#  Existence and smoothness of global bounded solutions to exponentially
stable nonautonomous linear hyperbolic systems

Irina Kmit Viktor Tkachenko Institute of Mathematics, Humboldt University of
Berlin. On leave from the Institute for Applied Problems of Mechanics and
Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine. E-mail:
<EMAIL_ADDRESS>Institute of Mathematics, National Academy of Sciences
of Ukraine, Kyiv, Ukraine. E-mail<EMAIL_ADDRESS>
###### Abstract

We investigate boundary value problems for the general linear nonautonomous
first order 1D hyperbolic system in a strip. Under the assumptions that the
problem is exponentially stable and a number of dissipativity conditions are
satisfied, we prove existence and uniqueness of $C^{1}$\- and $C^{2}$-global
bounded, in particular, almost periodic and periodic solutions. It should be
emphasized that the number of dissipativity conditions we impose is strongly
related to the order of regularity of solutions. Note that the last propety is
caused by the nonautonomity of the hyperbolic system and does not appear in
the autonomous setting. We show that the time-periodic solutions exit under
weaker assumptions.

##  1 Introduction

###  1.1 Problem setting and our results

We consider general first order linear hyperbolic systems of the type

$\partial_{t}u+a(x,t)\partial_{x}u+b(x,t)u=f(x,t),\quad x\in(0,1),$ (1.1)

where $u=(u_{1},\ldots,u_{n})$ and $f=(f_{1},\dots,f_{n})$ are vectors of
real-valued functions, $a=\mathop{\rm diag}\nolimits(a_{1},\dots,a_{n})$ and
$b=(b_{jk})$ are $n\times n$-matrices with real-valued entries.

Set

$\Pi=\\{(x,t)\in{\mathbb{R}}^{2}\,:\,0\leq x\leq 1\\}.$

Suppose that

$\displaystyle a_{1}(x,t)>\dots>a_{m}(x,t)>0>a_{m+1}(x,t)>\dots>a_{n}(x,t)$

for some integer $0\leq m\leq n$ and all $(x,t)\in\Pi$. Furthermore, there
exists a positive constant $a_{0}$ such that

$\begin{array}[]{ll}\inf\left\\{a_{j}(x,t)\,:\,(x,t)\in\Pi,\ 1\leq j\leq
m\right\\}\geq a_{0},\\\\[2.84526pt] \sup\left\\{a_{j}(x,t)\,:\,(x,t)\in\Pi,\
m+1\leq j\leq n\right\\}\leq-a_{0},\\\\[2.84526pt]
\inf\left\\{|a_{j}(x,t)-a_{k}(x,t)|\,:\,\,(x,t)\in\Pi,\ 1\leq j\neq k\leq
n\right\\}\geq a_{0}.\end{array}$ (1.2)

Now, we appropriately supplement the system (1.1) with the reflection boundary
conditions

$\begin{array}[]{ll}u_{j}(0,t)=(Ru)_{j}(t),\quad 1\leq j\leq m,\\\\[2.84526pt]
u_{j}(1,t)=(Ru)_{j}(t),\quad m<j\leq n,\end{array}$ (1.3)

where

$\displaystyle\displaystyle(Ru)_{j}(t)=\sum\limits_{k=m+1}^{n}r_{jk}u_{k}(0,t)+\sum\limits_{k=1}^{m}r_{jk}u_{k}(1,t),\quad
j\leq n,$ (1.4)

and $r_{jk}$ are real constants.

The purpose of the paper is to establish conditions for the existence and
uniqueness of global bounded classical ($C^{2}$-continuously differentiable)
solutions to the problem (1.1), (1.3) in the strip $\Pi.$ If the coefficients
$a,b,$ and $f$ are almost periodic in $t$ (in particular, $T$-periodic), then
the bounded solution is almost periodic (resp., $T$-periodic) also. Our main
assumptions are that the system is exponentialy stable and a number
dissipativity conditions are satisfied. It should be emphasized that in the
$T$-periodic case our solvability conditions are weaker. Our linear analysis
is an important part of our approach to nonlinear problems (see [18, 19])
describing various physical phenomena [3, 5, 6, 24].

We will use the same notation $\|\cdot\|$ for norms of vectors and matrices of
different sizes. Thus, if $A$ is an $n$-vector, then $\|A\|=\max_{j\leq
n}|A_{j}|$. If $B$ is an $k\times k$-matrix, then $\|B\|=\max_{i,j\leq
k}|B_{ij}|$.

Let $\Omega$ be a subdomain of $\Pi$ or ${\mathbb{R}}$ and let $X$ be a Banach
space. Then write $BC(\Omega;X)$ to denote the Banach space of all bounded and
continuous maps $u:\Omega\to X$, with norm

$\|u\|_{BC(\Omega;X)}=\sup\left\\{\|u(z)\|_{X}\,:\,z\in\Omega\right\\}.$

Similarly one can introduce the space $BC^{k}(\Omega;X),k=1,2$, of bounded and
$k$-times continuously differentiable functions. We also use the space
$BC^{k}_{t}(\Omega;X)$ of functions $u\in BC(\Omega;X)$ such that
$\partial_{t}u,\dots,\partial_{t}^{k}u\in BC(\Omega;X),$ with norm

$\|u\|_{BC_{t}^{k}(\Omega;X)}=\sum_{j=0}^{k}\|\partial_{t}^{j}u\|_{BC(\Omega;X)}.$

As usual, by ${\cal L}(X,Y)$ we will denote the space of linear bounded
operators from $X$ into $Y$, and write ${\cal L}(X)$ for ${\cal L}(X,X)$.
Also, by $H^{1}\left((0,1);{\mathbb{R}}^{n}\right)$ we will denote the Sobolev
space of all functions $v\in L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ whose
distributional derivatives $v^{\prime}$ belong to
$L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$.

For a given $s\in{\mathbb{R}}$, set $\Pi_{s}=\\{(x,t)\in\Pi:t\geq s\\}$. Along
with the system (1.1), in $\Pi_{s}$ consider its homogeneous version

$\partial_{t}u+a(x,t)\partial_{x}u+b(x,t)u=0,\quad x\in(0,1),$ (1.5)

and supplemet it with the boundary conditions (1.3) and the initial conditions

$\displaystyle u(x,s)=\varphi(x),\quad x\in[0,1],$ (1.6)

given at $t=s\in{\mathbb{R}}$.

We say that a function $\varphi\in C([0,1];{\mathbb{R}}^{n})$ satisfies the
zero order compatibility conditions if it fulfills the equations

$\begin{array}[]{ll}\displaystyle\varphi_{j}(0)=\sum\limits_{k=m+1}^{n}r_{jk}\varphi_{k}(0)+\sum\limits_{k=1}^{m}r_{jk}\varphi_{k}(1),\quad
1\leq j\leq m,\\\\[2.84526pt]
\displaystyle\varphi_{j}(1)=\sum\limits_{k=m+1}^{n}r_{jk}\varphi_{k}(0)+\sum\limits_{k=1}^{m}r_{jk}\varphi_{k}(1),\quad
m<j\leq n.\end{array}$ (1.7)

Our starting point is that the problem (1.5), (1.3), (1.6) generates the
evolution family on $L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ (see Theorem
1.2 below). To formulate this result, note that, if $\varphi\in
C_{0}^{1}([0,1];{\mathbb{R}}^{n})$, then the problem (1.5), (1.3), (1.6) has a
unique classical solution, see [12, Theorem 2.1].

######  Definition 1.1.

Let $\varphi\in L^{2}((0,1);{\mathbb{R}}^{n})$. A function $u\in
C\left([s,\infty);L^{2}\left((0,1);{\mathbb{R}}^{n}\right)\right)$ is called
an $L^{2}$-generalized solution to the problem (1.1), (1.3), (1.6) if, for any
sequence $\varphi^{l}\in C_{0}^{1}([0,1];{\mathbb{R}}^{n})$ converging to a
function $\varphi$ in $L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$, the sequence
$u^{l}\in C^{1}\left([0,1]\times[s,\infty);{\mathbb{R}}^{n}\right)$ of
classical solutions to (1.1), (1.3), (1.6) with $\varphi$ replaced by
$\varphi^{l}$ fulfills the convergence

$\|u(\cdot,t)-u^{l}(\cdot,t)\|_{L^{2}\left((0,1);{\mathbb{R}}^{n}\right)}\to
0\mbox{ as }l\to\infty,$

uniformly in $t$ varying in the range $\theta\leq t\leq\theta+\theta_{0}$, for
every $\theta_{0}>0$.

######  Theorem 1.2.

[16, Theorem 2.3] Suppose that the coefficients $a_{j}$ and $b_{jk}$ of the
system (1.5) belong to $BC^{1}(\Pi)$. If the conditions (1.2) are fulfilled,
then, for given $s\in{\mathbb{R}}$ and $\varphi\in
L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$, there exists a unique
$L^{2}$-generalized solution $u:{\mathbb{R}}^{2}\to{\mathbb{R}}^{n}$ to the
problem (1.5), (1.3), (1.6). Moreover, the map

$\varphi\mapsto U(t,s)\varphi:=u(\cdot,t)$

from $L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ to itself defines a strongly
continuous, exponentially bounded evolution family $U(t,s)\in{\cal
L}\left(L^{2}\left((0,1);{\mathbb{R}}^{n}\right)\right)$, which means that

(a) $U(t,t)=I$ and $U(t,s)=U(t,r)U(r,s)$ for all $t\geq r\geq s,$

(b) the map $(t,s)\in{\mathbb{R}}^{2}\mapsto U(t,s)\varphi\in
L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ is continuous for all $t\geq s$ and
each $\varphi\in L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$,

(c) there exist $K\geq 1$ and $\nu\in{\mathbb{R}}$ such that

$\|U(t,s)\|_{{\cal
L}\left(L^{2}\left((0,1);{\mathbb{R}}^{n}\right)\right)}\leq
Ke^{\nu(t-s)}\mbox{ for all }t\geq s.$

Next theorem describes regularity properties of the evolution family.

######  Theorem 1.3.

Suppose that the coefficients $a_{j}$ and $b_{jk}$ of the system (1.5) belong
to $BC^{1}(\Pi)$ and the conditions (1.2) are fulfilled. Then, for any
$\varphi\in H^{1}\left((0,1);{\mathbb{R}}^{n}\right)$ satisfying the zero-
order compatibility conditions (1.7), the following is true:

(a) The function $U(t,s)\varphi$ belongs to
$H^{1}\left((0,1);{\mathbb{R}}^{n}\right)$.

