\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 169, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2014/169\hfil Almost periodic solutions]
{Almost periodic solutions of anisotropic elliptic-parabolic equations with
 variable exponents of nonlinearity}

\author[M. Bokalo\hfil EJDE-2014/169\hfilneg]
{Mykola Bokalo}  % in alphabetical order

\address{Mykola Bokalo \newline
Department of Differential Equations\\
Ivan Franko National University of Lviv\\
Lviv, Ukraine}
\email{mm.bokalo@gmail.com}

\thanks{Submitted April 17, 2014. Published August 11, 2014.}
\subjclass[2000]{35K10, 35K55, 35K92}
\keywords{Fourier problem; problem without initial conditions;
\hfill\break\indent degenerate implicit equations; elliptic-parabolic equation;
periodic solution; 
\hfill\break\indent almost periodic solution; nonlinear evolution equation}

\begin{abstract}
 We prove the well-posedness of Fourier problems for anisotropic
 elliptic-parabolic equations  with variable exponents of nonlinearity
 without any restrictions at infinity. We obtain estimates of the weak solutions
 and conditions for the existence of periodic and almost periodic solutions.
 In addition, some properties of the weak solutions of the Fourier problem
 are considered.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}\label{Sect0}

We examine the question of well-posedness of the Fourier  problem
(the problem without initial conditions)
for anisotropic second order elliptic-parabolic equations
with variable exponents of nonlinearity.
These equations are defined on unbounded cylindrical domains which are
the Cartesian products of bounded space domains
and the whole time axis.
 Also the existence conditions of periodic and almost periodic
 solutions  are investigated. Moreover, we  examine the conditions on input data that
 guarantee the specific behavior of the   solutions at infinity.

The Fourier problem for evolution equations are examined in many papers; see, e.g.,
  \cite{Bokalo_Lorenzi, Bokalo90, Bokalo_Pauchok, Bokalo_Dmytryshyn, Bokalo_Dynam,
Lions, Oleinik_Iosifjan, Pankov, Showalter80, Showalter97, Tikhonov_1}.
A fairly good survey of results concerning these problems
can be found in \cite{Bokalo_Lorenzi}.
It is worth to mention that Fourier problem for linear and
a plenty of nonlinear evolution equations are correct only
under some restrictions on the growth
of  solutions and input data  as the time variable
converges to $-\infty$, in addition to boundary conditions
\cite{Bokalo_Lorenzi,Lions,Oleinik_Iosifjan,Pankov,Showalter80,Showalter97, Tikhonov_1}.
However, there are the nonlinear parabolic equations for which the Fourier
problem are uniquely  solvable with no conditions
at infinity \cite{Bokalo90} -- \cite{Bokalo_Dynam}.
This case for anisotropic elliptic-parabolic equations with variable exponents of
nonlinearity is considered here.

It is known that the problem to find time periodic
and almost periodic solutions of evolution
equations is close  to  Fourier problem for this equations
\cite{Bokalo_Dmytryshyn,Bor,Hu,Levitan78,Maqbul,Pankov,Ward}.
Note that the degenerated parabolic equations, in particular,
including elliptic-parabolic  are examined in
\cite{Bokalo_Dmytryshyn,Bokalo_Dynam,Showalter97,Showalter69,Showalter77} and other.
Differential equations with variable exponents of
nonlinearity are considered in many papers. Solutions of this equations belong
to the generalized Lebesgue and Sobolev spaces.
More information on these spaces and its applications can be received from
\cite{an-an-zyk-2009, Bokalo_Pauchok, fu-pan-2010, kov-rak, mash-buhr-2011,
mih-rar-ter-2011, Ruzicka}.

This article can be viewed as a natural continuation
of the paper \cite{Bokalo_Dmytryshyn} for
the case of equations with  variable exponents of nonlinearity.
It consists of three parts: in the first part the formulation
of  problem  and main results are presented, the second part encloses the auxiliary
 statements while the proofs of main results are in the third part.

 \section{Setting of the problem and main results}\label{Sect1}

Let $\Omega$ be a bounded domain in  $\mathbb{R}^n$ with the piecewise smooth
boundary  $\partial \Omega$.
Suppose that $\partial \Omega$ is divided into two subsets $\Gamma_0$ and $\Gamma_1$,
where $\Gamma_0$ is closed. The cases $\Gamma_0=\emptyset$ and
$\Gamma_0=\partial \Omega$  are also possible.
We denote by $\nu=(\nu_1,\dots ,\nu_n)$ the unit outward pointing normal vector
on  $\partial \Omega$.
  Set  $Q:=\Omega\times \mathbb{R}$, $\Sigma_0:=\Gamma_0\times \mathbb{R}$,
   $\Sigma_1:=\Gamma_1\times \mathbb{R}$, and $Q_{t_1,t_2}:=\Omega\times(t_1,t_2)$
for arbitrary real $t_1$ and $t_2$. Here and subsequently, we assume that $t_1<t_2$.

