $d_{t}^{\alpha}$ with the domain
$\mathcal{D}(d_{t}^{\alpha})={{}_{0}C^{1}[0,T]}$. Moreover, the domain of this
extension is the closure of ${{}_{0}C^{1}[0,T]}$ in the Sobolev-Slobodeckij
space $H^{\alpha}(0,T)$. Let us recall that the norm
$\|\cdot\|_{H^{\alpha}(0,T)}$ of the Sobolev-Slobodeckij space
$H^{\alpha}(0,T)$ is defined as follows (Ad ):

$\|v\|_{H^{\alpha}(0,T)}:=\left(\|v\|^{2}_{L^{2}(0,T)}+\int^{T}_{0}\int^{T}_{0}\frac{|v(t)-v(s)|^{2}}{|t-s|^{1+2\alpha}}dtds\right)^{\frac{1}{2}}.$

By setting

$H_{\alpha}(0,T):=\overline{{{}_{0}C^{1}[0,T]}}^{H^{\alpha}(0,T)},$

we obtain (KRY )

$H_{\alpha}(0,T)=\left\\{\begin{array}[]{rl}&H^{\alpha}(0,T),\quad
0<\alpha<\frac{1}{2},\\\ &\left\\{v\in
H^{\frac{1}{2}}(0,T);\,\int^{T}_{0}\frac{|v(t)|^{2}}{t}dt<\infty\right\\},\quad\alpha=\frac{1}{2},\\\
&\\{v\in
H^{\alpha}(0,T);\,v(0)=0\\},\quad\frac{1}{2}<\alpha<1,\end{array}\right.$

and

$\|v\|_{H_{\alpha}(0,T)}=\left\\{\begin{array}[]{rl}&\|v\|_{H^{\alpha}(0,T)},\quad\alpha\neq\frac{1}{2},\\\
&\left(\|v\|_{H^{\frac{1}{2}}(0,T)}^{2}+\int^{T}_{0}\frac{|v(t)|^{2}}{t}dt\right)^{\frac{1}{2}},\quad\alpha=\frac{1}{2}.\end{array}\right.$

In what follows, we also use the Riemann-Liouville fractional integral
operator $J^{\beta}$, $\beta>0$ defined by

$(J^{\beta}f)(t):=\frac{1}{\Gamma(\beta)}\int^{t}_{0}(t-s)^{\beta-1}f(s)ds,\quad
0<t<T.$

Then, according to GLY  and KRY ,

$H_{\alpha}(0,T)=J^{\alpha}L^{2}(0,T),\quad 0<\alpha<1.$

Next we define

$\partial_{t}^{\alpha}=(J^{\alpha})^{-1}\quad\mbox{with
$\mathcal{D}(\partial_{t}^{\alpha})=H_{\alpha}(0,T)$}.$

As have been shown in GLY  and KRY , there exists a constant $C>0$ depending
only on $\alpha$ such that

$C^{-1}\|v\|_{H_{\alpha}(0,T)}\leq\|\partial_{t}^{\alpha}v\|_{L^{2}(0,T)}\leq
C\|v\|_{H_{\alpha}(0,T)}\quad\mbox{for all }v\in H_{\alpha}(0,T).$

Now we can introduce a suitable form of initial condition for the problem
(2.2) as follows

$u(x,\cdot)-a(x)\in H_{\alpha}(0,T)\quad\mbox{for almost all }x\in\Omega$
(2.3)

and write down a complete formulation of an initial-boundary value problem for
the linear time-fractional diffusion equation (1.1):

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u(x,t)-a(x))+Au(x,t)=F(x,t),\quad
x\in\Omega,\thinspace 0<t<T,\\\
&\partial_{\nu_{A}}u(x,t)+\sigma(x)u(x,t)=0,\quad
x\in\partial\Omega,\,0<t<T,\\\ &u(x,\cdot)-a(x)\in
H_{\alpha}(0,T)\quad\mbox{for almost all }x\in\Omega.\end{array}\right.$ (2.4)

It is worth mentioning that the term $\partial_{t}^{\alpha}(u(x,t)-a(x))$ in
the first line of (2.4) is well-defined due to inclusion formulated in the
third line of (2.4). In particular, for $\frac{1}{2}<\alpha<1$, the Sobolev
embedding leads to the inclusions $H_{\alpha}(0,T)\subset
H^{\alpha}(0,T)\subset C[0,T]$. This means that $u\in
H_{\alpha}(0,T;L^{2}(\Omega))$ implies $u\in C([0,T];L^{2}(\Omega))$ and thus
in this case the initial condition can be formulated as $u(\cdot,0)=a$ in
$L^{2}$-sense. Moreover, for sufficiently smooth functions $a$ and $F$, the
solution to (2.4) can be proved to satisfy the initial condition in a usual
sense: $\lim_{t\to 0}u(\cdot,t)=a$ in $L^{2}(\Omega)$ (see Lemma 4 in Section
3). Consequently, the third line of (2.4) can be interpreted as a generalized
initial condition.

In the following theorem, a fundamental result regarding the unique existence
of the solution to the initial-boundary value problem (2.4) is presented.

###### Theorem 2.1

For $a\in H^{1}(\Omega)$ and $F\in L^{2}(0,T;L^{2}(\Omega))$, there exists a
unique solution $u(F,a)=u(F,a)(x,t)\in L^{2}(0,T;H^{2}(\Omega))$ to the
initial-boundary value problem (2.4) such that $u(F,a)-a\in
H_{\alpha}(0,T;L^{2}(\Omega))$.

Moreover, there exists a constant $C>0$ such that

$\displaystyle\|u(F,a)-a\|_{H_{\alpha}(0,T;L^{2}(\Omega))}+\|u(F,a)\|_{L^{2}(0,T;H^{2}(\Omega))}$
$\displaystyle\leq$ $\displaystyle
C(\|a\|_{H^{1}(\Omega)}+\|F\|_{L^{2}(0,T;L^{2}(\Omega))}).$

Before starting with a proof of Theorem 2.1, we introduce some notations and
derive several helpful results needed for the proof.

For an arbitrary constant $c_{0}>0$, we define an elliptic operator $A_{0}$ as
follows:

$\left\\{\begin{array}[]{rl}&(-A_{0}v)(x):=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v(x))-c_{0}v(x),\quad
x\in\Omega,\\\ &\mathcal{D}(A_{0})=\left\\{v\in
H^{2}(\Omega);\,\partial_{\nu_{A}}v+\sigma v=0\quad\mbox{on
}\partial\Omega\right\\}.\end{array}\right.$ (2.5)

We recall that in the definition (2.5), $\sigma$ is a smooth function, the
inequality $\sigma(x)\geq 0,\ x\in\partial\Omega$ holds true, and the
coefficients $a_{ij}$ satisfy the conditions (1.2).

Henceforth, by $\|\cdot\|$ and $(\cdot,\cdot)$ we denote the standard norm and
the scalar product in $L^{2}(\Omega)$, respectively. It is well-known that the
operator $A_{0}$ is self-adjoint and its resolvent is a compact operator.
Moreover, for a sufficiently large constant $c_{0}>0$, by Lemma 6 in Section
5, we can verify that $A_{0}$ is positive definite. Therefore, by choosing the
constant $c_{0}>0$ large enough, the spectrum of $A_{0}$ consists entirely of
discrete positive eigenvalues $0<\lambda_{1}\leq\lambda_{2}\leq\cdots$, which
are numbered according to their multiplicities and $\lambda_{n}\to\infty$ as
$n\to\infty$. Let $\varphi_{n}$ be an eigenvector corresponding to the
eigenvalue $\lambda_{n}$ such that $A\varphi_{n}=\lambda_{n}\varphi_{n}$ and
$(\varphi_{n},\varphi_{m})=0$ if $n\neq m$ and $(\varphi_{n},\varphi_{n})=1$.
Then the system $\\{\varphi_{n}\\}_{n\in\mathbb{N}}$ of the eigenvectors forms
an orthonormal basis in $L^{2}(\Omega)$ and for any $\gamma\geq 0$ we can
define the fractional powers $A_{0}^{\gamma}$ of the operator $A_{0}$ by the
following relation (see, e.g., Pa ):

$A_{0}^{\gamma}v=\sum_{n=1}^{\infty}\lambda_{n}^{\gamma}(v,\varphi_{n})\varphi_{n},$

where

$v\in\mathcal{D}(A_{0}^{\gamma}):=\left\\{v\in
L^{2}(\Omega):\thinspace\sum_{n=1}^{\infty}\lambda_{n}^{2\gamma}(v,\varphi_{n})^{2}<\infty\right\\}$

and

$\|A_{0}^{\gamma}v\|=\left(\sum_{n=1}^{\infty}\lambda_{n}^{2\gamma}(v,\varphi_{n})^{2}\right)^{\frac{1}{2}}.$

We note that $\mathcal{D}(A_{0}^{\gamma})\subset H^{2\gamma}(\Omega)$.

Our proof of Theorem 2.1 is similar to the one presented in GLY , KRY  for the
case of the homogeneous Dirichlet boundary condition. In particular, we employ
the operators $S(t)$ and $K(t)$ defined by (GLY , KRY )

$S(t)a=\sum_{n=1}^{\infty}E_{\alpha,1}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n},\quad
a\in L^{2}(\Omega),\thinspace t>0$ (2.6)

and

$K(t)a=-A_{0}^{-1}S^{\prime}(t)a=\sum_{n=1}^{\infty}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})(a,\varphi_{n})\varphi_{n},\quad
a\in L^{2}(\Omega),\thinspace t>0.$ (2.7)

In the above formulas, $E_{\alpha,\beta}(z)$ denotes the Mittag-Leffler
function defined by a convergent series as follows:

$E_{\alpha,\beta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(\alpha\,k+\beta)},\
\alpha>0,\ \beta\in\mathbb{C},\ z\in\mathbb{C}.$

It follows directly from the definitions given above that
$A_{0}^{\gamma}K(t)a=K(t)A_{0}^{\gamma}a$ and
$A_{0}^{\gamma}S(t)a=S(t)A_{0}^{\gamma}a$ for
$a\in\mathcal{D}(A_{0}^{\gamma})$. Moreover, the inequality (see, e.g.,
Theorem 1.6 (p. 35) in Po )

$\max\\{|E_{\alpha,1}(-\lambda_{n}t^{\alpha})|,\,|E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})|\\}\leq\frac{C}{1+\lambda_{n}t^{\alpha}}\quad\mbox{for
all $t>0$}$

implicates the estimations (GLY )

$\left\\{\begin{array}[]{l}\|A_{0}^{\gamma}S(t)a\|\leq
Ct^{-\alpha\gamma}\|a\|,\\\ \|A_{0}^{\gamma}K(t)a\|\leq
Ct^{\alpha(1-\gamma)-1}\|a\|,\quad a\in L^{2}(\Omega),\thinspace
t>0,\thinspace 0\leq\gamma\leq 1.\end{array}\right.$ (2.8)

In order to shorten the notations and to focus on the dependence on the time
variable $t$, henceforth we sometimes omit the variable $x$ in the functions
of two variables $x$ and $t$ and write, say, $u(t)$ instead of $u(\cdot,t)$.

Due to the inequalities (2.8), the estimations provided in the formulation of
Theorem 2.1 can be derived as in the case of the fractional powers of
generators of the analytic semigroups (He ). To do this, we first formulate
and prove the following lemma:

###### Lemma 1

Under the conditions formulated above, the following estimates hold true for
$F\in L^{2}(0,T;L^{2}(\Omega))$ and $a\in L^{2}(\Omega)$:

(i)

$\left\|\int^{t}_{0}A_{0}K(t-s)F(s)ds\right\|_{L^{2}(0,T;L^{2}(\Omega))}\leq
C\|F\|_{L^{2}(0,T;L^{2}(\Omega))},$

(ii)

$\left\|\int^{t}_{0}K(t-s)F(s)ds\right\|_{H_{\alpha}(0,T;L^{2}(\Omega))}\leq
C\|F\|_{L^{2}(0,T;L^{2}(\Omega))},$

(iii)

$\|S(t)a-a\|_{H_{\alpha}(0,T;L^{2}(\Omega))}+\|S(t)a\|_{L^{2}(0,T;H^{2}(\Omega))}\leq
C\|a\|.$

###### Proof

We start with proving the estimate (i). By (2.7), we have

$\displaystyle\left\|\int^{t}_{0}A_{0}K(t-s)F(s)ds\right\|^{2}$
$\displaystyle=$
$\displaystyle\left\|\sum_{n=1}^{\infty}\left(\int^{t}_{0}\lambda_{n}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(t-s)^{\alpha})(F(s),\varphi_{n})ds\right)\varphi_{n}\right\|^{2}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\left|\int^{t}_{0}\lambda_{n}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(t-s)^{\alpha})(F(s),\varphi_{n})ds\right|^{2}.$

Therefore, using the Parseval equality and the Young inequality for the
convolution, we obtain

$\displaystyle\left\|\int^{t}_{0}A_{0}K(t-s)F(s)ds\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\int^{T}_{0}|(\lambda_{n}s^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}s^{\alpha})\,*\,(F(s),\varphi_{n})|^{2}ds$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\|\lambda_{n}s^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}s^{\alpha})\,*\,(F(s),\varphi_{n})\|^{2}_{L^{2}(0,T)}$
$\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}\left(\lambda_{n}\int^{t}_{0}|t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})|dt\right)^{2}\|(F(t),\varphi_{n})\|^{2}_{L^{2}(0,T)}.$

Then we employ the representation

$\frac{d}{dt}E_{\alpha,1}(-\lambda_{n}t^{\alpha})=-\lambda_{n}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha}),$
(2.9)

and the complete monotonicity of the Mittag-Leffler function (GKMR )

$E_{\alpha,1}(-\lambda_{n}t^{\alpha})>0,\quad\frac{d}{dt}E_{\alpha,1}(-\lambda_{n}t^{\alpha})\leq
0,\quad t\geq 0,\quad 0<\alpha\leq 1$

to get the inequality

$\int^{T}_{0}|\lambda_{n}t^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})|dt=-\int^{T}_{0}\frac{d}{dt}E_{\alpha,1}(-\lambda_{n}t^{\alpha})dt$
(2.10) $=1-E_{\alpha,1}(-\lambda_{n}T^{\alpha})\leq 1\quad\mbox{for all
$n\in\mathbb{N}$}.$

Hence,

$\displaystyle\left\|\int^{t}_{0}A_{0}K(t-s)F(s)ds\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}\leq\sum_{n=1}^{\infty}\|(F(t),\varphi_{n})\|^{2}_{L^{2}(0,T)}$
$\displaystyle=$
$\displaystyle\int^{T}_{0}\sum_{n=1}^{\infty}|(F(t),\varphi_{n})|^{2}dt=\int^{T}_{0}\|F(\cdot,t)\|^{2}dt=\|F\|_{L^{2}(0,T;L^{2}(\Omega))}^{2}.$

Now we proceed with proving the estimate (ii). For $0<t<T,\,n\in\mathbb{N}$
and $f\in L^{2}(0,T)$, we set

$(L_{n}f)(t):=\int^{t}_{0}(t-s)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(t-s)^{\alpha})f(s)ds.$

Then

$\int^{t}_{0}K(t-s)F(s)ds=\sum_{n=1}^{\infty}(L_{n}f)(t)\varphi_{n}$

in $L^{2}(\Omega)$ for any fixed $t\in[0,T]$.

