# Parabolic Differential Equations with Bounded Delay

Marek Kryspin & Janusz Mierczyński
Faculty of Pure and Applied Mathematics
Wrocław University of Science and Technology
Wybrzeże Wyspiańskiego 27
PL-50-370 Wrocław, Poland

###### Abstract

We show the continuous dependence of solutions of linear nonautonomous second
order parabolic partial differential equations (PDEs) with bounded delay on
coefficients and delay. The assumptions are very weak: only convergence in the
weak-* topology of delay coefficients is required. The results are important
in the applications of the theory of Lyapunov exponents to the investigation
of PDEs with delay.

##  0 Introduction

The purpose of the present paper is to formulate and prove results on
existence and continuous dependence on parameters of solutions of linear
second order partial differential equations (PDEs) of parabolic type with
bounded time delay. To be more specific, consider a rather simplified example,
that is, an equation of the form

$\begin{cases}\displaystyle\frac{\partial u}{\partial
t}(t,x)={\Delta}u(t,x)+c_{1}(t,x)u(t-R(t),x),&t\in[0,T],\ x\in
D\\\\[6.45831pt] u(t,x)=0&t\in[0,T],\ x\in\partial D,\end{cases}$ (0.1)

where $D\subset\mathbb{R}^{N}$ is a bounded domain with boundary $\partial D$,
$\Delta$ is the Laplace operator in $x$, $T>0$, $c_{1}\colon(0,T)\times
D\to\mathbb{R}$ belongs to $L_{\infty}((0,T)\times D)$, and
$R\colon[0,T]\to[0,1]$ is a function in $L_{\infty}((0,T))$.

The theory of Lyapunov exponents (or rather, more generally, the theory of
skew-product dynamical systems) is a powerful tool in the applications of the
theory of dynamical systems to the investigation of evolution equations (in a
broad sense, containing but not excluded to, ordinary differential equations,
parabolic partial differential equations, hyperbolic partial differential
equations). That theory requires the (linear) equation to generate a skew-
product dynamical system on some bundle whose base is the closure of the set
of coefficients of the original equation.

Let us consider two cases. We will remain in the simplified framework of
(0.1).

  * $\bullet$

The _nonautonomous_ case,

$\begin{cases}\displaystyle\frac{\partial u}{\partial
t}(t,x)={\Delta}u(t,x)+c_{1}(t,x)u(t-R(t),x),&t>0,\ x\in D\\\\[6.45831pt]
u(t,x)=0&t>0,\ x\in\partial D,\end{cases}$ (0.2)

where $c_{1}$ is defined on $(-\infty,\infty)\times D$ and $R$ is defined on
$(-\infty,\infty)$. We take the closure, in an appropriate topology, of the
set of all time-translates of $c_{1}$ (the so-called _hull_). The topology
must be, on the one hand, coarse enough for the hull to be a compact
(metrizable) space, and, on the other hand, fine enough for, first, the time
translation operator on the hull to be continuous, and, second, the solution
operator to depend continuously on parameters, that is, members of the hull.
The paper [39] gives a survey of subsets of function spaces that can serve as
hulls.

For the theory of linear skew-product (semi)flows on bundles whose fibers are
Banach spaces and some of its applications, see, e.g., [40], [11], [12], [41],
[10], [35], [36], [4] for a very incomplete list arranged in chronological
order.

  * $\bullet$

The _random_ case,

$\begin{cases}\displaystyle\frac{\partial u}{\partial
t}(t,x)={\Delta}u(t,x)+c_{1}(\theta_{t}\omega,x)u(t-R(\theta_{t}\omega),x),&t>0,\
x\in D\\\\[6.45831pt] u(t,x)=0&t>0,\ x\in\partial D,\end{cases}$ (0.3)

where $c_{1}$ is now defined on $\Omega\times D$ and $R$ is defined on
$\Omega$, with $(\Omega,\mathfrak{F},\mathbb{P})$ a probability space on which
an ergodic measurable flow $\theta=(\theta_{t})_{t\in\mathbb{R}}$ acts. Here
the role of hull is played by $\Omega$, and the measurability of the flow
$\theta$ is one of the assumptions. In order to apply the theory of Lyapunov
exponents in the measurable setting, as presented in, e.g., [30], [25], [26],
[27], [9], one needs to show the measurable dependence of the solution
operators on $\omega\in\Omega$

In the present paper we address the problem of continuous dependence on
members of the hull. As the space of coefficients we take a closed and bounded
subset of the Banach space of essentially bounded (Lebesgue-)measurable
functions on $(0,T)\times D$, where $T>0$, with the weak-* topology induced by
the duality pairing between $L_{1}$ and $L_{\infty}$. Regarding the zero order
coefficients and delay terms, no additional assumption is made. In particular,
the dependence on $t$ can be quite weak.

Although there have been a lot of papers dealing with the issues of the
existence of solutions of delay PDEs (many of them nonlinear, and admitting
more general delay terms, employing various definitions of solutions, see,
e.g., [42], [43], [44], [20], [21], [31], [32], [46], [23], [24], [6], [7],
[8]), the only papers we are aware of dealing explicitly with continuous
dependence of solutions of delay PDEs on parameters are [37] and [38].

To give a flavor of our results, we formulate now some specializations of our
main results to the case of (0.1). We assume $1<p<\infty$.

The first, a specialization of Theorem 3.1, establishes the existence and
uniqueness of mild solutions.

######  Theorem.

Let $c_{1}\in L_{\infty}((0,T)\times D)$, $u_{0}\in C([-1,0],L_{p}(D))$ and
$R\in L_{\infty}((0,T))$ be such that $R(t)\in[0,1]$ for Lebesgue-a.e.
$t\in(0,T)$. Then there exists a unique solution $u(\cdot;c_{1},u_{0},R)\in
C([-1,T])$ of Eq. (0.1) with initial condition $u(t;c_{1},u_{0},R)=u_{0}$,
$t\in[-1,0]$. The solution is understood in a suitable integral sense (a mild
solution).

The second, a specialization of Theorem 5.1(ii), establishes the continuity,
in a suitable sense, of a solution with respect to initial conditions and
parameters.

######  Theorem.

Assume that $(c_{1,m})_{m=1}^{\infty}$, $(u_{0,m})_{m=1}^{\infty}$ and
$(R_{m})_{m=1}^{\infty}$ are sequences satisfying the following:

  * $\bullet$

$c_{1,m}$ have their $L_{\infty}((0,T)\times D)$-norms uniformly bounded, and
converge in the weak-* topology to $c_{1}\in L_{\infty}((0,T)\times D)$;

  * $\bullet$

$u_{0,m}$ converge in the norm topology of $C([-1,0],L_{p}(D)$ to $u_{0}$;

  * $\bullet$

$R_{m}$ converge for Lebesgue-a.e. $t\in(0,T)$ to $R$.

Then

$u(\cdot;c_{1,m},u_{0,m},R_{m})\to u(\cdot;c_{1},u_{0},R)$

in the $C([-1,0],L_{p}(D))$-norm.

The paper is organized as follows.

Section 1 presents the assumptions used throughout.

In Section 2 results concerning the existence and basic properties of (weak)
solutions to linear parabolic PDEs without delay terms are gathered. They are
for the most part taken from [35] and based on [13], though some of them
(Proposition 2.18, for example), perhaps belonging to the folk lore, appear in
print for the first time.

Section 3 is devoted to defining and proving the existence and uniqueness of
(mild) solutions of PDEs with delay terms. Section 4 provides estimates of the
solutions which are then used to prove the continuous dependence on initial
conditions.

Section 5 can be considered the main part of the paper. Here the continuous
dependence of solutions on coefficients and delay terms is proved under very
weak assumptions: coefficients are required to converge in the weak-* topology
only.

It should be mentioned that a similar approach has been successfully applied
in the case of ordinary differential equations with delay in [36], [33], [34],
see also [17, Chpt. 5], [18], [5].

###  0.1 General Notations

We write $\mathbb{R}^{+}$ for $[0,\infty)$, and $\mathbb{Q}$ for the set of
all rationals.

If $B\subset A$, we write $\mathbbm{1}_{\\!B}$ for the _indicator_ of $B$:
$\mathbbm{1}_{\\!B}(a)=1$ if $a\in B$ and $\mathbbm{1}_{\\!B}(a)=0$ if $a\in
A\setminus B$.

For a metric space $(Y,d)$, $\mathfrak{B}(Y)$ denotes the $\sigma$-algebra of
all Borel subsets of $Y$.

For Banach spaces $X_{1}$, $X_{2}$ with norms $\lVert\cdot\rVert_{X_{1}}$,
$\lVert\cdot\rVert_{X_{2}}$, we let $\mathcal{L}(X_{1},X_{2})$ stand for the
Banach space of bounded linear mappings from $X_{1}$ into $X_{2}$, endowed
with the standard norm $\lVert\cdot\rVert_{X_{1},X_{2}}$. Instead of
$\mathcal{L}(X,X)$ we write $\mathcal{L}(X)$, and instead of
$\lVert\cdot\rVert_{X,X}$ we write $\lVert\cdot\rVert_{X}$.
$\mathcal{L}_{\mathrm{s}}(X_{1},X_{2})$ denotes the space of bounded linear
mappings from $X_{1}$ into $X_{2}$ equipped with the strong operator topology.
Instead of $\mathcal{L}_{\mathrm{s}}(X,X)$ we write
$\mathcal{L}_{\mathrm{s}}(X)$.

Throughout the paper, $T>0$ will be fixed.

We set

$\Delta\vcentcolon=\\{\,(s,t)\in\mathbb{R}^{2}:0\leq s\leq t\leq
T\,\\},\qquad\dot{\Delta}\vcentcolon=\\{\,(s,t)\in\mathbb{R}^{2}:0\leq s<t\leq
T\,\\}.$

Throughout the paper, $D\subset\mathbb{R}^{N}$ stands for a bounded domain,
with boundary $\partial D$.

By $\mathfrak{L}((0,T))$ we understand the $\sigma$-algebra of all Lebesgue-
measurable subsets of $(0,T)$. The notations $\mathfrak{L}(D)$ and
$\mathfrak{L}((0,T)\times D)$ are defined in a similar way.

For $u$ belonging to a Banach space of (equivalence classes of) functions
defined on $D$ we will denote by $u[x]$ the value of $u$ at $x\in D$.

$L_{p}(D)=L_{p}(D,\mathbb{R})$ has the standard meaning, with the norm, for
$1\leq p<\infty$, given by

$\lVert u\rVert_{L_{p}(D)}\vcentcolon=\biggr{(}\int_{D}\lvert
u[x]\rvert^{p}\,\mathrm{d}x\biggr{)}^{\\!\frac{1}{p}},$

and for $p=\infty$ given by

$\lVert
u\rVert_{L_{\infty}(D)}\vcentcolon=\operatorname*{ess\,sup}\limits_{D}{u}.$

For $1\leq p\leq\infty$ let $p^{\prime}$ stand for the Hölder conjugate of
$p$. The duality pairing between $L_{p}(D)$ and $L_{p^{\prime}}(D)$ is given,
for $1<p<\infty$, or for $p=1$ and $p^{\prime}=\infty$, by

$\langle
u,v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=\int\limits_{D}u[x]\,v[x]\,\mathrm{d}x,\quad
u\in L_{p}(D),\ v\in L_{p^{\prime}}(D).$

Let $u$ be an equivalence class of functions defined for Lebesgue-a.e.
$t\in(0,T)$ and taking values in $L_{p}(D)$, $1\leq p<\infty$ (in the sequel
we will refer to such $u$ simply as a function).

  * $\bullet$

$u$ is said to be _measurable_ if it is
$(\mathfrak{L}((0,T)),\mathfrak{B}(L_{p}(D)))$-measurable, meaning that the
preimage under $u$ of any open subset of $L_{p}(D)$ belongs to
$\mathfrak{L}((0,T))$.

  * $\bullet$

$u$ is _strongly measurable_ (sometimes called _Bochner measurable_) if there
exists a sequence $(u_{m})_{m=1}^{\infty}$ of simple functions such that
$\lim\limits_{m\to\infty}\lVert u_{m}(t)-u(t)\rVert_{L_{p}(D)}=0$ for
Lebesgue-a.e. $t\in(0,T)$.

  * $\bullet$

$u$ is _weakly measurable_ if for any $v\in L_{p^{\prime}}(D)$ the function

$[\,t\mapsto\langle u(t),v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}\,]$

is $(\mathfrak{L}((0,T)),\mathfrak{B}(\mathbb{R}))$-measurable.

######  Theorem 0.1.

For $u\colon(0,T)\to L_{p}(D)$ measurability, strong measurability and weak
measurability are equivalent.

The equivalence of strong and weak measurability is a consequence of Pettis’s
Measurability Theorem (see, e.g., [16, Thm. 2.1.2]). For the fact that
measurability implies strong measurability see, e.g., [45, Thm. 1], whereas
the proof of the reverse implication is a simple exercise.

For our purposes we will use the following definitions (see, e.g. [2, Sect.
X.4]). A measurable $u\colon(0,T)\to L_{p}(D)$ belongs to
$L_{r}((0,T),L_{p}(D))$, $1\leq r<\infty$, if $\lVert
u(\cdot)\rVert_{L_{p}(D)}$ belongs to $L_{r}((0,T))$, with

$\displaystyle\lVert
u\rVert_{L_{r}((0,T),L_{p}(D))}=\biggl{(}\int\limits_{0}^{T}\lVert
u(t)\rVert_{L_{p}(D)}^{r}\,\mathrm{d}t\biggr{)}^{\\!\\!1/r}.$

Similarly, a measurable $u\colon(0,T)\to L_{p}(D)$ belongs to
$L_{\infty}((0,T),L_{p}(D))$, if $\lVert u(\cdot)\rVert_{L_{p}(D)}$ belongs to
$L_{\infty}((0,T))$, with

$\displaystyle\lVert
u\rVert_{L_{\infty}((0,T),L_{p}(D))}=\operatorname*{ess\,sup}_{t\in(0,T)}{\lVert
u(t)\rVert_{L_{p}(D)}}\ .$

The following result, a part of [19, Lemma III.11.16], will be used several
times.

######  Lemma 0.1.

  1. (a)

If $u\in L_{1}((0,T),L_{1}(D))$ then the function

$\bigl{[}\,(0,T)\times D\ni(t,x)\mapsto u(t)[x]\in\mathbb{R}\,\bigr{]}$

belongs to $L_{1}((0,T)\times D,\mathbb{R})$.

  2. (b)

If $w$ is $(\mathfrak{L}((0,T)\times D),\mathfrak{B}(\mathbb{R}))$-measurable,
and for Lebesgue-a.e. $t\in(0,T)$ the _$t$ -section_ $w(t,\cdot)$ belongs to
$L_{p}(D)$, where $1\leq p<\infty$, then the function

$\Bigl{[}\,(0,T)\ni t\mapsto\bigl{[}\,D\ni x\mapsto
w(t,x)\in\mathbb{R}\,\bigr{]}\,\Bigr{]}$

is $(\mathfrak{L}(0,T),\mathfrak{B}(L_{p}(D)))$-measurable.

######  Remark 0.1.

Regarding Lemma 0.1(b), we remark that in [19] the analog of $w$ is assumed to
be
$(\mathfrak{L}((0,T))\otimes\mathfrak{L}(D),\mathfrak{B}(\mathbb{R}))$-measurable
(no completion) rather than $(\mathfrak{L}((0,T)\times
D),\mathfrak{B}(\mathbb{R}))$-measurable. As an $(\mathfrak{L}((0,T)\times
D),\mathfrak{B}(\mathbb{R}))$-measurable function can be made into an
$(\mathfrak{L}((0,T))\otimes\mathfrak{L}(D),\mathfrak{B}(\mathbb{R}))$-measurable
function by changing its values on a set of $(N+1)$-dimensional Lebesgue
measure zero (see, e.g., [22, Prop. 2.12]), our formulation follows.

##  1 Assumptions and Definitions

###  1.1 Main Equation

Consider a linear second order partial differential equation with bounded
delay

$\displaystyle\frac{\partial u}{\partial t}$
$\displaystyle=\sum_{i=1}^{N}\frac{\partial}{\partial
x_{i}}\bigg{(}\sum_{j=1}^{N}a_{ij}(t,x)\frac{\partial u}{\partial
x_{j}}+a_{i}(t,x)u\bigg{)}$ (ME)
$\displaystyle+\sum_{i=1}^{N}b_{i}(t,x)\frac{\partial u}{\partial
x_{i}}+c_{0}(t,x)u$ $\displaystyle+c_{1}(t,x)u(t-R(t));\quad 0\leq t\leq T,\
x\in D.$

The delay map $R\colon[0,T]\to\mathbb{R}$ is bounded from below by $0$ and
from above by $1$, i.e.