    Consider the problem of finding a function $u:\overline Q\to\mathbb R$
satisfying (in some sense)  the equation
  \begin{equation}\label{EJDEp.4.2.5}
 (b(x)u)_t-\sum_{i=1}^{n}
 \big(a_i(x,t)|u_{x_i}|^{p_i(x)-2}u_{x_i}\big)_{x_i}+
  a_0(x,t)|u|^{p_0(x)-2}u=
 f(x,t),
 \end{equation}
for $(x,t)\in Q$,
 and the boundary conditions
  \begin{equation}\label{EJDEp.4.2.6}
  u\big|_{\Sigma_0}=0, \quad
  \frac{\partial u}{\partial \nu_a}\big|_{\Sigma_1}=0,
  \end{equation}
where ${\partial u(x,t)}/{\partial \nu_a}:=\sum_{i=1}^{n}
  a_i(x,t)|u_{x_i}|^{p_i(x)-2}u_{x_i}\,\nu_i(x)$ is the ``conormal''
derivative on $\Sigma_1$, and the functions $b:\Omega\to[0,+\infty)$,
 $p_j: \Omega\to (1,\infty)$, 
$a_j: Q\to (0,\infty)$ ($j=0,\ldots,n$),
$f: Q\to \mathbb{R}$ are given.

Next  we are going to define a weak solution of the problem
\eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6} and formulate the main result of our paper.
 For this, we need some functional  spaces
 and classes  of input data of the given problem.

First we introduce some functional spaces. Suppose that either  $G=\Omega$ or
   $G:=\Omega\times S$, where $S$ is an interval in $\mathbb{R}$.
    Consider a function $r\in L_{\infty}(\Omega)$ such
    that $r(x)\geq 1$ for almost each  $x\in \Omega$.
    Denote by $L_{r(\cdot)}(G)$  the generalized Lebesgue space consisting
 of the functions $v\in L_1(G)$  such that $\rho_{G,r}(v)<\infty$,
 where
 $ \rho_{G,r}(v):=\int_{\Omega}|v(x)|^{r(x)}\,dx$
 for $ G=\Omega$, $ \rho_{G,r}(v):=\int_{G} |v(x,t)|^{r(x)}\,dx\,dt$
  for $G=\Omega\times S$.
  The space is equipped with the norm
\[
 \|v\|_{L_{r(\cdot)}(G)}
 :=\inf\{{\lambda>0 : \rho_{G,r}(v/\lambda)\leq 1}\}
\]
\cite[p. 599]{kov-rak}.
 If $\operatorname{ess\,inf}_{x\in \Omega}r(x)>1$,
 then the dual space $[L_{r(\cdot)}(G)]'$
 can be identified with $L_{r'(\cdot)}(G)$,
 where  $r'$ is the function defined by the equality
  $\frac{1}{r(x)}+\frac{1}{r'(x)}=1$ for almost each $x\in \Omega$.

 Let $G=\Omega\times S$, where $S$ is  an unbounded interval in $\mathbb{R}$
 or $S=\mathbb{R}$.
 We denote by $L_{r(\cdot),\rm loc}(\overline G)$ the space of measurable
functions  $g: G\to\mathbb{R}$ such that the restriction of $g$ on $Q_{t_1,t_2}$
 belongs to   $L_{r(\cdot)}(Q_{t_1,t_2})$
 for each  $t_1,t_2\in S$. This space is complete locally convex
 with respect to the family of seminorms
 $\big\{\|\cdot\|_{L_{r(\cdot)}(Q_{t_1,t_2})}\mid t_1,t_2\in S\big\}$.
A sequence  $\{g_m\}$ is said to be convergent strongly (resp., weakly) in
 $L_{r(\cdot),\rm loc}(\overline G)$
 provided the sequences of restrictions $\{g_m|_{Q_{t_1,t_2}}\}$
 are convergent strongly (resp., weakly) in
 $L_{r(\cdot)}(Q_{t_1,t_2})$ for all $t_1,t_2\in S$.
  Similarly we can define  the space $L_{\infty,\rm loc}(\overline G)$.


Let $B$ be a linear space with a norm or a seminorm $\|\cdot\|_B$.
Let us denote by $C(S;B)$ the space of
functions $v: S\to B$ such that the restriction of $v$ on
any interval $[t_1,t_2]\subset S$ belongs to
$C([t_1,t_2];B)$.  The space $C(S;B)$  is complete
locally convex  with respect to  the family of seminorms
$\big\{\|v\|_{C([t_1,t_2];B)}:=
\max_{t\in[t_1,t_2]}\|v(t)\|_{B}\,\big|\,t_1,t_2\in S\big\}$.
Therefore a sequence  $\{g_m\}$  converges in $C(S;B)$ provided
 the sequences of restrictions $\{g_m|_{[t_1,t_2]}\}$
  converge in $C([t_1,t_2];B)$
for each  $t_1,t_2\in S$.


 Let $p=(p_0,\ldots, p_n):\Omega\to \mathbb{R}^{1+n}$ be
 a vector-function satisfying the following condition:
\begin{itemize}
\item[(P)] the function
 $ p_j:\Omega \to \mathbb{R}$ are measurable for all $j=0,1,\ldots, n$,
\begin{gather*}
 p_0^-:=\operatorname{ess\,inf} _{x\in \Omega}p_0(x)> 2, \quad
 p_i^-:=\operatorname{ess\,inf} _{x\in \Omega}p_i(x)\geq 2\quad\text{for }
 i=1,\dots, n,\\
 p_j^+:=\operatorname{ess\,sup}_{x\in \Omega}p_j(x)<+\infty\quad\text{for }
 j=0,1,\dots,n.
\end{gather*}
\end{itemize}
We also denote by $p':=({p_0}',\ldots, {p_n}')$  the vector-function
 whose components
are given by the equalities $1/p_j(x)+1/{p_j}'(x)=1$ for almost each
$x\in\Omega$.