First we prove that

$\left\\{\begin{array}[]{rl}&L_{n}f\in H_{\alpha}(0,T),\\\
&\partial_{t}^{\alpha}(L_{n}f)(t)=-\lambda_{n}L_{n}f(t)+f(t),\quad 0<t<T,\\\
&\|L_{n}f\|_{H_{\alpha}(0,T)}\leq C\|f\|_{L^{2}(0,T)},\quad
n\in\mathbb{N}\quad\mbox{for each }f\in L^{2}(0,T).\end{array}\right.$ (2.11)

In order to prove this, we apply the Riemann-Liouville fractional integral
$J^{\alpha}$ to $L_{n}f$ and get the representation

$\displaystyle
J^{\alpha}(L_{n}f)(t)=\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}(L_{n}f)(s)ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{0}(t-s)^{\alpha-1}\left(\int^{s}_{0}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(s-\xi)^{\alpha})f(\xi)d\xi\right)ds$
$\displaystyle=$
$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{0}f(\xi)\left(\int^{t}_{\xi}(t-s)^{\alpha-1}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(s-\xi)^{\alpha})ds\right)d\xi.$

By direct calculations, using (2.9), we obtain the formula

$\displaystyle\frac{1}{\Gamma(\alpha)}\int^{t}_{\xi}(t-s)^{\alpha-1}(s-\xi)^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}(s-\xi)^{\alpha})ds$
$\displaystyle=$
$\displaystyle-\frac{1}{\lambda_{n}}(t-\xi)^{\alpha-1}\left(E_{\alpha,\alpha}(-\lambda_{n}t^{\alpha})-\frac{1}{\Gamma(\alpha)}\right).$

Therefore, we have the relation

$\displaystyle
J^{\alpha}(L_{n}f)(t)=-\frac{1}{\lambda_{n}}(L_{n}f)(t)+\frac{1}{\lambda_{n}}\int^{t}_{0}(t-\xi)^{\alpha-1}\frac{1}{\Gamma(\alpha)}f(\xi)d\xi$
$\displaystyle=$
$\displaystyle-\frac{1}{\lambda_{n}}(L_{n}f)(t)+\frac{1}{\lambda_{n}}(J^{\alpha}f)(t),\quad
n\in\mathbb{N},$

that is,

$(L_{n}f)(t)=-\lambda_{n}J^{\alpha}(L_{n}f)(t)+(J^{\alpha}f)(t),\quad 0<t<T.$

Hence, $L_{n}f\in H_{\alpha}(0,T)=J^{\alpha}L^{2}(0,T)$. By definition,
$\partial_{t}^{\alpha}=(J^{\alpha})^{-1}$ (KRY ) and thus the last formula can
be rewritten in the form

$\partial_{t}^{\alpha}(L_{n}f)=-\lambda_{n}L_{n}f+f\quad\mbox{in }(0,T).$

Using the inequality (2.10), we obtain

$\lambda_{n}\|L_{n}f\|_{L^{2}(0,T)}\leq\lambda_{n}\|s^{\alpha-1}E_{\alpha,\alpha}(-\lambda_{n}s^{\alpha})\|_{L^{1}(0,T)}\|f\|_{L^{2}(0,T)}\leq\|f\|_{L^{2}(0,T)}.$

Therefore,

$\displaystyle\|L_{n}f\|_{H_{\alpha}(0,T)}\leq
C\|\partial_{t}^{\alpha}(L_{n}f)\|_{L^{2}(0,T)}\leq
C(\|-\lambda_{n}L_{n}f\|_{L^{2}(0,T)}+\|f\|_{L^{2}(0,T)})$ $\displaystyle\leq$
$\displaystyle C\|f\|_{L^{2}(0,T)},\quad n\in\mathbb{N},\quad f\in
L^{2}(0,T).$

Thus, the estimate (2.11) is proved.

Now we set $f_{n}(s):=(F(s),\,\varphi_{n})$ for $0<s<T$ and $n\in\mathbb{N}$.
Since

$\partial_{t}^{\alpha}\int^{t}_{0}K(t-s)F(s)ds=\sum_{n=1}^{\infty}\partial_{t}^{\alpha}(L_{n}f_{n})(t)\varphi_{n},$

we obtain

$\left\|\partial_{t}^{\alpha}\int^{t}_{0}K(t-s)F(s)ds\right\|^{2}_{L^{2}(\Omega)}=\sum_{n=1}^{\infty}|\partial_{t}^{\alpha}(L_{n}f_{n})(t)|^{2}.$

By applying (2.11), we get the following chain of inequalities and equations:

$\displaystyle\left\|\partial_{t}^{\alpha}\int^{t}_{0}K(t-s)F(s)ds\right\|^{2}_{H_{\alpha}(0,T;L^{2}(\Omega))}\leq
C\left\|\partial_{t}^{\alpha}\int^{t}_{0}K(t-s)F(s)ds\right\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}$
$\displaystyle=C\sum_{n=1}^{\infty}\|\partial_{t}^{\alpha}(L_{n}f_{n})\|^{2}_{L^{2}(0,T)}\leq
C\sum_{n=1}^{\infty}\|L_{n}f_{n}\|^{2}_{H_{\alpha}(0,T)}\leq
C\sum_{n=1}^{\infty}\|f_{n}\|^{2}_{L^{2}(0,T)}$
$\displaystyle=C\int^{T}_{0}\sum_{n=1}^{\infty}|(F(s),\varphi_{n})|^{2}ds=C\int^{T}_{0}\|F(s)\|_{L^{2}(\Omega)}^{2}ds=C\|F\|^{2}_{L^{2}(0,T;L^{2}(\Omega))}.$

Thus, the proof of the estimate (ii) is completed.

The estimate (iii) from Lemma 1 follows from the standard estimates of the
operator $S(t)$. It can be derived by the same arguments as those that were
employed in Section 6 of Chapter 4 in KRY  for the case of the homogeneous
Dirichlet boundary condition and we omit here the technical details.

Now we proceed to the proof of Theorem 2.1.

###### Proof

In the first line of the problem (2.4), we regard the expressions
$\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}u$ and $c(x,t)u$ as some non-homogeneous
terms. Then this problem can be rewritten in terms of the operator $A_{0}$ as
follows

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)+A_{0}u(x,t)=F(x,t)\\\
+&\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}u+(c_{0}+c(x,t))u,\quad
x\in\Omega,\,0<t<T,\\\ &\partial_{\nu_{A}}u+\sigma(x)u=0\quad\mbox{on
}\partial\Omega\times(0,T),\\\ &u(x,\cdot)-a(x)\in
H_{\alpha}(0,T)\quad\mbox{for almost all }x\in\Omega.\end{array}\right.$
(2.12)

In its turn, the first line of (2.12) can be represented in the form (GLY ,
KRY )

$u(t)=S(t)a+\int^{t}_{0}K(t-s)F(s)ds$
$+\int^{t}_{0}K(t-s)\left(\sum_{j=1}^{d}b_{j}(s)\partial_{j}u(s)+(c_{0}+c(s))u(s)\right)ds,\quad
0<t<T.$ (2.13)

Moreover, it is known that if $u\in L^{2}(0,T;H^{2}(\Omega))$ satisfies the
initial condition $u-a\in H_{\alpha}(0,T;L^{2}(\Omega))$ and the equation
(2.13), then $u$ is a solution to the problem (2.12). With the notations

$\left\\{\begin{array}[]{rl}&G(t):=\int^{t}_{0}K(t-s)F(s)ds+S(t)a,\\\
&Qv(t)=Q(t)v(t):=\sum_{j=1}^{d}b_{j}(\cdot,t)\partial_{j}v(t)+(c_{0}+c(\cdot,t))v(t),\\\
&Rv(t):=\int^{t}_{0}K(t-s)\left(\sum_{j=1}^{d}b_{j}(\cdot,s)\partial_{j}v(s)+(c_{0}+c(\cdot,s))v(s)\right)ds,\\\
&\qquad\qquad\qquad\mbox{for $0<t<T$},\end{array}\right.$ (2.14)

the equation (2.13) can be represented in form of a fixed point equation
$u=Ru+G$ on the space $L^{2}(0,T;H^{2}(\Omega))$.

Lemma 1 yields the inclusion $G\in L^{2}(0,T;H^{2}(\Omega))$. Moreover, since
$\|A_{0}^{\frac{1}{2}}a\|\leq C\|a\|_{H^{1}(\Omega)}$ and
$\mathcal{D}(A_{0}^{\frac{1}{2}})=H^{1}(\Omega)$ (see, e.g., Fu ), the
estimate (2.8) implies

$\|S(t)a\|_{H^{2}(\Omega)}\leq
C\|A_{0}S(t)a\|=C\|A_{0}^{\frac{1}{2}}S(t)A_{0}^{\frac{1}{2}}a\|\leq
Ct^{-\frac{1}{2}\alpha}\|a\|_{H^{1}(\Omega)}$

and thus

$\|S(t)a\|^{2}_{L^{2}(0,T;H^{2}(\Omega))}\leq
C\left(\int^{T}_{0}t^{-\alpha}dt\right)\|a\|^{2}_{H^{1}(\Omega)}\leq\frac{CT^{1-\alpha}}{1-\alpha}\|a\|^{2}_{H^{1}(\Omega)}.$

Consequently, the inclusion $S(t)a\in L^{2}(0,T;H^{2}(\Omega))$ holds valid.

For $0<t<T$, we next estimate $\|Rv(\cdot,t)\|_{H^{2}(\Omega)}$ for
$v(\cdot,t)\in\mathcal{D}(A_{0})$ as follows:

$\displaystyle\|Rv(\cdot,t)\|_{H^{2}(\Omega)}\leq
C\|A_{0}Rv(\cdot,t)\|_{L^{2}(\Omega)}$ $\displaystyle\leq$
$\displaystyle\int^{t}_{0}\left\|A_{0}^{\frac{1}{2}}K(t-s)A_{0}^{\frac{1}{2}}\left(\sum_{j=1}^{d}b_{j}(s)\partial_{j}v(s)+(c_{0}+c(s))v(s)\right)\right\|ds$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}\|A_{0}^{\frac{1}{2}}K(t-s)\|\left\|A_{0}^{\frac{1}{2}}\left(\sum_{j=1}^{d}b_{j}(s)\partial_{j}v(s)+(c_{0}+c(s))v(s)\right)\right\|ds$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|v(s)\|_{H^{2}(\Omega)}ds=C\left(\Gamma\left(\frac{1}{2}\alpha\right)J^{\frac{1}{2}\alpha}\|v\|_{H^{2}(\Omega)}\right)(t).$

For derivation of this estimate, we employed the inequalities

$\|A_{0}^{\frac{1}{2}}b_{j}(s)\partial_{j}v(t)\|\leq
C\|b_{j}(s)\partial_{j}v(s)\|_{H^{1}(\Omega)}\leq C\|v(s)\|_{H^{2}(\Omega)}$

and $\|(c(s)+c_{0})v(s)\|_{H^{1}(\Omega)}\leq C\|v(s)\|_{H^{2}(\Omega)}$ that
are valid because of the inclusions $b_{j}\in
C^{1}(\overline{\Omega}\times[0,T])$) and $c+c_{0}\in
C([0,T];C^{1}(\overline{\Omega}))$.

Since $(J^{\frac{1}{2}\alpha}w_{1})(t)\geq(J^{\frac{1}{2}\alpha}w_{2})(t)$ if
$w_{1}(t)\geq w_{2}(t)$ for $0\leq t\leq T$, and
$J^{\frac{1}{2}\alpha}J^{\frac{1}{2}\alpha}w=J^{\alpha}w$ for
$w_{1},w_{2},w\in L^{2}(0,T)$, we have

$\displaystyle\|R^{2}v(t)\|_{H^{2}(\Omega)}=\|R(Rv)(t)\|_{H^{2}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C\left(\Gamma\left(\frac{1}{2}\alpha\right)J^{\frac{1}{2}\alpha}\left(C\Gamma\left(\frac{1}{2}\alpha\right)J^{\frac{1}{2}\alpha}\|v\|_{H^{2}(\Omega)}\right)\right)(t)$
$\displaystyle=$
$\displaystyle\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{2}(J^{\alpha}\|v\|_{H^{2}(\Omega)})(t).$

Repeating this argumentation $m$-times, we obtain

$\displaystyle\|R^{m}v(t)\|_{H^{2}(\Omega)}\leq\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{m}\left(J^{\frac{1}{2}\alpha
m}\|v\|_{H^{2}(\Omega)}\right)(t)$ $\displaystyle\leq$
$\displaystyle\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{m}}{\Gamma\left(\frac{1}{2}\alpha
m\right)}\int^{t}_{0}(t-s)^{\frac{m}{2}\alpha-1}\|v(\xi)\|_{H^{2}(\Omega)}ds,\quad
0<t<T.$

Applying the Young inequality to the integral at the right-hand side of the
last estimate, we arrive to the inequality

$\displaystyle\|R^{m}v(t)\|_{L^{2}(0,T;H^{2}(\Omega))}^{2}\leq\left(\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{m}}{\Gamma\left(\frac{1}{2}\alpha
m\right)}\right)^{2}\|t^{\frac{\alpha
m}{2}-1}\|_{L^{1}(0,T)}^{2}\|v\|_{L^{2}(0,T;H^{2}(\Omega))}^{2}$
$\displaystyle=$
$\displaystyle\frac{\left(CT^{\frac{\alpha}{2}}\Gamma\left(\frac{1}{2}\alpha\right)\right)^{2m}}{\Gamma\left(\frac{1}{2}\alpha
m+1\right)^{2}}\|v\|_{L^{2}(0,T;H^{2}(\Omega))}^{2}.$

Employing the known asymptotic behavior of the gamma function, we obtain the
relation

$\lim_{m\to\infty}\frac{\left(CT^{\frac{\alpha}{2}}\Gamma\left(\frac{1}{2}\alpha\right)\right)^{m}}{\Gamma\left(\frac{1}{2}\alpha
m+1\right)}=0$

that means that for sufficiently large $m\in\mathbb{N}$, the mapping

$R^{m}:L^{2}(0,T;H^{2}(\Omega))\,\longrightarrow\,L^{2}(0,T;H^{2}(\Omega))$

is a contraction. Hence, by the Banach fixed point theorem, the equation
(2.13) possesses a unique fixed point. Therefore, by the first equation in
(2.4), we obtain the inclusion $\partial_{t}^{\alpha}(u-a)\in
L^{2}(0,T;L^{2}(\Omega))$. Since
$\|\eta\|_{H_{\alpha}(0,T)}\sim\|\partial_{t}^{\alpha}\eta\|_{L^{2}(0,T)}$ for
$\eta\in H_{\alpha}(0,T)$ (KRY ), we finally obtain the estimate

$\|u-a\|_{H_{\alpha}(0,T;L^{2}(\Omega))}+\|u\|_{L^{2}(0,T;H^{2}(\Omega))}\leq
C(\|a\|_{H^{1}(\Omega)}+\|F\|_{L^{2}(0,T;L^{2}(\Omega))}).$

The proof of Theorem 2.1 is completed.