(b) The map $(t,s)\in{\mathbb{R}}^{2}\mapsto U(t,s)\varphi\in
L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ is continuously differentiable for
all $t\geq s$. Furthermore,

$U(\cdot,s)\varphi\in C([s,T];H^{1}((0,1);\mathbb{R}^{n}))\cap
C^{1}([s,T];L^{2}((0,1);\mathbb{R}^{n})).$ (1.8)

To formulate our main result, we need to first define characteristic curves.
For given $j\leq n$, $x\in[0,1]$, and $t\in{\mathbb{R}}$, the $j$-th
characteristic of (1.5) passing through the point $(x,t)\in\Pi$ is defined as
the solution

$\xi\in[0,1]\mapsto\omega_{j}(\xi)=\omega_{j}(\xi,x,t)\in{\mathbb{R}}$

of the initial value problem

$\partial_{\xi}\omega_{j}(\xi,x,t)=\frac{1}{a_{j}(\xi,\omega_{j}(\xi,x,t))},\;\;\omega_{j}(x,x,t)=t.$

Due to the assumption (1.2), the characteristic curve
$\tau=\omega_{j}(\xi,x,t)$ reaches the boundary of $\Pi$ in two points with
distinct ordinates. Let $x_{j}$ denote the abscissa of that point whose
ordinate is smaller. Notice that the value of $x_{j}$ does not depend on
$x,t$, and it holds

$x_{j}=\left\\{\begin{array}[]{rl}0&\mbox{if}\ 1\leq j\leq m\\\ 1&\mbox{if}\
m<j\leq n.\end{array}\right.$

For $i=0,1,2$, we will use the notation

$\displaystyle
c_{j}^{i}(\xi,x,t)=\exp\int_{x}^{\xi}\left[\frac{b_{jj}}{a_{j}}-i\frac{\partial_{t}a_{j}}{a_{j}^{2}}\right](\eta,\omega_{j}(\eta))\,d\eta,\quad
d_{j}^{i}(\xi,x,t)=\frac{c_{j}^{i}(\xi,x,t)}{a_{j}(\xi,\omega_{j}(\xi))}.$

We write $c_{j}(\xi,x,t)$ and $d_{j}(\xi,x,t)$ for $c_{j}^{0}(\xi,x,t)$ and
$d_{j}^{0}(\xi,x,t)$ respectively. Introduce operators
$G_{0},G_{1},G_{2}\in{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))$ by

$\displaystyle[G_{i}\psi]_{j}(t)=c_{j}^{i}(x_{j},1-x_{j},t)(R\psi)_{j}(\omega_{j}(x_{j},1-x_{j},t)),\
\ j\leq n,\ \ i=0,1,2.$ (1.9)

A continuous function $w(x,t)$ defined on $[0,1]\times{\mathbb{R}}$ is Bohr
almost periodic in $t$ uniformly in $x$ if for every $\mu>0$ there exists a
relatively dense set of $\mu$-almost periods of $w$, i.e., for every $\mu>0$
there exists a positive number $l$ such that every interval of length $l$ on
${\mathbb{R}}$ contains a number $\tau$ such that

$|w(x,t+\tau)-w(x,t)|<\mu\ \ \mbox{ for all }(x,t)\in\Pi;$

see, e.g. [4, p. 55]. Let $AP({\mathbb{R}};\mathbb{R}^{n})$ be the space of
Bohr almost periodic vector-functions $u:{\mathbb{R}}\to{\mathbb{R}}^{n}$.
Analogously, let $AP(\Pi;\mathbb{R}^{n})$ be the space of Bohr almost periodic
functions in $t$ uniformly in $x\in[0,1]$. If $n=1$, then we will simply write
$AP({\mathbb{R}})$ for $AP(\Pi)$ and similarly for all other spaces introduced
above.

To state our main result, we need a notion of bounded classical solution.

######  Definition 1.4.

1. A function $u\in BC^{1}(\Pi;{\mathbb{R}}^{n})$ is called a bounded classical solution to (1.1), (1.3) in $\Pi$ if it satisfies (1.1), (1.3) pointwise.
2. A continuous function $u:\Pi_{s}\to{\mathbb{R}}^{n}$, which is continuously differentiable in $\Pi_{s}$ excepting at most countable number of characteristic curves of (1.1), and satisfies the problem (1.1), (1.3), (1.6) almost everywhere in $\Pi_{s}$ (excepting just mentioned characteristic curves), will be called a piecewise continuously differentiable solution. (Notice that the derivatives of a piecewise continuously differentiable solution restricted to a compact subset of $\Pi_{s}$ have not more than finite number of discontinuities of first order on certain characteristic curves.)

######  Theorem 1.5.

${\bf(i)}$ Assume that $a_{j},f_{j}\in
BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))$ and $b_{jk}\in
BC_{t}^{1}(\Pi)$ for all $j,k\leq n$. Let the evolution family $U(t,s)$
generated by the problem (1.5), (1.3) satisfy the exponential stability
estimates

$\displaystyle\|U(t,s)\varphi\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq
Me^{-\alpha(t-s)}\|\varphi\|_{L^{2}((0,1);{\mathbb{R}}^{n})},\quad t\geq s,$
(1.10) $\displaystyle\|U(t,s)\psi\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\leq
Me^{-\alpha(t-s)}\|\psi\|_{H^{1}((0,1);{\mathbb{R}}^{n})},\quad t\geq s,$
(1.11)

for all $\varphi\in L^{2}((0,1);{\mathbb{R}}^{n})$ and $\psi\in
H^{1}((0,1);{\mathbb{R}}^{n})$ fulfilling the zero order compatibility
conditions (1.7), and for some constants $M\geq 1$ and $\alpha>0$ not
depending on $\varphi$ and $\psi$. Moreover, assume that the conditions

$\displaystyle\|G_{i}\|_{{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))}<1$
(1.12)

are fulfilled for $i=0$ and $i=1$. Then the system (1.1), (1.3) has a unique
bounded classical solution $u^{*}(x,t)$. Moreover, the a priori estimate

$\|u^{*}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\leq
L_{1}\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}$ (1.13)

is fulfilled for a constant $L_{1}$ not depending on $f.$

${\bf(ii)}$ Assume, additionally, that $a_{j},b_{jk}\in BC_{t}^{2}(\Pi)$ and
$f_{j}\in BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))$ for all $j,k\leq
n$, and the condition (1.12) is satisfied for $i=2$. Then $u^{*}\in
BC^{2}_{t}(\Pi;\mathbb{R}^{n})$ and $\partial_{x}u^{*}\in
BC^{1}_{t}(\Pi;\mathbb{R}^{n})$. Moreover, the a priori estimate

$\|u^{*}\|_{BC^{2}(\Pi;{\mathbb{R}}^{n})}\leq
L_{2}\left(\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;\mathbb{R}^{n}\right)}\right).$
(1.14)

is fulfilled for a constant $L_{2}$ not depending on $f.$

${\bf(iii)}$ Assume that $a_{i},b_{ij},f_{i}\in AP(\Pi)$. Then the bounded
classical solution $u^{*}$ belongs to $AP(\Pi;{\mathbb{R}}^{n})$.

##  2 Proof of Theorem 1.3

The proof will follow from a sequence of Lemmas 2.1–2.3 below. In fact, those
Lemmas cover not only homogeneous but also nonhomogeneous case, which will be
used later on.

######  Lemma 2.1.

[12] Suppose that the functions $a_{j}$, $b_{jk}$ and $f_{j}$ of the system
(1.1) are continuous functions in both variables and are continuously
differentiable in $x$. Moreover, suppose that the condition (1.2) is
fulfilled. Let the initial function $\varphi$ belongs to
$C^{1}([0,1];{\mathbb{R}}^{n})$ and fulfills the zero order compatibility
conditions (1.7), Then the problem (1.1), (1.3), (1.6) has a unique piecewise
continuously differentiable solution in $\Pi_{s}$.

######  Lemma 2.2.

[19, Lemma 4.2] Let $a_{j}$, $b_{jk},$ $f_{j}\in BC^{1}(\Pi)$ and
$s\in{\mathbb{R}}$ be arbitrarily fixed. Moreover, the initial function
$\varphi$ belongs to $C^{1}([0,1];{\mathbb{R}}^{n})$ and fulfills the zero
order compatibility conditions (1.7). Then there exist constants $K_{1}$ and
$\nu_{1}$ such that the piecewise continuously differentiable solution $u$ to
the problem (1.1), (1.3), (1.6) fulfills the estimate

$\begin{array}[]{cc}\|u(\cdot,t)\|_{H^{1}\left((0,1);{\mathbb{R}}^{n}\right)}+\|\partial_{t}u(\cdot,t)\|_{L^{2}\left((0,1);{\mathbb{R}}^{n}\right)}\\\
\leq
K_{1}e^{\nu_{1}(t-s)}\left(\|\varphi\|_{H^{1}((0,1);{\mathbb{R}}^{n})}+\sup\limits_{t\in{\mathbb{R}}}\|f(\cdot,t)\|_{L^{2}\left((0,1);{\mathbb{R}}^{n}\right)}+\sup\limits_{t\in{\mathbb{R}}}\|\partial_{t}f(\cdot,t)\|_{L^{2}\left((0,1);{\mathbb{R}}^{n}\right)}\right)\end{array}$
(2.1)

for all $t\geq s$.

######  Lemma 2.3.

Let $a_{j}$, $b_{jk},$ $f_{j}\in BC^{1}(\Pi)$ and $s\in{\mathbb{R}}$ be
arbitrarily fixed. Moreover, let $\varphi$ belong to
$H^{1}\left((0,1);{\mathbb{R}}^{n}\right)$ and satisfy the zero order
compatibility conditions (1.7). Then the $L^{2}$-generalized solution $u$ to
the problem (1.1), (1.3), (1.6) belongs to
$C([s,t];H^{1}\left((0,1);{\mathbb{R}}^{n}\right))$ and to
$C^{1}([s,t];L^{2}\left((0,1);{\mathbb{R}}^{n}\right))$ for any $t\geq s$.

Proof. Define $\widetilde{\varphi}(x)=\varphi(0)+x(\varphi(1)-\varphi(0))$ for
$x\in[0,1].$ Note that $\varphi-\widetilde{\varphi}\in
H^{1}_{0}((0,1);\mathbb{R}^{n})$, see [7, p. 259]. Therefore, there exists a
sequence $\varphi^{l}_{0}\in C_{0}^{\infty}([0,1];{\mathbb{R}}^{n})$
approaching $\varphi-\widetilde{\varphi}$ in $H^{1}((0,1);{\mathbb{R}}^{n})$.
It follows that the sequence $\varphi^{l}=\varphi^{l}_{0}+\widetilde{\varphi}$
approaches $\varphi$ in $H^{1}((0,1);{\mathbb{R}}^{n})$.