$0\leq R(t)\leq 1,\qquad\forall\,t\in[0,T].$

Sometimes the function $\xi\mapsto\xi-R(\xi)$ will be denoted by $\Phi$. The
function $\Phi$ will be called relative time delay. Further,
$D\subset\mathbb{R}^{N}$ is a bounded domain with boundary $\partial D$. The
equation (ME) will be complemented with boundary conditions

$\mathcal{B}u=0,\qquad 0\leq t\leq T,\ x\in\partial D.$ (BC)

Later on, we will use the notation $\mathcal{B}_{a}$ in other to exhibit
dependence of the operator $\mathcal{B}$ on $a$. The boundary conditions
operator (BC) will be one of this form

$\mathcal{B}u=\left\\{\begin{array}[]{lll}u&\quad\mathrm{(Dirichlet)}\\\\[8.61108pt]
\displaystyle\sum_{i=1}^{N}\bigg{(}\sum_{j=1}^{N}a_{ij}(t,x)\frac{\partial
u(t)}{\partial x_{j}}+a_{i}(t,x)u\bigg{)}\nu_{i}&\quad\mathrm{(Neumann)}\\\
\displaystyle\sum_{i=1}^{N}\bigg{(}\sum_{j=1}^{N}a_{ij}(t,x)\frac{\partial
u(t)}{\partial
x_{j}}+a_{i}(t,x)u\bigg{)}\nu_{i}+d_{0}(t,x)u&\quad\mathrm{(Robin).}\end{array}\right.$

The vector $\boldsymbol{\nu}=(\nu_{1},\dots,\nu_{N})$ denotes the unit normal
on the boundary $\partial D$ pointing out of $D$, interpreted in a certain
weak sense (in the regular sense if $\partial D$ is sufficiently smooth [35]).

The initial condition is considered in the following way: for $u_{0}\in
C([-1,0],\linebreak L_{p}(D))$, where $1\leq p\leq\infty$, find a solution of
(ME)$+$(BC) satisfying

$u(t)=u_{0}(t)\text{ for }t\in[-1,0].$ (IC)

By (ME)$+$(BC) we understand equation (ME) equipped with boundary condition
(BC). Later on, we will also use $\eqref{main-eq}_{a}+\eqref{main-bc}_{a}$
notation to indicate that parameters of $\eqref{main-eq}+\eqref{main-bc}$ are
fixed to be $a$.

Note that, without any additional assumptions on the delay map $R$, the
initial data cannot be taken from $L_{p}(D)\oplus L_{r}((-1,0),L_{p}(D))$, as
in [33] or [34]. The reason for this is that the delay map $R$ can be
constructed in such way that $t\mapsto t-R(t)$ would be a constant function.
In such a situation the initial value problem $\eqref{main-eq}+\eqref{main-
bc}$ would be not meaningful. Under some additional assumptions the situation
can change, for example a constant delay map allows us to introduce
generalized initial data in $L_{p}(D)\oplus L_{r}((-1,0),L_{p}(D))$. However,
in this paper we will not focus on that.

In order to clearly define the problem $\eqref{main-eq}_{a}+\eqref{main-
bc}_{a}$, it is also necessary to set the delay map $R$. However, the
assumptions on $R$ will be given later. Moreover, we suppress the notation of
$R$ from $\eqref{main-eq}_{a}+\eqref{main-bc}_{a}$. We will present the
solutions of $\eqref{main-eq}_{a}+\eqref{main-bc}_{a}$ in the form of
$u(\cdot;a,u_{0},R)$ and often suppress the notation of $a,u_{0}$ or $R$ if it
does not lead to confusion.

###  1.2 Main Assumptions

We introduce some assumptions on the domain $D\subset\mathbb{R}^{N}$ and the
coefficients of the problem (ME)+(BC).

  1. $\mathrm{(DA\hskip 1.00006pt1)}$

(Boundary regularity) For Dirichlet boundary conditions, $D$ is a bounded
domain. For Neumann or Robin boundary conditions, $D$ is a bounded domain with
Lipschitz boundary.

In all expressions of the type “a.e.” we consider $1$-dimensional Lebesgue
measure on $(0,T)$, $N$-dimensional Lebesgue measure on $D$ and
$(N-1)$-dimensional Hausdorff measure on $\partial D$. The latter is, by
($\mathrm{DA\hskip 1.00006pt1}$), equal to surface measure on $\partial D$.

The notation $L_{\infty}(\partial D)$ [resp. $L_{\infty}((0,T)\times\partial
D)$] corresponds to surface measure on $\partial D$ [resp. to the product of
$1$-dimensional Lebesgue measure on $(0,T)$ and surface measure on $\partial
D$].

  1. $\mathrm{(DA\hskip 1.00006pt2)}$

(Boundedness) The functions

     * $\diamond$

$a_{ij}\colon(0,T)\times D\to\mathbb{R}$ ($i,j=1,\dots,N$),

     * $\diamond$

$a_{i}\colon(0,T)\times D\to\mathbb{R}$ ($i=1,\dots,N$),

     * $\diamond$

$b_{i}\colon(0,T)\times D\to\mathbb{R}$ ($i=1,\dots,N$),

     * $\diamond$

$c_{0}\colon(0,T)\times D\to\mathbb{R}$ ,

     * $\diamond$

$c_{1}\colon(0,T)\times D\to\mathbb{R}$

belong to $L_{\infty}((0,T)\times D)$. When the Robin boundary condition holds
the function $d_{0}\colon(0,T)\times\partial D\to\mathbb{R}$ belongs to
$L_{\infty}((0,T)\times\partial D)$.

It is worth noticing at this point that uniform $L_{\infty}(D)$-boundedness of
$a_{ij}(t,\cdot)$, $a_{i}(t,\cdot)$, $b_{i}(t,\cdot)$, $c_{0}(t,\cdot)$,
$c_{1}(t,\cdot)$ and uniform $L_{\infty}(\partial D)$-boundedness of
$d_{0}(t,\cdot)$ for a.e. $t\in(0,T)$ follow from the assumption
($\mathrm{DA\hskip 1.00006pt2}$) and Fubini’s theorem.

######  Definition 1.1 ($Y$ coefficients space).

Let $Y$ be a subset of the Banach space $L_{\infty}((0,T)\times
D,\mathbb{R}^{N^{2}+2N+2})\oplus L_{\infty}((0,T)\times\partial D,\mathbb{R})$
satisfying the following assumptions

  1. $\mathrm{(Y\hskip 1.00006pt1)}$

$Y$ is norm-bounded and, moreover, it is closed (hence compact, via the
Banach–Alaoglu theorem) in the weak-* topology,

  2. $\mathrm{(Y\hskip 1.00006pt2)}$

the function $d_{0}\geq 0$ if the Robin boundary condition holds. The function
$d_{0}$ is interpreted as the zero function in the Dirichlet or Neumann cases.

Elements of $Y$ will be denoted by

$a\vcentcolon=\big{(}(a_{ij})_{i,j=1}^{N},(a_{i})_{i=1}^{N},(b_{i})_{i=1}^{N},c_{0},c_{1},d_{0}\big{)}\in
Y.$

The weak-* topology of the space $Y$ is understood in the standard sense,
namely, as the weak-* topology induced via the isomorphism

$L_{\infty}((0,T)\times D,\mathbb{R}^{N^{2}+2N+2})\oplus
L_{\infty}((0,T)\times\partial D,\mathbb{R})\\\ \cong\big{(}L_{1}((0,T))\times
D,\mathbb{R}^{N^{2}+2N+2})\oplus L_{1}((0,T)\times\partial
D,\mathbb{R})\big{)}^{*}.$

######  Definition 1.2 (Flattening $Y$ to $Y_{0}$).

The mapping defined on $Y$ by

$\big{(}(a_{ij})_{i,j=1}^{N},(a_{i})_{i=1}^{N},(b_{i})_{i=1}^{N},c_{0},c_{1},d_{0}\big{)}\,\widetilde{}\vcentcolon=\big{(}(a_{ij})_{i,j=1}^{N},(a_{i})_{i=1}^{N},(b_{i})_{i=1}^{N},c_{0},0,d_{0}\big{)}$

will be called the _flattening_ of $a\in Y$.

The above mapping is obviously continuous. As a consequence, the image $Y_{0}$
of $Y$ under that mapping shares properties analogous to ($\mathrm{Y\hskip
1.00006pt1}$) and ($\mathrm{Y\hskip 1.00006pt2}$).

  1. $\mathrm{(DA\hskip 1.00006pt3)}$

(Ellipticity) There exists a constant $\alpha_{0}>0$ such that for any
$a_{0}\in Y_{0}$ the inequality

$\displaystyle\sum\limits_{i,j=1}^{N}a_{ij}(t,x)\xi_{i}\xi_{j}\geq\alpha_{0}\sum\limits_{i=1}^{N}\xi_{i}^{2},$

holds for a.e. $(t,x)\in(0,T)\times D$ and all $\xi\in\mathbb{R}^{N}$, and the
functions $a_{ij}(\cdot,\cdot)$ are symmetric in the indices, i.e.
$a_{ij}(\cdot,\cdot)\equiv a_{ji}(\cdot,\cdot)$ for all $i,j=1,\dots,N$.

  2. $\mathrm{(DA\hskip 1.00006pt4)}$

(Sequential compactness of $Y_{0}$ with respect to convergence a.e.)

Any sequence $(a_{0,m})_{m=1}^{\infty}$ of elements of $Y_{0}$, where

$a_{0,m}\vcentcolon=\bigl{(}(a_{ij,m})_{i,j=1}^{N},(a_{i,m})_{i=1}^{N},(b_{i,m})_{i=1}^{N},c_{0,m},0,d_{0,m}\bigr{)},$

convergent as $m\to\infty$ in the weak-* topology to $a_{0}\in Y_{0}$ has the
property that

     * $\bullet$

the sequence
$\bigl{(}(a_{ij,m})_{i,j=1}^{N},(a_{i,m})_{i=1}^{N},(b_{i,m})_{i=1}^{N}\bigr{)}$
converges to
$\bigl{(}(a_{ij})_{i,j=1}^{N},\linebreak(a_{i})_{i=1}^{N},(b_{i})_{i=1}^{N}\bigr{)}$
pointwise a.e. on $(0,T)\times D$,

     * $\bullet$

the sequence $d_{0,m}$ converges to $d_{0}$ pointwise a.e. on
$(0,T)\times\partial D$.

Occasionally we will use the following.

  3. $\mathrm{(DA\hskip 1.00006pt5)}$

$Y_{0}$ is a singleton.

For the purposes of studying continuous dependence on parameters and delay we
introduce now the delay class and the relative delay class equipped with
suitable topologies.

######  Definition 1.3.

The delay class is defined as follows

$\mathcal{R}\vcentcolon=\big{\\{}R\in L_{\infty}((0,T)):\,R(t)\in[0,1]\text{
for a.e. }t\in(0,T)\big{\\}}.$

The delay class is equipped with the weak-* topology.

At some moments we also use the following notation
$\widetilde{\mathcal{R}}\vcentcolon=\\{[\,t\mapsto
t-R(t)\,]:\,R\in\mathcal{R}\\}$ especially in more abstract lemmas when
general properties of the mapping $t\mapsto t-R(t)$ are important.

######  Remark 1.1.

The delay class $\mathcal{R}$ is a norm-bounded, convex and weak-* closed
subset of $L_{\infty}((0,T))$, hence, by the Banach–Alaoglu theorem, it is
compact in the weak-* topology.

The following assumption is a property of a subset
$\mathcal{R}_{0}\subset\mathcal{R}$.

  1. $\mathrm{(DA\hskip 1.00006pt6)}$

If $R\in\mathcal{R}_{0}$ is a weak-* limit of
$(R_{m})_{m=1}^{\infty}\subset\mathcal{R}_{0}$ then $(R_{m})_{m=1}^{\infty}$
converge pointwise a.e. on $(0,T)$ to $R$.

######  Remark 1.2.

Note that the assumption ($\mathrm{DA\hskip 1.00006pt6}$) is naturally
satisfied when $\mathcal{R}_{0}$ is compact in $\mathcal{R}$ with respect to
the norm topology. This fact follows from an observation that any Hausdorff
topology weaker than the norm topology (such as the weak-* topology) is equal
to the norm topology on a compact subset.

######  Remark 1.3.

Note that the weak-* topology on $Y$ is metrizable, see [35, (1.3.1)].
Similarly, the weak-* topology on $\mathcal{R}$ is metrizable, see [14, Thm.
3.6.17 and Cor. 3.6.18].

##  2 Weak Solutions

In the present section we assume ($\mathrm{DA\hskip 1.00006pt1}$),
($\mathrm{DA\hskip 1.00006pt2}$) and that the flattening $Y_{0}$ of $Y$ as in
Definition 1.1 satisfies ($\mathrm{DA\hskip 1.00006pt3}$). Occasionally we
will assume ($\mathrm{DA\hskip 1.00006pt4}$).

We start with a PDE parameterized by $a_{0}\in Y_{0}$

$\displaystyle\frac{\partial u}{\partial t}$
$\displaystyle=\sum_{i=1}^{N}\frac{\partial}{\partial
x_{i}}\bigg{(}\sum_{j=1}^{N}a_{ij}(t,x)\frac{\partial u}{\partial
x_{j}}+a_{i}(t,x)u\bigg{)}$ ($\widehat{\mathrm{ME}}$)
$\displaystyle+\sum_{i=1}^{N}b_{i}(t,x)\frac{\partial u}{\partial
x_{i}}+c_{0}(t,x)u;\quad 0\leq s\leq t\leq T,\ x\in D.$

Equation ($\widehat{\mathrm{ME}}$) is complemented with boundary conditions

$\mathcal{B}u=0,\quad 0\leq s\leq t\leq T,\ x\in\partial D,$
($\widehat{\mathrm{BC}}$)

where $\mathcal{B}$ is either the Dirichlet or the Neumann or else the Robin
boundary operator.

We are looking for solutions of the problem
($\widehat{\mathrm{ME}}$)${}_{a_{0}}$\+ ($\widehat{\mathrm{BC}}$)${}_{a_{0}}$
for initial condition $u_{0}\in L_{2}(D)$. To define a solution we introduce
the space $H$ as follows. Let

$V\vcentcolon=\left\\{\begin{array}[]{lll}H^{1}_{0}(D)&\quad\mathrm{for\
Dirichlet\ boundary\ condition}\\\\[4.30554pt] H^{1}(D)&\quad\mathrm{for\
Neumann\ or\ Robin\ boundary\ condition}\end{array}\right.$

and

$H=H(s,T;V)\vcentcolon=\\{\,v\in L_{2}((s,T),V):\dot{v}\in
L_{2}((s,T),V^{*})\,\\}$

equipped with the norm

$\lVert v\rVert_{H}\vcentcolon=\Bigl{(}\int_{s}^{T}\lVert
v(\zeta)\rVert_{V}^{2}\,\mathrm{d}\zeta+\int_{s}^{T}\lVert\dot{v}(\zeta)\rVert_{V^{*}}^{2}\,\mathrm{d}\zeta\Bigr{)}^{\frac{1}{2}},$

where $\dot{v}\vcentcolon=\mathrm{d}v/\mathrm{d}t$ is the time derivative in
the sense of distributions taking values in $V^{*}$ (see [15, Chpt. XVIII] for
definitions).

For $a_{0}\in Y_{0}$ define a bilinear form

$\displaystyle B_{a_{0}}[t;u,v]$
$\displaystyle\vcentcolon=\int\limits_{D}\biggl{(}\,\sum\limits_{i,j=1}^{N}a_{ij}(t,x)\frac{\partial
u}{\partial x_{i}}\frac{\partial v}{\partial
x_{j}}+\sum\limits_{i=1}^{N}a_{i}(t,x)u\frac{\partial v}{\partial x_{i}}$
$\displaystyle-\sum\limits_{i=1}^{N}b_{i}(t,x)\frac{\partial u}{\partial
x_{i}}v-c_{0}(t,x)uv\biggr{)}\,\mathrm{d}x$

in the Dirichlet or Neumann case, and

$\displaystyle B_{a_{0}}[t;u,v]$
$\displaystyle\vcentcolon=\int\limits_{D}\biggl{(}\,\sum\limits_{i,j=1}^{N}a_{ij}(t,x)\frac{\partial
u}{\partial x_{i}}\frac{\partial v}{\partial
x_{j}}+\sum\limits_{i=1}^{N}a_{i}(t,x)u\frac{\partial v}{\partial x_{i}}$
$\displaystyle-\sum\limits_{i=1}^{N}b_{i}(t,x)\frac{\partial u}{\partial
x_{i}}v-c_{0}(t,x)uv\biggr{)}\,\mathrm{d}x+\int\limits_{\partial
D}d_{0}(t,x)uv\,\mathrm{d}H$

in the Robin case, where $H_{N-1}$ stands for the $(N-1)$-dimensional
Hausdorff measure.