##  3 Key lemma

For derivation of the comparison principles for solutions to the initial-
boundary value problems for the linear and semilinear time-fractional
diffusion equations, we need some auxiliary results that are formulated and
proved in this section.

In addition to the operator $-A_{0}$ defined by (2.5), we define an elliptic
operator $-A_{1}$ with a positive zeroth-order coefficient:

$(-A_{1}(t)v)(x):=(-A_{1}v)(x)$ (3.1)
$:=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v(x))+\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}v-b_{0}(x,t)v,$

where $b_{0}\in C^{1}([0,T];C^{1}(\overline{\Omega}))\cap
C([0,T];C^{2}(\overline{\Omega}))$, $b_{0}(x,t)>0,\
(x,t)\in\overline{\Omega}\times[0,T]$, and
$\min_{(x,t)\in\overline{\Omega}\times[0,T]}b_{0}(x,t)$ is sufficiently large.

We also recall that for $y\in W^{1,1}(0,T)$, the pointwise Caputo derivative
$d_{t}^{\alpha}$ is defined by

$d_{t}^{\alpha}y(t)=\frac{1}{\Gamma(1-\alpha)}\int^{t}_{0}(t-s)^{-\alpha}\frac{dy}{ds}(s)ds.$
(3.2)

In what follows, we employ an extremum principle for the Caputo fractional
derivative formulated below.

######  Lemma 2 (Lu1 )

Let the inclusions $y\in C[0,T]$ and $t^{1-\alpha}y^{\prime}\in C[0,T]$ hold
true.

If the function $y=y(t)$ attains its minimum over the interval $[0,T]$ at the
point $t_{0}\in(0,\,T]$, then

$d_{t}^{\alpha}y(t_{0})\leq 0.$

In Lemma 2, the assumption $t_{0}>0$ is essential. This lemma was formulated
and proved in Lu1  under a weaker regularity condition posed on the function
$y$, but for our arguments we can assume that $y\in C[0,T]$ and
$t^{1-\alpha}y^{\prime}\in C[0,T]$.

Employing Lemma 2, we now formulate and prove our key lemma that is a basis
for further derivations in this paper.

######  Lemma 3 (Positivity of a smooth solution)

For $F\in L^{2}(0,T;L^{2}(\Omega))$ and $a\in H^{1}(\Omega)$, let $F(x,t)\geq
0,\ (x,t)\in\Omega\times(0,T)$, $a(x)\geq 0,\ x\in\Omega$, and
$\min_{(x,t)\in\overline{\Omega}\times[0,T]}b_{0}(x,t)$ be a sufficiently
large positive constant. Furthermore, we assume that there exists a solution
$u\in C([0,T];C^{2}(\overline{\Omega}))$ to the initial-boundary value problem

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)+A_{1}u=F(x,t),\quad
x\in\Omega,\,0<t<T,\\\ &\partial_{\nu_{A}}u+\sigma(x)u=0\quad\mbox{on
}\partial\Omega\times(0,T),\\\ &u(x,\cdot)-a(x)\in
H_{\alpha}(0,T)\quad\mbox{for almost all }x\in\Omega\end{array}\right.$ (3.3)

and $u$ satisfies the condition $t^{1-\alpha}\partial_{t}u\in
C([0,T];C(\overline{\Omega}))$.

Then the solution $u$ is non-negative:

$u(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T).$

For the partial differential equations of parabolic type with the Robin
boundary condition ($\alpha=1$ in (3.3)), a similar positivity property is
well-known. However, it is worth mentioning that the regularity of the
solution to the problem (3.3) at the point $t=0$ is a more delicate question
compared to the one in the case $\alpha=1$. In particular, we cannot expect
the inclusion $u(x,\cdot)\in C^{1}[0,T]$. This can be illustrated by a simple
example of the equation $\partial_{t}^{\alpha}y(t)=y(t)$ with $y(t)-1\in
H_{\alpha}(0,T)$ whose unique solution $y(t)=E_{\alpha,1}(t^{\alpha})$ does
not belong to the space $C^{1}[0,T]$.

###### Proof

First we introduce an auxiliary function $\psi\in
C^{1}([0,T];C^{2}(\overline{\Omega}))$ that satisfies the conditions

$\left\\{\begin{array}[]{rl}&A_{1}\psi(x,t)=1,\quad(x,t)\in\Omega\times[0,T],\\\
&\partial_{\nu_{A}}\psi+\sigma\psi=1\quad\mbox{on
}\partial\Omega\times[0,T].\end{array}\right.$ (3.4)

Proving existence of such function $\psi$ is non-trivial. In this section, we
focus on the proof of the lemma and then come back to the problem (3.4) in
Appendix.

Now, choosing $M>0$ sufficiently large and $\varepsilon>0$ sufficiently small,
we set

$w(x,t):=u(x,t)+\varepsilon(M+\psi(x,t)+t^{\alpha}),\quad x\in\Omega,\,0<t<T.$
(3.5)

For a fixed $x\in\Omega$, by the assumption on the regularity of $u$, we have
the inclusion

$t^{1-\alpha}\partial_{t}u(x,\cdot)\in C[0,T].$ (3.6)

Then, $\partial_{t}u(x,\cdot)\in L^{1}(0,T)$, that is, $u(x,\cdot)\in
W^{1,1}(0,T)$. Moreover,

$u(x,0)-a(x)=0,\quad x\in\Omega.$ (3.7)

On the other hand, for $w\in H_{\alpha}(0,T)\cap W^{1,1}(0,T)$ and $w(0)=0$,
the equality

$\partial_{t}^{\alpha}w=d_{t}^{\alpha}w=d_{t}^{\alpha}(w+c)$

holds true with any constant $c$ (see, e.g., Theorem 2.4 of Chapter 2 in KRY
).

Since $u(x,\cdot)-a\in H_{\alpha}(0,T)$ and $u(x,\cdot)\in W^{1,1}(0,T)$, by
(3.7), the relations
$\partial_{t}^{\alpha}(u-a)=d_{t}^{\alpha}(u-a)=d_{t}^{\alpha}u$ hold true for
almost all $x\in\Omega$.

Furthermore, since $\varepsilon(M+\psi(\cdot,t)+t^{\alpha})\in W^{1,1}(0,T)$,
we obtain

$\displaystyle
d_{t}^{\alpha}w=d_{t}^{\alpha}(u+\varepsilon(M+\psi(x,t)+t^{\alpha})))=d_{t}^{\alpha}u+\varepsilon
d_{t}^{\alpha}(M+\psi(x,t)+t^{\alpha})$ $\displaystyle=$
$\displaystyle\partial_{t}^{\alpha}(u-a)+\varepsilon(d_{t}^{\alpha}(\psi+t^{\alpha}))=\partial_{t}^{\alpha}(u-a)+\varepsilon(d_{t}^{\alpha}\psi+\Gamma(\alpha+1))$

and

$\displaystyle A_{1}w=A_{1}u+\varepsilon A_{1}\psi+\varepsilon
A_{1}t^{\alpha}+\varepsilon A_{1}M$ $\displaystyle=$ $\displaystyle
A_{1}u+\varepsilon+\varepsilon b_{0}(x,t)t^{\alpha}+b_{0}(x,t)\varepsilon M.$

Now we choose a constant $M>0$ such that $M+\psi(x,t)\geq 0$ and
$d_{t}^{\alpha}\psi(x,t)+b_{0}(x,t)M>0$ for
$(x,t)\in\overline{\Omega}\times[0,T]$, so that

$d_{t}^{\alpha}w+A_{1}w$ (3.8)
$=F+\varepsilon(\Gamma(\alpha+1)+d_{t}^{\alpha}\psi+1+b_{0}(x,t)t^{\alpha}+b_{0}(x,t)M)>0\quad\mbox{in
}\Omega\times(0,T).$

Moreover, because of the relation
$\partial_{\nu_{A}}w=\partial_{\nu_{A}}u+\varepsilon\partial_{\nu_{A}}\psi$,
we obtain the following estimate:

$\partial_{\nu_{A}}w+\sigma w=\partial_{\nu_{A}}u+\sigma
u+\varepsilon+\sigma\varepsilon t^{\alpha}+\varepsilon\sigma M$
$\geq\varepsilon+\sigma\varepsilon t^{\alpha}+\varepsilon\sigma
M\geq\varepsilon\quad\mbox{on }\partial\Omega\times(0,T).$ (3.9)

Evaluation of the representation (3.5) at the point $t=0$ immediately leads to
the formula

$w(x,0)=u(x,0)+\varepsilon(\psi(x,0)+M),\quad x\in\Omega.$

Let us assume that the inequality

$\min_{(x,t)\in\overline{\Omega}\times[0,T]}w(x,t)\geq 0$

does not hold valid, that is, there exists a point
$(x_{0},t_{0})\in\overline{\Omega}\times[0,T]$ such that

$w(x_{0},t_{0}):=\min_{(x,t)\in\overline{\Omega}\times[0,T]}w(x,t)<0.$ (3.10)

Since $M>0$ is sufficiently large and $u(x,0)$ is non-negative, we obtain the
inequality

$w(x,0)=u(x,0)+\varepsilon(\psi(x,0)+M)\geq u(x,0)\geq 0,\quad
x\in\overline{\Omega},$

and thus $t_{0}$ cannot be zero.

Next, we show that $x_{0}\not\in\partial\Omega$. Indeed, let us assume that
$x_{0}\in\partial\Omega$. Then the estimate (3.9) yields that
$\partial_{\nu_{A}}w(x_{0},t_{0})+\sigma(x_{0})w(x_{0},t_{0})\geq\varepsilon$.
By (3.10) and $\sigma(x_{0})\geq 0$, we obtain

$\partial_{\nu_{A}}w(x_{0},t_{0})\geq-\sigma(x_{0})w(x_{0},t_{0})+\varepsilon\geq\varepsilon>0,$

which implies

$\partial_{\nu_{A}}w(x_{0},t_{0})=\sum_{i,j=1}^{d}a_{ij}(x_{0})\nu_{j}(x_{0})\partial_{i}w(x_{0},t_{0})=\nabla
w(x_{0},t_{0})\cdot\mathcal{A}(x_{0})\nu(x_{0})$
$=\sum_{i=1}^{d}(\partial_{i}w)(x_{0},t_{0})[\mathcal{A}(x_{0})\nu(x_{0})]_{i}>0.$
(3.11)

Here $\mathcal{A}(x)=(a_{ij}(x))_{1\leq i,j\leq d}$ and $[b]_{i}$ means the
$i$-th element of a vector $b$.

For sufficiently small $\varepsilon_{0}>0$ and $x_{0}\in\partial\Omega$, we
now verify the inclusion

$x_{0}-\varepsilon_{0}\mathcal{A}(x_{0})\nu(x_{0})\in\Omega.$ (3.12)

Indeed, since the matrix $\mathcal{A}(x_{0})$ is positive-definite, the
inequality

$(\nu(x_{0})\,\cdot\,-\varepsilon_{0}\mathcal{A}(x_{0})\nu(x_{0}))=-\varepsilon_{0}(\mathcal{A}(x_{0})\nu(x_{0})\,\cdot\,\nu(x_{0}))<0$

holds true. In other words, the inequality

$\angle(\nu(x_{0}),\,(x_{0}-\varepsilon_{0}\mathcal{A}(x_{0})\nu(x_{0}))-x_{0})>\frac{\pi}{2}$

is satisfied. Because the boundary $\partial\Omega$ is smooth, the domain
$\Omega$ is locally located on one side of $\partial\Omega$. In a small
neighborhood of the point $x_{0}\in\partial\Omega$, the boundary
$\partial\Omega$ can be described in the local coordinates composed of its
tangential component in $\mathbb{R}^{d-1}$ and the normal component along
$\nu(x_{0})$. Consequently, if $y\in\mathbb{R}^{d}$ satisfies the inequality
$\angle(\nu(x_{0}),y-x_{0})>\frac{\pi}{2}$, then $y\in\Omega$. Therefore, for
a sufficiently small $\varepsilon_{0}>0$, the point
$x_{0}-\varepsilon_{0}\mathcal{A}(x_{0})\nu(x_{0})$ is located in $\Omega$ and
we have proved the inclusion (3.12).

Moreover, for sufficiently small $\varepsilon_{0}>0$, we can prove that

$w(x_{0}-\varepsilon_{0}\mathcal{A}(x_{0})\nu(x_{0}),\,t_{0})<w(x_{0},t_{0}).$
(3.13)

Indeed, the inequality (3.10) yields

$\sum_{i=1}^{d}(\partial_{i}w)(x_{0}-\eta\mathcal{A}(x_{0})\nu(x_{0}),\,t_{0})[\mathcal{A}(x_{0})\nu(x_{0})]_{i}>0\quad\mbox{if
}|\eta|<\varepsilon_{0}.$

Then, by the mean value theorem, we obtain the inequality

$\displaystyle
w(x_{0}-\xi\mathcal{A}(x_{0})\nu(x_{0}),\,t_{0})-w(x_{0},t_{0})$
$\displaystyle=$
$\displaystyle\xi\sum_{i=1}^{d}\partial_{i}w(x_{0}-\theta\mathcal{A}(x_{0})\nu(x_{0}),\,t_{0})(-[\mathcal{A}(x_{0})\nu(x_{0})]_{i})<0,$

where $\theta$ is a number between $0$ and $\xi\in(0,\varepsilon_{0})$. Thus,
the inequality (3.13) is verified.

By combining (3.13) with (3.12), we conclude that there exists a point
$\widetilde{x_{0}}\in\Omega$ such that the inequality
$w(\widetilde{x_{0}},t_{0})<w(x_{0},t_{0})$ holds true, which contradicts the
assumption (3.10). Thus, we have proved that $x_{0}\not\in\partial\Omega$.

According to (3.10), the function $w$ attains its minimum at the point
$(x_{0},t_{0})$. Because $0<t_{0}\leq T$, Lemma 2 yields the inequality

$d_{t}^{\alpha}w(x_{0},t_{0})\leq 0.$ (3.14)

Since $x_{0}\in\Omega$, the necessary condition for an extremum point leads to
the equality

$\nabla w(x_{0},t_{0})=0.$ (3.15)

Moreover, because the function $w$ attains its minimum at the point
$x_{0}\in\Omega$, in view of the sign of the Hessian, the inequality

$\sum_{i,j=1}^{d}a_{ij}(x_{0})\partial_{i}\partial_{j}w(x_{0},t_{0})\geq 0$
(3.16)

holds true (see, e.g., the proof of Lemma 1 in Section 1 of Chapter 2 in Fr ).

The inequalities $b(x_{0},t_{0})>0$, $w(x_{0},t_{0})<0$, and (3.14)-(3.16)
lead to the estimate

$\displaystyle d_{t}^{\alpha}w(x_{0},t_{0})+A_{1}w(x_{0},t_{0})$
$\displaystyle=$ $\displaystyle
d_{t}^{\alpha}w(x_{0},t_{0})-\sum_{i,j=1}^{d}a_{ij}(x_{0})\partial_{i}\partial_{j}w(x_{0},t_{0})-\sum_{i=1}^{d}(\partial_{i}a_{ij})(x_{0})\partial_{j}w(x_{0},t_{0})$
$\displaystyle-$
$\displaystyle\sum_{i=1}^{d}b_{i}(x_{0},t_{0})\partial_{i}w(x_{0},t_{0})+b(x_{0},t_{0})w(x_{0},t_{0})<0,$

which contradicts the inequality (3.8).