Let $W^1_{p(\cdot)}(\Omega)$ be the generalized Sobolev space
consisting of the functions  $v\in L_{p_0(\cdot)}(\Omega)$ such that
 $v_{x_i}\in L_{p_i(\cdot)}(\Omega)$ for all $i=1,\dots,n$.
 The space is equipped with the norm
\[
 \|v\|_{W^1_{p(\cdot)}(\Omega)}
 :=\|v\|_{L_{p_0(\cdot)}(\Omega)}
 +\sum_{i=1}^n\|v_{x_i}\|_{L_{p_i(\cdot)}(\Omega)}.
\]
 We denote by $\widetilde{W}^1_{p(\cdot)}(\Omega)$ the closure of the set
 $\{v\in C^1(\overline\Omega) \;\; | \;\; v|_{\Gamma_0}=0\big\}$
 in the space  $W^1_{p(\cdot)}(\Omega)$.

Next, for arbitrary $t_1,t_2\in\mathbb{R}$, we denote
by  $W^{1,0}_{p(\cdot)}(Q_{t_1,t_2})$
 the set of functions
 $w\in L_{p_0(\cdot)}(Q_{t_1,t_2})$ such that
  $w_{x_i}\in L_{p_i(\cdot)}(Q_{t_1,t_2})$ for all
  $i=1,\dots,n$.
 We  define the norm
\[
\|w\|_{W^{1,0}_{p(\cdot)}(Q_{t_1,t_2})}
 :=\|w\|_{L_{p_0(\cdot)}(Q_{t_1,t_2})}+
 \sum_{i=1}^n\|w_{x_i}\|_{L_{p_i(\cdot)}(Q_{t_1,t_2})}.
\]
 We denote by  $\widetilde{W}^{1,0}_{p(\cdot)}(Q_{t_1,t_2})$ the subspace of
  $W^{1,0}_{p(\cdot)}(Q_{t_1,t_2})$ consisting of functions
 $v$ such that $v(\cdot,t)\in \widetilde{W}^1_{p(\cdot)}(\Omega)$
 for a. e.  $t\in [t_1,t_2]$.

 Let $G=\Omega\times S$, where $S$ is either
 an unbounded $\mathbb{R}$ interval or the $\mathbb{R}$ axis.
  Let us denote
  by $\widetilde{W}^{1,0}_{p(\cdot),\rm loc}(\overline G)$
 the linear space of  measurable
  functions such that their restrictions on  $Q_{t_1,t_2}$ belong to
   $\widetilde{W}^{1,0}_{p(\cdot)}(Q_{t_1,t_2})$ for all $t_1,t_2\in S$.
   This space is  complete  locally convex
   with respect to the family of seminorms
 $\big\{\|\cdot\|_{{W}^{1,0}_{p(\cdot)}(Q_{t_1,t_2})}\,\big|\,
 t_1,t_2\in \mathbb{R}\big\}$.


 The following assumption on the function $b$ will be needed throughout the paper.
\begin{itemize}
\item[(B)]  $ b:\Omega \to \mathbb{R}$ is measurable and bounded,
 $b(x)\ge 0$  for a.e. $x\in \Omega$.
\end{itemize}
For each $x\in\Omega$ we define $\widetilde b(x)=b(x)$ if $b(x)>0$,
and $\widetilde b(x)=1$
if $b(x)=0$.
We denote by $\widetilde{H}^b (\Omega)$ the linear space of functions of the form
$w=\widetilde b^{-1/2}v$, where $v\in L_2(\Omega)$.
We introduce a seminorm on $\widetilde{H}^b (\Omega)$  by
${|\!|\!| w|\!|\!|}:=\|b^{1/2}w\|_{L_2(\Omega)}$.
It is easy to check that $\widetilde{H}^b (\Omega)$ is the completion
of  $\widetilde{W}^1_{p(\cdot)}(\Omega)$
 with respect to the seminorm  $|\!|\!| \cdot |\!|\!|$
 (see \cite[III.6, p. 141]{Showalter97}).

Set
 $$\mathbb{V}_p:=\widetilde{W}^1_{p(\cdot)}(\Omega),\quad
 \mathbb{U}_{p,\rm loc}^{\,b}:=
 \widetilde{W}^{1,0}_{p(\cdot),\rm loc}(\overline Q)
 \cap C(\mathbb{R};\widetilde{H}^b (\Omega)).
 $$
 The space $\mathbb{U}_{p,\rm loc}^{\,b}$ is a complete
 linear local convex space with respect to  the family of seminorms
\[
 \big\{\|w\|_{W^{1,0}_{p(\cdot)}(Q_{t_1,t_2})}
 +\max_{t\in [t_1,t_2]} \|w(\cdot,t)\|_{L_2(\Omega)}\,\big|\,
 t_1,t_2\in \mathbb{R}\big\}.
\]
For an interval $I$ we consider the space $ C^1_0(I)$
 of $C^1(I)$-functions of compact support.