######  Definition 2.1 (Local Weak Solution).

For $a_{0}\in Y_{0}$, $0\leq s\leq t\leq T$ and $u_{0}\in L_{2}(D)$ a function
$u\in L_{2}([s,t],V)$ such that $\dot{u}\in L_{2}([s,t],V^{*})$ is a weak
solution of ($\widehat{\mathrm{ME}}$)${}_{a_{0}}$\+
($\widehat{\mathrm{BC}}$)${}_{a_{0}}$ on $[s,t]$ with initial condition
$u(s)=u_{0}$ if

$-\int\limits_{s}^{t}(u(\zeta),v)_{L_{2}(D)}\,\dot{\psi}(\zeta)\,\mathrm{d}\zeta+\int\limits_{s}^{t}B_{a_{0}}[\zeta;u(\zeta),v]\psi(\zeta)\,\mathrm{d}\zeta=(u_{0},v)_{L_{2}(D)}\,\psi(s)$

for any $v\in V$ and any $\psi\in\mathcal{D}([s,t),\mathbb{R})$ where set
$\mathcal{D}([s,t),\mathbb{R})$ is the space of all smooth real functions
having compact support in $[s,t)$ and $(\cdot,\cdot)_{L_{2}(D)}$ denotes the
standard inner product in $L_{2}(D)$.

######  Definition 2.2 (Global Weak Solution).

When $t=T$ in definition 2.1 then a weak solution will be called a global weak
solution.

######  Proposition 2.1 (Existence of global weak solution).

For any initial condition $u_{0}\in L_{2}(D)$ there exists a unique global
weak solution of ($\widehat{\mathrm{ME}}$)$+$($\widehat{\mathrm{BC}}$).

###### Proof.

See [13, Thm. 2.4] for a proof and [35, Prop. 2.1.5] for a unified theory of
weak solutions. ∎∎

For $a_{0}\in Y_{0}$ and $0\leq s<T$ we write the unique global weak solution
of ($\widehat{\mathrm{ME}}$)${}_{a_{0}}$\+
($\widehat{\mathrm{BC}}$)${}_{a_{0}}$ with initial condition $u(s)=u_{0}$ as
$U_{a_{0}}(t,s)u_{0}\vcentcolon=u(t)$.

Below we present a couple of results from [35, Ch. 2].

######  Proposition 2.2.

The mappings

$U_{a_{0}}(t,s)u_{0}=u(t;a_{0},u_{0}),\quad 0\leq s\leq t\leq T,\ a_{0}\in
Y_{0},\ u_{0}\in L_{2}(D)$

have the following properties.

$\displaystyle U_{a_{0}}(s,s)=\mathrm{Id}_{L_{2}(D)},\quad a_{0}\in Y_{0},\
s\in[0,T],$ (2.1) $\displaystyle U_{a_{0}}(t_{2},t_{1})\circ
U_{a_{0}}(t_{1},s)=U_{a_{0}}(t_{2},s),\quad a_{0}\in Y_{0},\ 0\leq s\leq
t_{1}\leq t_{2}\leq T.$ (2.2)

###### Proof.

See [35, Props. 2.1.5 through 2.1.8]. ∎∎

######  Proposition 2.3.

  1. (i)

Let $1\leq p<\infty$ and $0\leq s<T$. For any $a_{0}\in Y_{0}$ there exists
$U_{a_{0},p}(t)\in\mathcal{L}(L_{p}(D))$ such that

$U_{a_{0},p}(t,s)u_{0}=U_{a_{0}}(t,s)u_{0},\quad u_{0}\in L_{2}(D)\cap
L_{p}(D).$

  2. (ii)

Let $1<p<\infty$ and $a_{0}\in Y_{0}$. Then the mapping

$\bigl{[}\,[s,T]\ni t\mapsto
U_{a_{0},p}(t,s)\in\mathcal{L}_{\mathrm{s}}(L_{p}(D))\,\bigr{]}$

is continuous.

###### Proof.

See [13, Cor. 7.2] for part (i) and [13, Thm. 5.1] for part (ii). ∎∎

For $p=1$ we have an analog of Proposition 2.3(ii).

######  Proposition 2.4.

Let $1\leq p<\infty$, $0\leq s<T$ and $a_{0}\in Y_{0}$. Then the mapping

$\big{[}\,(s,T]\ni t\mapsto
U_{a_{0}}(t,s)\in\mathcal{L}_{\mathrm{s}}(L_{p}(D))\,\big{]}$

is continuous.

###### Proof.

See [35, Prop. 2.2.6]. ∎∎

For $0\leq s<T$ we write $U_{a_{0},p}(s,s)=\mathrm{Id}_{L_{p}(D)}$ even if
$p=1,\infty$.

######  Proposition 2.5.

For any $a_{0}\in Y_{0}$, $0\leq s\leq t_{1}\leq t_{2}\leq T$ and any $1\leq
p\leq\infty$

$U_{a_{0},p}(t_{2},t_{1})\circ U_{a_{0},p}(t_{1},s)=U_{a_{0},p}(t_{2},s)$
(2.3)

###### Proof.

See [35, Prop. 2.1.7] for the proof of $p=2$ case. For $p\not=2$ it suffices
to use the fact that $U_{a_{0}}(t,s)\in\mathcal{L}(L_{2}(D))$ and the
continuity of the mappings $[u\mapsto U_{a_{0},p}(t_{2},t_{1})\circ
U_{a_{0},p}(t_{1},s)u]$ and $[u\mapsto U_{a_{0},p}(t_{2},s)u]$, which is
guaranteed by Proposition 2.3. ∎∎

######  Proposition 2.6.

For any $a_{0}\in Y_{0}$ and any $0\leq s\leq t_{1}\leq t_{2}\leq T$ the
operator $U_{a_{0}}(t_{2},t_{1})$ has an a.e. nonnegative kernel.

###### Proof.

See [3, Thm. 1.3] for the existence of a kernel, for nonnegativity see [13,
Cor. 8.2]. ∎∎

######  Proposition 2.7.

  1. (i)

For any $a_{0}\in Y_{0}$, any $(s,t)\in\dot{\Delta}$ and any $1\leq p\leq
q\leq\infty$ there holds $U_{a_{0}}(t,0)\in\mathcal{L}(L_{p}(D),L_{q}(D))$.

  2. (ii)

There are constants $M\geq 1$ and $\gamma\in\mathbb{R}$ such that

$\left\|U_{a_{0}}(t,s)\right\|_{\mathcal{L}(L_{p}(D),L_{q}(D))}\leq
M(t-s)^{-\frac{N}{2}\left(\frac{1}{p}-\frac{1}{q}\right)}e^{\gamma(t-s)}$
(2.4)

for $1\leq p\leq q\leq\infty$, $a_{0}\in Y_{0}$ and $(s,t)\in\dot{\Delta}$.

###### Proof.

See [13, Sect. 5 and Cor. 7.2]. ∎∎

In particular, setting $p=q$ we have

$\left\|U_{a_{0}}(t,s)\right\|_{\mathcal{L}(L_{p}(D))}\leq Me^{\gamma(t-s)}.$
(2.5)

In the sequel we will frequently assume that $\gamma\geq 0$ in Proposition 2.7
and its derivates.

######  Proposition 2.8.

Let $1\leq p\leq\infty$ and $0\leq s<T$. Then for any $T_{1}\in(s,T]$ there
exists $\alpha\in(0,1)$ such that for any $a_{0}\in Y_{0}$, any $u_{0}\in
L_{p}(D)$, and any compact subset $D_{0}\subset D$ the function
$\bigl{[}\,[T_{1},T]\times
D_{0}\ni(t,x)\mapsto(U_{a_{0}}(t)u_{0})[x]\,\bigr{]}$ belongs to
$C^{\alpha/2,\alpha}([T_{1},T]\times D_{0})$. Moreover, for fixed $T_{1}$, and
$D_{0}$, the $C^{\alpha/2,\alpha}([T_{1},T]\times D_{0})$-norm of the above
restriction is bounded above by a constant depending on $\lVert
u_{0}\rVert_{L_{p}(D)}$ only.

###### Proof.

It follows from Proposition 2.7 and from [29, Chpt. III, Thm. 10.1]. ∎∎

######  Proposition 2.9.

For any $(s,T_{1})\in\dot{\Delta}$, $1\leq p<\infty$ and a bounded $E\subset
L_{p}(D)$ the set

$\bigl{\\{}\,\bigl{[}\,[T_{1},T]\ni t\mapsto U_{a_{0}}(t,s)u_{0}\in
L_{p}(D)\bigr{]}\vcentcolon a_{0}\in Y_{0},u_{0}\in E\,\bigr{\\}}$

is precompact in $C([T_{1},T],L_{p}(D))$.

###### Proof.

Fix $(s,T_{1})\in\dot{\Delta}$, $1\leq p<\infty$ and a bounded $E\subset
L_{p}(D)$. Let $(a_{0,m})_{m=1}^{\infty}\subset Y_{0}$ and
$(u_{0,m})_{m=1}^{\infty}\subset E$. Put, for $m=1,2,\dots$,

$u_{m}(t)\vcentcolon=U_{a_{0,m}}(t)u_{0,m},\quad t\in[T_{1},T].$

It follows from Proposition 2.8 via the Ascoli–Arzelà theorem by diagonal
process that, after possibly taking a subsequence, $(u_{m})_{m=1}^{\infty}$
converges as $m\to\infty$ to some function $\tilde{u}$ defined on $[T_{1},T]$
and taking values in the set of continuous real functions on $D$ in such a way
that for any compact $D_{0}\subset D$ the functions $[\,t\mapsto
u_{m}(t)\\!\\!\restriction_{D_{0}}\,]$ converge to
$[\,t\mapsto\tilde{u}(t)\\!\\!\restriction_{D_{0}}\,]$ in
$C([T_{1},T],C(D_{0}))$.

We claim that $u_{m}$ converge to $\tilde{u}$ in the
$C([T_{1},T],L_{p}(D))$-norm. By Proposition 2.7, there is $M>0$ such that
$\lVert u_{m}(t)\rVert_{L_{\infty}(D)}\leq M$ and
$\lVert\tilde{u}(t)\rVert_{L_{\infty}(D)}\leq M$ for all $m=1,2,\ldots$ and
all $t\in[T_{1},T]$. For $\epsilon>0$ take a compact $D_{0}\subset D$ such
that $\lambda(D\setminus D_{0})<(\epsilon/(4M))^{p}$, where $\lambda$ denotes
the $N$-dimensional Lebesgue measure. We have

$\lVert(u_{m}(t)-\tilde{u}(t))\,\mathbbm{1}_{\\!D\setminus
D_{0}}\rVert_{L_{p}(D)}\leq\frac{\epsilon}{2}$

for all $m=1,2,\ldots$ and all $t\in[T_{1},T]$. Further, since $[\,t\mapsto
u_{m}(t)\\!\\!\restriction_{D_{0}}\,]$ converge to
$[\,t\mapsto\tilde{u}(t)\\!\\!\restriction_{D_{0}}\,]$ in the
$C([T_{1},T],C(D_{0}))$-norm, there is $m_{0}$ such that

$\lVert(u_{m}-\tilde{u})\mathbbm{1}_{\\!D_{0}}\rVert_{C([T_{1},T],L_{p}(D))}\leq\frac{\epsilon}{2}$

for all $m\geq m_{0}$ (here $\mathbbm{1}_{\\!D_{0}}$ stands for the function
constantly equal to $\mathbbm{1}_{\\!D_{0}}$). Consequently,

$\lVert u_{m}-\tilde{u}\rVert_{C([T_{1},T],L_{p}(D))}\leq\epsilon$

for all $m\geq m_{0}$. ∎∎

######  Corollary 2.1.

Let $1\leq p<\infty$, $0\leq s<T$, $a_{0}\in Y_{0}$. Then the mapping

$[\,(s,T]\times L_{p}(D)\ni(t,u_{0})\mapsto U_{a_{0}}(t,s)u_{0}\in
L_{p}(D)\,]$

is continuous.

###### Proof.

Let $(u_{m})_{m=1}^{\infty}$ converge in $L_{p}(D)$ to $u_{0}$ and let
$(t_{m})_{m=1}^{\infty}$ converge to $t>s$. Take $\epsilon>0$. It follows from
Proposition 2.4 that there is $m_{1}$ such that $\lVert
U_{a_{0}}(t_{m},s)u_{0}-U_{a_{0}}(t,s)u_{0}\rVert_{L_{p}(D)}<\epsilon/2$ for
$m\geq m_{1}$, and it follows from Proposition 2.7(ii) that there is $m_{2}$
such that $\lVert
U_{a_{0}}(t_{m},s)u_{m}-U_{a_{0}}(t_{m},s)u_{0}\rVert_{L_{p}(D)}<\epsilon/2$
for $m\geq m_{2}$. Consequently,

$\lVert U_{a_{0}}(t_{m},s)u_{m}-U_{a_{0}}(t,s)u_{0}\rVert_{L_{p}(D)}\\\
\leq\lVert U_{a_{0}}(t_{m},s)u_{m}-U_{a_{0}}(t_{m},s)u_{0}\rVert_{L_{p}(D)}\\\
+\lVert U_{a_{0}}(t_{m},s)u_{0}-U_{a_{0}}(t,s)u_{0}\rVert_{L_{p}(D)}<\epsilon$

for $m\geq\max\\{m_{1},m_{2}\\}$. ∎∎

###  2.1 The Adjoint Operator

For a fixed $0<s\leq T$ together with ($\widehat{\mathrm{ME}}$)${}_{a_{0}}$\+
($\widehat{\mathrm{BC}}$)${}_{a_{0}}$ we consider the adjoint equations, that
is the backward parabolic equations

$\displaystyle-\frac{\partial u}{\partial t}=$
$\displaystyle\sum_{i=1}^{N}\frac{\partial}{\partial
x_{i}}\Bigl{(}\sum_{j=1}^{N}a_{ji}(t,x)\frac{\partial u}{\partial
x_{j}}-b_{i}(t,x)u\Bigr{)}$
$\displaystyle-\sum_{i=1}^{N}a_{i}(t,x)\frac{\partial u}{\partial
x_{i}}+c_{0}(t,x)u,\quad 0\leq t<s,\ x\in D,$ (2.6)

complemented with the boundary conditions:

$\mathcal{B}^{*}_{a_{0}}u=0,\quad 0\leq t<s,\ x\in\partial D,$ (2.7)

where $\mathcal{B}^{*}_{a_{0}}u=\mathcal{B}_{a_{0}^{*}}u$ with
$a_{0}^{*}\vcentcolon=((a_{ji})_{i,j=1}^{N},-(b_{i})_{i=1}^{N},-(a_{i})_{i=1}^{N},c_{0},d_{0})$
and $\mathcal{B}_{a_{0}^{*}}$ is as in ($\widehat{\mathrm{BC}}$)${}_{a_{0}}$
with $a_{0}$ replaced by $a_{0}^{*}$.

Since all analogs of the assumptions ($\mathrm{DA\hskip 1.00006pt1}$) and
($\mathrm{DA\hskip 1.00006pt2}$) are satisfied for (2.1)+(2.7), we can define,
for $u_{0}\in L_{2}(D)$, a global (weak) solution of
(2.1)${}_{a^{*}_{0}}$+(2.7)${}_{a^{*}_{0}}$, defined on $[0,s]$, with the
final condition $u(s)=u_{0}$. The following analog of Proposition 2.2 holds.

######  Proposition 2.10.

For $a_{0}\in Y_{0}$, $0<s\leq T$ and $u_{0}\in L_{2}(D)$ there is precisely
one global weak solution

$\bigl{[}\,[0,s]\ni t\mapsto U^{*}_{a_{0}}(t,s)u_{0}\in L_{2}(D)\,\bigr{]}$

of (2.1)${}_{a^{*}_{0}}$+(2.7)${}_{a^{*}_{0}}$ satisfying the final condition
$u^{*}(s;a_{0},u_{0})=u_{0}$. This mapping has the following properties

$\displaystyle U^{*}_{a_{0}}(t,t)=\mathrm{Id}_{L_{2}(D)},$ $\displaystyle\quad
a_{0}\in Y_{0},\ t\in[0,s],$ (2.8) $\displaystyle
U^{*}_{a_{0}}(t_{1},t_{2})\circ
U^{*}_{a_{0}}(t_{2},s)=U^{*}_{a_{0}}(t_{1},s),$ $\displaystyle\quad a_{0}\in
Y_{0},\ 0\leq t_{1}\leq t_{2}\leq s.$ (2.9)

From now on $s$ and $t$ will play a role as in the
($\widehat{\mathrm{ME}}$)${}_{a_{0}}$\+ ($\widehat{\mathrm{BC}}$)${}_{a_{0}}$.