Thus, we have proved that

$u(x,t)+\varepsilon(M+\psi(x,t)+t^{\alpha})=w(x,t)\geq
0,\quad(x,t)\in\Omega\times(0,T).$

Since $\varepsilon>0$ is arbitrary, we let $\varepsilon\downarrow 0$ to obtain
the inequality $u(x,t)\geq 0$ for $(x,t)\in\Omega\times(0,T)$ and the proof of
Lemma 3 is completed.

Let us finally mention that the positivity of the function $b_{0}$ from the
definition of the operator $-A_{1}$ is an essential condition for validity of
our proof of Lemma 3. However, in the next section, we remove this condition
while deriving the comparison principles for the solutions to the initial-
boundary value problem (2.4).

##  4 Comparison principles

According to the results formulated in Theorem 2.1, in this section, we
consider the solutions to the initial-boundary value problem (2.4) that belong
to the following space of functions:

$\mathcal{Y}_{\alpha}:=\\{u;\,u-a\in H_{\alpha}(0,T;L^{2}(\Omega)),\,u\in
L^{2}(0,T;H^{2}(\Omega))\\}.$ (4.1)

In what follows, by $u(F,a)$ we denote the solution to the problem (2.4) with
the initial data $a$ and the source function $F$.

Our first result concerning the comparison principles for the solutions to the
initial-boundary value problems for the linear time-fractional diffusion
equation is presented in the next theorem.

###### Theorem 4.1

Let the functions $a\in H^{1}(\Omega)$ and $F\in L^{2}(\Omega\times(0,T))$
satisfy the inequalities $F(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$ and
$a(x)\geq 0,\ x\in\Omega$, respectively.

Then the solution $u(F,a)\in\mathcal{Y}_{\alpha}$ to the initial-boundary
value problem (2.4) is non-negative, e.g., the inequality

$u(F,a)(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$

holds true.

Let us emphasize that the non-negativity of the solution $u$ to the problem
(2.4) holds true for the space $\mathcal{Y}_{\alpha}$ and thus $u$ does not
necessarily satisfy the inclusions $u\in C([0,T];C^{2}(\overline{\Omega}))$
and $t^{1-\alpha}\partial_{t}u\in C([0,T];C(\overline{\Omega}))$. Therefore,
Theorem 4.1 is widely applicable. Before presenting its proof, let us discuss
one of its corollaries in form of a comparison property:

###### Corollary 1

Let $a_{1},a_{2}\in H^{1}(\Omega)$ and $F_{1},F_{2}\in
L^{2}(\Omega\times(0,T))$ satisfy the inequalities $a_{1}(x)\geq a_{2}(x),\
x\in\Omega$ and $F_{1}(x,t)\geq F_{2}(x,t),\ (x,t)\in\Omega\times(0,T)$,
respectively.

Then the inequality

$u(F_{1},a_{1})(x,t)\geq u(F_{2},a_{2})(x,t),\ (x,t)\in\Omega\times(0,T)$

holds true.

###### Proof

Setting $a:=a_{1}-a_{2}$, $F:=F_{1}-F_{2}$ and
$u:=u(F_{1},a_{1})-u(F_{2},a_{2})$, we immediately obtain the inequalities
$a(x)\geq 0,\ x\in\Omega$ and $F(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$ and

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)+Au=F\geq
0\quad\mbox{in }\Omega\times(0,T),\\\ &\partial_{\nu_{A}}u+\sigma
u=0\quad\mbox{on }\partial\Omega.\end{array}\right.$

Therefore, Theorem 4.1 implies that $u(x,t)\geq 0,\
(x,t)\in\Omega\times(0,T)$, that is, $u(F_{1},a_{1})(x,t)\geq
u(F_{2},a_{2})(x,t),\ (x,t)\in\Omega\times(0,T)$.

In its turn, Corollary 1 can be applied for derivation of the lower and upper
bounds for the solutions to the initial-boundary value problem (2.4) by
suitably choosing the initial values and the source functions. Let us
demonstrate this technique on an example.

###### Example 1

Let the coefficients $a_{ij},b_{j}$, $1\leq i,j\leq d$ of the operator

$-Av(x)=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v(x))+\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}v(x)$

from the initial-boundary value problem (2.4) satisfy the conditions (1.2).
Now we consider the homogeneous initial condition $a(x)=0,\ x\in\Omega$ and
assume that the source function $F\in L^{2}(0,T;L^{2}(\Omega))$ satisfies the
inequality

$F(x,t)\geq\delta t^{\beta},\quad x\in\Omega,\,0<t<T$

with certain constants $\beta\geq 0$ and $\delta>0$.

Then the solution $u(F,0)$ can be estimated from below as follows:

$u(F,0)(x,t)\geq\frac{\delta\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}t^{\alpha+\beta},\quad
x\in\Omega,\,0\leq t\leq T.$ (4.2)

Indeed, it is easy to verify that the function

$\underline{u}(x,t):=\frac{\delta\Gamma(\beta+1)}{\Gamma(\alpha+\beta+1)}t^{\alpha+\beta},\quad
x\in\Omega,\,t>0$

is a solution to the following problem:

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}\underline{u}+A\underline{u}=\delta
t^{\beta}\quad\mbox{in $\Omega\times(0,T)$},\\\
&\partial_{\nu_{A}}\underline{u}=0\quad\mbox{on
$\partial\Omega\times(0,T)$},\\\ &\underline{u}(x,\cdot)\in
H_{\alpha}(0,T).\end{array}\right.$

Due to the inequality $F(x,t)\geq\delta t^{\beta},\
(x,t)\in\Omega\times(0,T)$, we can apply Corollary 1 to the solutions $u$ and
$\underline{u}$ and the inequality (4.2) immediately follows.

In particular, for the spatial dimensions $d\leq 3$, the Sobolev embedding
theorem leads to the inclusion $u\in L^{2}(0,T;H^{2}(\Omega))\subset
L^{2}(0,T;C(\overline{\Omega}))$ and thus the strict inequality
$u(F,0)(x,t)>0$ holds true for almost all $t>0$ and all
$x\in\overline{\Omega}$.

Now we proceed to the proof of Theorem 4.1.

###### Proof

In the proof, we employ the operators $Qv(t)$ and $G(t)$ defined by (2.14). In
terms of these operators, the solution $u(t):=u(F,a)(t)$ to the initial-
boundary problem (2.4) satisfies the integral equation

$u(F,a)(t)=G(t)+\int^{t}_{0}K(t-s)Qu(s)ds,\quad 0<t<T.$ (4.3)

For readers’ convenience, we split the proof into three parts.

I. First part of the proof: existence of a smoother solution.

In the formulation of Lemma 3, we assumed existence of a solution $u\in
C([0,T];C^{2}(\overline{\Omega}))$ to the initial-boundary value problem (3.3)
satisfying the inclusion $t^{1-\alpha}\partial_{t}u\in
C([0,T];C(\overline{\Omega}))$. On the other hand, Theorem 2.1 asserts the
unique existence of solution $u$ to the initial-boundary value problem (2.4)
from the space $\mathcal{Y}_{\alpha}$, i.e., of the solution $u$ that
satisfies the inclusions $u\in L^{2}(0,T;H^{2}(\Omega))$ and $u-a\in
H_{\alpha}(0,T;L^{2}(\Omega))$.

In this part of the proof, we show that for $a\in C^{\infty}_{0}(\Omega)$ and
$F\in C^{\infty}_{0}(\Omega\times(0,T))$, the solution to the problem (2.4)
satisfies the regularity assumptions formulated in Lemma 3.

More precisely, we first prove the following lemma:

###### Lemma 4

Let $a_{ij}$, $b_{j}$, $c$ satisfy the conditions (1.2) and the inclusions
$a\in C^{\infty}_{0}(\Omega)$, $F\in C^{\infty}_{0}(\Omega\times(0,T))$ hold
true.

Then the solution $u=u(F,a)$ to the problem (2.4) satisfies the inclusions

$u\in C([0,T];C^{2}(\overline{\Omega})),\quad t^{1-\alpha}\partial_{t}u\in
C([0,T];C(\overline{\Omega}))$

and $\lim_{t\to 0}\|u(t)-a\|_{L^{2}(\Omega)}=0$.

###### Proof

We recall that $c_{0}>0$ is a positive fixed constant and

$-A_{0}v=\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v)-c_{0}v,\
\mathcal{D}(A_{0})=\\{v\in H^{2}(\Omega);\,\partial_{\nu_{A}}v+\sigma
v=0\,\,\mbox{on }\partial\Omega\\}.$

Then $\mathcal{D}(A_{0}^{\frac{1}{2}})=H^{1}(\Omega)$ and
$\|A_{0}^{\frac{1}{2}}v\|\sim\|v\|_{H^{1}(\Omega)}$ (Fu ). Moreover, for the
operators $S(t)$ and $K(t)$ defined by (2.6) and (2.7), the estimates (2.8)
hold true.

In what follows, we denote $\frac{\partial u}{\partial t}(\cdot,t)$ by
$u^{\prime}(t)=\frac{du}{dt}(t)$ if there is no fear of confusion.

The solution $u$ to the integral equation (4.3) can be constructed as a fixed
point of the equation

$A_{0}u(t)=A_{0}G(t)+\int^{t}_{0}A_{0}^{\frac{1}{2}}K(t-s)A_{0}^{\frac{1}{2}}Qu(s)ds,\quad
0<t<T.$ (4.4)

As already proved, this fixed point satisfies the inclusion $u\in
L^{2}(0,T;H^{2}(\Omega))$ $\cap$ $(H_{\alpha}(0,T;L^{2}(\Omega))+\\{a\\}).$

Now we derive some estimates for the norms $\|A_{0}^{\kappa}u(t)\|$,
$\kappa=1,2$ and $\|A_{0}u^{\prime}(t)\|$ for $0<t<T$. First we set

$D:=\sup_{0<t<T}(\|A_{0}F(t)\|+\|A_{0}F^{\prime}(t)\|+\|A_{0}^{2}F(t)\|)+\|a\|_{H^{4}(\Omega)}.$

Since $F\in C^{\infty}_{0}(\Omega\times(0,T))$, we obtain the inclusion $F\in
L^{\infty}(0,T;\mathcal{D}(A_{0}^{2}))$ and the inequality $D<+\infty$.
Moreover, in view of (2.8), for $\kappa=1,2$, we get the estimates

$\displaystyle\left\|A_{0}^{\kappa}\int^{t}_{0}K(t-s)F(s)ds\right\|\leq
C\int^{t}_{0}\|K(t-s)\|\|A_{0}^{\kappa}F(s)\|ds$ $\displaystyle\leq$
$\displaystyle
C\left(\int^{t}_{0}(t-s)^{\alpha-1}ds\right)\sup_{0<s<T}\|A_{0}^{\kappa}F(s)\|\leq
CD,$
$\displaystyle\left\|A_{0}\frac{d}{dt}\int^{t}_{0}K(t-s)F(s)ds\right\|=\left\|A_{0}\frac{d}{dt}\int^{t}_{0}K(s)F(t-s)ds\right\|$
$\displaystyle=$
$\displaystyle\left\|A_{0}K(t)F(0)+A_{0}\int^{t}_{0}K(s)F^{\prime}(t-s)ds\right\|$
$\displaystyle\leq$ $\displaystyle
C\left\|A_{0}\int^{t}_{0}K(s)F^{\prime}(t-s)ds\right\|\leq
C\int^{t}_{0}s^{\alpha-1}\|A_{0}F^{\prime}(t-s)\|ds<CD.$

The regularity conditions (1.2) lead to the estimates

$\|A_{0}^{\frac{1}{2}}Q(s)u(s)\|\leq
C\|Q(s)u(s)\|_{H^{1}(\Omega)}=C\left\|\sum_{j=1}^{d}b_{j}(s)\partial_{j}u(s)+(c_{0}+c(s))u(s)\right\|_{H^{1}(\Omega)}$
$\leq C\|u(s)\|_{H^{2}(\Omega)}\leq C\|A_{0}u(s)\|,\quad 0<s<T.$ (4.5)

Moreover,

$\|A_{0}S(t)a\|=\|S(t)A_{0}a\|\leq C\|a\|_{H^{2}(\Omega)}\leq CD$

by using the inequalities (2.8). Then

$\displaystyle\|A_{0}u(t)\|\leq
CD+\int^{t}_{0}\|A_{0}^{\frac{1}{2}}K(t-s)\|\|A_{0}^{\frac{1}{2}}Q(s)u(s)\|ds$
$\displaystyle\leq$ $\displaystyle
CD+C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}u(s)\|ds,\quad 0<s<T.$

The generalized Gronwall inequality yields the estimate

$\|A_{0}u(t)\|\leq CD+C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}Dds\leq CD,\quad
0<t<T,$

which implies the inequality

$\|A_{0}u\|_{L^{\infty}(0,T;H^{2}(\Omega))}\leq CD.$

Next, for the space $C([0,T];L^{2}(\Omega))$, we can repeat the same arguments
as the ones employed for the iterations $R^{n}$ of the operator $R$ in the
proof of Theorem 2.1 and apply the fixed point theorem to the equation (4.3)
that leads to the inclusion $A_{0}u\in C([0,T];L^{2}(\Omega))$. The obtained
results implicate

$u\in C([0,T];H^{2}(\Omega)),\quad\|u\|_{C([0,T];H^{2}(\Omega))}\leq CD.$
(4.6)

Choosing $\varepsilon_{0}>0$ sufficiently small, we have the equation

$A_{0}^{\frac{3}{2}}u(t)=A_{0}^{\frac{3}{2}}G(t)+\int^{t}_{0}A_{0}^{\frac{3}{4}+\varepsilon_{0}}K(t-s)A_{0}^{\frac{3}{4}-\varepsilon_{0}}Q(s)u(s)ds,\quad
0<t<T.$ (4.7)

Next, according to Fu , the inclusion

$\mathcal{D}(A_{0}^{\frac{3}{4}-\varepsilon_{0}})\subset
H^{\frac{3}{2}-2\varepsilon_{0}}(\Omega)$

holds true. Now we proceed to the proof of the inclusion
$Q(s)u(s)\in\mathcal{D}(A_{0}^{\frac{3}{4}-\varepsilon_{0}})$. By (2.8), we
obtain the inequality

$\|A_{0}^{\frac{3}{2}}u(t)\|\leq
CD+\int^{t}_{0}(t-s)^{(\frac{1}{4}-\varepsilon_{0})\alpha-1}\|A_{0}^{\frac{3}{4}-\varepsilon_{0}}Q(s)u(s)\|ds,$

which leads to the estimate

$\|u(t)\|_{H^{3}(\Omega)}\leq
CD+\int^{t}_{0}(t-s)^{(\frac{1}{4}-\varepsilon_{0})\alpha-1}\|u(s)\|_{H^{3}(\Omega)}ds,\quad
0<t<T$ (4.8)

because of the inequality

$\|A_{0}^{\frac{3}{4}-\varepsilon_{0}}Q(s)u(s)\|\leq
C\|Q(s)u(s)\|_{H^{\frac{3}{2}}(\Omega)}\leq C\|Q(s)u(s)\|_{H^{2}(\Omega)}\leq
C\|u(s)\|_{H^{3}(\Omega)},$

which follows from the regularity conditions (1.2) posed on the coefficients
$b_{j},c$.