By Lemma 2.2, the piecewise continuously differentiable solution $u^{l}$ to
the problem (1.1), (1.3), (1.6) with $\varphi^{l}$ in place of $\varphi$
satisfies the estimate (2.1) with $u=u^{l}$ and $\varphi=\varphi^{l}$. This
entails the convergence

$\max\limits_{s\leq\tau\leq
t}\|u^{m}(\cdot,\tau)-u^{l}(\cdot,\tau)\|_{H^{1}\left((0,1);{\mathbb{R}}^{n}\right)}+\max\limits_{s\leq\tau\leq
t}\|\partial_{t}u^{m}(\cdot,\tau)-\partial_{t}u^{l}(\cdot,\tau)\|_{L^{2}\left((0,1);{\mathbb{R}}^{n}\right)}\to
0$

as $m,l\to\infty$ for each $t>s$. Consequently, the sequence $u^{l}$ converges
in $C([s,t];H^{1}\left((0,1);{\mathbb{R}}^{n}\right))\cap
C^{1}([s,t];L^{2}\left((0,1);{\mathbb{R}}^{n}\right))$ and, hence $u\in
C([s,t];H^{1}\left((0,1);{\mathbb{R}}^{n}\right))\cap
C^{1}([s,t];L^{2}\left((0,1);{\mathbb{R}}^{n}\right))$, which finishes the
proof of the lemma. $\Box$

Finally, let us consider the homogeneous problem, namely the problem (1.5),
(1.3), (1.6). For given $\varphi\in H^{1}\left((0,1);{\mathbb{R}}^{n}\right)$
satisfying the zero order compatibility conditions (1.7), let $\varphi^{l}$ be
an arbitrary fixed sequence from $C^{1}([0,1];{\mathbb{R}}^{n})$ satisfying
the zero order compatibility conditions (1.7) and approaching $\varphi$ in
$C^{1}([0,1];{\mathbb{R}}^{n})$. Let, for any $l\in{\mathbb{N}}$, $u^{l}$ be
the piecewise continuously differentiable solution to the problem (1.5),
(1.3), (1.6) with $\varphi=\varphi^{l}$. Then passing to the limit as
$l\to\infty$ implies that the limit function $u$ belongs to
$C([s,t],H^{1}\left((0,1);{\mathbb{R}}^{n}\right))\cap
C^{1}([s,t],L^{2}\left((0,1);{\mathbb{R}}^{n}\right))$ and satisfies (1.5)
almost everywhere and (1.3) and (1.6) pointwise. On account of the definition
of $U(t,s)$, this means that (1.8) is true and, furthermore, that
$U(t,s)\varphi$ satisfies the system (1.5) in
$L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ and the conditions (1.3) and (1.6)
pointwise.

##  3 Proof of Theorem 1.5

###  3.1 Abstract problem setting

Now we rewrite the problem (1.1), (1.3), (1.6) as an abstract evolution
equation in $L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$. To this end, define a
one-parameter family of operators ${\cal A}(t)$ (with the parameter
$t\in{\mathbb{R}}$) from $L^{2}\left((0,1);{\mathbb{R}}^{n}\right)$ into
itself, by

$[{\cal A}(t)u](x)=\left(-a(x,t)\frac{\partial}{\partial x}-b(x,t)\right)u,$

with the domain

$\begin{array}[]{ll}\displaystyle D({\cal A}(t))=\left\\{u\in
H^{1}\left((0,1);{\mathbb{R}}^{n}\right)\,:\,u_{j}(x_{j},t)=(Ru)_{j}(t),j\leq
n\right\\}\subset L^{2}\left((0,1);{\mathbb{R}}^{n}\right).\end{array}$

Note that $D({\cal A}(t))=D({\cal A})$ is independent of $t$.

Writing $u(t)$ and $f(t)$, we mean bounded and continuous maps
$u:{\mathbb{R}}\to L^{2}((0,1);{\mathbb{R}}^{n})$ and $f:{\mathbb{R}}\to
L^{2}((0,1);{\mathbb{R}}^{n})$ defined by $[u(t)](x)=u(x,t)$ and
$[f(t)](x)=f(x,t)$, respectively. In this notation, the abstract version of
the problem (1.1), (1.3), (1.6) reads

$\displaystyle\frac{d}{dt}u={\cal A}(t)u+f(t),$ (3.1) $\displaystyle
u(s)=\varphi\in L^{2}((0,1);{\mathbb{R}}^{n}).$ (3.2)

######  Definition 3.1.

1. A function $u\in C\left([s,\infty),L^{2}((0,1);{\mathbb{R}}^{n})\right)$ is called a classical solution to the abstract problem (3.1)–(3.2) if $u$ is continuously differentiable in $L^{2}((0,1);{\mathbb{R}}^{n})$ for $t>s$, $u(t)\in D({\cal A})$ for $t>s$, and (3.1)–(3.2) is satisfied in $L^{2}((0,1);{\mathbb{R}}^{n})$.
2. A function $u\in BC\left({\mathbb{R}},L^{2}((0,1);{\mathbb{R}}^{n})\right)$ is called a bounded classical solution to the abstract equation (3.1) if $u$ is continuously differentiable in $t$ with values in $L^{2}((0,1);{\mathbb{R}}^{n})$, $u(t)\in D({\cal A})$ for all $t\in{\mathbb{R}}$, and (3.1) is satisfied in $L^{2}((0,1);{\mathbb{R}}^{n})$.

We will use equivalence between the original and the abstract problem settings
given by the next theorem.

######  Theorem 3.2.

[19, Theorem 4.1] Suppose that $a_{j},b_{jk},f_{j}\in BC^{1}(\Pi)$ and the
condition (1.2) is fulfilled. If $\varphi$ belongs to $D({\cal A})$ and
$u(x,t)$ is the $L^{2}$-generalized solution to the problem (1.1), (1.3),
(1.6), then the function $u(t)$ such that $[u(t)](x):=u(x,t)$, is a classical
solution to the abstract problem (3.1). Vice versa, if $u(t)$ is a classical
solution to the abstract problem (3.1), then $u(x,t):=[u(t)](x)$ is an
$L^{2}$-generalized solution to the problem (1.1), (1.3), (1.6).

######  Theorem 3.3.

Assume that $a_{j},b_{jk},f_{j}\in BC^{1}(\Pi)$ for all $j,k\leq n$ and the
evolution operator $U(t,s)$ satisfies the exponential estimate (1.10). Then
the equation (3.1) has a unique bounded classical solution

$\displaystyle v^{*}(t)=\int_{-\infty}^{t}U(t,s)f(s)\\!\;\mathrm{d}s.$ (3.3)

Moreover, the following apriori estimate is fulfilled:

$\displaystyle\|v^{*}\|_{BC\left({\mathbb{R}};L^{2}((0,1);{\mathbb{R}}^{n})\right)}\leq\frac{M}{\alpha}\|f\|_{BC\left({\mathbb{R}};L^{2}((0,1);{\mathbb{R}}^{n})\right)},$
(3.4)

where $M$ and $\alpha$ are as in (1.10).

Proof. Since the boundedness estimate (3.4) is a straightforward consequence
of the representation formula (3.3) and the exponential stability estimate
(1.10), our task is reduced to prove (3.3). We devide the proof of (3.3) into
two claims.

Claim 1. There exists a (sufficiently large) $\lambda>0$ such that, for any
$t\in{\mathbb{R}}$, the operator ${\cal A}(t)+\lambda I$ is invertible, with
the inverse $({\cal A}(t)+\lambda I)^{-1}:L^{2}((0,1);{\mathbb{R}}^{n})\mapsto
H^{1}((0,1);{\mathbb{R}}^{n})$ being continuous and uniformly bounded in
$t\in{\mathbb{R}}$. Furthermore,

$\left\|({\cal A}(t)+\lambda
I)^{-1}g\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq\frac{M_{0}}{\sqrt{\lambda}}\|g\|_{L^{2}((0,1);{\mathbb{R}}^{n})}$
(3.5)

and

$\left\|({\cal A}(t)+\lambda
I)^{-1}g\right\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\leq
M_{0}\sqrt{\lambda}\|g\|_{L^{2}((0,1);{\mathbb{R}}^{n})}$ (3.6)

for all $g\in L^{2}((0,1);{\mathbb{R}}^{n})$ and some $M_{0}>0$ independent of
$\lambda$, $t$, and $g$.

Proof of Claim. Set $u(x,t)=(u^{1}(x,t),u^{2}(x,t)),$ where

$u^{1}(x,t)=(u_{1}(x,t),\dots,u_{m}(x,t)),\quad
u^{2}(x,t)=(u_{m+1}(x,t),\dots,u_{n}(x,t)).$

Moreover, set

$R_{11}=(r_{jk})_{j,k=1}^{m},\ R_{12}=(r_{jk})_{j=1,k=m+1}^{m,n},\
R_{21}=(r_{jk})_{j=m+1,k=1}^{n,m},\ R_{22}=(r_{jk})_{j,k=m+1}^{n},$

where $r_{jk}$ are the reflection boundary coefficients in (1.3).

We now prove that there exists $\lambda>0$ such that, for each
$t\in{\mathbb{R}}$ and $g\in L^{2}((0,1);{\mathbb{R}}^{n})$, the linear ODE
system

$\displaystyle\frac{du}{dx}+a^{-1}(x,t)(b(x,t)+\lambda I)u=g(x)$ (3.7)

with the boundary conditions

$\displaystyle u^{1}(0,t)=R_{11}u^{1}(1,t)+R_{12}u^{2}(0,t),\quad
u^{2}(1,t)=R_{21}u^{1}(1,t)+R_{22}u^{2}(0,t)$ (3.8)

has a unique solution $u(\cdot,t)\in H^{1}((0,1);{\mathbb{R}}^{n})$, which is
continuous and uniformly bounded in the parameter $t$. Rewrite (3.7) in the
form of an operator equation in $L^{2}((0,1);{\mathbb{R}}^{n})$, namely

$\displaystyle{\cal A}_{1}(t)u+{\cal A}_{2}(t)u=g,$

where

$\begin{array}[]{cc}[{\cal A}_{1}(t)u](x)=\displaystyle\frac{du}{dx}+\lambda
a^{-1}(x,t)u,\\\\[8.53581pt] D({\cal A}_{1})=\left\\{v\in
H^{1}((0,1);{\mathbb{R}}^{n})\,:\,v^{1}(0)=R_{11}v^{1}(1)+R_{12}v^{2}(0),\
v^{2}(1)=R_{21}v^{1}(1)+R_{22}v^{2}(0)\right\\}\end{array}$

and

$\begin{array}[]{cc}[{\cal A}_{2}(t)u](x)=a^{-1}(x,t)b(x,t)u,\\\\[8.53581pt]
D({\cal A}_{2})=L^{2}((0,1);{\mathbb{R}}^{n}).\end{array}$