Below we formulate an analog of Proposition 2.3.

######  Proposition 2.11.

  1. (i)

Let $1\leq p<\infty$ and $(s,t)\in\dot{\Delta}$. Then $U^{*}_{a_{0}}(s,t)$
extends to a linear operator in $\mathcal{L}(L_{p}(D)).$

  2. (ii)

Let $1<p<\infty$, $0<s\leq T$ and $a_{0}\in Y_{0}$. Then the mapping

$\bigl{[}\,[0,s]\ni t\mapsto
U^{*}_{a_{0}}(s,t)\in\mathcal{L}_{\mathrm{s}}(L_{p}(D))\,\bigr{]}$

is continuous.

The following analog of Proposition 2.7(i) holds.

######  Proposition 2.12.

For any $a_{0}\in Y_{0}$, any $0\leq t<s\leq T$ and any $1\leq p\leq
q\leq\infty$ there holds
$U^{*}_{a_{0}}(s,t)\in\mathcal{L}_{\mathrm{s}}(L_{p}(D),L_{q}(D))$.

######  Proposition 2.13.

For $a_{0}\in Y_{0}$ there holds

$\langle U_{a_{0}}(t,s)u_{0},v_{0}\rangle_{L_{2}(D)}=\langle
u_{0},U^{*}_{a_{0}}(s,t)v_{0}\rangle_{L_{2}(D)}\\\ \text{for any }0\leq s\leq
t\leq T,\ u_{0},v_{0}\in L_{2}(D).$ (2.10)

Proposition 2.13 states that the linear operator
$U^{*}_{a_{0}}(s,t)\in\mathcal{L}(L_{2}(D))$ is the dual (in the functional-
analytic sense) of $U_{a_{0}}(t,s)\in\mathcal{L}(L_{2}(D))$. For a proof, see
[35, Prop. 2.3.3].

######  Proposition 2.14.

For $1<p<\infty$ and $a_{0}\in Y_{0}$ there holds

$\langle U_{a_{0}}(t,s)u_{0},v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=\langle
u_{0},U^{*}_{a_{0}}(s,t)v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}$ (2.11)

for any $(s,t)\in\dot{\Delta}$, $u_{0}\in L_{p}(D)$ and $v_{0}\in
L_{p^{\prime}}(D)$.

###### Proof.

Fix $(s,t)\in\dot{\Delta}$, $u_{0}\in L_{p}(D)$ and $v_{0}\in
L_{q^{\prime}}(D)$. From Propositions 2.7(i) and 2.12 it follows that
$U_{a_{0}}(\zeta,s)u_{0},U^{*}_{a_{0}}(\zeta,t)v_{0}\in L_{2}(D)$ for all
$\zeta\in(s,t)$, consequently $\langle
U_{a_{0}}(\zeta,s)u_{0},U_{a_{0}}(\zeta,t)v_{0}\rangle_{L_{2}(D)}$ is well
defined for such $\zeta$. An application of (2.2), Proposition 2.13 and (2.9)
gives that for any $s<\zeta_{1}\leq\zeta_{2}<t$ there holds

$\langle
U_{a_{0}}(\zeta_{2},s)u_{0},U^{*}_{a_{0}}(\zeta_{2},t)v_{0}\rangle_{L_{2}(D)}\\\
=\langle
U_{a_{0}}(\zeta_{2},\zeta_{1})U_{a_{0}}(\zeta_{1},s)u_{0},U^{*}_{a_{0}}(\zeta_{2},t)v_{0}\rangle_{L_{2}(D)}\\\
=\langle
U_{a_{0}}(\zeta_{1},s)u_{0},U^{*}_{a_{0}}(\zeta_{1},\zeta_{2})U^{*}_{a_{0}}(\zeta_{2},t)v_{0}\rangle_{L_{2}(D)}\\\
=\langle
U_{a_{0}}(\zeta_{1},s)u_{0},U^{*}_{a_{0}}(\zeta_{1},t)v_{0}\rangle_{L_{2}(D)}.$

Therefore the assignment

$\displaystyle(s,t)\ni\zeta\mapsto\langle$ $\displaystyle
U_{a_{0}}(\zeta,s)u_{0},U^{*}_{a_{0}}(\zeta,t)v_{0}\rangle_{L_{2}(D)}$
$\displaystyle={}$ $\displaystyle\langle
U_{a_{0}}(\zeta,s)u_{0},U^{*}_{a_{0}}(\zeta,t)v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}$
$\displaystyle={}$ $\displaystyle\langle
U_{a_{0}}(\zeta,s)u_{0},U^{*}_{a_{0}}(\zeta,t)v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}$

is constant (denote its value by $A$). If we let $\zeta\nearrow t$, then
$U_{a_{0}}(\zeta,s)u_{0}$ converges, by Proposition 2.4, in the
$L_{p}(D)$-norm to $U_{a_{0}}(t,s)u_{0}$ and $U^{*}_{a_{0}}(\zeta,t)v_{0}$
converges, by Proposition 2.11(ii), in the $L_{p^{\prime}}(D)$-norm to
$v_{0}$, consequently $\langle
U_{a_{0}}(t,s)u_{0},v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=A$. If we let
$\zeta\searrow s$, then $U_{a_{0}}(\zeta,s)u_{0}$ converges, by Proposition
2.3(ii), in the $L_{p}(D)$-norm to $u_{0}$ and $U^{*}_{a_{0}}(\zeta,t)v_{0}$
converges, by Propositions 2.12 and 2.11(ii), in the $L_{p^{\prime}}(D)$-norm
to $U^{*}_{a_{0}}(s,t)v_{0}$, consequently $\langle
u_{0},U^{*}_{a_{0}}(s,t)v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=A$. This
concludes the proof. ∎∎

It follows from Proposition 2.14 that the linear operator $\allowbreak
U^{*}_{a_{0}}(s,t)\in\mathcal{L}(L_{p^{\prime}}(D))$ is the dual (in the
functional-analytic sense) of $U_{a_{0}}(t,s)\in\mathcal{L}(L_{p}(D))$.

In the light of the above, the following counterpart to Proposition 2.7(ii)
holds.

######  Proposition 2.15.

There are constants $M\geq 1$ and $\gamma\in\mathbb{R}$, the same as in
Proposition 2.7, such that

$\lVert U^{*}_{a_{0}}(s,t)\rVert_{\mathcal{L}(L_{p}(D),L_{q}(D))}\leq
M(t-s)^{-\tfrac{N}{2}\bigl{(}\tfrac{1}{p}-\tfrac{1}{q}\bigr{)}}e^{\gamma(t-s)}$

for $1\leq p\leq q\leq\infty$, $a_{0}\in Y_{0}$ and $(s,t)\in\dot{\Delta}$.

###  2.2 Continuous Dependence of Weak Solutions

######  Lemma 2.1.

Let $1<p<\infty$ and $a_{0}\in Y_{0}$. Then the mapping

$\big{[}\,\dot{\Delta}\ni(s,t)\mapsto
U_{a_{0}}(t,s)\in\mathcal{L}_{\mathrm{s}}(L_{p}(D))\,\big{]}$

is continuous.

###### Proof.

If $t_{m}\to t>s$ as $m\to\infty$, $U_{a_{0}}(t_{m},s)u_{0}\to
U_{a_{0}}(t,s)u_{0}$ in $L_{p}(D)$, by Proposition 2.4.

Assume that $s_{m}\to s<t$ as $m\to\infty$. Fix $u_{0}\in L_{p}(D)$ and
$v_{0}\in L_{p^{\prime}}(D)$. We have, by Proposition 2.14 and the adjoint
equation analog of Proposition 2.4,

$\langle
U_{a_{0}}(t,s_{m})u_{0},v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=\langle
u_{0},U_{a_{0}}^{*}(s_{m},t)v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}\\\
\to\langle
u_{0},U_{a_{0}}^{*}(s,t)v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=\langle
U_{a_{0}}(t,s)u_{0},v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)},$

so $U_{a_{0}}(t,s_{m})u_{0}\rightharpoonup U_{a_{0}}(t,s)u_{0}$ in $L_{p}(D)$.
As $\\{\,U_{a_{0}}(t,s_{m})u_{0}:m\in\mathbb{N}\,\\}$ is, by Proposition 2.9,
precompact in $L_{p}(D)$, the convergence is in the norm.

Finally, assume that $s_{m}\to s$ and $t_{m}\to t$ with $s<t$, and fix
$u_{0}\in L_{p}(D)$. We can assume that $s_{m}<(s+t)/2<t$ for all $m$. By the
previous paragraph, $U_{a_{0}}((s+t)/2,s_{m})u_{0}\to
U_{a_{0}}((s+t)/2,s)u_{0}$ in $L_{p}(D)$. Corollary 2.1 implies that

$U_{a_{0}}(t_{m},s_{m})u_{0}=U_{a_{0}}(t_{m},\tfrac{1}{2}(s+t))(U_{a_{0}}(\tfrac{1}{2}(s+t),s_{m})u_{0}\\\
\to
U_{a_{0}}(t,\tfrac{1}{2}(s+t))(U_{a_{0}}(\tfrac{1}{2}(s+t),s)u_{0}=U_{a_{0}}(t,s)u_{0},$

where the convergence is in $L_{p}(D)$, too. ∎∎

######  Proposition 2.16.

Let $1<p<\infty$ and $a_{0}\in Y_{0}$. Then the mapping

$\big{[}\,\dot{\Delta}\ni(s,t)\mapsto
U_{a_{0}}(t,s)\in\mathcal{L}(L_{p}(D))\,\big{]}$

is continuous.

###### Proof.

Let $s_{m}\to s$ and $t_{m}\to t$ with $s<t$. Suppose to the contrary that
there are $\epsilon>0$ and $(u_{m})_{m=1}^{\infty}\subset L_{p}(D)$, $\lVert
u_{m}\rVert_{L_{p}(D)}=1$, such that

$\lVert
U_{a_{0}}(t_{m},s_{m})u_{m}-U(t,s)u_{m}\rVert_{L_{p}(D)}\geq\epsilon,\quad
m=1,2,3,\dots.$

It follows from Proposition 2.9 that, after possibly taking a subsequence and
relabelling, we can assume that $U_{a_{0}}(t_{m},s_{m})u_{m}$ converge to
$\tilde{u}$ and $U_{a_{0}}(t,s)u_{m}$ converge to $\hat{u}$, both in
$L_{p}(D)$. For any $v_{0}\in L_{p^{\prime}}(D)$ we have, by Proposition 2.14,

$\langle(U_{a_{0}}(t_{m},s_{m})-U_{a_{0}}(t,s))u_{m},v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}\\\
=\langle
u_{m},(U_{a_{0}}^{*}(s_{m},t_{m})-U_{a_{0}}^{*}(s,t))v_{0}\rangle_{L_{p}(D),L_{p^{\prime}}(D)}.$

Since $\lVert u_{m}\rVert_{L_{p}(D)}=1$, we conclude from the adjoint equation
analog of Lemma 2.1 that the above expression converges to zero as
$m\to\infty$. Consequently $\tilde{u}=\hat{u}$, a contradiction. ∎∎

######  Proposition 2.17.

Assume, in addition, ($\mathrm{DA\hskip 1.00006pt4}$). For $1<p<\infty$ the
mapping

$\bigl{[}\,Y_{0}\times\dot{\Delta}\times L_{p}(D)\ni(a_{0},s,t,u_{0})\mapsto
U_{a_{0}}(t,s)u_{0}\in L_{p}(D)\,\bigl{]}$

is continuous.

###### Proof.

It follows from [35, Props. 2.2.12 and 2.2.13] that, for $2\leq p<\infty$, the
mapping

$\big{[}\,Y_{0}\times\dot{\Delta}\times L_{2}(D)\ni(a_{0},s,t,u_{0})\mapsto
U_{a_{0}}(t,s)u_{0}\in L_{p}(D)\,\big{]}$

is continuous, too.

To conclude the proof it suffices to show that for any $1<p<2$ the mapping

$\big{[}\,Y_{0}\times\dot{\Delta}\times L_{p}(D)\ni(a_{0},s,t,u_{0})\mapsto
U_{a_{0}}(t,s)u_{0}\in L_{2}(D)\,\big{]}$

is continuous. Observe that if we have $a_{0,m}\to a_{0}\in Y_{0}$, $s_{m}\to
s$, $t_{m}\to t$ with $s_{m}<t_{n}$ and $s<t$, and $u_{0,m}\to u_{0}\in
L_{p}(D)$, then from Proposition 2.9 it follows that, after possibly choosing
a subsequence, there is $w\in L_{2}(D)$ such that
$U_{a_{0,m}}(t_{m},s_{m})u_{0,m}\to w$ in $L_{2}(D)$. Consequently, $\langle
U_{a_{0,m}}(t_{m},s_{m})u_{0,m},v\rangle_{L_{2}(D)}\to\langle
w,v\rangle_{L_{2}(D)}$ as $m\to\infty$, for any $v\in L_{2}(D)$. On the other
hand, one has, by Proposition 2.14,

$\langle U_{a_{0,m}}(t_{m},s_{m})u_{0,m},v\rangle_{L_{2}(D)}=\langle
u_{0,m},U^{*}_{a_{0,m}}(s_{m},t_{m})v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}.$

As $2<p^{\prime}<\infty$, an application of the result already obtained to the
adjoint equation yields that $U^{*}_{a_{0,m}}(s_{m},t_{m})v$ converges, as
$m\to\infty$, to $U^{*}_{a_{0}}(s,t)v$ in $L_{p^{\prime}}(D)$. As $u_{0,m}$
converges to $u_{0}$ in $L_{p}(D)$, we have that $\langle
u_{0,m},U^{*}_{a_{0,m}}(s_{m},t_{m})v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}$
converges to $\langle
u_{0},U^{*}_{a_{0}}(s,t)v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}$, which is, by
Proposition 2.14, equal to $\langle U_{a_{0}}(t,s)u_{0},v\rangle_{L_{2}(D)}$.
As $v\in L_{2}(D)$ is arbitrary, we have $w=U_{a_{0}}(t,s)u_{0}$. ∎∎

######  Proposition 2.18.

Assume, in addition, $\eqref{a: compactness of Y }$. For $1<p<\infty$ the
mapping

$\bigl{[}\,Y_{0}\times\dot{\Delta}\ni(a_{0},s,t)\mapsto
U_{a_{0}}(t,s)\in\mathcal{L}(L_{p}(D))\,\bigr{]}$

is continuous.

###### Proof.

In order not to overburden the notation we assume $s=0$.

Let $(a_{0,m})_{m=1}^{\infty}\subset Y_{0}$ be a sequence converging to
$a_{0}$ as $m\to\infty$, and let $(t_{m})_{m=1}^{\infty}\subset(0,T]$ be a
sequence converging to $t>0$ as $m\to\infty$. Suppose to the contrary that
$\lVert U_{a_{0,m}}(t_{m},0)-U_{a_{0}}(t,0)\rVert_{\mathcal{L}(L_{p}(D))}$
does not converge to $0$, that is, there exist $\epsilon>0$ and a sequence
$(u_{m})_{m=1}^{\infty}\subset L_{p}(D)$, $\lVert u_{m}\rVert_{L_{p}(D)}=1$
for all $m$, such that

$\lVert
U_{a_{0,m}}(t_{m},0)u_{m}-U_{a_{0}}(t,0)u_{m}\rVert_{L_{p}(D)}\geq\epsilon$

for all $m$.

It follows from Proposition 2.9 that, after possibly extracting a subsequence,
we can assume that $U_{a_{0,m}}(t_{m}/2,0)u_{m}$ and $U_{a_{0}}(t/2,0)u_{m}$
converge, as $m\to\infty$, in the $L_{p}(D)$-norm. We claim that both converge
to the same $\tilde{u}$. Indeed, it suffices to check that the difference
$(U_{a_{0,m}}(t_{m}/2,0)-U_{a_{0}}(t/2,0))u_{m}$ converges to zero in
$L_{p}(D)$, which is, in light of the equalities

$\langle(U_{a_{0,m}}(t_{m}/2,0)-U_{a_{0}}(t/2,0))u_{m},v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}\\\
=\langle
u_{m},(U^{*}_{a_{0,m}}(0,t_{m}/2)-U^{*}_{a_{0}}(0,t/2))v\rangle_{L_{p}(D),L_{p^{\prime}}(D)},\quad
v\in L_{p^{\prime}}(D),$

a consequence of the analog for the adjoint equation of Proposition 2.17.