Let us denote by $\mathbb{A}$  the set of ordered {arrays}
  of functions
 $(a_0,a_1,\ldots,a_n)$ satisfying the condition
\begin{itemize}
\item[(A)] for each $j\in\{0,1,\ldots,n\}$ the function $a_{j}$ belongs
    to the space $L_{\infty,\rm loc}(\overline Q)$ and the following holds
    \begin{equation}\label{EJDEajk}
    a_j(x,t)\geq K_1\quad\textrm{for almost each}\quad (x,t)\in Q
    \end{equation}
 with some constant $K_1>0$ being dependent on $(a_0,a_1,\dots ,a_n)$.
\end{itemize}

\begin{definition}\label{def1}\rm
   Suppose that $b$, $p$ satisfy conditions (B), (P),
 respectively,
    $(a_0,a_1$, $\dots ,a_n)\in \mathbb A$,
 and $f\in L_{{p_0}'(\cdot),\rm loc}(\overline{Q})$.
  A function  $u$ is called a weak solution of
   \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6} provided
   $u\in \mathbb{U}_{p,\rm loc}^b$ and the
    following integral identity holds
  \begin{equation}\label{EJDEtotozh}
   \iint_Q \Bigl\{\sum_{i=1}^n \big(a_i|u_{x_i}|^{p_i(x)-2}u_{x_i}
   \psi_{x_i}+a_0|u|^{p_0(x)-2}u\psi\big)\varphi-
   bu\psi\varphi'\Bigr\}\,dx\,dt
   = \iint_Q  f\psi \varphi\,dx\,dt
    \end{equation}
 for all $\psi\in \mathbb{V}_p$, 
$\varphi\in C^1_0(\mathbb{R})$.
\end{definition}

We say that the weak solution of
\eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6} \textit{continuously depends on input data},
 if  for each sequence
 $\{f_k\}_{k=1}^\infty\subset L_{{p_0}'(\cdot),\rm loc}(\overline{Q})$ such that
    $f_k \to f$ as $k\to\infty$
  in $L_{{p_0}'(\cdot),\rm loc}(\overline{Q})$
  we have $u_k\to u$ as $k\to\infty$ in
$\mathbb{U}_{p,{\rm loc}}^b$.  Here $u_k$ and $u$ are
weak solutions of \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}
 with the right-hand sides $f_{k}$ and $f$, respectively.

 \begin{theorem}\label{thm1}
 Suppose that $b$ and $p$ satisfy conditions {\rm (B)} and {\rm (P)},
 respectively,
    $(a_0,a_1,\dots, a_n)\in \mathbb A$, and
     $f\in L_{{p_0}'(\cdot),\rm loc}(\overline{Q})$.
  Then there exists a unique weak solution
  of  \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}, and it
  continuously depends on the input data. In addition, the estimate
 \begin{equation} \label{EJDEotsin_rozv}
\begin{aligned}
&\max_{t\in [t_0-R_0,t_0]}\int_{\Omega}b(x)|u(x,t)|^2\,dx +
    \int_{t_0-R_0}^{t_0}\int_{\Omega}
    \Bigr[\sum_{i=1}^n |u_{x_i}(x,t)|^{p_i(x)}+
     |u(x,t)|^{p_{0}(x)}\Bigr]\,dx\,dt \\
&\leq C_1\Bigl\{R^{-2/(p_0^+-2)} +
    \int_{t_0-R}^{t_0}
    \int_{\Omega}|f(x,t)|^{{p_0}'(x)}\,dx\,dt \Bigr\}
\end{aligned}
\end{equation}
holds for each $R$, $R_0$  such that $R\geq 1$, $0<R_0<R/2$,
and $t_0\in \mathbb{R}$.
 Here   $C_1$ is a positive constant which  depends
  on  $K_1$
 and  $p_j^\pm\ (j=0,\ldots,n)$ only.
\end{theorem}

\begin{remark}\label{rmk1} \rm
  Note that Theorem \ref{thm1} has no conditions
 imposed on the behaviour of the solution and
the growth of the functions $a_j$ $(j=0,\ldots,n)$ as well as on
the behaviour of $f$ as $t\to - \infty$.
However, the theorem is not true for the case
when $p_0(x)=p_1(x)=\cdots=p_n(x)=2$ for almost each
$x\in \Omega$ (see, for example, \cite{Bokalo_Lorenzi}).
Therefore  the condition (P) is essential.
\end{remark}

  A solution $u$ of \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6} is called
{\it bounded},
   if $\sup_{t\in \mathbb{R}}  \int_{\Omega}b(x)|u(x,t)|^2\,dx<\infty$.


\begin{corollary}\label{coro1}
 Under the assumptions of Theorem \ref{thm1},
 if $f\in L_{{p_0}'(\cdot)}(Q)$ then
 the weak solution of \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}
is bounded; it belongs to  $ \widetilde W^{1,0}_{p(\cdot)}(Q)$
and the  estimate 
\begin{equation} \label{EJDE3.4.q.7}
\begin{aligned}
&\sup_{t\in \mathbb{R}}  \int_{\Omega}b(x)|u(x,t)|^2\,dx +
\iint_Q \Bigr[\sum_{i=1}^n |u_{x_i}(x,t)|^{p_i(x)}+
     |u(x,t)|^{p_{0}(x)}\Bigr]\,dx\,dt \\
& \leq C_1 \iint_{Q}|f(x,t)|^{{p_0}'(x)}\,dx\,dt
\end{aligned}
\end{equation}
holds.
\end{corollary}