To prove the solvability of (3.7)–(3.8), we first prove the invertibility of
${\cal A}_{1}$. To this end, take arbitrary $g=(g^{1},g^{2})$ with $g^{1}\in
L^{2}((0,1);{\mathbb{R}}^{m})$ and $g^{2}\in L^{2}((0,1);{\mathbb{R}}^{n-m})$
and consider the two decoupled systems

$\displaystyle\frac{du^{1}}{dx}=\lambda
A_{1}(x,t)u^{1}+g^{1}(x)\quad\mbox{and}\quad\frac{du^{2}}{dx}=\lambda
A_{2}(x,t)u^{2}+g^{2}(x),$

where $A_{1}=-\mathop{\rm diag}\nolimits(a_{1}^{-1},\dots,a_{m}^{-1}),\
A_{2}=-\mathop{\rm diag}\nolimits(a_{m+1}^{-1},\dots,a_{n}^{-1}).$ Let

$V_{1}(x,\xi;t,\lambda)=\exp\int_{\xi}^{x}\lambda
A_{1}(\xi,t)\\!\;\mathrm{d}\xi,\quad
V_{2}(x,\xi;t,\lambda)=\exp\int_{\xi}^{x}\lambda
A_{2}(\xi,t)\\!\;\mathrm{d}\xi$

be fundamental matrices of the corresponding homogeneous systems,
respectively. Notice that $V_{j}$ are continuous in $t$. This yields the
solution formulas

$\displaystyle\begin{array}[]{ll}\displaystyle
u^{1}(x;t,\lambda)=V_{1}(x,0;t,\lambda)u^{1}(0;t,\lambda)+\int_{0}^{x}V_{1}(x,\xi;t,\lambda)g^{1}(\xi)\\!\;\mathrm{d}\xi,\\\\[8.53581pt]
\displaystyle
u^{2}(x;t,\lambda)=V_{2}(x,1;t,\lambda)u^{2}(1;t,\lambda)+\int_{1}^{x}V_{2}(x,\xi;t,\lambda)g^{2}(\xi)\\!\;\mathrm{d}\xi.\end{array}$
(3.11)

Since $a_{j}>0$ for $j\leq m$ and $a_{j}<0$ for $j>m,$ we have

$\displaystyle\|V_{1}(x,\xi;t,\lambda)\|\leq
e^{-a\lambda(x-\xi)}\quad\mbox{and}\quad\|V_{2}(x,\xi;t,\lambda)\|\leq
e^{a\lambda(x-\xi)}\quad\mbox{for}\ x\geq\xi,$ (3.12)

where the constant $a$ is positive and is independent of $\lambda$ and $t$. By
(3.11),

$\displaystyle u^{1}(1)=V_{1}u^{1}(0)+J_{1},\quad
u^{2}(0)=V_{2}u^{2}(1)+J_{2},$ (3.13)

where we use the following notation: $u^{1}(x)=u^{1}(x;t,\lambda)$,
$u^{2}(x)=u^{2}(x;t,\lambda)$, $V_{1}=V_{1}(1,0;t,\lambda),$
$V_{2}=V_{2}(0,1;t,\lambda)$,

$J_{1}=\int_{0}^{1}V_{1}(1,\xi;t,\lambda)g^{1}(\xi)\\!\;\mathrm{d}\xi,\quad
J_{2}=-\int_{0}^{1}V_{2}(0,\xi;t,\lambda)g^{2}(\xi)\\!\;\mathrm{d}\xi.$

Substituting (3.13) into (3.8) implies the equalities

$\begin{array}[]{rcl}(I-R_{11}V_{1})u^{1}(0)-R_{12}V_{2}u^{2}(1)&=&R_{11}J_{1}+R_{12}J_{2},\\\\[5.69054pt]
-R_{21}V_{1}u^{1}(0)+(I-R_{22}V_{2})u^{2}(1)&=&R_{21}J_{1}+R_{22}J_{2}.\end{array}$
(3.14)

Note that, due to (3.12), it holds

$\|V_{j}\|\leq
e^{-a\lambda}\quad\mbox{and}\quad\|J_{j}\|\leq\frac{1}{\sqrt{2a\lambda}}\|g\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\quad\mbox{for
}j=1,2,$ (3.15)

the bounds being uniform in $t$. We, therefore, can fix $\lambda>0$ such that
the operator $I-R_{11}V_{1}$ is invertible, and the following estimate is
fulfilled:

$\displaystyle\|(I-R_{11}V_{1})^{-1}\|\leq(1-\|R_{11}\|e^{-a\lambda})^{-1}.$

Hence,

$u^{1}(0)=(I-R_{11}V_{1})^{-1}R_{12}V_{2}u^{2}(1)+(I-R_{11}V_{1})^{-1}(R_{11}J_{1}+R_{12}J_{2})$

and, therefore,

$\begin{array}[]{ll}\left[(I-R_{22}V_{2}-R_{21}V_{1}(I-R_{11}V_{1})^{-1}R_{12}V_{2}\right]u^{2}(1)\\\\[2.84526pt]
\hskip
28.45274pt=R_{21}J_{1}+R_{22}J_{2}+R_{21}V_{1}(I-R_{11}V_{1})^{-1}(R_{11}J_{1}+R_{12}J_{2}).\end{array}$

Because of (3.15), we can fix (probably new) sufficiently large $\lambda$ such
that the matrix $(I-R_{22}V_{2}-R_{21}V_{1}(I-R_{11}V_{1})^{-1}R_{12}V_{2}$ is
invertible and there exists a constant $M_{1}>0$ fulfilling the estimate

$\|\left[(I-R_{22}V_{2}-R_{21}V_{1}(I-R_{11}V_{1})^{-1}R_{12}V_{2}\right]^{-1}\|\leq
M_{1},$

uniformly in $t\in{\mathbb{R}}$. Therefore, for every $g$ there exists a
unique solution $(u^{1}(0),u^{2}(1))$ to the system (3.14), which is
continuous in $t$ and, moreover,

$\|u_{1}(0)\|+\|u_{2}(1)\|\leq\frac{M_{2}}{\sqrt{\lambda}}\|g\|_{L^{2}((0,1);{\mathbb{R}}^{n})},$

the constant $M_{2}$ being independent of $t$ and $\lambda$.

Therefore, the operator ${\cal A}_{1}$ is invertible and, by (3.11), ${\cal
A}_{1}^{-1}g$ belongs to $H^{1}((0,1);{\mathbb{R}}^{n})$ and is a continuous
function in the parameter $t$. Moreover, there exists $M_{3}>0$ such that
$\|{\cal A}_{1}^{-1}g\|_{{\cal
L}(L^{2}((0,1);{\mathbb{R}}^{n}))}\leq\frac{M_{3}}{\sqrt{\lambda}}\|g\|_{L^{2}((0,1);{\mathbb{R}}^{n})}$.
Furthermore, since the operator ${\cal A}_{2}$ is bounded and does not depend
on $\lambda$, we have $\|{\cal A}_{1}^{-1}{\cal A}_{2}\|_{{\cal
L}(L^{2}((0,1);{\mathbb{R}}^{n}))}\leq\frac{M_{4}}{\sqrt{\lambda}}<1$ for some
$M_{4}>0$ and for sufficiently large $\lambda.$ Summarizing, there exists a
constant $M_{0}>0$ independent of $\lambda$ and $t$ such that

$\begin{array}[]{ll}\|({\cal A}_{1}+{\cal A}_{2})^{-1}\|_{{\cal
L}(L^{2}((0,1);{\mathbb{R}}^{n}))}\leq\|(I+{\cal A}_{1}^{-1}{\cal
A}_{2})^{-1}{\cal A}_{1}^{-1}\|_{{\cal L}(L^{2}((0,1);{\mathbb{R}}^{n}))}\\\
\displaystyle\qquad\leq\|{\cal A}_{1}^{-1}\|_{{\cal
L}(L^{2}((0,1);{\mathbb{R}}^{n}))}\sum_{j=0}^{\infty}\|{\cal A}_{1}^{-1}{\cal
A}_{2}\|_{{\cal
L}(L^{2}((0,1);{\mathbb{R}}^{n}))}^{j}\leq\frac{M_{0}}{\sqrt{\lambda}}.\end{array}$

To finish the proof of this claim, note that the estimate (3.6) now easily
follows from (3.7). ∎

Claim 2. The function $v^{*}$ given by (3.3) is a classical solution to the
abstract equation (3.1).

Proof of Claim. Note that the integral in (3.3) exists due to the continuity
of $U(t,s)f(s)$ in $s$ and the exponential bound (1.10).

On account of Claim 1, we can rewrite (3.3) in the form

$\displaystyle v^{*}(t)$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{t}U(t,s)({\cal{A}}(s)+\lambda
I)({\cal{A}}(s)+\lambda I)^{-1}f(s)\\!\;\mathrm{d}s$ (3.16) $\displaystyle=$
$\displaystyle\int_{-\infty}^{t}U(t,s){\cal{A}}(s)({\cal{A}}(s)+\lambda
I)^{-1}f(s)\\!\;\mathrm{d}s+\lambda\int_{-\infty}^{t}U(t,s)({\cal{A}}(s)+\lambda
I)^{-1}f(s)\\!\;\mathrm{d}s$ $\displaystyle=$
$\displaystyle-\int_{-\infty}^{t}\frac{\partial U(t,s)}{\partial
s}({\cal{A}}(s)+\lambda
I)^{-1}f(s)\\!\;\mathrm{d}s+\lambda\int_{-\infty}^{t}U(t,s)({\cal{A}}(s)+\lambda
I)^{-1}f(s)\\!\;\mathrm{d}s$ $\displaystyle=$
$\displaystyle-U(t,s)({\cal{A}}(s)+\lambda
I)^{-1}f(s)\biggl{|}_{s=-\infty}^{t}+\int_{-\infty}^{t}U(t,s)\frac{\partial}{\partial
s}\left[({\cal{A}}(s)+\lambda I)^{-1}f(s)\right]\\!\;\mathrm{d}s$
$\displaystyle+\lambda\int_{-\infty}^{t}U(t,s)({\cal{A}}(s)+\lambda
I)^{-1}f(s)\\!\;\mathrm{d}s.$

Here we used the property that, if $u\in D({\cal A})$, then

$\frac{\partial}{\partial s}U(t,s)u(s)=-U(t,s){\cal A}(s)u(s).$

Since

$\frac{\partial}{\partial s}\left[({\cal A}(s)+\lambda
I)^{-1}f(s)\right]=-({\cal{A}}(s)+\lambda
I)^{-1}\left(\frac{d{\cal{A}}(s)}{ds}({\cal{A}}(s)+\lambda
I)^{-1}f(s)-f^{\prime}(s)\right)$ (3.17)

and the function $\left(\frac{d{\cal A}(s)}{ds}({\cal{A}}(s)+\lambda
I)^{-1}f(s)-f^{\prime}(s)\right)$ is continuous and bounded in $s$, the
function $\frac{\partial}{\partial s}\left[({\cal A}(s)+\lambda
I)^{-1}f(s)\right]$ is also continuous and bounded in $s$, with values in
$D({\cal A})$. Further, taking into account the exponential stability estimate
(1.10), the function $\int_{-\infty}^{t}\frac{\partial U(t,s)}{\partial
s}({\cal{A}}(s)+\lambda I)^{-1}f(s)ds$ is continuously differentiable in $t$,
like in the proof of [20, Theorem 3.2] . The last integral in (3.16) is
continuously differentiable in $t$ as well, by the same argument.