Proposition 2.17 implies that

$\lVert
U_{a_{0,m}}(t_{m},0)u_{m}-U_{a_{0}}(t,t/2)\tilde{u}\rVert_{L_{p}(D)}\\\
=\lVert
U_{a_{0,m}}(t_{m},t_{m}/2)(U_{a_{0,m}}(t_{m}/2,0)u_{m})-U_{a_{0}}(t,t/2)\tilde{u}\rVert_{L_{p}(D)}\to
0,$

and

$\lVert U_{a_{0}}(t,0)u_{m}-U_{a_{0}}(t,t/2)\tilde{u}\rVert_{L_{p}(D)}\\\
=\lVert
U_{a_{0}}(t,t/2)(U_{a_{0}}(t/2,0)u_{m})-U_{a_{0}}(t,t/2)\tilde{u}\rVert_{L_{p}(D)}\\\
\leq\lVert U_{a_{0}}(t,t/2)\rVert_{\mathcal{L}(L_{p}(D))}\lVert
U_{a_{0}}(t/2,0)u_{m}-\tilde{u}\rVert_{L_{p}(D)}\to 0$

therefore $\lVert
U_{a_{0,m}}(t_{m},0)u_{m}-U_{a_{0}}(t,0)u_{m}\rVert_{L_{p}(D)}$, converges to
zero, a contradiction. ∎∎

##  3 Mild Solutions

In the present section we assume ($\mathrm{DA\hskip 1.00006pt1}$),
($\mathrm{DA\hskip 1.00006pt2}$) and that $Y$ as in Definition 1.1 is such
that its flattening $Y_{0}$ satisfies ($\mathrm{DA\hskip 1.00006pt3}$).
Occasionally we will assume ($\mathrm{DA\hskip 1.00006pt5}$).

######  Definition 3.1 (Multiplication Operator).

For $a\in Y$, $1\leq p\leq\infty$ and $0\leq t\leq T$ we define multiplication
operator $C^{1}_{a}(t)\colon L_{p}(D)\to L_{p}(D)$ as follows

$C_{a}^{1}(t)v=c_{1}(t,\cdot)v.$

The $C^{1}_{a}(t)$ operator is well defined as long as assumption
($\mathrm{DA\hskip 1.00006pt2}$) holds. To be more precise we use a corollary
from assumption ($\mathrm{DA\hskip 1.00006pt2}$) on $t$-sections of $c_{1}$.

######  Lemma 3.1 (Boundedness of Multiplication Operator).

The multiplication operator $C_{a}^{1}(t)$ is linear and bounded uniformly
with respect to a.e. $0<t<T$ and $a\in Y$.

It should be remarked that the exceptional sets can be different for different
$a\in Y$.

###### Proof.

Let $K$ be the norm bound of $Y$ (see assumption $\mathrm{Y\hskip
1.00006pt1}$). For any $v\in L_{p}(D)$ by the Hölder inequality we get

$\displaystyle\lVert C_{a}^{1}(t)v\rVert_{L_{p}(D)}$ $\displaystyle=\lVert
c_{1}(t,\cdot)v\rVert_{L_{p}(D)}$ $\displaystyle\leq\lVert
c_{1}(t,\cdot)\rVert_{L_{\infty}(D)}\lVert v\rVert_{L_{p}(D)}$
$\displaystyle\leq K\lVert v\rVert_{L_{p}(D)}$

where above inequality holds for a.e. $0<t<T$, so the operator norm of
$C^{1}_{a}(t)$ is bounded a.e. by $\lVert
c_{1}(t,\cdot)\rVert_{L_{\infty}(D)}$ what can be bounded uniformly with
respect to a.e. $0<t<T$ by virtue of assumption ($\mathrm{DA\hskip
1.00006pt2}$). Since $Y$ is bounded by $K$ we also have uniform boundedness in
$a\in Y$. ∎∎

Below we present a series of lemmas to prove the measurability of individual
parts of the mild solution. We will make frequent use of the Lemma 0.1 in this
part of the work, in particular, Remark 0.1 on measurability will also be
useful.

######  Lemma 3.2.

For any $1<p<\infty$ and any norm bounded set $E\subset C([-1,T],L_{p}(D))$
the set

$\big{\\{}\,[\,(0,T)\ni t\mapsto(u\circ\Phi)(t)\in L_{p}(D)\,]:u\in
E,\Phi\in\widetilde{\mathcal{R}}\,\big{\\}}$ (3.1)

is bounded in $L_{\infty}((0,T),L_{p}(D))$.

###### Proof.

Let $u\in E$ and $\Phi\in\widetilde{\mathcal{R}}$. The mapping
$u\,\circ\,\Phi$ is $(\mathfrak{L}((0,T)),\mathfrak{B}(L_{p}(D)))$-measurable,
since for any fixed open set $V\subset L_{p}(D)$ the preimage $u^{-1}[V]$ is
open, hence $(u\circ\Phi)^{-1}[V]\in\mathfrak{L}((0,T))$. Moreover, the
$L_{\infty}((0,T),L_{p}(D))$-norm of the map $u\circ\Phi$ is uniformly bounded
with respect to $u$ and $\Phi$ by the same constant as $E$ due to the
inequality

$\displaystyle\lVert u\circ\Phi\rVert_{L_{\infty}((0,T),L_{p}(D))}$
$\displaystyle=\operatorname*{ess\,sup}_{t\in(0,T)}{\lVert(u\circ\Phi)(t)\rVert_{L_{p}(D)}}$
$\displaystyle\leq\sup_{t\in[-1,T]}\lVert u(t)\rVert_{L_{p}(D)}=\lVert
u\rVert_{C([-1,T],L_{p}(D))}.$

∎∎

######  Lemma 3.3.

For any $1<p<\infty$ and any norm bounded set $\tilde{E}\subset
L_{\infty}((0,T),L_{p}(D))$ the set

$\big{\\{}\,[\,(0,T)\ni t\mapsto C^{1}_{a}(t)\tilde{u}(t)\in L_{p}(D)\,]:a\in
Y,\tilde{u}\in\tilde{E}\,\big{\\}}$

is bounded in $L_{\infty}((0,T),L_{p}(D))$.

###### Proof.

Let $\tilde{u}\in\tilde{E}$ and $a\in Y$. From Lemma 0.1(a) follows that the
mapping

$\big{[}\,(0,T)\times D\ni(t,x)\mapsto\tilde{u}(t)[x]\in\mathbb{R}\,\big{]}$
(3.2)

is
$(\mathfrak{L}((0,T))\otimes\mathfrak{L}(D),\mathfrak{B}(\mathbb{R}))$-measurable.
Hence the function

$\big{[}\,(0,T)\times
D\ni(t,x)\mapsto\big{(}C_{a}(t)\tilde{u}(t)\big{)}[x]\in\mathbb{R}\,\big{]}$
(3.3)

for any $a\in Y$ is
$(\mathfrak{L}((0,T))\otimes\mathfrak{L}(D),\mathfrak{B}(\mathbb{R}))$-measurable,
since it can be rewritten as the product of
$(\mathfrak{L}((0,T))\otimes\mathfrak{L}(D),\mathfrak{B}(\mathbb{R}))$-measurable
functions, namely

$\big{[}\,(0,T)\times D\ni(t,x)\mapsto
c_{1}(t,x)\,\tilde{u}(t)[x]\in\mathbb{R}\,\big{]}.$ (3.4)

It suffices now to notice that for a.e. $t\in(0,T)$ the $t$-section of (3.3)
belongs (by the definition of the multiplication operator) to $L_{p}(D)$. So
from Lemma 0.1(b) it follows that the mapping

$\big{[}\,(0,T)\ni t\mapsto C_{a}(t)\tilde{u}(t)\in L_{p}(D)\,\big{]}$

is $(\mathfrak{L}((0,T)),\mathfrak{B}(L_{p}(D)))$-measurable. By the norm
estimate,

$\displaystyle\lVert
C^{1}_{a}(\cdot)\tilde{u}\rVert_{L_{\infty}((0,T),L_{p}(D))}$
$\displaystyle=\operatorname*{ess\,sup}_{t\in(0,T)}{\lVert
c_{1}(t,\cdot)\tilde{u}(t)\rVert_{L_{p}(D)}}$ $\displaystyle\leq
K\lVert\tilde{u}\rVert_{L_{\infty}((0,T),L_{p}(D))}$

we obtain the statement. ∎∎

######  Lemma 3.4.

Assume $1<p<\infty$, $a_{0}\in Y_{0}$, and $u\in L_{\infty}((0,T),L_{p}(D))$.
Then

  1. (i)

for any $0<t\leq T$ the function

$[\,\zeta\mapsto U_{a_{0}}(t,\zeta)u(\zeta),\text{ for a.e. }\zeta\in(0,t)\,]$
(3.5)

belongs to $L_{\infty}((0,t),L_{p}(D))$ moreover, the linear operator
assigning (3.5) to $u$ belongs to
$\mathcal{L}(L_{\infty}((0,1),L_{p}(D)),L_{\infty}((0,1),L_{p}(D)))$, with the
norm bounded uniformly in $a_{0}\in Y_{0}$,

  2. (ii)

the mapping

$\biggl{[}\,[0,T]\ni
t\mapsto\int_{0}^{t}U_{a_{0}}(t,\zeta)u(\zeta)\,\mathrm{d}\zeta\,\biggr{]}$
(3.6)

belongs to $C([0,T],L_{p}(D))$.

###### Proof.

Fix $0<t\leq T$. We show first that (3.5) defines a
$(\mathfrak{L}((0,t)),\mathfrak{B}(L_{p}(D)))$-measurable function. It is
equivalent, by Theorem 0.1, to showing that for each $v\in L_{p^{\prime}}(D)$
the function

$[\,\zeta\mapsto\langle
U_{a_{0}}(t,\zeta)u(\zeta),v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}]$

is $(\mathfrak{L}((0,t)),\mathfrak{B}(\mathbb{R}))$-measurable. By Proposition
2.14, for Lebesgue-a.e. $\zeta\in[0,t)$ there holds

$\langle
U_{a_{0}}(t,\zeta)u(\zeta),v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}=\langle
u(\zeta),U_{a_{0}}^{*}(\zeta,t)v\rangle_{L_{p}(D),L_{p^{\prime}}(D)}$

It suffices now to notice that $u$ is
$(\mathfrak{L}((0,t)),\mathfrak{B}(L_{p}(D)))$-measurable, by assumption, and
that the function

$\bigl{[}\,[0,t)\ni\zeta\mapsto U_{a_{0}}^{*}(\zeta,t)v\in
L_{p^{\prime}}(D)\,\bigr{]}$

is continuous, by the adjoint equation analog of Lemma 2.1. It follows from
Proposition 2.7(ii) that the function

$\big{[}\,(0,t)\ni\zeta\mapsto\lVert
U_{a_{0}}(t,\zeta)\rVert_{\mathcal{L}(L_{p}(D))}\,\big{]}$

belongs to $L_{\infty}((0,t))$. Then the membership of (3.5) in
$L_{\infty}((0,t),L_{p}(D))$ as well as the bound on its norm follow from the
generalized Hölder inequality. The proof of part (i) is thus completed.

We proceed to the proof of part (ii). By part (i), the function (3.6) is well
defined. Let $0\leq t_{1}\leq t_{2}\leq T$. We write

$\int_{0}^{t_{2}}U_{a_{0}}(t_{2},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}-\int_{0}^{t_{1}}U_{a_{0}}(t_{1},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}\\\
=\int_{0}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}+\int_{t_{1}}^{t_{2}}U_{a_{0}}(t_{2},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}.$

Let $\epsilon>0$. As (3.5) belongs to $L_{\infty}((0,t),L_{p}(D))$, it is a
consequence of [16, Thm. II.2.4(i)] that the $L_{p}(D)$-norm of the second
term on the right-hand side can be made $<\epsilon/3$ by taking $t_{1},t_{2}$
sufficiently close to each other. Regarding the first term, we write

$\int_{0}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}\\\
=\int_{0}^{t_{1}-\eta}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}+\int_{t_{1}-\eta}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}.$

Again by [16, Thm. II.2.4(i)], for $\eta>0$ sufficiently small there holds

$\biggl{\lVert}\int_{t_{1}-\eta}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}\biggr{\rVert}_{L_{p}(D)}<\frac{\epsilon}{3}.$

It follows from Proposition 2.16 that the assignment

$\bigl{[}\,\\{\,\dot{\Delta}:\eta\leq\zeta+\eta\leq t\leq
T\,\\}\ni(\zeta,t)\mapsto
U_{a_{0}}(t,\zeta)\in\mathcal{L}(L_{p}(D))\,\bigr{]}$

is uniformly continuous, consequently there exists $\delta>0$ such that if
$\eta\leq\zeta+\eta\leq t_{1}\leq t_{2}$, $t_{2}-t_{1}<\delta$, then

$\lVert
U_{a_{0}}(t_{2},\zeta)-U_{a_{0}}(t_{1},\zeta)\rVert_{\mathcal{L}(L_{p}(D))}<\frac{\epsilon}{3\lVert
u\rVert_{L_{1}((0,T),L_{p}(D))}}.$

Therefore

$\biggl{\lVert}\int_{0}^{t_{1}-\eta}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}\biggr{\rVert}_{L_{p}(D)}<\frac{\epsilon}{3}.$

This concludes the proof of part (ii). ∎∎

######  Definition 3.2 (Mild Solution).

For $1\leq p<\infty$, $a\in Y$, $0\leq s<T_{0}\leq T$ and $u_{0}\in
C([s-1,s],L_{p}(D))$ and $R\in\mathcal{R}$ the function $u\in
C([s-1,T_{0}],L_{p}(D))$ such that

$u(t)=u_{0}(t)\text{ for }t\in[s-1,s],$ (3.7)

holds and the integral equation

$u(t)=U_{\tilde{a}}(t,s)u_{0}(s)+\int_{s}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)u(\zeta-R(\zeta))\,\mathrm{d}\zeta$
(3.8)

is satisfied in $L_{p}(D)$ on $[s,T_{0}]$ will be called a _mild solution_ of
(ME)${}_{a}+$(BC)a.

For $T_{0}=T$ we have a _global mild solution_.

At first note that the concept of mild solution, especially part (3.8), is
well defined based on Lemma 3.3 and Lemma 3.4. At some moments we use the name
“mild solution” to describe function $u\upharpoonright_{[0,T_{0}]}$ instead of
$u\in C([-1,T_{0}],L_{p}(D))$ satisfying (3.7) and (3.8). This convention
seems more natural especially in the context of continuous dependence on
coefficients. A similar convention can be found in the literature [28].

###  3.1 Existence and Uniqueness of Global Mild Solutions

######  Proposition 3.1.

There exists $\Theta_{0}\in(0,T]$ such that for any $1<p<\infty$, $a\in Y$,
$R\in\mathcal{R}$, $0\leq s\leq T-\Theta_{0}$, $u_{0}\in C([s-1,s],L_{p}(D))$
and any $0<\Theta\leq\Theta_{0}$ there exist unique solution of (ME)a+(BC)a on
$[s-1,s+\Theta]$ with initial condition $u_{0}$.

###### Proof.

The idea of the proof runs as follows. The solution is obtained as a fixed
point of the contraction mapping $\mathfrak{G}$ of $C([s,s+\Theta],L_{p}(D))$
into itself (see 2.3(ii), 3.4(ii) and 3.3) defined as

$\displaystyle(\mathfrak{G}u)[t]\vcentcolon={}$ $\displaystyle
U_{\tilde{a}}(t,s)u_{0}(s)+\int_{s}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)u(\zeta-R(\zeta))\,\mathrm{d}\zeta,$
(3.9)

where $s\leq t\leq s+\Theta$ and $\Theta\in(0,T]$ is sufficiently small. Until
revoking, $u$ and $v$ stand for generic functions in
$C([s,s+\Theta],L_{p}(D))$. For such a $u$ we interpret $u(\zeta-R(\zeta))$
(similarly $v$) as $u_{0}(\zeta-R(\zeta))$ when $\zeta-R(\zeta)\in(-1,0)$.

$\displaystyle\lVert(\mathfrak{G}u)[t]-(\mathfrak{G}v)[t]\rVert_{L_{p}(D)}$
$\displaystyle\leq\int_{s}^{t}\lVert
U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\big{(}u(\zeta-R(\zeta))-v(\zeta-R(\zeta))\big{)}\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$
$\displaystyle\leq MKe^{\gamma t}\int_{s}^{t}\lVert
u(\zeta-R(\zeta))-v(\zeta-R(\zeta))\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$
$\displaystyle\leq MKe^{\gamma t}\int_{s}^{t}\sup_{s\leq\xi\leq t}\lVert
u(\xi-R(\xi))-v(\xi-R(\xi))\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$
$\displaystyle\leq MKe^{\gamma t}\int_{s}^{t}\sup_{s-1\leq\xi\leq t}\lVert
u(\xi)-v(\xi)\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$ $\displaystyle=MKe^{\gamma
t}\int_{s}^{t}\sup_{s\leq\xi\leq t}\lVert
u(\xi)-v(\xi)\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$ $\displaystyle\leq
MKe^{\gamma T}\Theta\lVert u-v\rVert_{C([s,s+\Theta],L_{p}(D))}.$

By taking $\displaystyle 0<\Theta\leq\Theta_{0}\coloneqq 1/(2MKe^{\gamma T})$
we obtain that the contraction coefficient is less than $1$. ∎∎

The Contraction Mapping Principle guarantees the existence and uniqueness of
the fixed point $u$ of $\mathfrak{G}$, which is then the unique mild
$L_{p}$-solution of (ME)a+(BC)a on $[s-1,\Theta_{0}]$ satisfying the initial
condition (IC).