\begin{corollary}\label{coro2}
Under the assumptions of Theorem \ref{thm1}, if
$$
\sup_{\tau\in \mathbb{R}}\int_{\tau-1}^{\tau}
\int_{\Omega}|f(x,t)|^{{p_0}'(x)}\,dx\,dt \leq C_2
$$
for some positive constant $C_2$,
then the weak solution $u$ of  \eqref{EJDEp.4.2.5},
\eqref{EJDEp.4.2.6}  is bounded. In addition,
$$ 
\sup_{\tau\in \mathbb{R}}\int_{\tau-1}^{\tau}
\int_\Omega \Bigr[\sum_{i=1}^n |u_{x_i}(x,t)|^{p_i(x)}+
     |u(x,t)|^{p_0(x)}\Bigr]\,dx\,dt \leq C_3 
$$
with some positive constant  $C_3$ being dependent  on
      $K_1, p_j^{\pm}\,(j=0,\ldots,n)$ and $C_2$  only.
\end{corollary}

\begin{corollary}\label{coro3}
Under the assumptions of Theorem \ref{thm1}, if moreover
$$
\lim_{\tau\to \pm\infty }
\int_{\tau-1}^{\tau}
\int_{\Omega}|f(x,t)|^{{p_0}'(x)}\,dx\,dt =0, 
$$
then for the weak solution $u$ of problem
\eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6} the following relations hold:
\begin{gather*}
\lim_{t\to \pm\infty} \|b(\cdot)u(\cdot,t)\|_{L_2(\Omega)} =0,\\
\lim_{\tau\to \pm\infty}\int_{\tau-1}^{\tau}
\int_\Omega \Bigr[\sum_{i=1}^n |u_{x_i}(x,t)|^{p_i(x)}+
     |u(x,t)|^{p_{0}(x)}\Bigr]\,dx\,dt=0.
\end{gather*}

\end{corollary}


 \begin{theorem}\label{thm2}
Under the assumptions of Theorem \ref{thm1}, if
 $f$, $a_0$,\dots,$a_n$
are periodic in time with period $\sigma>0$,
then the weak solution
of  \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}
is also $\sigma$-periodic in time.
\end{theorem}


A set $X\subset \mathbb{R}$ is called {\it relatively dense},
if there exists a  positive $l$ such
that the interval $[a,a+l]$ contains at least one element
of the set $X$ for any  $a \in\mathbb{R}$, i.e.
$X\cap[a,a+l]\ne \emptyset$.

 Let $B$ be a linear space with a norm or a seminorm $\|\cdot\|_B$.
 A function $v\in C(\mathbb{R};B)$
 is  {\it Borh almost periodic }, if for each $\varepsilon>0$ the set
 $ \{\sigma\mid \sup_{t\in \mathbb{R}}\Vert v(t+\sigma)-
v(t)\Vert_{B}\leq\varepsilon\}$
 is relatively dense.
  A function  $f\in L_{p_0(\cdot),\rm loc}(\overline Q)$
  is  {\it Stepanov almost periodic} provided
the set $\{\sigma\mid \sup_{\tau\in \mathbb{R}} \int^\tau_{\tau-1}
\int_{\Omega}|f(x,t+\sigma)-f(x,t)|^{p_{0}(x)}\,dx\,dt\leq
\varepsilon\}$
 is relatively dense for each positive $\varepsilon$.
We say that $w\in \widetilde W^{1,0}_{p(\cdot),\rm loc}(\overline Q)$ is
 {\it Stepanov almost periodic}, if for each $\varepsilon>0$ the set
$\{\sigma:\sup_{\tau\in\mathbb{R}}
\int^\tau_{\tau-1}\int_{\Omega}
    \bigr[\sum_{i=1}^n |w_{x_i}(x,t+\sigma)-w_{x_i}(x,t)|^{p_i(x)}$
    $+|w(x,t+\sigma)-w(x,t)|^{p_{0}(x)}\bigr]\,dx\,dt
\leq\varepsilon\}$
 is relatively dense.
We  refer to \cite{Bor, Levitan78, Pankov} for the detailed information
on the theory of  almost periodic functions.


\begin{theorem}\label{thm3}
Let  the hypotheses of Theorem \ref{thm1} hold. In addition,  suppose that
 $a_0,\ldots,a_n$ are Borh almost periodic
  functions in $C(\mathbb{R};L_\infty(\Omega))$. Assume also that $f$
is Stepanov almost periodic  in $L_{p_0(\cdot),\rm loc}(\overline Q)$. Moreover,
the set
\begin{align*}
F_{\varepsilon} := \Big\{&\sigma: \sup_{\tau\in \mathbb{R}}
\int^\tau_{\tau-1}\int_{\Omega}|f(x,t+\sigma)-f(x,t)|^{p_{0}'(x)}\,dx\,dt
\leq\varepsilon,\\
 &\max_{j\in\{0,\ldots,n\}}\sup_{t\in \mathbb{R}}
\Vert a_j(\cdot,t+\sigma)-a_j(\cdot,t)\Vert_{L_\infty(\Omega)} \leq \varepsilon
\Big\}
\end{align*}
 is relatively dense for each   $\varepsilon>0$.
Then the (unique) weak solution of
 \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6} is
Borh almost periodic  in  $C(\mathbb{R};\widetilde{H}^b (\Omega))$ and Stepanov almost periodic
 in $\widetilde W^{1,0}_{p(\cdot),\rm loc}(\overline Q)$.
\end{theorem}