It remains to show that $v^{*}$ satisfies (3.1) in the classical sense. To
this end, set

$\widetilde{f}(s)=-({\cal{A}}(s)+\lambda
I)^{-1}\left(\frac{d{\cal{A}}(s)}{ds}({\cal{A}}(s)+\lambda
I)^{-1}f(s)-f^{\prime}(s)-\lambda f(s)\right).$ (3.18)

By (3.16), the function $v^{*}(t)$ reads

$v^{*}(t)=-({\cal{A}}(t)+\lambda
I)^{-1}f(t)+\int_{-\infty}^{t}U(t,s)\widetilde{f}(s)ds.$ (3.19)

As $\widetilde{f}(s)\in D({\cal A})$, then the right hand side of (3.3) is
continuously differentiable and, due to (3.17), we compute

$\displaystyle\begin{array}[]{rcl}\displaystyle\frac{dv^{*}(t)}{dt}&=&\displaystyle({\cal{A}}(t)+\lambda
I)^{-1}\left(\frac{d{\cal{A}}(t)}{dt}({\cal{A}}(t)+\lambda
I)^{-1}f(t)-f^{\prime}(t)\right)+\widetilde{f}(t)+\int_{-\infty}^{t}\frac{\partial
U(t,s)}{\partial t}\widetilde{f}(s)\\!\;\mathrm{d}s\\\
&=&\displaystyle\lambda({\cal{A}}(s)+\lambda
I)^{-1}f(t)+\int_{-\infty}^{t}\frac{\partial U(t,s)}{\partial
t}\widetilde{f}(s)\\!\;\mathrm{d}s.\end{array}$

On the other hand,

$\begin{array}[]{rcl}{\cal A}(t)v^{*}(t)+f(t)&=&\displaystyle-{\cal
A}(t)({\cal{A}}(t)+\lambda I)^{-1}f(t)+{\cal
A}(t)\int_{-\infty}^{t}U(t,s)\widetilde{f}(s)ds+f(t)\\\
&=&\displaystyle\lambda({\cal{A}}(s)+\lambda
I)^{-1}f(t)+\int_{-\infty}^{t}\frac{\partial U(t,s)}{\partial
t}\widetilde{f}(s)\\!\;\mathrm{d}s,\end{array}$

as desired. ∎

The proof of Theorem 3.3 is complete. $\Box$

###  3.2 Proof of Part ${\bf(i)}$: $BC$-solution

First show that the solution $v^{*}$ given by (3.3) belongs to
$BC({\mathbb{R}},H^{1}((0,1);{\mathbb{R}}^{n}))$. Since all the coefficients
of the initial system and their first order derivatives are globally bounded
in $\Pi$, on account of (3.5), there exists $M_{5}>0$ such that

$\|\widetilde{f}\|_{BC({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n}))}\leq
M_{5}\|f\|_{BC^{1}\left({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n})\right)}.$
(3.21)

Introduce operator ${\cal A}_{d}:H^{1}((0,1);{\mathbb{R}}^{n})\to
L^{2}((0,1);{\mathbb{R}}^{n})$ by

${\cal
A}_{d}u=\left(\partial_{x}u_{1},\partial_{x}u_{2},\dots,\partial_{x}u_{n}\right)$

and apply it to both sides of (3.19). The resulting equation reads

${\cal A}_{d}v^{*}(t)=-{\cal A}_{d}({\cal{A}}(t)+\lambda
I)^{-1}f(t)+\int_{-\infty}^{t}{\cal
A}_{d}U(t,s)\widetilde{f}(s)\\!\;\mathrm{d}s.$ (3.22)

The first summand in the right-hand side can easily be estimated by using
(3.6), namely

$\begin{array}[]{ll}\|{\cal A}_{d}({\cal{A}}(t)+\lambda
I)^{-1}f(t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\\\[8.53581pt] \qquad\leq\|{\cal
A}_{d}\|_{{\cal L}(H^{1},L^{2})}\left\|({\cal A}(t)+\lambda
I)^{-1}\right\|_{{\cal
L}(L^{2},H^{1})}\|f(t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq
M_{6}\|f(t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\end{array}$ (3.23)

for some $M_{6}$ not depending on $t$. For the second summand, we use (3.21)
and the assumption about the uniform exponential stability of $U$ in $H^{1}$
and, therefore, get the estimate

$\begin{array}[]{ll}\displaystyle\int_{-\infty}^{t}\left\|{\cal
A}_{d}U(t,s)\widetilde{f}(s)\right\|_{L^{2}(\Pi;{\mathbb{R}}^{n})}\\!\;\mathrm{d}s\leq\displaystyle\int_{-\infty}^{t}\left\|{\cal
A}_{d}\right\|_{{\cal
L}(H^{1},L^{2})}\bigl{\|}U(t,s)\widetilde{f}(s)\bigr{\|}_{H^{1}(\Pi;{\mathbb{R}}^{n})}\\!\;\mathrm{d}s\\\
\displaystyle\qquad\leq\displaystyle
M\int_{-\infty}^{t}e^{-\alpha(t-s)}\bigl{\|}\widetilde{f}(s)\bigr{\|}_{H^{1}(\Pi;{\mathbb{R}}^{n})}\\!\;\mathrm{d}s\leq
M_{7}\|f\|_{BC^{1}\left({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n})\right)}.\end{array}$
(3.24)

Combining (3.22), (3.23), and (3.24) gives the bound

$\displaystyle\|v^{*}\|_{BC\left({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n})\right)}\leq
M_{8}\|f\|_{BC^{1}\left({\mathbb{R}};H^{1}([0,1];{\mathbb{R}}^{n})\right)},$
(3.25)

being true for some $M_{8}>0$, which does not depend on $f$. Due to the
continuous embedding of $H^{1}(0,1)$ into $C([0,1])$, we have $v^{*}\in
BC\left({\mathbb{R}};C([0,1];{\mathbb{R}}^{n})\right)$. Moreover, there exist
constants $M_{9}$ and $M_{10}$ such that

$\displaystyle\|v^{*}\|_{BC\left({\mathbb{R}};C([0,1];{\mathbb{R}}^{n})\right)}\leq
M_{9}\|v^{*}\|_{BC\left({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n})\right)}\leq
M_{8}M_{9}\|f\|_{BC^{1}\left({\mathbb{R}};H^{1}([0,1];{\mathbb{R}}^{n})\right)}.$
(3.26)

Define $u^{*}(x,t)=[v^{*}(t)](x)$. Hence, $u^{*}$ belongs to
$BC(\Pi;\mathbb{R}^{n})$. Now, since $v^{*}$ is a bounded classical solution
to (3.1), the function $u^{*}$ satisfies (1.1), (1.3) almost everywhere, the
partial derivatives in (1.1) being understood in the distributional sense and
the boundary conditions (1.3) pointwise.

Next, the function $u^{*}$ satisfies (1.1) in the distributional sense and the
boundary conditions (1.3) pointwise if and only if $u^{*}$ satisfies the
following system of integral equations:

$\displaystyle u^{*}_{j}(x,t)=c_{j}(x_{j},x,t)(Ru^{*})_{j}(\omega_{j}(x_{j}))$
$\displaystyle\
-\displaystyle\int_{x_{j}}^{x}d_{j}(\xi,x,t)\left(\sum_{k\not=j}b_{jk}(\xi,\omega_{j}(\xi))u^{*}_{k}(\xi,\omega_{j}(\xi))-f_{j}(\xi,\omega_{j}(\xi))\right)d\xi,\
j\leq n,$ (3.27)

pointwise (for details see, e.g. [18, p. 4192–4193]). In this case, $u^{*}$
will be called bounded continuous solution to (1.1), (1.3).

###  3.3 Proof of Part ${\bf(i)}$: $BC^{1}$-solution

Rewrite the system (3.2) in an operator form, as follows: First we introduce
operators $C,D,F\in{\cal L}(BC(\Pi;{\mathbb{R}}^{n}))$ by

$\displaystyle(Cu)_{j}(x,t)$ $\displaystyle=$ $\displaystyle
c_{j}(x_{j},x,t)(Ru)_{j}(\omega_{j}(x_{j},x,t)),$ $\displaystyle(Du)_{j}(x,t)$
$\displaystyle=$ $\displaystyle-\int_{x_{j}}^{x}d_{j}(\xi,x,t)\sum_{k\neq
j}b_{jk}(\xi,\omega_{j}(\xi,x,t))u_{k}(\xi,\omega_{j}(\xi,x,t))d\xi,$
$\displaystyle(Ff)_{j}(x,t)$ $\displaystyle=$
$\displaystyle\int_{x_{j}}^{x}d_{j}(\xi,x,t)f_{j}(\xi,\omega_{j}(\xi,x,t))d\xi.$

and rewrite the system (3.2) in the operator form

$\displaystyle u=Cu+Du+Ff.$ (3.28)

After iteration of this equality we get

$\displaystyle u=Cu+(DC+D^{2})u+(I+D)Ff.$ (3.29)

Next we prove that $I-C$ is a bijective operator from
$BC_{t}^{1}(\Pi;\mathbb{R}^{n})$ to itself. We are done if we show that the
system

$u_{j}(x,t)=c_{j}(x_{j},x,t)(Ru)_{j}(\omega_{j}(x_{j},x,t))+h_{j}(x,t),\quad
j\leq n,$ (3.30)

is uniquely solvable in $BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ for any $h\in
BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$. Putting $x=0$ for $m<j\leq n$ and $x=1$ for
$1\leq j\leq m$ in (3.30), we get the following system of $n$ equations with
respect to
$z(t)=(z_{1}(t),\dots,z_{n}(t))=(u_{1}(1,t),\dots,u_{m}(1,t),u_{m+1}(0,t),\dots,u_{n}(0,t))$,
with

$\begin{array}[]{ll}z_{j}(t)=c_{j}(x_{j},1-x_{j},t)(Rz)_{j}(\omega_{j}(x_{j},1-x_{j},t))+h_{j}(1-x_{j},t),\quad
j\leq n.\end{array}$ (3.31)

On the account of (1.9), the system (3.30) is uniquelly solvable in
$BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ if and only if

$I-G_{0}\mbox{ is bijective from }BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})\mbox{
to }BC^{1}({\mathbb{R}};{\mathbb{R}}^{n}).$ (3.32)

To prove (3.32), let us norm the space $BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$
with

$\|v\|_{\sigma}=\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\|\partial_{t}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})},$
(3.33)

where a positive constant $\sigma$ will be defined later. Note that the norms
(3.33) are equivalent for all $\sigma>0$.