######  Lemma 3.5.

For any $1<p<\infty$, $0<s_{1}<s_{2}\leq T$, $a\in Y$, $R\in\mathcal{R}$,
$u_{0}\in C([-1,0],L_{p}(D))$ and $v\colon[-1,s_{2}]\to L_{p}(D)$ the
following statements are equivalent:

  1. (i)

a function $v$ is the mild solution of (ME)a+(BC)a on $[-1,s_{2}]$ with
initial condition $u_{0}$,

  2. (ii)

a function $v\\!\\!\restriction_{[-1,s_{1}]}$ is the mild solution of
(ME)a+(BC)a with initial condition $u_{0}$ and
$v\\!\\!\restriction_{[s_{1}-1,s_{2}]}$ is the mild solution of (ME)a+(BC)a
with initial condition $v\\!\\!\restriction_{[s_{1}-1,s_{1}]}$.

###### Proof.

Let $p,s_{1},s_{2},a,R,u_{0}$ and $v$ be as in the statement. To prove
$\ref{glue}\Rightarrow\ref{unglue}$ it suffices to see that for any $s_{1}\leq
t\leq s_{2}$, in view of (2.2) and [1, Lemma 11.45] there holds

$\displaystyle v(t)={}$ $\displaystyle
U_{\tilde{a}}(t,0)u_{0}(0)+\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)v(\zeta-R(\zeta))\,\mathrm{d}\zeta$
(3.10) $\displaystyle={}$ $\displaystyle
U_{\tilde{a}}(t,s_{1})\Bigg{(}U_{\tilde{a}}(s_{1},0)u_{0}(0)+\int\limits_{0}^{s_{1}}U_{\tilde{a}}(s_{1},\zeta)C^{1}_{a}(\zeta)v(\zeta-R(\zeta))\,\mathrm{d}\zeta\Bigg{)}$
$\displaystyle+\int\limits_{s_{1}}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)v(\zeta-R(\zeta))\,\mathrm{d}\zeta$
$\displaystyle={}$ $\displaystyle
U_{\tilde{a}}(t,s_{1})v{\restriction_{[s_{1}-1,s_{1}]}}(s_{1})+\int\limits_{s_{1}}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)v{\restriction_{[s_{1}-1,s_{2}]}}(\zeta-R(\zeta))\,\mathrm{d}\zeta.$

In order to prove $\ref{unglue}\Rightarrow\ref{glue}$ fix $t\in[-1,s_{2}]$ and
consider the cases: $t\in[-1,0]$, $t\in[0,s_{1}]$ or $t\in[s_{1},s_{2}]$.
Since the first two cases are straightforward and the third is a similar
calculation to (3.10), the proof is finished. ∎∎

######  Theorem 3.1.

For any $1<p<\infty$, $a\in Y$, $u_{0}\in C([-1,0],L_{p}(D))$ and
$R\in\mathcal{R}$ equation (ME)a+(BC)a has a unique global mild solution on
$[0,T]$.

###### Proof.

Fix $a$, $u_{0}$ and $R$ as in the statement. Let

$Q=\\{q\in[0,T]:\text{\eqref{main-eq}${}_{a}${+}\eqref{main-bc}${}_{a}$ has a
unique mild solution on }[-1,q]\\}.$

It suffices to prove that $T\in Q$. Suppose to the contrary that $T\not\in Q$.
Since $\Theta_{0}\in Q$ (where $\Theta_{0}$ stands for constant obtained in
Proposition 3.1) and $Q\subset[0,T]$, $\sup Q<\infty$. It is straightforward
that $0\leq\sup Q-\Theta_{0}/2<\sup Q\leq T$ hence there exists $s\in Q$ such
that $s>\sup Q-\Theta_{0}/2$.

Let $v_{1}\colon[-1,s]\to L_{p}(D)$ be the unique mild solution with initial
condition $u_{0}$. From the definition of $\Theta$ it follows that there is a
mild solution $v_{2}\colon[s-1,\min\\{s+\Theta_{0},T\\}]\to L_{p}(D)$ with
initial condition $v_{1}\\!\\!\restriction_{[s-1,s]}$. Let

$v(t)=\begin{cases}v_{1}(t)&\text{for }t\in[-1,s]\\\ v_{2}(t)&\text{for
}t\in[s-1,\min\\{s+\Theta_{0},T\\}].\end{cases}$

We claim that $v$ is a unique mild solution of (ME)a+(BC)a on
$[-1,\min\\{s+\Theta_{0},T\\}]$ with initial condition $u_{0}$. From Lemma 3.5
it follows that $v$ is in fact a mild solution. For uniqueness, assume
$w\colon[-1,\min\\{s+\Theta_{0},T\\}]\to L_{p}(D)$ is any mild solution. Then
clearly $w\\!\\!\restriction_{[-1,s]}=v_{1}$. Moreover, by Lemma 3.5, the
function $w\\!\\!\restriction_{[s-1,\min\\{s+\Theta_{0},T\\}]}$ is a mild
solution with initial condition
$w\\!\\!\restriction_{[s-1,s]}=v_{1}\\!\\!\restriction_{[s-1,s]}$, so by the
uniqueness of $v_{2}$ we have that
$w\\!\\!\restriction_{[s-1,\min\\{s+\Theta_{0},T\\}]}=v_{2}$. Hence $v=w$. The
proof is completed by the following observation: if
$\min\\{s+\Theta_{0},T\\}=s+\Theta_{0}$ then $s+\Theta_{0}\in Q$, so we get a
contradiction with the fact that $s+\Theta_{0}>\sup Q$: otherwise $T\in Q$,
which contradicts the assumption. ∎∎

The above result allows us to define a mild solution of (ME)a+(BC)a on the
whole of $[-1,T]$ or $[s-1,T]$ if necessary.

For $s=0$, to stress the dependence of the solution on $a$, $u_{0}$, $R$ we
write $u(\cdot;a,u_{0},R)$. For $t\in[-1,0]$, $u(t;a,u_{0},R)$ is interpreted
as $u_{0}(t)$. Moreover, when it does not lead to confusion, we sometimes
write $u(t;a,u_{0},\Phi)$ instead of $u(t;a,u_{0},R)$.

###  3.2 Compactness of Solution Operator

######  Lemma 3.6.

Assume $1<p<\infty$ and $0<T_{1}\leq T$. Then for any bounded $F\subset
L_{\infty}((0,T),L_{p}(D))$ the set

$\widehat{F}\vcentcolon=\biggl{\\{}\int\limits_{0}^{t}U_{a_{0}}(t,{\zeta})u({\zeta})\,\mathrm{d}{\zeta}:a_{0}\in
Y_{0},\ u\in F,\ t\in[T_{1},T]\biggr{\\}}$

is precompact in $L_{p}(D)$.

###### Proof.

Compare [35, Thm. 6.1.3]. Fix $p$, $T_{1}$ and $F$ as in the statement. Let
$(t_{m})_{m=1}^{\infty}\subset[T_{1},T]$, $(a_{0,m})_{m=1}^{\infty}\subset
Y_{0}$, $(u_{m})_{m=1}^{\infty}\subset F$. We claim that for any fixed
$l\in\mathbb{N}$ the set

$\displaystyle\widetilde{F}_{l}$
$\displaystyle\vcentcolon=\biggl{\\{}\int\limits_{0}^{t_{m}-\frac{1}{l}}U_{a_{0,m}}(t_{m},{\zeta})u_{m}({\zeta})\,\mathrm{d}\zeta:{m}\in\mathbb{N}\biggr{\\}}.$

is precompact in $L_{p}(D)$. Denote by $M_{0}>0$ the supremum of the
$L_{\infty}((0,T),\allowbreak L_{p}(D))$-norms of $u_{m}$, and put
$\widecheck{F}_{l}$ to be the closure in $L_{p}(D)$ of the set

$\left\\{U_{a_{0}}(s,0){\tilde{u}}:a_{0}\in
Y_{0},s\in\Bigl{[}\frac{1}{l},T\Bigr{]},\rVert\tilde{u}\lVert_{L_{p}(D)}\leq
M_{0}\right\\}$

The set $\widecheck{F}_{l}$ is balanced. We have
$\widecheck{F}_{l}\subset\widecheck{F}_{l+1}$. By Proposition 2.9,
$\widecheck{F}_{l}$ is compact. [16, Cor. II.2.8] implies that

$\int\limits_{0}^{t_{m}-\frac{1}{l}}U_{a_{0,m}}(t_{m},\zeta)u_{m}(\zeta)\,\mathrm{d}\zeta\in
T\cdot\operatorname{\overline{co}}{\widecheck{F}_{l}}$

where $\operatorname{\overline{co}}$ denotes the closed convex hull in
$L_{p}(D)$. As, by Mazur’s theorem ([16, Thm. II.2.12]),
$T\cdot\operatorname{\overline{co}}{\widecheck{F}_{l}}$ is compact for any
$l\in\mathbb{N}$, this proves our claim that $\widetilde{F}_{l}$ are
precompact in $L_{p}(D)$. By a diagonal process we can assume without loss of
generality that for each $l\in\mathbb{N}$ the integrals

$\int\limits_{0}^{{t_{m}}-\frac{1}{l}}U_{a_{0,m}}(t_{m},\zeta)u_{m}(\zeta)\,\mathrm{d}\zeta$

converge, as $m\to\infty$, in $L_{p}(D)$.

Lemma 3.4(i) guarantees that the functions
$U_{a_{0,m}}(t_{m},\cdot)u_{m}(\cdot)$ belong to
$L_{\infty}((0,t_{m}),L_{p}(D))$, with their
$L_{\infty}((0,t_{m}),L_{p}(D))$-norms bounded uniformly in $m$. We estimate,
via Hölder’s inequality,

$\biggl{\lVert}\int_{t_{m}-\frac{1}{l}}^{t_{m}}U_{a_{0,m}}(t_{m},\zeta)u_{m}(\zeta)\,\mathrm{d}{\zeta}\biggr{\rVert}_{L_{p}(D)}\\\
\leq\int_{t_{m}-\frac{1}{l}}^{t_{m}}\lVert
U_{a_{0,m}}(t_{m},\zeta)u_{m}(\zeta)\rVert_{L_{p}(D)}\,\mathrm{d}{\zeta}\\\
\leq\lVert
U_{a_{0,m}}(t_{m},\cdot)u_{m}(\cdot)\rVert_{L_{\infty}((0,t_{m}),L_{p}(D))}\cdot(1/l).$

It follows from Proposition 2.7(ii) that for any $\epsilon>0$ there is
$l_{0}\in\mathbb{N}$ such that

$\biggl{\lVert}\,\int\limits_{t_{m}-\frac{1}{l_{0}}}^{t_{m}}U_{a_{0,m}}(t_{m},\zeta)u_{m}(\zeta)\,\mathrm{d}\zeta\,\biggr{\rVert}_{L_{p}(D)}<\frac{\epsilon}{3}$

uniformly in $m\in\mathbb{N}$. By the previous paragraph, there is $m_{0}$
such that if $m_{1},m_{2}\geq m_{0}$ then

$\biggl{\lVert}\int\limits_{0}^{t_{{m}_{1}}-\frac{1}{l_{0}}}U_{a_{0,m_{1}}}(t_{{m}_{1}},\zeta)u_{{m}_{1}}(\zeta)\,\mathrm{d}\zeta-\int\limits_{0}^{t_{{m}_{2}}-\frac{1}{l_{0}}}U_{a_{0,m_{2}}}(t_{{m}_{2}},\zeta)u_{{m}_{2}}(\zeta)\,\mathrm{d}\zeta\,\biggr{\rVert}_{L_{p}(D)}<\frac{\epsilon}{3}.$

Therefore

$\biggl{\lVert}\int\limits_{0}^{t_{{m}_{1}}}U_{a_{0,m_{1}}}(t_{{m}_{1}},\zeta)u_{{m}_{1}}(\zeta)\,\mathrm{d}\zeta-\int\limits_{0}^{t_{{m}_{2}}}U_{a_{0,m_{2}}}(t_{{m}_{2}},\zeta)u_{{m}_{2}}(\zeta)\,\mathrm{d}\zeta\,\biggr{\rVert}_{L_{p}(D)}<\epsilon$

for any $m_{1},m_{2}\geq m_{0}$.

From this it follows that

$\biggl{(}\int\limits_{0}^{t_{m}}\widetilde{U}_{a_{0,m}}(t_{m},\zeta)u_{m}(\zeta)\,\mathrm{d}\zeta\biggr{)}_{\\!\\!m=1}^{\\!\\!\infty}$

is a Cauchy sequence in $L_{p}(D)$. Therefore $\widehat{F}$ is precompact in
$L_{p}(D)$. ∎∎

######  Lemma 3.7.

For any $1<p<\infty$ and any bounded $F\subset L_{\infty}((0,T),L_{p}(D))$ the
set

$\biggl{\\{}\Bigl{[}\,[0,T]\ni
t\mapsto\int\limits_{0}^{t}U_{a_{0}}(t,\zeta)u(\zeta)\,\mathrm{d}\zeta\in
L_{p}(D)\,\Bigr{]}:a_{0}\in Y_{0},\ u\in F\biggr{\\}}$

is precompact in $C([0,T],L_{p}(D))$.

###### Proof.

By the Ascoli–Arzelà theorem, it suffices, taking Lemma 3.6 into account, to
show that for any $\epsilon>0$ there is $\delta>0$ such that, if $0\leq
t_{1}\leq t_{2}\leq T$, $t_{2}-t_{1}<\delta$, then

$\biggl{\lVert}\int_{0}^{t_{2}}U_{a_{0}}(t_{2},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}-\int_{0}^{t_{1}}U_{a_{0}}(t_{1},{\zeta})u({\zeta})\,\mathrm{d}\zeta\biggr{\rVert}_{L_{p}(D)}<\epsilon$

for all $a_{0}\in Y_{0}$ and all $u\in F$. In order not to introduce too many
constants we assume that $F$ equals the unit ball in
$L_{\infty}((0,T),L_{p}(D))$.

We write

$\int_{0}^{t_{2}}U_{a_{0}}(t_{2},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}-\int_{0}^{t_{1}}U_{a_{0}}(t_{1},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}\\\
=\int_{0}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}+\int_{t_{1}}^{t_{2}}U_{a_{0}}(t_{2},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}$

By Proposition 2.7(ii),

$\biggl{\lVert}\int_{t_{1}}^{t_{2}}U_{a_{0}}(t_{2},{\zeta})u({\zeta})\,\mathrm{d}{\zeta}\biggr{\rVert}_{L_{p}(D)}<\frac{\epsilon}{3},$
(3.11)

provided $t_{2}-t_{1}<\epsilon/(3Me^{\gamma T})$.

Further, we write

$\int_{0}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}\\\
=\int_{0}^{t_{1}-\eta}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}+\int_{t_{1}-\eta}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}.$

By Proposition 2.7(ii), if $0<\eta<\epsilon/(6Me^{\gamma T})$ then

$\biggl{\lVert}\int_{t_{1}-\eta}^{t_{1}}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}\biggr{\rVert}_{L_{q}(D)}<\frac{\epsilon}{3}.$
(3.12)

It follows from Proposition 2.18 that the assignment

$\bigl{[}\,Y_{0}\times[\eta,T]\ni(a_{0},t)\mapsto
U_{a_{0}}(t,0)\in\mathcal{L}(L_{p}(D))\,\bigr{]}$

is uniformly continuous, consequently there exists $\delta>0$ such that if
$\eta\leq s_{1}<s_{2}$, $s_{2}-s_{1}<\delta$, then

$\lVert
U_{a_{0}}(s_{2},0)-U_{a_{0}}(s_{1},0)\rVert_{\mathcal{L}(L_{p}(D))}<\frac{\epsilon}{3T}.$

Therefore

$\biggl{\lVert}\int_{0}^{t_{1}-\eta}\bigl{(}U_{a_{0}}(t_{2},{\zeta})-U_{a_{0}}(t_{1},{\zeta})\bigr{)}u({\zeta})\,\mathrm{d}{\zeta}\biggr{\rVert}_{L_{q}(D)}<\frac{\epsilon}{3}.$

The estimates (3.11), (3.12) and (3.2) do not depend on the choice of
$a_{0}\in Y_{0}$, so gathering them gives the required property. ∎∎

######  Theorem 3.2.

For any $0<T_{1}\leq T$, any $1<p<\infty$ and any bounded $E\subset
C([-1,0],L_{p}(D))$ the set

$\bigl{\\{}\,\bigl{[}\,[T_{1},T]\ni t\mapsto u(t;a,u_{0},R)\,\bigr{]}:a\in
Y,u_{0}\in E,R\in\mathcal{R}\,\bigr{\\}}$

is precompact in $C([T_{1},T],L_{p}(D))$.