 \section{Auxiliary statements}\label{sect2}

We start with some auxiliary results, which will be  used below.

\begin{lemma}\label{lem1}
Suppose that $b$, $p$ satisfy conditions {\rm (B), (P)},
 respectively. Given $t_1, t_2\in \mathbb{R}$, we assume that
a function  $ w\in \widetilde W^{1,0}_{p(\cdot)}{(Q_{t_1,t_2})}$
 satisfies the equality
 \begin{equation} \label{EJDEtotlemq}
\int_{t_1}^{t_2}\int_{\Omega}\Bigl\{\Bigl(\sum_{i=1}^n
 g_i\psi_{x_i} + g_0\psi\Bigr)\varphi -
 bw\psi\varphi'\Bigr\}\,dx\,dt=0,
 \quad \psi\in \mathbb{V}_p,
 \;\varphi\in C^1_0(t_1,t_2),
 \end{equation}
for some functions
  $g_j\in L_{{p_j}'(\cdot)}\big(Q_{t_1,t_2}\big)$
 $(j=0,\ldots,n)$.
 Then  $w\in C([t_1,t_2];\widetilde{H}^b (\Omega))$ and
  the  equality
  \begin{equation}\label{EJDEozhlemq}
\begin{aligned}
&\theta(t)\int_{\Omega}b(x)|w(x,t)|^2\,dx \Big|_{t=\tau_1}^{t=\tau_2} -
\int_{\tau_1}^{\tau_2}
 \int_{\Omega}b|w|^2\theta'\,dx\,dt \\
&+  2\int_{\tau_1}^{\tau_2} \int_{\Omega}
 \Bigl(\sum_{i=1}^n g_iw_{x_i} +
 g_0w\Bigr)\theta\,dx\,dt =0
\end{aligned}
 \end{equation}
 holds for all $\tau_1,\tau_2\in[t_1,t_2]\, (\tau_1<\tau_2)$,
  $\theta\in C^1([t_1,t_2])$.
\end{lemma}

 This statement can be proved similarly to \cite[Lemma 1]{Bokalo_Pauchok}.

\begin{lemma}\label{lem2}
 Suppose that $b$ and $p$ satisfy conditions {\rm (B)} and {\rm (P)},
 respectively. Given $t_1, t_2\in \mathbb{R}$ such
   that $t_2-t_1\geq 1$ and $a\in \mathbb{A}$, we  suppose that functions
  $u_1$ and $u_2$ from
  $\widetilde{W}^{1,0}_{p(\cdot)}(Q_{t_1,t_2})\cap C([t_1,t_2];\widetilde{H}^b (\Omega))$
  satisfy  the equality
  \begin{equation} \label{EJDEtotozhl}
\begin{aligned}
&\int_{t_1}^{t_2}\int_{\Omega}
   \Bigl\{\Bigl(\sum_{i=1}^n a_i|u_{l,x_i}|^{p_i(x)-2}u_{l,x_i}\psi_{x_i}
   +a_0|u_l|^{p_0(x)-2}u_l\psi\Big)\varphi-
   bu_l\psi\varphi'\Bigr\}\,dx\,dt
   \\
&=  \int_{t_1}^{t_2}\int_{\Omega}
   \Bigl(\sum_{i=1}^n f_{i,l}\psi_{x_i}+
   f_{0,l}\psi\Bigr)\varphi\,dx\,dt,\quad \psi\in \mathbb{V}_p,
 \; \varphi\in C^1_0(t_1,t_2)
\end{aligned}
\end{equation}
with the functions $f_{j,l}\in L_{{p_j}'(\cdot)}(Q_{t_1,t_2})$
 ($j=0,\ldots,n; l=1,2$), respectively.
Then  the  inequality
  \begin{equation} \label{rizn_rozv}
\begin{aligned}
&\max_{t\in [t_0-R_0,t_0]}
  \int_\Omega b(x)|u_1(x,t)-u_2(x,t)|^2\,dx
 \\
&\quad + \int_{t_0-R_0}^{t_0}\int_{\Omega}
 \Big(\sum_{i=1}^n |u_{1,x_i}-u_{2,x_i}|^{p_i(x)}+
  |u_1-u_2|^{p_0(x)}\Big)\,dx\,dt
  \\
&\leq C_{4}\Big\{ R^{-2/(p_0^+-2)}
 + \int_{t_0-R}^{t_0}\int_{\Omega}
 \sum_{j=0}^n |f_{j,1}(x,t)-f_{j,2}(x,t)|^{{p_j}'(x)}\,dx\,dt\Big\}
 \end{aligned}
\end{equation}
 holds  for each $R$, $R_0$ and $t_0$  such
 that $R\geq 1$, $0<R_0<R/2$, and $t_1\leq t_0-R< t_0\leq t_2$.
 Here   $C_4$ is a positive constant which  depends on   $K_1$ and
 $p_j^\pm$  $(j=0,\ldots,n)$ only.
\end{lemma}