For $0<t<T$, the generalized Gronwall inequality applied to the integral
inequality (4.8) yields the estimate

$\|u(t)\|_{H^{3}(\Omega)}\leq
C\left(1+t^{\alpha\left(\frac{1}{4}-\varepsilon_{0}\right)}\right)D.$

For the relation (4.7), we repeat the same arguments as the ones employed in
the proof of Theorem 2.1 to estimate $A_{0}^{\frac{3}{2}}u(t)$ in the norm
$C([0,T];L^{2}(\Omega))$ by the fixed point theorem arguments and thus we
obtain the inclusion $A_{0}^{\frac{3}{2}}u\in C([0,T];L^{2}(\Omega))$.

Summarising the estimates derived above, we have shown that

$\left\\{\begin{array}[]{rl}&u\in
C([0,T];\mathcal{D}(A_{0}^{\frac{3}{2}}))\subset C([0,T];H^{3}(\Omega)),\\\
&\|u(t)\|_{H^{3}(\Omega)}\leq
C\left(1+t^{\alpha\left(\frac{1}{4}-\varepsilon_{0}\right)}\right)D,\quad
0<t<T.\end{array}\right.$ (4.9)

Next we estimate the norm $\|Au^{\prime}(t)\|$. First, $u^{\prime}(t)$ is
represented in the form

$\displaystyle
u^{\prime}(t)=G^{\prime}(t)+\frac{d}{dt}\int^{t}_{0}K(t-s)Q(s)u(s)ds$
$\displaystyle=$ $\displaystyle
G^{\prime}(t)+\frac{d}{dt}\int^{t}_{0}K(s)Q(t-s)u(t-s)ds\,=\,G^{\prime}(t)+K(t)Q(0)u(0)$
$\displaystyle+$
$\displaystyle\int^{t}_{0}K(s)(Q(t-s)u^{\prime}(t-s)+Q^{\prime}(t-s)u(t-s))ds,\quad
0<t<T,$

so that

$A_{0}u^{\prime}(t)=A_{0}G^{\prime}(t)+A_{0}K(t)Q(0)u(0)$ (4.10)
$+\int^{t}_{0}A_{0}^{\frac{1}{2}}K(s)A_{0}^{\frac{1}{2}}(Q(t-s)u^{\prime}(t-s)+Q^{\prime}(t-s)u(t-s))ds,\quad
0<t<T.$

Similarly to the arguments applied for derivation of (4.5), we obtain the
inequality

$\|A_{0}^{\frac{1}{2}}(Q(t-s)u^{\prime}(t-s)+Q^{\prime}(t-s)u(t-s))\|\leq
C\|A_{0}u^{\prime}(t-s)\|,\quad 0<t<T.$

The inclusion $Q(0)u(0)=Q(0)a\in C^{2}_{0}(\Omega)\subset\mathcal{D}(A_{0})$
follows from the regularity conditions (1.2) and the inclusion $a\in
C^{\infty}_{0}(\Omega)$. Furthermore, by (2.7) and (2.8), we obtain

$\|A_{0}S^{\prime}(t)a\|=\|A_{0}^{2}K(t)a\|=\|K(t)A_{0}^{2}a\|\leq
Ct^{\alpha-1}\|A_{0}^{2}a\|\leq Ct^{\alpha-1}\|a\|_{H^{4}(\Omega)}$

and

$\|K(t)A_{0}(Q(0)a)\|\leq Ct^{1-\alpha}\|A_{0}(Q(0)a)\|\leq
Ct^{\alpha-1}\|a\|_{H^{3}(\Omega)}.$

Hence, the representation (4.10) leads to the estimate

$\|A_{0}u^{\prime}(t)\|\leq
Ct^{\alpha-1}D+C\int^{t}_{0}s^{\frac{1}{2}\alpha-1}\|A_{0}u^{\prime}(t-s)\|ds,\quad
0<t<T.$

Now we consider a vector space

$\widetilde{X}:=\\{v\in C([0,T];L^{2}(\Omega))\cap
C^{1}((0,T];L^{2}(\Omega));\,t^{1-\alpha}\partial_{t}v\in
C([0,T];L^{2}(\Omega))\\}$

with the norm

$\|v\|_{\widetilde{X}}:=\max_{0\leq t\leq
T}\|t^{1-\alpha}\partial_{t}v(\cdot,t)\|_{L^{2}(\Omega)}+\max_{0\leq t\leq
T}\|v(\cdot,t)\|_{L^{2}(\Omega)}.$

It is easy to verify that $\widetilde{X}$ with the norm
$\|v\|_{\widetilde{X}}$ defined above is a Banach space.

Arguing similarly to the proof of Theorem 2.1 and applying the fixed point
theorem in the Banach space $\widetilde{X}$, we conclude that
$A_{0}u\in\widetilde{X}$, that is, $t^{1-\alpha}A_{0}u^{\prime}\in
C([0,T];L^{2}(\Omega))$. Using the inclusion $\mathcal{D}(A_{0})\subset
C(\overline{\Omega})$ in the spatial dimensions $d=1,2,3$, the Sobolev
embedding theorem yields

$u^{\prime}\in C(\overline{\Omega}\times(0,T]),\quad\|A_{0}u^{\prime}(t)\|\leq
CDt^{\alpha-1},\quad 0\leq t\leq T.$ (4.11)

Now we proceed to the estimation of $A_{0}^{2}u(t)$. Since
$\frac{d}{ds}(-A_{0}^{-1}S(s))=K(s)$ for $0<s<T$ by (2.7), the integration by
parts yields

$\displaystyle\int^{t}_{0}K(t-s)Q(s)u(s)ds=\int^{t}_{0}K(s)Q(t-s)u(t-s)ds$
$\displaystyle=$
$\displaystyle\left[-A_{0}^{-1}S(s)Q(t-s)u(t-s)\right]^{s=t}_{s=0}$
$\displaystyle-$
$\displaystyle\int^{t}_{0}A_{0}^{-1}S(s)(Q^{\prime}(t-s)u(t-s)+Q(t-s)u^{\prime}(t-s))ds$
$\displaystyle=$ $\displaystyle A_{0}^{-1}Q(t)u(t)-A_{0}^{-1}S(t)Q(0)u(0)$
$-\int^{t}_{0}A_{0}^{-1}S(s)(Q^{\prime}(t-s)u(t-s)+Q(t-s)u^{\prime}(t-s))ds,\quad
0<t<T.$ (4.12)

Applying the Lebesgue convergence theorem and the estimate
$|E_{\alpha,1}(\eta)|\leq\frac{C}{1+\eta},\ \eta>0$ (Theorem 1.6 in Po ), we
readily reach

$\|S(t)a-a\|^{2}=\sum_{n=1}^{\infty}|(a,\varphi_{n})|^{2}(E_{\alpha,1}(-\lambda_{n}t^{\alpha})-1)^{2}\,\longrightarrow\,0$

as $t\to\infty$ for $a\in L^{2}(\Omega)$.
Hence, $u\in C([0,T];L^{2}(\Omega))$ and $\lim_{t\downarrow 0}\|(S(t)-1)a\|=0$
and thus

$\lim_{s\downarrow 0}S(s)Q(t-s)u(t-s)=S(0)Q(t)u(t)\quad\mbox{in
}L^{2}(\Omega)$

and

$\lim_{s\uparrow t}S(s)Q(t-s)u(t-s)=S(t)Q(0)u(0)\quad\mbox{in }L^{2}(\Omega),$

which justify the last equality in the formula (4.12).

Thus, in terms of (4.12), the representation (2.13) can be rewritten in the
form

$A_{0}^{2}(u(t)-A_{0}^{-1}Q(t)u(t))=A_{0}^{2}G(t)-A_{0}S(t)Q(0)u(0)$
$-\int^{t}_{0}A_{0}^{\frac{1}{2}}S(s)A_{0}^{\frac{1}{2}}(Q^{\prime}(t-s)u(t-s)+Q(t-s)u^{\prime}(t-s))ds,\quad
0<t<T.$ (4.13)

Since $u(0)=a\in C^{\infty}_{0}(\Omega)$ and $F\in
C^{\infty}_{0}(\Omega\times(0,T))$, in view of (1.2) we have the inclusions

$A_{0}^{2}G(\cdot)\in C([0,T];L^{2}(\Omega)),\
A_{0}S(t)Q(0)u(0)=S(t)(A_{0}Q(0)a)\in C([0,T];L^{2}(\Omega)).$

Now we use the conditions (1.2) and (2.8) and repeat the arguments employed
for derivation of (4.5) by means of (4.6) and (4.11) to obtain the estimates

$\displaystyle\left\|\int^{t}_{0}A_{0}^{\frac{1}{2}}S(s)A_{0}^{\frac{1}{2}}(Q^{\prime}(t-s)u(t-s)+Q(t-s)u^{\prime}(t-s))ds\right\|$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}s^{-\frac{1}{2}\alpha}\|Q^{\prime}(t-s)u(t-s)+Q(t-s)u^{\prime}(t-s)\|_{H^{1}(\Omega)}ds$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}s^{-\frac{1}{2}\alpha}(\|A_{0}u^{\prime}(t-s)\|+\|A_{0}u(t-s)\|)ds\leq
Ct^{\frac{1}{2}\alpha}D$

and the inclusion

$-\int^{t}_{0}A_{0}^{\frac{1}{2}}S(s)A_{0}^{\frac{1}{2}}(Q^{\prime}(t-s)u(t-s)+Q(t-s)u^{\prime}(t-s))ds\in
C([0,T];L^{2}(\Omega)).$

Therefore,

$A_{0}^{2}(u(t)-A_{0}^{-1}Q(t)u(t))=A_{0}(A_{0}u(t)-Q(t)u(t))\in
C([0,T];L^{2}(\Omega)),$

that is,

$A_{0}u(t)-Q(t)u(t)\in C([0,T];\mathcal{D}(A_{0}))\subset
C([0,T];H^{2}(\Omega)).$

On the other hand, the estimate (4.9) implies $Q(t)u(t)\in
C([0,T];H^{2}(\Omega))$ and we obtain

$A_{0}u(t)\in C([0,T];H^{2}(\Omega)).$ (4.14)

For further arguments, we define the Schauder spaces
$C^{\theta}(\overline{\Omega})$ and $C^{2+\theta}(\overline{\Omega})$ with
$0<\theta<1$ (see, e.g., GT , LU ) as follows: A function $w$ is said to
belong to the space $C^{\theta}(\overline{\Omega})$ if

$\sup_{x,x^{\prime}\in\Omega,\,x\neq
x^{\prime}}\frac{|w(x)-w(x^{\prime})|}{|x-x^{\prime}|^{\theta}}<\infty.$

For $w\in C^{\theta}(\overline{\Omega})$, we define the norm

$\|w\|_{C^{\theta}(\overline{\Omega})}:=\|w\|_{C(\overline{\Omega})}+\sup_{x,x^{\prime}\in\Omega,\,x\neq
x^{\prime}}\frac{|w(x)-w(x^{\prime})|}{|x-x^{\prime}|^{\theta}}$

and for $w\in C^{2+\theta}(\overline{\Omega})$, the norm is given by

$\|w\|_{C^{2+\theta}(\overline{\Omega})}:=\|w\|_{C^{2}(\overline{\Omega})}+\sum_{|\tau|=2}\sup_{x,x^{\prime}\in\Omega,\,x\neq
x^{\prime}}\frac{|\partial_{x}^{\tau}w(x)-\partial_{x}^{\tau}w(x^{\prime})|}{|x-x^{\prime}|^{\theta}}.$

In the last formula, the notations
$\tau:=(\tau_{1},...,\tau_{d})\in(\mathbb{N}\cup\\{0\\})^{d}$,
$\partial_{x}^{\tau}:=\partial_{1}^{\tau_{1}}\cdots\partial_{d}^{\tau_{d}}$,
and $|\tau|:=\tau_{1}+\cdots+\tau_{d}$ are employed.

For $d=1,2,3$, the Sobolev embedding theorem says that $H^{2}(\Omega)\subset
C^{\theta}(\overline{\Omega})$ with some $\theta\in(0,1)$ (Ad ).

Therefore, in view of (4.14), we obtain the inclusion $h:=A_{0}u(\cdot,t)\in
C^{\theta}(\overline{\Omega})$ for each $t\in[0,T]$. Now we apply the Schauder
estimate (see, e.g., GT  or LU ) for solutions to the elliptic boundary value
problem

$A_{0}u(\cdot,t)=h\in C^{\theta}(\overline{\Omega})\quad\mbox{in }\Omega$

with the boundary condition
$\partial_{\nu_{A}}u(\cdot,t)+\sigma(\cdot)u(\cdot,t)=0$ on $\partial\Omega$
to reach the inclusion

$u\in C([0,T];C^{2+\theta}(\overline{\Omega})).$

This inclusion and (4.11) yield the conclusion $u\in
C([0,T];C^{2}(\overline{\Omega}))$ and $t^{1-\alpha}\partial_{t}u\in
C([0,T];C(\overline{\Omega}))$ of the lemma.

Finally we prove that $\lim_{t\to 0}\|u(t)-a\|=0$. By (2.8), we have

$\displaystyle\left\|\int^{t}_{0}K(t-s)h(s)ds\right\|\leq\int^{t}_{0}\|K(t-s)h(s)\|ds\leq
C\int^{t}_{0}(t-s)^{\alpha-1}\|h(s)\|ds$ $\displaystyle\leq$
$\displaystyle\frac{Ct^{\alpha}}{\alpha}\|h\|_{L^{\infty}(0,T;L^{2}(\Omega))},$

and so

$\lim_{t\to 0}\int^{t}_{0}K(t-s)h(s)ds=0\quad\mbox{in $L^{2}(\Omega)$}$ (4.15)

for each $h\in L^{\infty}(0,T;L^{2}(\Omega))$. Therefore by the regularity
$u\in C([0,T];C^{2}(\overline{\Omega}))$, we see that

$\lim_{t\to 0}\left(\int^{t}_{0}K(t-s)F(s)ds+Ru(t)\right)=0\quad\mbox{in
$L^{2}(\Omega)$},$

where $R$ is defined in (2.14). Moreover, for justifying (4.12), we have
already proved $\lim_{t\to 0}\|S(t)a-a\|=0$ for $a\in L^{2}(\Omega)$. Thus the
proof of Lemma 4 is complete.

II. Second part of the proof.

In this part, we weaken the regularity conditions posed on the solution $u$ to
(3.3) in Lemma 3 and prove the same results provided that $u\in
L^{2}(0,T;H^{2}(\Omega))$ and $u-a\in H_{\alpha}(0,T;L^{2}(\Omega))$, under
the assumption that
$\min\limits_{(x,t)\in\overline{\Omega}\times[0,T]}b_{0}(x,t)>0$ is
sufficiently large.

Let $F\in L^{2}(0,T;L^{2}(\Omega))$ and $a\in H^{1}(\Omega)$ satisfy the
inequalities $F(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$ and $a(x)\geq 0,\
x\in\Omega$.