###### Proof.

We will use the notation $I_{i}(t;a,u_{0},R)$, $i=0,1$ where

$\displaystyle I_{0}(t)$
$\displaystyle\vcentcolon=U_{\tilde{a}}(t,0)u_{0}(0),$ (3.13) $\displaystyle
I_{1}(t)$
$\displaystyle\vcentcolon=\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)u(\zeta-R(\zeta))\,\mathrm{d}\zeta,$

taking account of the parameter $a$ and the initial value $u_{0}$. The
precompactness of the set

$\bigl{\\{}\,\bigl{[}\,[T_{1},T]\ni t\mapsto I_{0}(t;a,u_{0},R)\,\bigr{]}:a\in
Y,u_{0}\in E,R\in\mathcal{R}\,\bigr{\\}}$

in $C([T_{1},T],L_{p}(D))$ is a consequence of Proposition 2.9. In order to
prove the precompactness in $C([T_{1},T],L_{p}(D))$ of

$\bigl{\\{}\,\bigl{[}\,[T_{1},T]\ni t\mapsto I_{1}(t;a,u_{0},R)\,\bigr{]}:a\in
Y,u_{0}\in E,R\in\mathcal{R}\,\bigr{\\}},$

it suffices to use results from Lemma 3.3 and Lemma 3.7. ∎∎

Theorem 3.2 leads to the following conclusion about precompactness of the
solutions up to zero. Since under additional assumption ($\mathrm{DA\hskip
1.00006pt5}$) for a fixed $u_{0}\in C([-1,0],L_{p}(D))$ the set

$\bigl{\\{}\,\bigl{[}\,[0,T]\ni t\mapsto
U_{\tilde{a}}(t,0)u_{0}(0)\,\bigr{]}:a\in Y\,\bigr{\\}}$

is simply a singleton, this observation combined with Lemma 3.7 leads to the
following result.

######  Theorem 3.3.

Assume additionally ($\mathrm{DA\hskip 1.00006pt5}$). For any $1<p<\infty$ and
any $u_{0}\in C([-1,0],L_{p}(D))$ the set

$\bigl{\\{}\,\bigl{[}\,[0,T]\ni t\mapsto u(t;a,u_{0},R)\,\bigr{]}:a\in
Y,R\in\mathcal{R}\,\bigr{\\}}$

is precompact in $C([0,T],L_{p}(D))$.

##  4 Continuous Dependence on Initial Conditions

In the present section we assume ($\mathrm{DA\hskip 1.00006pt1}$),
($\mathrm{DA\hskip 1.00006pt2}$) and that $Y$ as in Definition 1.1 is such
that its flattening $Y_{0}$ satisfies ($\mathrm{DA\hskip 1.00006pt3}$).
Further, $1<p<\infty$.

######  Definition 4.1.

For $t\in[0,T]$, $a\in Y$, $u_{0}\in C([-1,0],L_{p}(D))$ and $R\in\mathcal{R}$
we define

$\displaystyle\delta(t;a,u_{0},R)$
$\displaystyle=\sup_{\vartheta\in[-1,0]}\lVert
u(t+\vartheta;a,u_{0},R)\rVert_{L_{p}(D)}$ $\displaystyle=\lVert
u(t+\cdot\,;a,u_{0},R)\upharpoonright_{[-1,0]}\rVert_{C([-1,0],L_{p}(D))}.$

For notational simplicity we often write $u(t+\,\cdot)$ instead of
$u(t+\,\cdot\,;a,u_{0},R)$ and $\delta(t)$ instead of $\delta(t;a,u_{0},R)$
when $a\in Y$ and $u_{0}\in C([-1,0],L_{p}(D))$ are fixed and this does not
lead to confusion.

######  Lemma 4.1.

For any $a\in Y$, $u_{0}\in C([-1,0],L_{p}(D))$ and $R\in\mathcal{R}$ the
function $\delta(\cdot;a,u_{0},R)\colon[0,T]\to\mathbb{R}^{+}$ is continuous.

###### Proof.

First note that the mapping

$\big{[}\,[0,T]\times[-1,0]\ni(t,\vartheta)\mapsto\lVert
u(t+\vartheta;a,u_{0},R)\rVert_{L_{p}(D)}\in\mathbb{R}^{+}\,\big{]}$

is continuous as a composition of continuous mappings. Due to the compactness
of $[-1,0]$ the $\delta(\cdot,a,u_{0},R)$ function is continuous when $a\in
Y$, $R\in\mathcal{R}$ and $u_{0}\in C([-1,0],L_{p}(D))$ are fixed. ∎∎

######  Lemma 4.2.

There are constants $M_{1},M_{2}$ such that for any $\rho\in[0,T]$ the
inequality

$\lVert u(\rho;a,u_{0},R)\rVert_{L_{p}(D)}\leq
M_{1}\delta(0;a,u_{0},R)+M_{2}\int_{0}^{\rho}\delta(\zeta;a,u_{0},R)\,\mathrm{d}\zeta$

holds for all $a\in Y$, $u_{0}\in C([-1,0],L_{p}(D))$ and $R\in\mathcal{R}$.

###### Proof.

Fix $\rho\in[0,T]$ and note that

$\displaystyle\lVert u(\rho)\rVert_{L_{p}(D)}$ $\displaystyle\leq\lVert
U_{\tilde{a}}(\rho)u_{0}(0)\rVert_{L_{p}(D)}+\int_{0}^{\rho}\lVert
U_{\tilde{a}}(\rho,\zeta)C^{1}_{a}(\zeta)(u\circ\Phi)(\zeta)\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$
$\displaystyle\leq Me^{\gamma\rho}\lVert
u_{0}(0)\rVert_{L_{p}(D)}+Me^{\gamma\rho}K\int_{0}^{\rho}\lVert(u\circ\Phi)(\zeta)\rVert_{L_{p}(D)}\,\mathrm{d}\zeta$
$\displaystyle\leq Me^{\gamma\rho}\lVert
u_{0}\rVert_{C([-1,0],L_{p}(D))}+Me^{\gamma\rho}K\int_{0}^{\rho}\lVert
u(\zeta+\cdot)\upharpoonright_{[-1,0]}\rVert_{C([-1,0],L_{p}(D))}\,\mathrm{d}\zeta$
$\displaystyle=M_{1}\delta(0)+M_{2}\int_{0}^{\rho}\delta(\zeta)\,\mathrm{d}\zeta,$

where $M$ is a uniform bound of the operator $U_{\tilde{a}}(t)$ with respect
to $a\in Y$ and $0\leq t\leq T$ (see Proposition 2.7(ii)), the constant $K$ is
a uniform bound of the operator $C^{1}_{a}(\zeta)$ with respect to $a\in Y$
and $0\leq t\leq T$ (see Lemma 3.1). Moreover, the bounds $M$ and $K$ are
independent on initial condition $u_{0}$. By setting $M_{1}=Me^{\gamma T}$,
$M_{2}=MKe^{\gamma T}$ we end the proof. ∎∎

From now on, throughout this section the constants $M_{1}$ and $M_{2}$ will be
defined as in Lemma 4.2.

######  Proposition 4.1.

For any sequence $(u_{0,m})_{m=1}^{\infty}\subset C([-1,0],L_{p}(D))$
convergent to zero, any $t\in[0,T]$, $R\in\mathcal{R}$ and $a\in Y$ the
sequence $\delta_{m}(t)\vcentcolon=\delta(t;a,u_{0,m},R)$ converges to zero.

###### Proof.

Fix $t\in[0,T]$ and $-1\leq\vartheta\leq 0$ and let us consider two cases.

  * $\bullet$

If $0\leq t+\vartheta\leq T$ then from Lemma 4.2 there holds

$\displaystyle\lVert u(t+\vartheta)\rVert_{L_{p}(D)}$ $\displaystyle\leq
M_{1}\delta(0)+M_{2}\int_{0}^{t+\vartheta}\delta(\zeta)\,\mathrm{d}\zeta$
$\displaystyle\leq
M_{1}\delta(0)+M_{2}\int_{0}^{t}\delta(\zeta)\,\mathrm{d}\zeta.$

  * $\bullet$

If $-1\leq t+\vartheta\leq 0$ then the inequality

$\displaystyle\lVert u(t+\vartheta)\rVert_{L_{p}(D)}\leq
M_{1}\delta(0)+M_{2}\int_{0}^{t}\delta(\zeta)\,\mathrm{d}\zeta$

is straightforward, as even the stronger one $\lVert
u(t+\vartheta)\rVert_{L_{p}(D)}\leq M_{1}\delta(0)$ is true.

Applying $\sup$ with respect to $\vartheta$ on both sides give us that

$\sup_{\vartheta\in[-1,0]}\lVert u(t+\vartheta,a,u_{0},R)\rVert_{L_{p}(D)}\leq
M_{1}\delta(0)+M_{2}\int_{0}^{t}\delta(\zeta)\,\mathrm{d}\zeta,$

what can be rewritten in terms of the $\delta$ function as

$\delta(t)\leq
M_{1}\delta(0)+M_{2}\int_{0}^{t}\delta(\zeta)\,\mathrm{d}\zeta.$

The function $\delta$ is nonnegative and continuous on the compact domain,
hence it is integrable. Using the Grönwall lemma we get

$\delta(t)\leq
M_{1}\delta(0)\exp\Big{(}M_{2}\int_{0}^{t}\,\mathrm{d}\zeta\Big{)}.$ (4.1)

∎∎

The above Lemma 4.2 and Proposition 4.1 lead to global $L_{p}$-norm estimation
of the mild solution of $\eqref{main-eq}_{a}+\eqref{main-bc}_{a}$ in terms of
initial conditions.

######  Proposition 4.2.

There is constant $\overline{M}>0$ such that inequality

$\lVert u(t;a,u_{0},R)\rVert_{L_{p}(D)}\leq\overline{M}\lVert
u_{0}\rVert_{C([-1,0],L_{p}(D))}$

holds for any $1<p<\infty$, $t\in[0,T]$, $a\in Y$, $R\in\mathcal{R}$ and
$u_{0}\in C([-1,0],L_{p}(D))$.

###### Proof.

Let $\overline{M}=M_{1}\exp(M_{2}T)$, where $M_{1},M_{2}$ are constants as in
Lemma 4.2. Then by Proposition 4.1 we can write

$\displaystyle\lVert u(t;a,u_{0},R)\rVert_{L_{p}(D)}\leq{}$
$\displaystyle\delta(t;a,u_{0},R)$ $\displaystyle\leq{}$
$\displaystyle\overline{M}\lVert u_{0}\rVert_{C([-1,0],L_{p}(D))}.$

∎∎

######  Proposition 4.3.

For any $a\in Y$ and $R\in\mathcal{R}$ the mapping

$\big{[}\,C([-1,0],L_{p}(D))\ni u_{0}\mapsto u(\cdot;a,u_{0},R)\in
C([-1,T],L_{p}(D))\,\big{]}$

is continuous.

###### Proof.

Let $a\in Y$, $R\in\mathcal{R}$ be fixed. Then in the spirit of Cauchy’s
definition we can find that

$\lVert
u(\cdot;a,u_{0,1},R)-u(\cdot;a,u_{0,2},R)\rVert_{C([-1,T],L_{p}(D))}\\\
\leq\sup_{t\in[0,T]}\delta(t;a,u_{0,1}-u_{0,2},R)\\\ \leq
M_{1}\exp(M_{2}T)\lVert u_{0,1}-u_{0,2}\rVert_{C([-1,0],L_{p}(D))}$

for any initial conditions $u_{0,1},u_{0,2}\in C([-1,0],L_{p}(D))$. The first
inequality results from the linearity of the problem $\eqref{main-
eq}_{a}+\eqref{main-bc}_{a}$ and the second inequality follows from (4.1). ∎∎

##  5 Continuous Dependence on Coefficients and Delay

In the present section we assume ($\mathrm{DA\hskip 1.00006pt1}$),
($\mathrm{DA\hskip 1.00006pt2}$) and that $Y$ as in Definition 1.1 is such
that its flattening $Y_{0}$ satisfies ($\mathrm{DA\hskip 1.00006pt3}$) and
($\mathrm{DA\hskip 1.00006pt4}$). As in Section 4, $1<p<\infty$.

######  Proposition 5.1.

For any $0<T_{1}\leq T$, $R\in\mathcal{R}$ and $u_{0}\in C([-1,0],L_{p}(D))$
the mapping

$\bigl{[}\,Y\ni a\mapsto u(\cdot;a,u_{0},R)\\!\\!\restriction_{[T_{1},T]}\in
C([T_{1},T],L_{p}(D))\,\bigr{]}$

is continuous.

###### Proof.

Fix $p$, $T_{1}$, $R$ and $u_{0}$ as in the Proposition. Let
$(a_{m})_{m=1}^{\infty}\subset Y$ converge to $a$. Put $u_{m}(\cdot)$ for
$u(\cdot;a_{m},u_{0},R)$ and $u(\cdot)$ for $u(\cdot;a,u_{0},R)$. It suffices
to prove that there is a subsequence $(a_{m_{k}})_{k=1}^{\infty}\subset Y$
such that $u_{m_{k}}(\cdot)$ converges to $u(\cdot)$ on $[T_{1},T]$ uniformly.
By Theorem 3.2 via diagonal process, we can find a subsequence $u_{m_{k}}$
such that $u_{m_{k}}\\!\\!\restriction_{(0,T]}$ converge to some continuous
$\hat{u}\colon(0,T]\to L_{p}(D)$ and the convergence is uniform on compact
subsets of $(0,T]$. The function $\hat{u}$ is clearly bounded by Proposition
4.2. Moreover, we extend the map $\hat{u}$ to the whole $[-1,T]$ by $u_{0}$ on
$[-1,0]$, i.e., now

$\tilde{u}(t)\vcentcolon=\left\\{\begin{array}[]{lll}u_{0}(t)&\quad\text{for
}t\in[-1,0]\\\\[4.30554pt] \lim\limits_{k\to\infty}u_{m_{k}}(t)&\quad\text{for
}t\in(0,T].\end{array}\right.$

It remains to prove that $\tilde{u}=u$. In order not to overburden the
notation we write $u_{m}$ instead of $u_{m_{k}}$.

Our first step is to show that, for each $t\in(0,T]$,

$\displaystyle U_{\tilde{a}_{m}}(t,0)u_{0}(0)$ $\displaystyle\to
U_{\tilde{a}}(t,0)u_{0}(0)$ (5.1)
$\displaystyle\int\limits_{0}^{t}U_{\tilde{a}_{m}}(t,\zeta)C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-R(\zeta))\,\mathrm{d}\zeta$
$\displaystyle\to\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta.$
(5.2)

in the $L_{p}(D)$-norm as $m\to\infty$. The convergence in (5.1) is a
consequence of Proposition 2.17. The convergence in (5.2) can be shown by
showing the convergence of the difference

$\displaystyle\int\limits_{0}^{t}U_{\tilde{a}_{m}}(t,\zeta)$ $\displaystyle
C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-R(\zeta))\,\mathrm{d}\zeta-\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta$
(5.3) $\displaystyle=$
$\displaystyle\int\limits_{0}^{t}(U_{\tilde{a}_{m}}(t,\zeta)-U_{\tilde{a}}(t,\zeta))C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-R(\zeta))\,\mathrm{d}\zeta$
$\displaystyle+\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-R(\zeta))-\tilde{u}(\zeta-R(\zeta)))\,\mathrm{d}\zeta$
$\displaystyle+\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)(C^{1}_{a_{m}}(\zeta)-C^{1}_{a}(\zeta))\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta$

to zero. Write $J^{(i)}_{m}(t)$, $i=1,2,3$, for the $i$-th term on the right-
hand side of (5.3). The convergence of $J^{(1)}_{m}(t)$ follows from the
Lebesgue dominated convergence theorem for Bochner integral: since the
integrand

$(0,t)\ni\zeta\mapsto(U_{\tilde{a}_{m}}(t,\zeta)-U_{\tilde{a}}(t,\zeta))C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-R(\zeta))\in
L_{p}(D)$

is $(\mathfrak{L}((0,t)),\mathfrak{B}(L_{p}(D)))$-measurable for all
$m\in\mathbb{N}$ (see Lemmas 3.4(i) and 3.3) and bounded uniformly (see
Proposition 4.2) in $m\in\mathbb{N}$:

$\lVert(U_{\tilde{a}_{m}}(t,\zeta)-U_{\tilde{a}}(t,\zeta))C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-R(\zeta))\rVert_{L_{p}(D)}\\\
\leq 2Me^{\gamma T}K\overline{M}\lVert u_{0}\rVert_{C([-1,0],L_{p}(D))}$

it suffices to check that for a.e. $\zeta\in(0,t)$ the integrand converges to
zero, which follows from the estimate

$\lVert(U_{\tilde{a}_{m}}(t,\zeta)-U_{\tilde{a}}(t,\zeta))C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-R(\zeta))\rVert_{L_{p}(D)}\\\
\leq\lVert(U_{\tilde{a}_{m}}(t,\zeta)-U_{\tilde{a}}(t,\zeta))\rVert_{\mathcal{L}(L_{p}(D))}K\overline{M}\lVert
u_{0}\rVert_{C([-1,0],L_{p}(D))}$

and Proposition 2.18.