\begin{proof}
 Let  $R, R_0, t_0$ be such as
 in the formulation of the lemma,
and $\eta(t):=t-t_0+R,\, t\in\mathbb{R}$ (see \cite{Bernis}).
For given $\psi\in \mathbb{V}_p$, $ \varphi\in C^1_0(t_1,t_2) $
 we subtract equality \eqref{EJDEtotozhl} when $l=1$,
  and the same equality when $l=2$. Then, putting
\begin{gather*}
 u_{12}(x,t):=u_1(x,t)-u_2(x,t),\quad f_{j,12}(x,t):= f_{j,1}(x,t)-
 f_{j,2}(x,t),\\
 a_{0,12}(x,t):=a_0(x,t)\big(|u_1(x,t)|^{p_0(x)-2}u_1(x,t)-
 |u_{2}(x,t)|^{p_0(x)-2}u_{2}(x,t)\big),\\
  a_{i,12}(x,t):=a_i(x,t)\big(|u_{1,x_i}(x,t)|^{p_i(x)-2}u_{1,x_i}(x,t)-
 |u_{2,x_i}(x,t)|^{p_i(x)-2}u_{2,x_i}(x,t)\big)\\
 (i=1,\ldots,n;\, j=0,\ldots,n;\, (x,t)\in Q),
\end{gather*}
 we have an equality. From this equality using Lemma \ref{lem1}
  with  $w=u_{12}$,
 $g_j=a_{j,12}-f_{j,12}$
 ($j=0,\ldots,n$),
 $\theta=\eta^s$, $s:=p_0^-/(p_0^--2)$,
  $\tau_1= t_0-R$, $\tau_2=\tau\in(t_0-R,t_0]$, we obtain the equality
 \begin{equation}\label{EJDEeq19z}
\begin{aligned}
&\eta^s(\tau)\int_{\Omega}b(x)|u_{12}(x,\tau)|^2dx+
 2\int_{t_0-R}^{\tau}\int_\Omega
 \Big\{\sum_{i=1}^n  a_{i,12}(u_{12})_{x_i}
 +a_{0,12}u_{12}\Big\}\eta^s\,dx\,dt
 \\
 &=s\int_{t_0-R}^{\tau}\int_\Omega b|u_{12}|^2\eta^{s-1}\, dx\,dt
 +2\int_{t_0-R}^{\tau}\int_\Omega
 \Big(\sum_{i=1}^n f_{i,12}(u_{12})_{x_i}+f_{0,12}u_{12}\Big)\eta^s\, dx\,dt.
 \end{aligned}
\end{equation}
 We make the corresponding estimates of the
 integrals of equality \eqref{EJDEeq19z}.
 First we note if $r\in L_\infty(\Omega)$ and
 $\operatorname{ess\,inf} _{x\in \Omega}r(x)\geq 2$,
 then on the basis of \cite[Lemma 1.2]{Bokalo90} we have the  inequality
 $$
 (|s_1|^{r(x)-2}s_1-|s_2|^{r(x)-2}s_2)(s_1-s_2)\geq 2^{2-r^+}|s_1-s_2|^{r(x)}
 $$
 for each $s_1, s_2\in \mathbb{R}$ and for almost each
  $x\in \Omega$ (here $r^+:=
 \operatorname{ess\,sup} _{x\in \Omega}r(x)$).
Using this inequality we obtain
  \begin{equation} \label{EJDEeq20}
\begin{aligned}
&\int_{t_0-R}^{\tau}\int_\Omega
 \Big\{\sum_{i=1}^{n}a_{i,12}(u_{12})_{x_{i}}+
 a_{0,12}\,u_{12}\Big\}\eta^sdx\,dt
 \\
&\geq  C_5\int_{t_0-R}^{\tau}\int_\Omega
 \Big(\sum_{i=1}^{n}|(u_{12})_{x_i}|^{p_i(x)}+
 |u_{12}|^{p_0(x)}\Big)\eta^s\,dx\,dt,
 \end{aligned}
\end{equation}
 where  $C_5>0$ is a constant depending only on
  $K_1$ and $p_j^+\,(j=0,\ldots,n)$.

 Further we need the  inequality
 \begin{equation}\label{EJDEYunga_ab}
 a\,c\leq\varepsilon |a|^{q}+\,{\varepsilon^{-1/(q-1)}}\,|c|^{\,q'},\quad
  a,c\in \mathbb{R},\; q>1,\; 1/q+1/q'=1,\; \varepsilon>0,
 \end{equation}
 which is a corollary from  standard
  Young's inequality:
 $a\,c\leq  {|a|^{q}}/{q}+{|c|^{q'}}/{q'}$.

 Putting (for almost each  $x\in \Omega$)
 $q=p_0(x)/2$, $q'=p_0(x)/(p_0(x)-2)$,
 $a=|u_{12}|^{2}\eta^{s/q}$,
 $c=b\eta^{s/q'-1}$,
 $\varepsilon=\varepsilon_1>0$,
 under \eqref{EJDEYunga_ab} we obtain
\begin{equation} \label{EJDEeq21}
\begin{aligned}
&\int_{t_0-R}^{\tau}\int_\Omega b|u_{12}|^{2}\eta^{s-1}dx\,dt\\
&\leq  \varepsilon_1
  \int_{t_0-R}^{\tau}\int_\Omega
  |u_{12}|^{p_0(x)}\eta^s\,dx\,dt
  + \varepsilon_1^{-2/(p_0^--2)} \\
&\quad\times   \Big(\operatorname{ess\,sup}_{x\in\Omega}
  |b(x)|^{p_0(x)/(p_0(x)-2)}\Big)
   \int_{t_0-R}^{\tau}\int_\Omega
 \eta^{s-p_0(x)/(p_0(x)-2)}\,dx\,dt,
 \end{aligned}
\end{equation}
 where  $\varepsilon_1\in (0,1)$ is an arbitrary number.