Now we apply the standard mollification procedure (see, e.g., Ad ) and
construct the sequences $F_{n}\in C^{\infty}_{0}(\Omega\times(0,T))$ and
$a_{n}\in C^{\infty}_{0}(\Omega)$, $n\in\mathbb{N}$ such that $F_{n}(x,t)\geq
0,\ (x,t)\in\Omega\times(0,T)$ and $a_{n}(x)\geq 0,\ x\in\Omega$,
$n\in\mathbb{N}$ and
$\lim_{n\to\infty}\|F_{n}-F\|_{L^{2}(0,T;L^{2}(\Omega))}=0$ and
$\lim_{n\to\infty}\|a_{n}-a\|_{H^{1}(\Omega)}=0$. Then Lemma 4 yields the
inclusion

$u(F_{n},a_{n})\in C([0,T];C^{2}(\overline{\Omega})),\quad
t^{1-\alpha}\partial_{t}u(F_{n},a_{n})\in C([0,T];C(\overline{\Omega})),\quad
n\in\mathbb{N}$

and thus Lemma 3 ensures the inequalities

$u(F_{n},a_{n})(x,t)\geq 0,\ \ (x,t)\in\Omega\times(0,T),\,n\in\mathbb{N}.$
(4.16)

Since Theorem 2.1 holds true for the initial-boundary value problem (3.3) with
$F$ and $a$ replaced by $F-F_{n}$ and $a-a_{n}$, respectively, we have

$\|u(F,a)-u(F_{n},a_{n})\|_{L^{2}(0,T;H^{2}(\Omega))}$ $\leq
C(\|a-a_{n}\|_{H^{1}(\Omega)}+\|F-F_{n}\|_{L^{2}(0,T;L^{2}(\Omega))})\,\to\,0$

as $n\to\infty$. Therefore, we can choose a subsequence $m(n)\in\mathbb{N}$
such that $u(F,a)(x,t)=\lim_{m(n)\to\infty}u(F_{m(n)},a_{m(n)})(x,t)$ for
almost all $(x,t)\in\Omega\times(0,T)$. Then the inequality (4.16) leads to
the desired result, namely, to the inequality $u(F,a)(x,t)\geq 0$ for almost
all $(x,t)\in\Omega\times(0,T)$.

III. Third part of the proof.

Let the inequalities $a(x)\geq 0,\ x\in\Omega$ and $F(x,t)\geq 0,\
(x,t)\in\Omega\times(0,T)$ hold true for $a\in H^{1}(\Omega)$ and $F\in
L^{2}(0,T;L^{2}(\Omega))$ and let $u=u(F,a)\in L^{2}(0,T;H^{2}(\Omega))$ is a
solution to the problem (2.4). In order to complete the proof of Theorem 2, we
have to demonstrate the non-negativity of the solution without any assumptions
on the sign of the zeroth-order coefficient.

First, the zeroth-order coefficient $b_{0}(x,t)$ in the definition (3.1) of
the operator $-A_{1}$ is set to a constant $b_{0}>0$ that is assumed to be
sufficiently large. In this case, the initial-boundary value problem (2.4) can
be rewritten as follows:

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)+A_{1}u=(b_{0}+c(x,t))u+F(x,t),\quad(x,t)\in\Omega\times(0,T),\\\
&\partial_{\nu_{A}}u+\sigma u=0\quad\mbox{on
$\partial\Omega\times(0,T)$}.\end{array}\right.$ (4.17)

In what follows, we choose sufficiently large $b_{0}>0$ such that
$b_{0}\geq\|c\|_{C(\overline{\Omega}\times[0,T])}$.

In the previous parts of the proof, we already interpreted the solution $u$ as
a unique fixed point for the equation (4.3). Now let us construct an
appropriate approximating sequence $u_{n}$, $n\in\mathbb{N}$ for the fixed
point $u$. First we set $u_{0}(x,t):=0$ for $(x,t)\in\Omega\times(0,T)$ and
$u_{1}(x,t)=a(x)\geq 0,\ (x,t)\in\Omega\times(0,T)$. Then we define a sequence
$u_{n+1},\ n\in\mathbb{N}$ of solutions to the following initial-boundary
value problems with the given $u_{n}$:

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u_{n+1}-a)+A_{1}u_{n+1}=(b_{0}+c(x,t))u_{n}+F(x,t)\quad\mbox{in
}\Omega\times(0,T),\\\ &\partial_{\nu_{A}}u_{n+1}+\sigma
u_{n+1}=0\quad\mbox{on }\partial\Omega\times(0,T),\\\ &u_{n+1}-a\in
H_{\alpha}(0,T;L^{2}(\Omega)),\quad n\in\mathbb{N}.\end{array}\right.$ (4.18)

First we show that

$u_{n}(x,t)\geq 0,\quad(x,t)\in\Omega\times(0,T),\quad n\in\mathbb{N}.$ (4.19)

Indeed, the inequality (4.19) holds for $n=1$. Now we assume that
$u_{n}(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$. Then
$(b_{0}+c(x,t))u_{n}(x,t)+F(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$, and thus
by the results established in the second part of the proof of Theorem 4.1, we
obtain the inequality $u_{n+1}(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$. By the
principle of mathematical induction, the inequality (4.19) holds true for all
$n\in\mathbb{N}$.

Now we rewrite the problem (4.18) as

$\partial_{t}^{\alpha}(u_{n+1}(t)-a)+A_{0}u_{n+1}(t)=(Q(t)u_{n+1}-(c(t)+b_{0})u_{n+1})+(b_{0}+c(t))u_{n}+F,$

where $A_{0}$ and $Q(t)$ are defined by (2.5) and (2.14), respectively. Next
we estimate $w_{n+1}:=u_{n+1}-u_{n}$. By the relation (4.18), $w_{n+1}$ is a
solution to the problem

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}w_{n+1}+A_{0}w_{n+1}=(Q(t)w_{n+1}-(c(t)+b_{0})w_{n+1})+(b_{0}+c(x,t))w_{n}\\\
&\qquad\qquad\quad\mbox{in }\Omega\times(0,T),\\\
&\partial_{\nu_{A}}w_{n+1}+\sigma w_{n+1}=0\quad\mbox{on
}\partial\Omega\times(0,T),\\\ &w_{n+1}\in H_{\alpha}(0,T;L^{2}(\Omega)),\quad
n\in\mathbb{N}.\end{array}\right.$

In terms of the operator $K(t)$ defined by (2.7), acting similarly to our
analysis of the fixed point equation (4.3), we obtain the integral equation

$\displaystyle
w_{n+1}(t)=\int^{t}_{0}K(t-s)(Qw_{n+1})(s)ds-\int^{t}_{0}K(t-s)(c(s)+b_{0})w_{n+1}(s)ds$
$\displaystyle+$ $\displaystyle\int^{t}_{0}K(t-s)(b_{0}+c(s))w_{n}(s)ds,\quad
0<t<T,$

which leads to the inequalities

$\displaystyle\|A_{0}^{\frac{1}{2}}w_{n+1}(t)\|\leq\int^{t}_{0}\|A_{0}^{\frac{1}{2}}K(t-s)\|\|Q(s)w_{n+1}(s)\|ds$
$\displaystyle+$
$\displaystyle\int^{t}_{0}\|A_{0}^{\frac{1}{2}}K(t-s)\|\|(c(s)+b_{0})w_{n+1}(s)\|ds+\int^{t}_{0}\|A_{0}^{\frac{1}{2}}K(t-s)\|\|(b_{0}+c(s))w_{n}(s)\|ds$
$\displaystyle\leq$ $\displaystyle
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n+1}(s)\|ds+C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(s)\|ds\quad\mbox{for
$0<t<T$.}$

For their derivation, we used the norm estimates

$\|Q(s)w_{n+1}(s)\|\leq C\|w_{n+1}(s)\|_{H^{1}(\Omega)}\leq
C\|A_{0}^{\frac{1}{2}}w_{n+1}(s)\|$

and

$\|(c(s)+b_{0})w_{\ell}(s)\|\leq C\|w_{\ell}(s)\|_{H^{1}(\Omega)}\leq
C\|A_{0}^{\frac{1}{2}}w_{\ell}(s)\|,\quad\ell=n,n+1$

that hold true under the conditions (1.2). Thus we arrive at the integral
inequality

$\displaystyle\|A_{0}^{\frac{1}{2}}w_{n+1}(t)\|\leq
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n+1}(s)\|ds$
$\displaystyle+$ $\displaystyle
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(s)\|ds,\quad
0<t<T.$

The generalized Gronwall inequality yields now the estimate

$\displaystyle\|A_{0}^{\frac{1}{2}}w_{n+1}(t)\|\leq
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(s)\|ds$
$\displaystyle+$ $\displaystyle
C\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\left(\int^{s}_{0}(s-\xi)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(\xi)\|d\xi\right)ds.$

The second term at the right-hand side of the last inequality can be
represented as follows:

$\displaystyle\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\left(\int^{s}_{0}(s-\xi)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(\xi)\|d\xi\right)ds$
$\displaystyle=$
$\displaystyle\int^{t}_{0}\|A_{0}^{\frac{1}{2}}w_{n}(\xi)\|\left(\int^{t}_{\xi}(t-s)^{\frac{1}{2}\alpha-1}(s-\xi)^{\frac{1}{2}\alpha-1}ds\right)d\xi$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{1}{2}\alpha\right)\Gamma\left(\frac{1}{2}\alpha\right)}{\Gamma(\alpha)}\int^{t}_{0}(t-\xi)^{\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(\xi)\|d\xi$
$\displaystyle=$
$\displaystyle\frac{\Gamma\left(\frac{1}{2}\alpha\right)^{2}}{\Gamma(\alpha)}T^{\frac{1}{2}\alpha}\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(s)\|ds.$

Thus, we can choose a constant $C>0$ depending on $\alpha$ and $T$, such that

$\|A_{0}^{\frac{1}{2}}w_{n+1}(t)\|\leq
C\int^{t}_{0}(t-\xi)^{\frac{1}{2}\alpha-1}\|A_{0}^{\frac{1}{2}}w_{n}(s)\|ds,\quad
0<t<T,\,n\in\mathbb{N}.$ (4.20)

Recalling that

$\int^{t}_{0}(t-s)^{\frac{1}{2}\alpha-1}\eta(s)ds=\Gamma\left(\frac{1}{2}\alpha\right)(J^{\frac{1}{2}\alpha}\eta)(t),\quad
t>0,$

and setting $\eta_{n}(t):=\|A_{0}^{\frac{1}{2}}w_{n}(t)\|$, we can rewrite
(4.20) in the form

$\eta_{n+1}(t)\leq
C\Gamma\left(\frac{1}{2}\alpha\right)(J^{\frac{1}{2}\alpha}\eta_{n})(t),\quad
0<t<T,\,n\in\mathbb{N}.$ (4.21)

Since the Riemann-Liouville integral $J^{\frac{1}{2}\alpha}$ preserves the
sign and the semi-group property
$J^{\beta_{1}}(J^{\beta_{2}}\eta)(t)=J^{\beta_{1}+\beta_{2}}\eta(t)$ is valid
for any $\beta_{1},\beta_{2}>0$, applying the inequality (4.21) repeatedly, we
obtain the estimates

$\displaystyle\eta_{n}(t)\leq\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{n-1}(J^{(n-1)\frac{\alpha}{2}}\eta_{1})(t)$
$\displaystyle=$
$\displaystyle\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{n-1}}{\Gamma\left(\frac{\alpha}{2}(n-1)\right)}\left(\int^{t}_{0}(t-s)^{(n-1)\frac{1}{2}\alpha-1}ds\right)\|A_{0}^{\frac{1}{2}}a\|$
$\displaystyle=$
$\displaystyle\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)\right)^{n-1}}{\Gamma\left(\frac{\alpha}{2}(n-1)\right)}\frac{t^{(n-1)\frac{1}{2}\alpha}}{(n-1)\frac{1}{2}\alpha}\|A_{0}^{\frac{1}{2}}a\|\leq
C_{1}\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)T^{\frac{\alpha}{2}}\right)^{n-1}}{\Gamma\left(\frac{\alpha}{2}(n-1)\right)}.$

The known asymptotic behavior of the gamma function justifies the relation

$\lim_{n\to\infty}\frac{\left(C\Gamma\left(\frac{1}{2}\alpha\right)T^{\frac{\alpha}{2}}\right)^{n-1}}{\Gamma\left(\frac{\alpha}{2}(n-1)\right)}=0.$

Thus we have proved that the sequence $u_{N}=w_{0}+\cdots+w_{N}$ converges to
the solution $u$ in $L^{\infty}(0,T;H^{1}(\Omega))$ as $N\to\infty$.
Therefore, we can choose a subsequence $m(n)\in\mathbb{N}$ such that
$\lim_{m(n)\to\infty}u_{m(n)}(x,t)=u(x,t)$ for almost all
$(x,t)\in\Omega\times(0,T)$. This statement in combination with the inequality
(4.19) means that $u(x,t)\geq 0$ for almost all $(x,t)\in\Omega\times(0,T)$.
The proof of Theorem 4.1 is completed.

Now let us fix a source function $F=F(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$
and an initial value $a\in H^{1}(\Omega)$ in the initial-boundary value
problem (2.4) and denote by $u(c,\sigma)=u(c,\sigma)(x,t)$ the solution to the
problem (2.4) with the functions $c=c(x,t)$ and $\sigma=\sigma(x)$. Then the
following comparison property regarding the coefficients $c$ and $\sigma$ is
valid:

###### Theorem 4.2

Let $a\in H^{1}(\Omega)$ and $F\in L^{2}(\Omega\times(0,T))$ and the
inequalities $a(x)\geq 0,\ x\in\Omega$ and $F(x,t)\geq 0,\
(x,t)\in\Omega\times(0,T)$ hold true.
(i) Let $c_{1},c_{2}\in C^{1}([0,T];C^{1}(\overline{\Omega}))\cap
C([0,T];C^{2}(\overline{\Omega}))$ and $c_{1}(x,t)\geq c_{2}(x,t)$ for
$(x,t)\in\Omega$. Then $u(c_{1},\sigma)(x,t)\geq u(c_{2},\sigma)(x,t)$ in
$\Omega\times(0,T)$.
(ii) Let $c(x,t)<0,\ (x,t)\in\Omega\times(0,T)$ and a constant $\sigma_{0}>0$
be arbitrary and fixed. If the smooth functions $\sigma_{1},\sigma_{2}$ on
$\partial\Omega$ satisfy the conditions

$\sigma_{2}(x)\geq\sigma_{1}(x)\geq\sigma_{0},\ x\in\partial\Omega,$

then the inequality $u(c,\sigma_{1})\geq u(c,\sigma_{2}),\
x\in\Omega\times(0,T)$ holds true.