We, therefore, have to prove that there exist constants $\sigma<1$ and
$\gamma<1$ such that

$\|G_{0}v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\left\|\frac{d}{dt}G_{0}v\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\leq\gamma\left(\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\|v^{\prime}\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\right)$

for all $v\in BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$. For given $v\in
BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$, we have

$\displaystyle\frac{d}{dt}(G_{0}v)_{j}(t)=\partial_{t}c_{j}(x_{j},1-x_{j},t)(Rv)_{j}(\omega_{j}(x_{j},1-x_{j},t))$
$\displaystyle\hskip
71.13188pt+c_{j}^{1}(x_{j},1-x_{j},t)\left[(Rv^{\prime})_{j}\right](\omega_{j}(x_{j},1-x_{j},t)),\,1\leq
j\leq n.$

Here we used the equality

$\displaystyle\partial_{t}\omega_{j}(\xi,x,t)=\exp\int_{\xi}^{x}\left[\frac{\partial_{t}a_{j}}{a_{j}^{2}}\right](\eta,\omega_{j}(\eta,x,t))d\eta$

and $c_{j}^{1}(\xi,x,t)=c_{j}(\xi,x,t)\partial_{t}\omega_{j}(\xi,x,t).$ Define
an operator $W\in\mathcal{L}(BC(\mathbb{R};\mathbb{R}^{n}))$ by

$(Wv)_{j}(t)=\partial_{t}c_{j}(x_{j},1-x_{j},t)(Rv)_{j}(\omega_{j}(x_{j},1-x_{j},t)),\quad
j\leq n.$

Due to the assumption (1.12) for $i=0$, we have $\left\|G_{0}\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}<1$. Hence, we can fix $\sigma<1$ such that
$\left\|G_{0}\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}+\sigma\left\|W\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}<1$. Set

$\gamma=\max\left\\{\left\|G_{0}\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}+\sigma\left\|W\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))},\left\|G_{1}\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}\right\\}.$

Due to the assumption (1.12) for $i=1$, $\left\|G_{1}\right\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}<1$. This yields that $\gamma<1$. It follows
that

$\begin{array}[]{rcl}\|G_{0}v\|_{\sigma}&\leq&\left\|G_{0}v\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\|Wv\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\left\|G_{1}v^{\prime}\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\\\
&\leq&\gamma\left(\|v\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}+\sigma\left\|v^{\prime}\right\|_{BC({\mathbb{R}};{\mathbb{R}}^{n})}\right)=\gamma\|v\|_{\sigma}.\end{array}$

Furthermore, $\|(I-G_{0})^{-1}v\|_{\sigma}\leq(1-\gamma)^{-1}\|v\|_{\sigma}$
and, hence

$\|(I-G_{0})^{-1}v\|_{BC_{t}^{1}({\mathbb{R}};{\mathbb{R}}^{n})}\leq\frac{1}{\sigma}\|(I-G_{0})^{-1}v\|_{\sigma}\leq\frac{1}{\sigma(1-\gamma)}\|v\|_{\sigma}\leq\frac{1}{\sigma(1-\gamma)}\|v\|_{BC_{t}^{1}({\mathbb{R}};{\mathbb{R}}^{n})}.$
(3.34)

Now, the system (3.31) can be rewritten in the form

$\displaystyle z=(I-G_{0})^{-1}\tilde{h},$ (3.35)

where
$\tilde{h}(t)=\left(h_{1}(1,t),\dots,h_{m}(1,t),h_{m+1}(0,t),\dots,h_{n}(0,t)\right)$.
Substituting (3.35) into (3.30), we obtain

$u^{*}_{j}(x,t)=\left[(I-C)^{-1}h\right]_{j}(x,t)=c_{j}(x_{j},x,t)\left[R(I-G_{0})^{-1}\tilde{h}\right]_{j}(\omega_{j}(x_{j},x,t))+h_{j}(x,t)$
(3.36)

for $j\leq n$. Combining (3.36) with (3.34) gives

$\|(I-C)^{-1}\|_{{\cal L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}\leq
1+\frac{1}{\sigma(1-\gamma)}\|C\|_{{\cal
L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}.$ (3.37)

As we showed in [18, p. 4188], the operators $DC$ and $D^{2}$ are smoothing
and, more precisely, map continuously $BC(\Pi;\mathbb{R}^{n})$ into
$BC^{1}_{t}(\Pi;\mathbb{R}^{n})$. This means that there exists a positive
constant $M_{10}$ such that for all $u\in BC(\Pi;{\mathbb{R}}^{n})$ we have

$\displaystyle\left\|\partial_{t}\left[(DC+D^{2})u\right]\right\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
M_{10}\|u\|_{BC(\Pi;{\mathbb{R}}^{n})}.$ (3.38)

Now, from (3.29) follows that

$\|u^{*}\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}\leq\displaystyle\|(I-C)^{-1}\|_{{\cal
L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}\|(DC+D^{2})u^{*}+(I+D)Ff\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}.$
(3.39)

Combining (3.39) with (3.26), (3.37), and (3.38), we obtain

$\begin{array}[]{rcl}\|u^{*}\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}\leq
M_{11}\|f\|_{BC^{1}({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n}))},\end{array}$

the constant $M_{11}$ being independent of $f$. From (1.1) we get

$\|\partial_{x}u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq\frac{1}{\Lambda_{0}}\left(\|f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|bu^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\right)\leq
M_{12}\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}$

for some $M_{12}>0$ not depending on $f$. Summarizing, the estimate (1.13) is
fulfilled.

###  3.4 Proof of Part ${\bf(ii)}$: $BC^{2}$-solution

Putting $u=u^{*}$ and differentiating the system (1.1) in a distributional
sense in $t$ and the boundary conditions (1.3) pointwise, we get,
respectively,

$(\partial_{t}+a\partial_{x})\partial_{t}u^{*}+\left(b-a^{-1}\partial_{t}a\right)\partial_{t}u^{*}+\left(\partial_{t}b-a^{-1}\partial_{t}a\,b\right)u^{*}=\partial_{t}f-a^{-1}\partial_{t}a\,f$
(3.40)

and

$\begin{array}[]{ll}\displaystyle\partial_{t}u^{*}_{j}(0,t)=\left(R\partial_{t}u^{*}\right)_{j}(t),\
1\leq j\leq m,\;t\in{\mathbb{R}},\\\\[5.69054pt]
\displaystyle\partial_{t}u^{*}_{j}(1,t)=\left(R\partial_{t}u^{*}\right)_{j}(t),\;\;\;m<j\leq
n,\;t\in{\mathbb{R}}.\end{array}$ (3.41)

Using a similar argument as above, we state that the function $u^{*}\in
BC^{1}(\Pi,\mathbb{R}^{n})$ satisfies both (3.40) in distributional sense and
(3.41) pointwise if and only if the function $w=\partial_{t}u^{*}$ satisfies
the following system pointwise:

$\displaystyle
w_{j}(x,t)=c_{j}^{1}(x_{j},x,t)\left(Rw\right)_{j}(\omega_{j}(x_{j}))$ (3.42)
$\displaystyle-\int_{x_{j}}^{x}d_{j}^{1}(\xi,x,t)\Biggl{(}\sum_{k\not=j}b_{jk}(\xi,\omega_{j}(\xi))w_{k}(\xi,\omega_{j}(\xi))$
$\displaystyle+\Bigl{[}\sum_{k=1}^{n}(\partial_{t}b_{jk}-a^{-1}_{j}\partial_{t}a_{j}\,b_{jk})u^{*}_{k}-\partial_{t}f_{j}+a^{-1}_{j}\partial_{t}a_{j}\,f_{j}\Bigr{]}(\xi,\omega_{j}(\xi))\Biggl{)}\\!\;\mathrm{d}\xi,\quad
j\leq n.$

Then the system (3.42) reads

$\displaystyle w=C_{1}w+D_{1}w+F_{1}(u^{*},f),$ (3.43)

where the operators $C_{1},D_{1},F_{1}\in{\cal L}(BC(\Pi;{\mathbb{R}}^{n}))$
are defined, respectively, by

$\displaystyle(C_{1}w)_{j}(x,t)=c_{j}^{1}(x_{j},x,t)(Rw)_{j}(\omega_{j}(x_{j})),$
$\displaystyle(D_{1}w)_{j}(x,t)=-\int_{x_{j}}^{x}d_{j}^{1}(\xi,x,t)\sum_{k\neq
j}b_{jk}(\xi,\omega_{j}(\xi))w_{k}(\xi,\omega_{j}(\xi))\\!\;\mathrm{d}\xi,$
$\displaystyle[F_{1}(u,f)]_{j}(x,t)=-\int_{x_{j}}^{x}d_{j}^{1}(\xi,x,t)\left[\sum_{k=1}^{n}(\partial_{t}b_{jk}-a^{-1}_{j}\partial_{t}a_{j}\,b_{jk})u_{k}-\partial_{t}f_{j}+a^{-1}_{j}\partial_{t}a_{j}\,f_{j}\right](\xi,\omega_{j}(\xi))\\!\;\mathrm{d}\xi.$

Iterating (3.43), we obtain

$\displaystyle
w=C_{1}w+(D_{1}C_{1}+D_{1}^{2})w+(I+D_{1})F_{1}(u^{*},f).{\tiny}$ (3.44)

Using the above argument once again, we conclude that the operators
$D_{1}C_{1}$ and $D_{1}^{2}$ are smoothing and map continuously
$BC(\Pi;\mathbb{R}^{n})$ into $BC^{1}_{t}(\Pi;\mathbb{R}^{n}).$ Moreover, the
following estimate is true:

$\left\|(D_{1}C_{1}+D_{1}^{2})w\right\|_{BC^{1}_{t}(\Pi;\mathbb{R}^{n})}\leq
M_{14}\|w\|_{BC(\Pi;\mathbb{R}^{n})}$ (3.45)

for some $M_{14}>0$ not depending on $w\in BC(\Pi;{\mathbb{R}}^{n})$.