In order to prove $J_{m}^{(2)}(t)\to 0$ as $m\to\infty$ we proceed similarly.
We see that the mapping

$\big{[}\,(0,t)\ni\zeta\mapsto
U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-R(\zeta))-\tilde{u}(\zeta-R(\zeta)))\in
L_{p}(D)\,\big{]}$

is $(\mathfrak{L}((0,t)),\mathfrak{B}(L_{p}(D)))$-measurable for all
$m\in\mathbb{N}$, as a consequence of Lemmas 3.4(i) and 3.3, and bounded
uniformly in $m\in\mathbb{N}$, since, by Proposition 2.7(ii), Lemma 3.1 and
Proposition 4.2,

$\lVert
U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-R(\zeta))-\tilde{u}(\zeta-R(\zeta)))\rVert_{L_{p}(D)}\\\
\leq 2MKe^{\gamma T}\overline{M}\lVert u_{0}\rVert_{C([-1,0],L_{p}(D))}.$

Further, the convergence, for a.e. $\zeta\in(0,t)$,

$U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-R(\zeta))-\tilde{u}(\zeta-R(\zeta)))\to
0$

in $L_{p}(D)$ is due to the pointwise convergence of $u_{m}$ to $\tilde{u}$ on
$[-1,T]$ and the estimate (by Proposition 2.7(ii) and Lemma 3.1)

$\lVert
U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-R(\zeta))-\tilde{u}(\zeta-R(\zeta)))\rVert_{L_{p}(D)}\\\
\leq MKe^{\gamma T}\lVert
u_{m}(\zeta-R(\zeta))-\tilde{u}(\zeta-R(\zeta))\rVert_{L_{p}(D)}.$

The convergence of $J^{(3)}_{m}(t)$ follows from the facts that the set

$\\{\,J^{(3)}_{m}(t):m\in\mathbb{N}\,\\}$

is precompact in $L_{p}(D)$ (see Lemma 3.6, Lemma 3.3, Proposition 4.2) and
that $J^{(3)}_{m}(t)$ converge weakly to zero, i.e.,

$\displaystyle\langle J^{(3)}_{m}(t),v\rangle$ $\displaystyle\to 0\text{ as
}m\to\infty,$ (5.4)

for any $v\in L_{{p^{\prime}}}(D)$, where $\langle\cdot,\cdot\rangle$ stands
for the duality pairing between $L_{p}(D)$ and $L_{p^{\prime}}(D)$. By the
Hille theorem ([16, Thm. II.2.6]) and Proposition 2.14,

$\langle J^{(3)}_{m}(t),v\rangle=\int_{0}^{t}\langle
U_{\tilde{a}}(t,\zeta)(C^{1}_{a_{m}}(\zeta)-C^{1}_{a}(\zeta))\tilde{u}(\zeta-R(\zeta)),v\rangle\,\mathrm{d}\zeta\\\
=\int_{0}^{t}\langle(C^{1}_{a_{m}}(\zeta)-C^{1}_{a}(\zeta))\tilde{u}(\zeta-R(\zeta)),U^{*}_{\tilde{a}}(t,\zeta)v\rangle\,\mathrm{d}\zeta.$

Now we need to use a subtler approach based on the convergence $c_{1,m}$ to
$c_{1}$ in the weak-* topology of $L_{\infty}((0,t)\times D)$. First note that
the mappings

$\displaystyle\big{[}\,(0,t)\ni\zeta\mapsto\tilde{u}(\zeta-R(\zeta))\in
L_{p}(D)\,\big{]}\quad\&\quad\big{[}\,(0,t)\ni\zeta\mapsto
U^{*}_{\tilde{a}}(t,\zeta)v\in L_{p^{\prime}}(D)\,\big{]}$

belong to $L_{\infty}((0,t),L_{p}(D))$ and
$L_{\infty}((0,t),L_{p^{\prime}}(D))$ respectively. Therefore the mapping (the
product of the above maps)

$\big{[}\,(0,t)\times
D\ni(\zeta,x)\mapsto\tilde{u}(\zeta-R(\zeta))[x](U^{*}_{\tilde{a}}(t,\zeta)v)[x]\in\mathbb{R}\,\big{]}$

belong to $L_{1}((0,t)\times D)$ see Lemma 0.1(a). It suffices now to note
that from Fubini’s theorem we have

$\int_{0}^{t}\langle(C^{1}_{a_{m}}(\zeta)-C^{1}_{a}(\zeta))\tilde{u}(\zeta-R(\zeta)),U^{*}_{\tilde{a}}(t,\zeta)v\rangle\,\mathrm{d}\zeta\\\
=\int_{0}^{t}\\!\\!\int_{D}(c_{1,m}(\zeta,x)-c_{1}(\zeta,x))\,\tilde{u}(\zeta-R(\zeta))[x]\,(U^{*}_{\tilde{a}}(\zeta,t)v)[x]\,\mathrm{d}x\,\mathrm{d}\zeta,$

so the integral tends to zero as $m\to\infty$.

We have thus proved that

$\displaystyle\tilde{u}(t)={}$ $\displaystyle
U_{\tilde{a}}(t,0)u_{0}(0)+\int_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta,\quad
t\in[0,T].$

Now we prove the continuity of the extension $\tilde{u}$. Note that the only
point where it can fail is $t=0$. However, this is not the case since the
mappings

$\displaystyle\Big{[}\,[0,T]\ni t\mapsto U_{\tilde{a}}(t,0)u_{0}(0)\in
L_{p}(D)\,\Big{]}$ $\displaystyle\Big{[}\,[0,T]\ni
t\mapsto\int_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta\in
L_{p}(D)\,\Big{]}$

are continuous (see Lemmas 2.3(ii) and 3.4(ii)), so the mapping $\tilde{u}$ is
continuous on the whole $[-1,T]$. Also, $\tilde{u}=u_{0}$ on $[-1,0]$, hence
$\tilde{u}$ is in fact the mild solution of $\eqref{main-
eq}_{\tilde{a}}+\eqref{main-bc}_{\tilde{a}}$, therefore, by uniqueness,
$\tilde{u}(t)=u(t;\tilde{a},u_{0},R)$ for any $t\in[-1,T]$. ∎∎

######  Proposition 5.2.

Assume additionally ($\mathrm{DA\hskip 1.00006pt5}$). For any $u_{0}\in
C([-1,0],L_{p}(D))$ and $\mathcal{R}_{0}\subset\mathcal{R}$ satisfying the
assumption ($\mathrm{DA\hskip 1.00006pt6}$) the mapping

$\bigl{[}\,Y\times\mathcal{R}_{0}\ni(a,R)\mapsto
u(\cdot;a,u_{0},R)\\!\\!\restriction_{[0,T]}\in C([0,T],L_{p}(D))\,\bigr{]}$

is continuous.

###### Sketch of proof.

Fix $p$, $u_{0}$ and $\mathcal{R}_{0}$. Let $(a_{m})_{m=1}^{\infty}\subset Y$
converge to $a$ and $(R_{m})_{m=1}^{\infty}\subset\mathcal{R}_{0}$ converge to
$R$, and put $u_{m}(\cdot)$ for $u(\cdot;a_{m},u_{0},R_{m})$ and $u(\cdot)$
for $u(\cdot;a,u_{0},R)$. We will proceed as in Proposition 5.1. In
particular, $\tilde{u}$ has the same meaning. However, we have in fact more:
as we assume ($\mathrm{DA\hskip 1.00006pt5}$), we can apply Theorem 3.3 to
show that $u_{m}$ converge to $\hat{u}$ uniformly on $[0,T]$, from which it
follows in particular that $\tilde{u}$ is continuous on the whole of $[-1,T]$.

It follows again from ($\mathrm{DA\hskip 1.00006pt5}$) that
$\tilde{a}_{m}=\tilde{a}$ for all $m\in\mathbb{N}$, so

$U_{\tilde{a}_{m}}(t,0)u_{0}(0)\to U_{\tilde{a}}(t,0)u_{0}(0)$

holds trivially. We start by showing that, for each $t\in[0,T]$,

$\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-
R_{m}(\zeta))\,\mathrm{d}\zeta\to\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta.$
(5.5)

in the $L_{p}(D)$-norm as $m\to\infty$. The above convergence can be proved by
showing convergence of the terms

$\displaystyle\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)$ $\displaystyle
C^{1}_{a_{m}}(\zeta)u_{m}(\zeta-
R_{m}(\zeta))\,\mathrm{d}\zeta-\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a}(\zeta)\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta$
(5.6)
$\displaystyle=\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-
R_{m}(\zeta))-\tilde{u}(\zeta-R(\zeta)))\,\mathrm{d}\zeta$
$\displaystyle+\int\limits_{0}^{t}U_{\tilde{a}}(t,\zeta)(C^{1}_{a_{m}}(\zeta)-C^{1}_{a}(\zeta))\tilde{u}(\zeta-R(\zeta))\,\mathrm{d}\zeta.$

to zero. Write $K^{(i)}_{m}(t)$, $i=1,2$, for the $i$-th term on the right-
hand side of (5.6).

Regarding the convergence of $K^{(1)}_{m}(t)$ to zero, we show the
$(\mathfrak{L}((0,t)),\mathfrak{B}(L_{p}(D)))$-measurability, for all
$m\in\mathbb{N}$, of the integrand

$\bigl{[}\,(0,t)\ni\zeta\mapsto
U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-
R_{m}(\zeta))-\tilde{u}(\zeta-R(\zeta)))\in L_{p}(D)\,\bigr{]}$

in the same way as in the proof of the convergence of $J^{(2)}_{m}(t)$ in
Proposition 5.1. The fact that for a.e. $\zeta\in(0,T)$ we have that

$U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-
R_{m}(\zeta))-\tilde{u}(\zeta-R(\zeta)))\to 0$

in $L_{p}(D)$ is due, in view of ($\mathrm{DA\hskip 1.00006pt6}$), to the
uniform convergence of $u_{m}$ to $\tilde{u}$ on $[-1,T]$ together with the
estimate

$\lVert U_{\tilde{a}}(t,\zeta)C^{1}_{a_{m}}(\zeta)(u_{m}(\zeta-
R_{m}(\zeta))-\tilde{u}(\zeta-R(\zeta)))\rVert_{L_{p}(D)}\\\ \leq MKe^{\gamma
T}\lVert u_{m}(\zeta-
R_{m}(\zeta))-\tilde{u}(\zeta-R(\zeta))\rVert_{L_{p}(D)}.$

The proof of the convergence of $K^{(2)}_{m}(t)$ to zero is just a copy, word
for word, of the proof of the convergence of $J^{(3)}_{m}(t)$ in Proposition
5.1. ∎∎

######  Theorem 5.1.

  1. (i)

For any $0<T_{1}\leq T$ and $R\in\mathcal{R}$ the mapping

$\Bigl{[}\,Y\times C([-1,0],L_{p}(D))\ni(a,u_{0})\\\ \mapsto
u(\cdot;a,u_{0},R)\\!\\!\restriction_{[T_{1},T]}\in
C([T_{1},T],L_{p}(D))\,\Bigr{]}$

is continuous.

  2. (ii)

Under ($\mathrm{DA\hskip 1.00006pt5}$), if $\mathcal{R}_{0}\subset\mathcal{R}$
is a subset such that the assumption ($\mathrm{DA\hskip 1.00006pt6}$) holds
then the mapping

$\Bigl{[}\,Y\times C([-1,0],L_{p}(D))\times\mathcal{R}_{0}\ni(a,u_{0},R)\\\
\mapsto u(\cdot;a,u_{0},R)\\!\\!\restriction_{[0,T]}\in
C([0,T],L_{p}(D))\,\Bigr{]}$

is continuous.

###### Proof.

Fix $1<p<\infty$. We start by proving (i), so fix also $T_{1}$, $R$ as in the
statement. Let a sequence $(a_{m})_{m=1}^{\infty}\subset Y$ converge to $a\in
Y$ and $(u_{0,m})_{m=1}^{\infty}\subset C([-1,0],L_{p}(D))$ converge to
$u_{0}\in C([-1,0],L_{p}(D))$. The main idea of the proof is based on the
estimation

$\begin{multlined}\lVert
u(\cdot;a_{m},u_{0,m},R){\restriction}_{[T_{1},T]}-u(\cdot;a,u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}\\\
\quad\leq\lVert
u(\cdot;a_{m},u_{0,m},R){\restriction}_{[T_{1},T]}-u(\cdot;a_{m},u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}\\\
+\lVert
u(\cdot;a_{m},u_{0},R){\restriction}_{[T_{1},T]}-u(\cdot;a,u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}.\end{multlined}\lVert
u(\cdot;a_{m},u_{0,m},R){\restriction}_{[T_{1},T]}-u(\cdot;a,u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}\\\
\quad\leq\lVert
u(\cdot;a_{m},u_{0,m},R){\restriction}_{[T_{1},T]}-u(\cdot;a_{m},u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}\\\
+\lVert
u(\cdot;a_{m},u_{0},R){\restriction}_{[T_{1},T]}-u(\cdot;a,u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}.$
(5.7)

Proposition 4.2 implies

$\lVert
u(\cdot;a_{m},u_{0,m},R){\restriction}_{[T_{1},T]}-u(\cdot;a_{m},u_{0},R){\restriction}_{[T_{1},T]}\rVert_{C([T_{1},T],L_{p}(D))}\\\
\leq\overline{M}\lVert u_{0,m}-u_{0}\rVert_{C([-1,0],L_{p}(D))}.$

Therefore the first part of the right-hand side of (5.7) converges to zero as
$m\to\infty$. The second part of (5.7) converges to zero by Proposition 5.1.
Item (ii) can be proved similarly. So, assume additionally ($\mathrm{DA\hskip
1.00006pt5}$) and, instead of fixing the delay $R\in\mathcal{R}$ take a
sequence $(R_{m})_{m=1}^{\infty}\subset\mathcal{R}_{0}$ convergent to some
$R\in\mathcal{R}_{0}$. By similar estimation,

$\begin{multlined}\lVert
u(\cdot;a_{m},u_{0,m},R_{m}){\restriction}_{[0,T]}-u(\cdot;a,u_{0},R){\restriction}_{[0,T]}\rVert_{C([0,T],L_{p}(D))}\\\
\quad\leq\lVert
u(\cdot;a_{m},u_{0,m},R_{m}){\restriction}_{[0,T]}-u(\cdot;a_{m},u_{0},R_{m}){\restriction}_{[0,T]}\rVert_{C([0,T],L_{p}(D))}\\\
+\lVert
u(\cdot;a_{m},u_{0},R_{m}){\restriction}_{[0,T]}-u(\cdot;a,u_{0},R){\restriction}_{[0,T]}\rVert_{C([0,T],L_{p}(D))}\end{multlined}\lVert
u(\cdot;a_{m},u_{0,m},R_{m}){\restriction}_{[0,T]}-u(\cdot;a,u_{0},R){\restriction}_{[0,T]}\rVert_{C([0,T],L_{p}(D))}\\\
\quad\leq\lVert
u(\cdot;a_{m},u_{0,m},R_{m}){\restriction}_{[0,T]}-u(\cdot;a_{m},u_{0},R_{m}){\restriction}_{[0,T]}\rVert_{C([0,T],L_{p}(D))}\\\
+\lVert
u(\cdot;a_{m},u_{0},R_{m}){\restriction}_{[0,T]}-u(\cdot;a,u_{0},R){\restriction}_{[0,T]}\rVert_{C([0,T],L_{p}(D))}$
(5.8)

together with Propositions 4.2 and 5.2 concludes the proof. ∎∎

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