###### Proof

We start with a proof of the statement (i). Because $a(x)\geq 0,\ x\in\Omega$
and $F(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$, Theorem 4.1 yields the
inequality $u(c_{2},\sigma)(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$. Setting
$u(x,t):=u(c_{1},\sigma)(x,t)-u(c_{2},\sigma)(x,t)$ for
$(x,t)\in\Omega\times(0,T)$, we obtain

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}u-\sum_{i,j=1}^{d}\partial_{i}(a_{ij}\partial_{j}u)-\sum_{j=1}^{d}b_{j}\partial_{j}u\\\
-&c_{1}(x,t)u=(c_{1}-c_{2})u(c_{2},\sigma)(x,t)\quad\mbox{in
}\Omega\times(0,T),\\\ &\partial_{\nu_{A}}u+\sigma u=0\quad\mbox{on
}\partial\Omega,\\\ &u\in H_{\alpha}(0,T;L^{2}(\Omega)).\end{array}\right.$

Since $u(c_{2},\sigma)(x,t)\geq 0$ and $(c_{1}-c_{2})(x,t)\geq 0$ for
$(x,t)\in\Omega\times(0,T)$, Theorem 4.1 leads to the estimate $u(x,t)\geq 0$
for $(x,t)\in\Omega\times(0,T)$, which is equivalent to the inequality
$u(c_{1},\sigma)(x,t)\geq u(c_{2},\sigma)(x,t)$ for
$(x,t)\in\Omega\times(0,T)$ and the statement (i) is proved.

Now we proceed to the proof of the statement (ii). Similarly to the procedure
applied for the second part of the proof of Theorem 4.1, we choose the
sequences $F_{n}\geq 0$, $F_{n}\in C^{\infty}_{0}(\Omega\times(0,T))$ and
$a_{n}\geq 0$, $a_{n}\in C^{\infty}_{0}(\Omega)$, $n\in\mathbb{N}$ such that
$F_{n}\to F$ in $L^{2}(\Omega\times(0,T))$ and $a_{n}\to a$ in
$H^{1}(\Omega)$. Let $u_{n}$, $v_{n}$ be the solutions to the initial-boundary
value problem (2.4) with $F=F_{n}$, $a=a_{n}$ and with the coefficients
$\sigma_{1}$ and $\sigma_{2}$ in the boundary condition, respectively.
According to Lemma 4, the inclusions $v_{n},u_{n}\in
C(\overline{\Omega}\times[0,T])$ and
$t^{1-\alpha}\partial_{t}v_{n},\,t^{1-\alpha}\partial_{t}u_{n}\in
C([0,T];C(\overline{\Omega}))$, $n\in\mathbb{N}$ hold true and thus Theorem
4.1 yields

$v_{n}(x,t)\geq 0,\quad(x,t)\in\partial\Omega\times(0,T).$ (4.22)

Moreover, the relation

$\lim_{n\to\infty}\|u_{n}-u(c,\sigma_{1})\|_{L^{2}(0,T;L^{2}(\Omega))}=\lim_{n\to\infty}\|v_{n}-u(c,\sigma_{2})\|_{L^{2}(0,T;L^{2}(\Omega))}=0$
(4.23)

follows from Theorem 2.1. Let us now define an auxiliary function
$w_{n}:=u_{n}-v_{n}$. For this function, the inclusions

$t^{1-\alpha}\partial_{t}w_{n}\in C([0,T];C(\overline{\Omega})),\quad w_{n}\in
C([0,T];C^{2}(\overline{\Omega})),\quad n\in\mathbb{N}$ (4.24)

hold true. Furthermore, it is a solution to the initial-boundary value problem

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}w_{n}+Aw_{n}=0\quad\mbox{in
}\Omega\times(0,T),\\\
&\partial_{\nu_{A}}w_{n}+\sigma_{1}w_{n}=(\sigma_{2}-\sigma_{1})v_{n}\quad\mbox{on
}\partial\Omega\times(0,T),\\\ &w_{n}(x,\cdot)\in
H_{\alpha}(0,T)\quad\mbox{for almost all }x\in\Omega.\end{array}\right.$
(4.25)

The inequalities (4.22) and $\sigma_{2}(x)\geq\sigma_{1}(x),\
x\in\partial\Omega$ lead to the estimate

$\partial_{\nu_{A}}w_{n}+\sigma_{1}w_{n}\geq 0\quad\mbox{on
$\partial\Omega\times(0,T)$}.$ (4.26)

To finalize the proof of the theorem, a variant of Lemma 3 formulated below
will be employed.

###### Lemma 5

Let the elliptic operator $-A$ be defined by (2.1) and the conditions (1.2) be
satisfied. Moreover, let the inequality $c(x,t)<0$ for $x\in\overline{\Omega}$
and $0\leq t\leq T$ hold true and there exist a constant $\sigma_{0}>0$ such
that

$\sigma(x)\geq\sigma_{0}\quad\mbox{for all $x\in\partial\Omega$}.$

For $a\in H^{1}(\Omega)$ and $F\in L^{2}(\Omega\times(0,T))$, we further
assume that there exists a solution $u\in C([0,T];C^{2}(\overline{\Omega}))$
to the initial-boundary value problem

$\left\\{\begin{array}[]{rl}&\partial_{t}^{\alpha}(u-a)+Au=F\quad\mbox{in
$\Omega\times(0,T)$},\\\ &\partial_{\nu_{A}}u+\sigma(x)u\geq 0\quad\mbox{on
$\partial\Omega\times(0,T)$},\\\ &u(x,\cdot)-a\in
H_{\alpha}(0,T)\quad\mbox{for almost all $x\in\Omega$}\end{array}\right.$

that satisfies the inclusion $t^{1-\alpha}\partial_{t}u\in
C([0,T];C(\overline{\Omega}))$. Then the inequalities $F(x,t)\geq 0,\
(x,t)\in\Omega\times(0,T)$ and $a(x)\geq 0,\ \Omega$ implicate the inequality
$u(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$.

In the formulation of this lemma, at the expense of the extra condition
$\sigma(x)>0$ on $\partial\Omega$, we do not assume that
$\min\limits_{(x,t)\in\overline{\Omega}\times[0,T]}(-c(x,t))$ is sufficiently
large. This is the main difference between the conditions supposed in Lemma 5
and in Lemma 3. The proof of Lemma 5 is much simpler compared to the one of
Lemma 3; it will be presented at the end of this section.

Now we complete the proof of Theorem 4.2. Since $c(x,t)<0$ for
$(x,t)\in\Omega\times(0,T)$ and $\sigma_{1}(x)\geq\sigma_{0}>0$ on
$\partial\Omega$ and taking into account the conditions (4.24) and (4.26), we
can apply Lemma 5 to the initial-boundary value problem (4.25) and deduce the
inequality $w_{n}(x,t)\geq 0,\ (x,t)\in\Omega\times(0,T)$, that is,
$u_{n}(x,t)\geq v_{n}(x,t),\ (x,t)\in\Omega\times(0,T)$ for $n\in\mathbb{N}$.
Due to the relation (4.23), we can choose a suitable subsequence of $w_{n},\
n\in\mathbb{N}$ and pass to the limit as $n$ tends to infinity thus arriving
at the inequality $u(c,\sigma_{1})(x,t)\geq u(c,\sigma_{2})(x,t)$ in
$\Omega\times(0,T)$. The proof of Theorem 4.2 is completed.

At this point, let us mention a direction for further research in connection
with the results formulated and proved in this sections. In order to remove
the negativity condition posed on the coefficient $c=c(x,t)$ in Theorem 4.2
(ii), one needs a unique existence result for solutions to the initial-
boundary value problems of type (2.4) with non-zero Robin boundary condition
similar to the one formulated in Theorem 2.1. There are several works that
treat the case of the initial-boundary value problems with non-homogeneous
Dirichlet boundary conditions (see, e.g., Ya18  and the references therein).
However, to the best of the authors’ knowledge, analogous results are not
available for the initial-boundary value problems with the non-homogeneous
Neumann or Robin boundary conditions. Thus, in Theorem 4.2 (ii), we assumed
the condition $c(x,t)<0,\ (x,t)\in\Omega\times(0,T)$, although our conjecture
is that this result holds true for an arbitrary coefficient $c=c(x,t)$.

We conclude this section with a proof of Lemma 5 that is simple because in
this case we do not need the function $\psi$ defined as in (3.4).

###### Proof

First we introduce an auxiliary function as follows:

$\widetilde{w}(x,t):=u(x,t)+\varepsilon(1+t^{\alpha}),\quad
x\in\Omega,\,0<t<T.$

The inequalities $c(x,t)<0,\ (x,t)\in\overline{\Omega}\times[0,T]$ and
$\sigma(x)\geq\sigma_{0}>0,\ x\in\partial\Omega$ and the calculations similar
to the ones done in the proof of Lemma 3 implicate the inequalities

$d_{t}^{\alpha}\widetilde{w}+A\widetilde{w}=F+\varepsilon\Gamma(\alpha+1)-c(x,t)\varepsilon(1+t^{\alpha})>0\quad\mbox{in
$\Omega\times(0,T)$},$
$\partial_{\nu_{A}}\widetilde{w}+\sigma\widetilde{w}=\partial_{\nu_{A}}u+\sigma
u+\sigma\varepsilon(1+t^{\alpha})\geq\sigma_{0}\varepsilon\quad\mbox{on
$\partial\Omega\times(0,T)$}$

and

$\widetilde{w}(x,0)=a(x)+\varepsilon\geq\varepsilon\quad\mbox{in $\Omega$}.$

Based on these inequalities, the same arguments that were employed after the
formula (3.10) in the proof of Lemma 3 readily complete the proof of Lemma 5.

##  5 Appendix

In the proof of Lemma 3 that is a basis for all other derivations presented in
this paper, we essentially used an auxiliary function that satisfies the
conditions (3.4). Thus, ensuring existence of such function is an important
problem worth for detailed considerations. In this Appendix, we present a
solution to this problem.

For the readers’ convenience, we split our existence proof into three parts.

I. First part of the proof.

In this part, we prove the following lemma:

###### Lemma 6

Let the conditions (1.2) be satisfied and the constant

$M:=\min_{(x,t)\in\overline{\Omega}\times[0,T]}b_{0}(x,t)>0$

is sufficiently large.

Then there exists a constant $\kappa_{1}>0$ such that

$(A_{1}(t)v,\,v)\geq\kappa_{1}\|v\|_{H^{1}(\Omega)}^{2}$ (5.1)

for all $v\in H^{2}(\Omega)$ satisfying $\partial_{\nu_{A}}v+\sigma v=0$ on
$\partial\Omega$ for each $t\in[0,T]$.

In particular, Lemma 6 implies that all of the eigenvalues of the operator
$A_{0}$ defined by (2.5) are positive if the constant $c_{0}>0$ is
sufficiently large. Henceforth we employ the notation $b=(b_{1},...,b_{d})$.

###### Proof

By using the conditions (1.2) and the boundary condition
$\partial_{\nu_{A}}v+\sigma v=0$ on $\partial\Omega$, integration by parts
yields

$\displaystyle(A_{1}(t)v,v)$ $\displaystyle=$
$\displaystyle-\int_{\Omega}\sum_{i,j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}v)vdx-\frac{1}{2}\int_{\Omega}\sum_{j=1}^{d}b_{j}(x,t)\partial_{j}(|v|^{2})dx+\int_{\Omega}b_{0}(x,t)|v|^{2}dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}\sum_{i,j=1}^{d}a_{ij}(x)(\partial_{i}v)(\partial_{j}v)dx-\int_{\partial\Omega}(\partial_{\nu_{A}}v)vdS$
$\displaystyle+$
$\displaystyle\frac{1}{2}\int_{\Omega}(\mbox{div}\,b)|v|^{2}dx-\frac{1}{2}\int_{\partial\Omega}(b\cdot\nu)|v|^{2}dS+\int_{\Omega}b_{0}(x,t)|v|^{2}dx$
$\displaystyle\geq$ $\displaystyle\kappa\int_{\Omega}|\nabla
v|^{2}dx+\int_{\Omega}\left(\min_{(x,t)\in\overline{\Omega}\times[0,T]}b_{0}(x,t)-\frac{1}{2}|\mbox{div}\,b|\right)|v|^{2}dx$
$\displaystyle+$
$\displaystyle\int_{\partial\Omega}\left(\sigma-\frac{1}{2}(b\cdot\nu)\right)|v|^{2}dS$
$\geq\kappa\int_{\Omega}|\nabla
v|^{2}dx+\left(M-\frac{1}{2}\|\mbox{div}\,b\|_{C(\overline{\Omega}\times[0,T])}\right)\int_{\Omega}|v|^{2}dx-C\int_{\Omega}|v|^{2}dS.$
(5.2)

Here and henceforth $C>0$, $C_{\varepsilon},C_{\delta}>0$, etc. denote generic
constants which are independent of the function $v$.

By the trace theorem (Theorem 9.4 (pp. 41-42) in LM ), for
$\delta\in(0,\frac{1}{2}]$, there exists a constant $C_{\delta}>0$ such that

$\|v\|_{L^{2}(\partial\Omega)}\leq
C_{\delta}\|v\|_{H^{\delta+\frac{1}{2}}(\Omega)}\quad\mbox{for all $v\in
H^{1}(\Omega)$.}$

Now we fix $\delta\in\left(0,\,\frac{1}{2}\right)$. The interpolation
inequality for the Sobolev spaces implicates that for any $\varepsilon>0$
there exists a constant $C_{\varepsilon,\delta}>0$ such that the following
inequality holds true (see, e.g., Chapter IV in Ad  or Sections 2.5 and 11 of
Chapter 1 in LM ):

$\|v\|_{H^{\delta+\frac{1}{2}}(\Omega)}\leq\varepsilon\|\nabla
v\|_{L^{2}(\Omega)}+C_{\varepsilon,\delta}\|v\|_{L^{2}(\Omega)}\quad\mbox{for
all $v\in H^{1}(\Omega)$}.$

Therefore, we obtain the estimate

$\displaystyle\|v\|^{2}_{L^{2}(\partial\Omega)}\leq(\varepsilon
C_{\delta}\|\nabla
v\|_{L^{2}(\Omega)}+C_{\delta}C_{\varepsilon,\delta}\|v\|_{L^{2}(\Omega)})^{2}$
$\displaystyle\leq$ $\displaystyle 2(\varepsilon C_{\delta})^{2}\|\nabla
v\|_{L^{2}(\Omega)}^{2}+2(C_{\delta}C_{\varepsilon,\delta})^{2}\|v\|_{L^{2}(\Omega)}^{2}$

for all $v\in H^{1}(\Omega)$. Substituting this inequality into (5.2), we
obtain

$\displaystyle(\kappa-2C(\varepsilon C_{\delta})^{2})\|\nabla
v\|_{L^{2}(\Omega)}^{2}$ $\displaystyle+$
$\displaystyle(M-\frac{1}{2}\|\mbox{div}\,b\|_{C(\overline{\Omega}\times[0,T])}-2(CC_{\delta}C_{\varepsilon,\delta})^{2})\|v\|_{L^{2}(\Omega)}^{2}\leq(A_{1}(t)v,v).$

Choosing a sufficiently small $\varepsilon>0$ such that $\kappa-2C(\varepsilon
C_{\delta})^{2}>0$ and a sufficiently large $M>0$ such that

$M>\frac{1}{2}\|\mbox{div}\,b\|_{C(\overline{\Omega}\times[0,T])}+2(CC_{\delta}C_{\varepsilon,\delta})^{2}$

completes the proof of Lemma 6.

II. Second part of the proof.

Due to the estimate (5.1), we can apply Theorem 3.2 (p. 137) in LU  that
implicates existence of a constant $\theta\in(0,1)$ such that for each
$t\in[0,T]$, a solution $\psi(\cdot,t)\in C^{2+\theta}(\overline{\Omega})$ to
the problem (3.4) exists uniquely, where $C^{2+\theta}(\overline{\Omega})$ is
the Schauder space defined in the proof of Lemma 4 in Section 4.