Next, similarly to the bijectivity of the operator $I-C$, we prove that
$I-C_{1}$ is a bijective operator from $BC^{1}_{t}(\Pi;\mathbb{R}^{n})$ to
itself. In other words, we have to show that the system

$w_{j}(x,t)=c_{j}^{1}(x_{j},x,t)(Rw)_{j}(\omega_{j}(x_{j},x,t))+h_{j}(x,t),\quad
j\leq n,$

is uniquely solvable in $BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ for each $h\in
BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})$ or, the same, if and only if $I-G_{1}$ is a
bijective operator from $BC^{1}({\mathbb{R}};{\mathbb{R}}^{n})$ to itself,
where the operator $G_{1}\in{\cal L}(BC({\mathbb{R}};{\mathbb{R}}^{n}))$ is
given by (1.9). To this end, we use the assumptions that $\|G_{1}\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}<1$ and $\|G_{2}\|_{{\cal
L}(BC(\mathbb{R};\mathbb{R}^{n}))}<1$.

Similarly to (3.37), the inverse to $I-C_{1}$ fulfills the bound

$\|(I-C_{1})^{-1}\|_{{\cal L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}\leq
1+M_{15}\|C_{1}\|_{{\cal L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}.$ (3.46)

for some $M_{15}>0$. Moreover, from (3.44) we get the estimate

$\|w\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}\leq\displaystyle\|(I-C_{1})^{-1}\|_{{\cal
L}(BC^{1}_{t}(\Pi;{\mathbb{R}}^{n}))}\|(D_{1}C_{1}+D_{1}^{2})w+(I+D_{1})F_{1}(u^{*},f)\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}.$
(3.47)

Combining (3.47) with (1.13), (3.45), and (3.46), we conclude that there
exists $M_{16}>0$ such that

$\begin{array}[]{rcl}\|w\|_{BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})}\leq
M_{16}\left(\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;\mathbb{R}^{n}\right)}\right).\end{array}$

Therefore,

$\|\partial_{t}^{2}u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
M_{16}\left(\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;\mathbb{R}^{n}\right)}\right).$
(3.48)

Hence, from (3.40) we immediately get the bound

$\|\partial^{2}_{xt}u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
M_{17}\left(\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;\mathbb{R}^{n}\right)}\right).$

Differentiating now the system (1.1) in $x$, we easily obtain the estimate for
$\partial_{x}^{2}u^{*}$, namely

$\|\partial^{2}_{x}u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
M_{18}\left(\|f\|_{BC^{1}({\mathbb{R}};C^{1}([0,1];{\mathbb{R}}^{n}))}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;\mathbb{R}^{n}\right)}\right).$

The estimate (1.14) now follows. The proof of Part ${\bf(ii)}$ of the theorem
is complete.

###  3.5 Proof of Part ${\bf(iii)}$: Almost periodic solution

Now we assume that the coefficients of the initial problem are almost periodic
in $t$ and prove that the solution $u^{*}$ constructed above belongs to
$AP(\Pi;{\mathbb{R}}^{n})$.

Fix $\mu>0$ and let $h$ be a common $\mu$-almost period of the functions
$a_{i}$, $b_{ij}$, and $f_{i}$. Then

$\displaystyle\|({\cal A}(t+h)-{\cal
A}(t))\varphi\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq\mu
M_{19}\|\varphi\|_{H^{1}((0,1);{\mathbb{R}}^{n})},\quad t\in\mathbb{R},$
(3.49)

where $\varphi\in D({\cal A})$ and constant $M_{19}>0$ is independent of $\mu$
and $t.$

For a fixed $\varphi\in D({\cal A}),$ the function $U(t+h,\tau+h)\varphi$
satisfies the equation

$\frac{d}{dt}U(t+h,\tau+h)\varphi={\cal A}(t+h)U(t+h,\tau+h)\varphi.$

Then

$\displaystyle\frac{d}{dt}U(t+h,\tau+h)\varphi={\cal
A}(t)U(t+h,\tau+h)\varphi+\left({\cal A}(t+h)-{\cal
A}(t)\right)U(t+h,\tau+h)\varphi$

and, using (3.49) as well as the variation of constant formula and the
exponential decay conditions (1.10) and (1.11), we have

$\displaystyle\|U(t+h,\tau+h)\varphi-U(t,\tau)\varphi\|_{L^{2}((0,1);{\mathbb{R}}^{n})}$
$\displaystyle\leq\int_{\tau}^{t}\|U(t,s)\|_{{\cal L}(L^{2})}\|{\cal
A}(s+h)-{\cal A}(s)\|_{{\cal
L}(H^{1},L^{2})}\|U(s+h,\tau+h)\varphi\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\\!\;\mathrm{d}s$
$\displaystyle\leq\mu
M_{19}M^{2}e^{-\alpha(t-\tau)}(t-\tau)\|\varphi\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\leq\mu
M_{20}e^{-\alpha_{1}(t-\tau)}\|\varphi\|_{H^{1}((0,1);{\mathbb{R}}^{n})}$
(3.50)

for some $\alpha_{1}<\alpha$ and constant $M_{20}$ independent of $\mu$ and
$t,\tau.$

Further, using (3.6) and (3.49), we get the estimate

$\begin{array}[]{ll}\left\|\left(\left({\cal{A}}(t+h)+\lambda
I\right)^{-1}-\left({\cal{A}}(t)+\lambda
I\right)^{-1}\right)f(t)\right\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\\\
\qquad\displaystyle=\left\|\left({\cal{A}}(t+h)+\lambda
I\right)^{-1}\left[{\cal{A}}(t+h)-{\cal{A}}(t)\right]\left({\cal{A}}(t)+\lambda
I\right)^{-1}f(t)\right\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\\\
\qquad\displaystyle\leq\left\|\left({\cal{A}}(t+h)+\lambda
I\right)^{-1}\|_{{\cal
L}(L^{2},H^{1})}\|\left[{\cal{A}}(t+h)-{\cal{A}}(t)\right]\|_{{\cal
L}(H^{1},L^{2})}\|\left({\cal{A}}(t)+\lambda
I\right)^{-1}f(t)\right\|_{H^{1}((0,1);{\mathbb{R}}^{n})}\\\
\qquad\displaystyle\leq\mu\lambda M_{0}^{2}M_{19},\end{array}$ (3.51)

which immediately implies that

$\begin{array}[]{ll}\left\|\left(\left({\cal{A}}(t+h)+\lambda
I\right)^{-1}-\left({\cal{A}}(t)+\lambda
I\right)^{-1}\right)f(t)\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq\mu\lambda
M_{0}^{2}M_{19}.\end{array}$ (3.52)

Moreover, combining the formula (3.18) with the almost periodicity of the
coefficients of the original problem and with the estimate (3.51), we conclude
that there is a constant $M_{21}$ such that

$\left\|\widetilde{f}(s+h)-\widetilde{f}(s))\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\leq\mu
M_{21}.$ (3.53)

Combining now the formulas (3.19), (3.49), (3.5), (3.52), and (3.53) results
in the desired estimate

$\begin{array}[]{ll}\|v^{*}(t+h)-v^{*}(t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\\\[5.69054pt]
\qquad\leq\displaystyle\|-({\cal{A}}(t+h)+\lambda
I)^{-1}f(t+h)+({\cal{A}}(t)+\lambda
I)^{-1}f(t)\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\\\[8.53581pt]
\qquad\displaystyle+\left\|\int_{-\infty}^{t+h}U(t+h,s)\widetilde{f}(s)\\!\;\mathrm{d}s-\int_{-\infty}^{t}U(t,s)\widetilde{f}(s)\\!\;\mathrm{d}s\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\\\[8.53581pt]
\qquad\displaystyle\leq\left\|({\cal{A}}(t+h)+\lambda
I)^{-1}(f(t+h)-f(t))\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\\\[8.53581pt]
\qquad+\displaystyle\left\|\left(\left({\cal{A}}(t+h)+\lambda
I\right)^{-1}-\left({\cal{A}}(t)+\lambda
I\right)^{-1}\right)f(t)\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\\
\qquad\displaystyle+\int_{-\infty}^{t}\left\|(U(t+h,s+h)-U(t,s))\widetilde{f}(s+h)\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\!\;\mathrm{d}s\\\
\qquad\displaystyle+\int_{-\infty}^{t}\left\|U(t,s)(\widetilde{f}(s+h)-\widetilde{f}(s))\right\|_{L^{2}((0,1);{\mathbb{R}}^{n})}\\!\;\mathrm{d}s\leq\mu
M_{22}\end{array}$ (3.54)

for some $M_{22}$ not depending on $\mu$. Therefore, the function
$v^{*}:\mathbb{R}\to L^{2}((0,1);\mathbb{R}^{n})$ is almost periodic in $t$.
Due to the definition of $u^{*}(x,t)$, this means that $u^{*}$ is almost
periodic in $t$ in the mean in $x$, see [4, Theorem 2.15].

Further, since $u^{*}\in BC^{2}(\Pi,\mathbb{R}^{n}),$ then we conclude that
$\partial_{t}u^{*}(t,x)$ is uniformly continuous in $t$ in the mean in $x$.
Hence, by [4, Theorem 2.13], $\partial_{t}u^{*}(t,x)$ is almost periodic in
$t$ in the mean in $x$. Now, the almost periodicity of
$\partial_{x}u_{*}(x,t)$ follows directly from the equation (1.1).

Summarising, the function $u^{*}(x,t)$ is almost periodic in $t$ with values
in $H^{1}((0,1);{\mathbb{R}}^{n})$ and, hence, due to the continuous embedding
of $H^{1}((0,1);{\mathbb{R}}^{n})$ into $C([0,1];{\mathbb{R}}^{n})$,
$u^{*}(x,t)$ is almost periodic in $t$ with values in
$C([0,1];{\mathbb{R}}^{n})$.

##  4 Periodic problems

By $C_{T}(\Omega;{\mathbb{R}}^{n})$ we denote the vector space of all
continuous maps $u:\Omega\to{\mathbb{R}}^{n}$ which are $T$-periodic in $t$,
endowed with the supremum norm. Similarly, one introduces the space
$C_{T}^{k}(\Omega;{\mathbb{R}}^{n})$ of bounded and $k$-times continuously
differentiable functions.