Now we introduce an auxiliary function

$\eta(t):=\|\psi(\cdot,t)\|_{C^{2+\theta}(\overline{\Omega})},\quad 0\leq
t\leq T.$ (5.3)

Because of the inclusion $\psi(\cdot,t)\in C^{2+\theta}(\overline{\Omega})$,
the value of the function $\eta(t)$ is finite for each $t\in[0,T]$.

Furthermore, for an arbitrary $G\in C^{\theta}(\overline{\Omega})$, there
exists a unique solution $w=w(\cdot,t)$ to the problem

$\left\\{\begin{array}[]{rl}&A_{1}(t)w=G\quad\mbox{in $\Omega$},\\\
&\partial_{\nu_{A}}w+\sigma w=0\quad\mbox{on
$\partial\Omega$}\end{array}\right.$ (5.4)

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

Now we prove that for each $t\in[0,T]$ there exists a constant $C_{t}>0$ such
that

$\|w(\cdot,t)\|_{C^{2+\theta}(\overline{\Omega})}\leq
C_{t}\|G\|_{C^{\theta}(\overline{\Omega})}$ (5.5)

for all solutions $w$ of the problem (5.4). In the inequality (5.5), the
constant $C_{t}>0$ depends on the norms
$\|a_{ij}\|_{C^{1}(\overline{\Omega})}$, $1\leq i,j\leq d$,
$\|b_{k}\|_{C([0,T];C^{2}(\overline{\Omega}))}$, $0\leq k\leq d$ of the
coefficients, but not on the coefficients by themselves.

Indeed, for each $t\in[0,T]$, the inequality

$\|w(\cdot,t)\|_{C^{2+\theta}(\overline{\Omega})}\leq
C_{t}(\|G\|_{C^{\theta}(\overline{\Omega})}+\|w(\cdot,t)\|_{C(\overline{\Omega})})$
(5.6)

holds true (see, e.g., the formula (3.7) on p. 137 in LU ). To obtain the
desired estimate we have to eliminate the term
$\|w(\cdot,t)\|_{C(\overline{\Omega})}$ on the right-hand side of the last
inequality. This can be done by the standard compactness-uniqueness arguments.
More precisely, let us assume that (5.5) does not hold. Then there exist the
sequences $w_{n}\in C^{2+\theta}(\overline{\Omega}),\ n\in\mathbb{N}$ and
$G_{n}\in C^{\theta}(\overline{\Omega}),\ n\in\mathbb{N}$ such that
$\|w_{n}\|_{C^{2+\theta}(\overline{\Omega})}=1$ and
$\lim_{n\to\infty}\|G_{n}\|_{C^{\theta}(\overline{\Omega})}=0$. By the Ascoli-
Arzelà theorem, we can extract a subsequence $w_{k(n)}$ from the sequence
$w_{n}$ such that $w_{k(n)}\longrightarrow\widetilde{w}$ in
$C(\overline{\Omega})$ as $n\to\infty$. Applying the estimation (5.6) to the
equation

$A_{1}(t)(w_{k(n)}-w_{k(m)})=G_{k(n)}-G_{k(m)}\quad\mbox{in $\Omega$}$

equipped with the homogeneous boundary condition
$\partial_{\nu_{A}}(w_{k(n)}-w_{k(m)})+\sigma(w_{k(n)}-w_{k(m)})=0$ on
$\partial\Omega$, we we arrive at the relation

$\displaystyle\|w_{k(n)}-w_{k(m)}\|_{C^{2+\theta}(\overline{\Omega})}$
$\displaystyle\leq$ $\displaystyle
C_{t}(\|G_{k(n)}-G_{k(m)}\|_{C^{\theta}(\overline{\Omega})}+\|w_{k(n)}-w_{k(m)}\|_{C(\overline{\Omega})})\,\to
0$

as $n,m\to\infty$. Hence, there exists a function $w_{0}\in
C^{2+\theta}(\overline{\Omega})$ such that $w_{k(n)}\to w_{0}$ in
$C^{2+\theta}(\overline{\Omega})$. Moreover, we obtain the relations

$\|w_{0}\|_{C^{2+\theta}(\overline{\Omega})}=\lim_{n\to\infty}\|w_{k(n)}\|_{C^{2+\theta}(\overline{\Omega})}=1$

and $G_{k(n)}=A_{1}(t)w_{k(n)}\to A_{1}(t)w_{0}$ in
$C^{\theta}(\overline{\Omega})$.

Since $\lim_{n\to\infty}\|G_{k(n)}\|_{C^{\theta}(\overline{\Omega})}=0$, we
obtain $A_{1}(t)w_{0}=0$ in $\Omega$ with $\partial_{\nu_{A}}w_{0}+\sigma
w_{0}=0$ on $\partial\Omega$. Then Lemma 6 yields $w_{0}(x,t)=0$ in $\Omega$
that contradicts the relation $\|w_{0}\|_{C^{2+\theta}(\overline{\Omega})}=1$
that we established above. The obtained contradiction implicates the desired
norm estimate (5.5).

III. Third part of the proof.

The last missing detail of the proof is the inclusion $\psi\in
C^{1}([0,T];C^{2}(\overline{\Omega}))$ for the function $\psi$ constructed in
the previous part of the proof.

To show this inclusion, we first verify that for an arbitrarily but fixed
$t\in[0,T]$ the function $d(x,s):=\psi(x,t)-\psi(x,s)$ satisfies the equations

$\left\\{\begin{array}[]{rl}&-A_{1}(t)d(\cdot,s)=(b_{0}(t)-b_{0}(s))\psi(\cdot,s)\\\
-&\sum_{j=1}^{d}(b_{j}(t)-b_{j}(s))\partial_{j}\psi(\cdot,s)\quad\mbox{in
$\Omega$ for $0\leq s,t\leq T$},\\\ &\partial_{\nu_{A}}d+\sigma
d=0\quad\mbox{on $\partial\Omega$, $\,\,$ $0\leq s,t\leq
T$}.\end{array}\right.$ (5.7)

For an arbitrarily but fixed $\delta>0$ we set
$I_{\delta,t}:=[0,T]\cap\\{s;\,|t-s|\leq\delta\\}$.

Applying the relation (5.3) and the estimate (5.5) to the solution $d$ of
(5.7) yields

$\displaystyle\|d(\cdot,s)\|_{C^{2+\theta}(\overline{\Omega})}$
$\displaystyle\leq$ $\displaystyle
C\left(\left\|\sum_{j=1}^{d}(b_{j}(t)-b_{j}(s))\partial_{j}\psi(\cdot,s)\right\|_{C^{\theta}(\overline{\Omega})}+\|(b_{0}(t)-b_{0}(s))\psi(\cdot,s)\|_{C^{\theta}(\overline{\Omega})}\right)$
$\leq
C\sum_{j=0}^{d}\|b_{j}(s)-b_{j}(t)\|_{C^{1}(\overline{\Omega})}\eta(s)\leq
C\max_{0\leq j\leq
d}\|b_{j}(s)-b_{j}(t)\|_{C^{1}(\overline{\Omega})}\,\sup_{s\in
I_{\delta,t}}\eta(s).$ (5.8)

For the function

$h(\delta):=\max_{0\leq j\leq
d}\sup_{|s-t|\leq\delta}\|b_{j}(s)-b_{j}(t)\|_{C^{1}(\overline{\Omega})},$

the inclusions $b_{j}\in C([0,T];C^{1}(\overline{\Omega}))$, $0\leq j\leq d$
imply $\lim_{\delta\downarrow 0}h(\delta)=0$.

Now we rewrite the estimate (5.8) in terms of the function $\eta$ defined by
(5.3) as

$|\eta(s)-\eta(t)|\leq Ch(\delta)\sup_{s\in I_{\delta,t}}\eta(s)\quad\mbox{for
$s\in I_{\delta,t}$},$

and thus obtain the inequality

$\eta(s)\leq\eta(t)+Ch(\delta)\sup_{s\in I_{\delta,t}}\eta(s)\quad\mbox{for
$s\in I_{\delta,t}$}.$

Choosing $\delta:=\delta(t)>0$ sufficiently small, for a given $t\in[0,T]$,
the estimate $\sup_{s\in I_{\delta(t),t}}\eta(s)\leq C_{1}\eta(t)$ holds true.
Varying $t\in[0,T]$, we now choose a finite number of the intervals
$I_{\delta(t),t}$ that cover the whole interval $[0,T]$ and thus obtain the
norm estimate

$\|\psi\|_{L^{\infty}(0,T;C^{2+\theta}(\overline{\Omega}))}\leq C_{2}$ (5.9)

with some constant $C_{2}>0$.

For $s\in I_{\delta(t),t}$, substitution of (5.9) into (5.8) yields the
estimate

$\|d(\cdot,s)\|_{C^{2+\theta}(\overline{\Omega}))}=\|\psi(\cdot,t)-\psi(\cdot,s)\|_{C^{2+\theta}(\overline{\Omega}))}\leq
Ch(\delta)C_{2}.$

Consequently, $\lim_{s\to
t}\|\psi(\cdot,s)-\psi(\cdot,t)\|_{C^{2+\theta}(\overline{\Omega})}=0$ and we
have shown the inclusion

$\psi\in C([0,T];C^{2+\theta}(\overline{\Omega})).$ (5.10)

To complete the proof, the now verify the inclusion $\psi\in
C^{1}([0,T];C^{2+\theta}(\overline{\Omega}))$. Since
$-A_{1}(\xi)\psi(x,\xi)=1$ in $\Omega$, differentiating this formula with
respect to $\xi$ leads to the representation

$\sum_{j=1}^{d}\partial_{i}(a_{ij}(x)\partial_{j}\partial_{\xi}\psi(x,\xi))+\sum_{j=1}^{d}b_{j}(\xi)\partial_{j}\partial_{\xi}\psi(x,\xi)-b_{0}(\xi)\partial_{\xi}\psi(x,\xi)$
(5.11)
$=-\sum_{j=1}^{d}\partial_{\xi}b_{j}(\xi)\partial_{j}\psi(x,\xi)+(\partial_{\xi}b_{0})(\xi)\psi(x,\xi)\quad\mbox{in
$\Omega$}.$

By subtracting the equation (5.11) with $\xi=s$ from the one with $\xi=t$, we
deduce that the function
$d_{1}(x,s):=(\partial_{t}\psi)(x,t)-(\partial_{t}\psi)(x,s)$ satisfies the
relation

$-A_{1}(t)d_{1}(x,s)$ (5.12) $\displaystyle=$
$\displaystyle\biggl{[}-\sum_{i,j=1}^{d}(b_{j}(t)-b_{j}(s))\partial_{j}\partial_{s}\psi(x,s)+(b_{0}(t)-b_{0}(s))\partial_{s}\psi(x,s)$
$\displaystyle-$
$\displaystyle\sum_{j=1}^{d}(\partial_{t}b_{j}(t)-\partial_{s}b_{j}(s))\partial_{j}\psi(x,s)+(\partial_{t}b_{0}(t)-\partial_{s}b_{0}(s))\psi(x,s)\biggr{]}$
$\displaystyle+$
$\displaystyle\left[-\sum_{j=1}^{d}\partial_{t}b_{j}(t)(\partial_{j}\psi(x,t)-\partial_{j}\psi(x,s))+\partial_{t}b_{0}(t)(\psi(x,t)-\psi(x,s))\right]$
$\displaystyle=:$ $\displaystyle H_{1}(x,t,s)+H_{2}(x,t,s)\quad\mbox{in
$\Omega$}$

and the boundary condition $\partial_{\nu_{A}}d_{1}+\sigma d_{1}=0$ on
$\partial\Omega$ for all $s,t\in[0,T]$. Thus, the inclusion $\psi\in
C^{1}([0,T];C^{2+\theta}(\overline{\Omega}))$ will follow from the relation
$\lim_{s\to t}\|d_{1}(\cdot,s)\|_{C^{2+\theta}(\overline{\Omega})}=0$ that we
now prove. To this end, by applying Theorem 3.2 (p. 137) in LU  to the
equation (5.12), it suffices to prove that

$\lim_{s\to
t}\|H_{\ell}(\cdot,t,s)\|_{C^{\theta}(\overline{\Omega})}=0,\quad\ell=1,2.$
(5.13)

Applying Theorem 3.2 in LU  to the equation (5.11), in view of the regularity
conditions (1.2) and the equation (5.11), we obtain the estimates

$\|\partial_{t}\psi(\cdot,t)\|_{C^{2+\theta}(\overline{\Omega})}$ (5.14)
$\displaystyle\leq$ $\displaystyle
C\left(\left\|\sum_{j=1}^{d}(\partial_{t}b_{j})(\cdot,t)\partial_{j}\psi(\cdot,t)\right\|_{C^{\theta}(\overline{\Omega})}+\|(\partial_{t}b_{0})(\cdot,t)\psi(\cdot,t)\|_{C^{\theta}(\overline{\Omega})}\right)$
$\displaystyle\leq$ $\displaystyle
C\sum_{j=0}^{d}\|b_{j}(\cdot,t)\|_{C^{1}([0,T];C^{1}(\overline{\Omega}))}\|\psi(\cdot,t)\|_{C^{2+\theta}(\overline{\Omega})}\leq
C_{3}\quad\mbox{for $0\leq t\leq T$.}$

The inequalities (5.9) and (5.14) lead to the norm estimate

$\|H_{1}(\cdot,t,s)\|_{C^{\theta}(\overline{\Omega})}\leq
C_{4}\sum_{k=0}^{1}\sum_{j=0}^{d}\|(\partial_{t}^{k}b_{j})(\cdot,t)-(\partial_{t}^{k}b_{j})(\cdot,s)\|_{C^{1}(\overline{\Omega})}.$
(5.15)

By employing the arguments similar to those used above, the estimate

$\|H_{2}(\cdot,t,s)\|_{C^{\theta}(\overline{\Omega})}\leq
C_{4}\sum_{j=0}^{d}\|\partial_{t}b_{j}\|_{C([0,T];C^{1}(\overline{\Omega}))}\sum_{k=0}^{1}\|\nabla^{k}\psi(\cdot,t)-\nabla^{k}\psi(\cdot,s)\|_{C^{\theta}(\overline{\Omega})}$
(5.16)

can be derived. Since $\partial_{t}^{k}b_{j}\in
C([0,T];C^{1}(\overline{\Omega}))$ for $k=0,1$ and $0\leq j\leq d$ by the
conditions (1.2) and $\nabla^{k}\psi\in
C([0,T];C^{1+\theta}(\overline{\Omega}))$ with $k=0,1$ by the inclusion
(5.10), the relation $\lim_{s\to
t}\|H_{\ell}(\cdot,t,s)\|_{C^{\theta}(\overline{\Omega})}=0$ with $\ell=1,2$
immediately follows from the norm estimates (5.15) and (5.16). As mentioned
above, this completes the proof of existence of a function satisfying the
conditions (3.4).

###### Acknowledgements.

The second author was supported by Grant-in-Aid for Scientific Research Grant-
in-Aid (A) 20H00117 of Japan Society for the Promotion of Science.

## Conflict of interest

The authors declare that they have no conflict of interest.

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