######  Theorem 4.1.

${\bf(i)}$ Assume that the coefficients $a_{j},b_{jk},f_{j}$ are $T$-periodic
in $t$ and belong to $C^{1}(\Pi)$ for all $j,k\leq n$. Moreover, suppose that
the evolution family $U(t,s)$ generated by the problem (1.5), (1.3) satisfies
the estimate (1.10). If the conditions (1.12) are fulfilled for $i=0$ and
$i=1$, then the system (1.1), (1.3) has a unique classical $T$-periodic
solution $u^{*}(x,t)$. Moreover, the a priori estimate

$\|u^{*}\|_{BC^{1}(\Pi;{\mathbb{R}}^{n})}\leq
L_{3}\|f\|_{BC_{t}^{1}(\Pi;{\mathbb{R}}^{n})}$ (4.1)

is fulfilled for a constant $L_{3}$ not depending on $f.$

${\bf(ii)}$ Assume, additionally, that $b_{jk}$ and $f_{j}$ belong to
$C_{t}^{2}(\Pi)$ for all $j,k\leq n$ and that the condition (1.12) is true for
$i=2$. Then $u^{*}\in C^{2}_{t}(\Pi,\mathbb{R}^{n})$ and $\partial_{x}u^{*}\in
C^{1}_{t}(\Pi,\mathbb{R}^{n})$. Moreover, the a priori estimate

$\|u^{*}\|_{BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})}+\|\partial_{x}u^{*}\|_{BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})}\leq
L_{4}\|\partial_{t}^{2}f\|_{BC\left(\Pi,\mathbb{R}^{n}\right))}$ (4.2)

is fulfilled for a constant $L_{4}$ not depending on $f.$

Note that all conditions of Theorem 3.3 are fulfilled. This implies that the
problem (3.1) has a unique bounded classical solution $v^{*}$, and this
solution is given by the formula (3.3). As the coefficients $a_{j}$ and
$b_{jk}$ are $T$-periodic in $t$, we have ${\cal A}(t+T)={\cal A}(t)$ and,
hence $U(t+T,s+T)=U(t,s)$ for all $t\geq s$. This, in its turn, implies that
$v^{*}(t+T)=v^{*}(t)$ for $t\in{\mathbb{R}}$. Moreover, as $v^{*}$ is the
classical solution, then it is continuous in $t\in{\mathbb{R}}$ with values in
$H^{1}((0,1);{\mathbb{R}}^{n})$. Consequently, $v^{*}\in
BC({\mathbb{R}};H^{1}((0,1);{\mathbb{R}}^{n}))$ and, hence $v^{*}\in
BC({\mathbb{R}};C([0,1];{\mathbb{R}}^{n}))$. This means that the function
$u^{*}(x,t)=[v^{*}(t)](x)$ belongs to $BC(\Pi;\mathbb{R}^{n})$. Moreover, due
to the proof of Theorem 1.5 (see Section 3.2), $u^{*}$ satisfies the equation
(3.2) or, the same, its operator form (3.28) pointwise.

Now our aim is to prove that the map $f\in C_{T}(\Pi;{\mathbb{R}}^{n})\to
u^{*}\in C_{T}(\Pi;{\mathbb{R}}^{n})$ is continuous. To this end, we prove
that the operator $I-C-D$, first, is Fredholm of index zero from
$C_{T}(\Pi;{\mathbb{R}}^{n})$ to itself and, second, is injective from
$C_{T}(\Pi;{\mathbb{R}}^{n})$ to itself. This together will imply the
bijectivity of $I-C-D$ and the boundedness of the inverse $(I-C-D)^{-1}$, as
desired.

The Fredholmness of $I-C-D$ follows from [17, Theorem 1.2] and [15, Theorem
1.2]. Nevertheless we give a short proof of this result, using the notion of
parametrix.

First note that the operator $I-C$ is bijective from
$C_{T}(\Pi;{\mathbb{R}}^{n})$ to itself. Indeed, similarly to the proof of
$BC^{1}$-smoothness in Section 3.3, it is equivalent to the unique solvability
in $C_{T}(\Pi;{\mathbb{R}}^{n})$ of the system (3.30) with respect to $u$ or,
the same, to the unique solvability in $C_{T}({\mathbb{R}};{\mathbb{R}}^{n})$
of the system (3.31) with respect to $z$. The last, in its turn, is equivalent
to the bijectivity of $I-G_{0}$ from $C_{T}({\mathbb{R}};{\mathbb{R}}^{n})$ to
itself. But this is true due to the assumption (1.12) for $i=0$ and the Banach
fixed-point theorem.

It follows that the operator $I-C-D$ is Fredholm of index zero from
$C_{T}(\Pi;{\mathbb{R}}^{n})$ to itself if and only if the operator
$I-(I-C)^{-1}D$ is Fredholm of index zero from $C_{T}(\Pi;{\mathbb{R}}^{n})$
to itself. Now we use a Fredholmness criterion given by [27, Proposition 1,
pp. 298–299] stating that, if there exist linear continuous operators
$P_{l},P_{r}:C_{T}(\Pi;{\mathbb{R}}^{n})\to C_{T}(\Pi;{\mathbb{R}}^{n})$
(called parametrices) and compact operators
$C_{l},C_{r}:C_{T}(\Pi;{\mathbb{R}}^{n})\to C_{T}(\Pi;{\mathbb{R}}^{n})$ such
that

$P_{l}\left[I-(I-C)^{-1}D\right]=I+C_{l}\quad\mbox{and}\quad\left[I-(I-C)^{-1}D\right]P_{r}=I+C_{r},$

then the operator $I-(I-C)^{-1}D$ is Fredholm of index zero. Take

$P_{l}=P_{r}=I+(I-C)^{-1}D.$

Hence,

$\begin{array}[]{ll}P_{l}\left[I-(I-C)^{-1}D\right]=\left[I-(I-C)^{-1}D\right]P_{r}=I-(I-C)^{-1}D(I-C)^{-1}D.\end{array}$

In order to use [27, Proposition 1, pp. 298–299], we prove the compactness of
the operator $(I-C)^{-1}D(I-C)^{-1}D$ or, on the account of the boundedness of
$(I-C)^{-1}$, the compactness of the operator $D(I-C)^{-1}D$. Indeed,

$D(I-C)^{-1}D=D\left[I+C(I-C)^{-1}\right]D=D^{2}+DC(I-C)^{-1}D.$

Due to the boundedness of the operator $(I-C)^{-1}D$, we are done if we prove
the compactness of the operators $D^{2}$ and $DC$. But this is the scope of
the statement of [17, Equation (4.2)]. Actually, in [17] it was shown that the
operators $D^{2}$ and $DC$ map continuously $C_{T}(\Pi;{\mathbb{R}}^{n})$ into
$C_{T}^{1}(\Pi;{\mathbb{R}}^{n})$. Due to the compact embedding of
$C_{T}^{1}(\Pi;{\mathbb{R}}^{n})$ into $C_{T}(\Pi;{\mathbb{R}}^{n})$ (by the
Arzela-Ascoli theorem), the compactness of the operators $D^{2}$ and $DC$
straightforwardly follows. The desired Fredholmness of $I-C-D$ is therewith
proved.

To prove the injectivitiy of $I-C-D$, suppose that the equation (3.28) has two
continuous solutions, say $u^{1}$ and $u^{2}$. Then the difference
$u^{1}-u^{2}$ satisfies the equation (3.28) with $f=0$ and, consequently, the
equation (3.29) with $f=0$. Since the operators $D^{2}$ and $DC$ map
continuously $C_{T}(\Pi;{\mathbb{R}}^{n})$ into
$C_{T}^{1}(\Pi;{\mathbb{R}}^{n})$, we have $u^{1}-u^{2}\in
C_{T}^{1}(\Pi;{\mathbb{R}}^{n})$ (see also the estimate (3.38)). This
immediately yields that $u^{1}-u^{2}$ satisfies the problem (1.5), (1.3)
pointwise or, the same, is a classical solution to (1.5), (1.3).

Further, fix arbitrary $s\in{\mathbb{R}}$. Notice that the function
$u^{1}(\cdot,s)-u^{2}(\cdot,s)$ belongs to $D({\cal A})$. Then, by [19, Lemma
4.4], the function $u^{1}-u^{2}$ is the $L^{2}$-generalized solution to the
problem (1.5), (1.3), (1.6), which is $T$-periodic in $t$. Taking now into
account the assumption (1.10) about exponential decay of the
$L^{2}$-generalized solution, we conclude that $u^{1}(x,s)=u^{2}(x,s)=0$ for
all $x\in[0,1]$ and $t\geq s$. As $u^{1}-u^{2}$ is $T$-periodic, this yields
that $u^{1}(x,t)\equiv u^{2}(x,t)=0$ for all $x\in[0,1]$ and
$t\in{\mathbb{R}}$. The injectivity is therewith proved.

It follows that $u^{*}=(I-C-D)^{-1}Ff$ and, hence that

$\|u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
L_{5}\|f\|_{BC(\Pi;{\mathbb{R}}^{n})}$ (4.3)

for some $L_{5}>0$ not depending on $f\in BC(\Pi;{\mathbb{R}}^{n})$. Moreover,
combining (3.39) with (3.37), (3.38), and (4.3), we obtain the apriori
estimate

$\|u^{*}\|_{BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})}\leq
L_{6}\|f\|_{BC^{1}_{t}(\Pi;{\mathbb{R}}^{n})}.$

The final estimate (4.1) now straightforwardly follows from (1.1). The proof
of Claim ${\bf(i)}$ of Theorem 4.1 is therefore complete.

To prove Claim ${\bf(ii)}$ of Theorem 4.1, we follow a similar argument as in
the proof of Claim ${\bf(ii)}$ of Theorem 1.5, the changes being of the
following kind. Combining (3.39) with (3.45) and (3.46), we get, instead of
(3.48), the estimate

$\|u^{*}\|_{BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})}\leq
L_{7}\|f\|_{BC^{2}_{t}(\Pi;{\mathbb{R}}^{n})}.$

Hence, from (3.40) we immediately get the desired bound (4.2).

######  Remark 4.2.

If $f\in BC({\mathbb{R}},C^{1}([0,1]);{\mathbb{R}}^{n})$, then differentiating
the system (1.1) in $x$, we get also the bound for $\partial_{x}^{2}u^{*}$,
namely

$\|\partial^{2}_{x}u^{*}\|_{BC(\Pi;{\mathbb{R}}^{n})}\leq
L_{8}\left(\|\partial_{x}f\|_{BC(\Pi;{\mathbb{R}}^{n})}+\|\partial_{t}^{2}f\|_{BC\left(\Pi;\mathbb{R}^{n}\right)}\right).$

Acknowledgements. Both authors were supported by the Volkswagen Foundation
grant ”From Modelling and Analysis to Approximation”. V. Tkachenko was
supported by the project SFB/TRR 109 of German Research Fundation DFG and by
the Volkswagen Foundation grant ”Dynamic Phenomena in Elasticity Problems”.

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