  Again using inequality \eqref{EJDEYunga_ab}, we obtain
   \begin{equation} \label{EJDEeq22}
\begin{aligned}
&\int_{t_0-R}^{\tau}\int_\Omega
 \Big(\sum_{i=1}^{n}f_{i,12}(u_{12})_{x_i}+
 f_{0,12}u_{12}\Big)\eta^sdx\,dt \\
&\leq   \varepsilon_{2}
 \int_{t_0-R}^{\tau}\int_\Omega
   \Big(\sum_{i=1}^{n}|(u_{12})_{x_i}|^{p_i(x)}+
 |u_{12}|^{p_0(x)}\Big)\eta^s\,dx\,dt \\
&\quad+ \int_{t_0-R}^{\tau}\int_\Omega
 \Big(\sum_{j=0}^{n}
 \varepsilon_{2}^{-1/(p_j^--1)}|f_{j,12}|^{{p_j}'(x)}\Big) \eta^sdx\,dt,
 \end{aligned}
\end{equation}
where $\varepsilon_{2}\in (0,1)$ is an arbitrary number.

 From \eqref{EJDEeq19z} using \eqref{EJDEeq20}, \eqref{EJDEeq21},
 \eqref{EJDEeq22},
 if  $\varepsilon_1$ and $\varepsilon_2$ are sufficiently small,
 we obtain 
 \begin{equation} \label{EJDE19}
\begin{aligned}
&\eta^s(\tau)\int_{\Omega_R}b(x)|u_{12}(x,\tau)|^2\,dx+
 \int_{t_0-R}^{\tau}\int_\Omega
 \Big\{\sum_{i=1}^{n}|(u_{12})_{x_{i}}|^{p_i(x)}+
 |u_{12}|^{p_0(x)}\Big\}\eta^s\,dx\,dt \\
&\leq  C_{6}\Big[ \int_{t_0-R}^{\tau}\int_\Omega
  \eta^{s-p_0(x)/(p_0(x)-2)}\,dx\,dt +
 \int_{t_0-R}^{\tau}\int_\Omega
 \Big(\sum_{j=0}^{n}|f_{j,12}|^{{p_j}'(x)}\Big)\eta^s\,dx\,dt\Big],
\end{aligned}
\end{equation}
 where $C_{6}$ is a positive constant depending only
  on $K_1$ and $p_j^{\pm}\,(j=0,\ldots,n)$.

 Note that  $ 0\leq \eta(t)\leq R$, if $t\in[t_0-R,t_0]$,
 and $\eta(t)\geq R-R_0$,
 if  $t\in [t_0-R_0,t_0]$,
 where $R_0\in(0,R)$ is an arbitrary number.
 Using this and that  $R\geq\max\{1;2R_0\}$ (then, in particular, we have
  $R/(R-R_0)=1+R_0/(R-R_0)\leq2$),  from \eqref{EJDE19}
  we obtain the required statement.
 \end{proof}

 \section{Proof of the main results}\label{sect3}

\begin{proof}[Proof of Theorem \ref{thm1}]
 First we prove that there exists at most one weak solution of problem
 \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}.
 Assume the contrary. Let $u_1,\,u_2$ be (distinct)
 weak solutions of this problem.
 Using Lemma \ref{lem2}  we obtain
 \begin{equation}\label{EJDEyed}
\int_{t_0-R_0}^{t_0} \int_\Omega |u_1-u_2|^{p_0(x)}\,dx\,dt
  \leq
 C_{4}R^{-2/(p_0^+-2)},
 \end{equation}
where $R,\, R_0,\,t_0$ are  arbitrary numbers such that $R\geq 1$,
  $0<R_0<R/2$, $t_0\in\mathbb{R}$.

We fix arbitrary numbers $R_0>0$, $t_0\in \mathbb{R}$,
 and  take the limit when $R\to
 +\infty$  in \eqref{EJDEyed}. As a result we receive that
 $u_1=u_2$ almost everywhere on $Q_{t_0-R_0,t_0}$.
 Since $R_0$ and $t_0$ are arbitrary numbers, we obtain $u_1=u_2$
 almost everywhere on $Q$.
 The obtained contradiction proves our statement.

Now we are turn to the proof
of the existence of a weak solution of problem
\eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}. For each $m\in \mathbb{N}$
we consider an initial-boundary
 value problem for equation  \eqref{EJDEp.4.2.5}
in the domain $Q_m=\Omega \times (-m,+\infty)$
with a homogeneous initial condition and
boundary conditions \eqref{EJDEp.4.2.6}, namely:
we are searching a function
$u_m\in \widetilde{W}^{1,0}_{p(\cdot),loc}(\overline{Q_m})\cap
C([-m,+\infty);\widetilde{H}^b (\Omega))$ which satisfies the initial condition:
$b^{1/2}u_m|_{t=-m}=0$ and the integral equality
\begin{equation} \label{eq18}
\begin{aligned}
&\iint_{Q_m} \Big\{\big( \sum_{i=1}^n a_i|u_{m,x_i}|^{p_i(x)-2} u_{m,x_i} \psi_{x_i}
+ a_0|u_m|^{p_0(x)-2}u_m\psi\big)\varphi -bu_m\psi\varphi' \Big\}\, dx\,dt \\
&= \iint_{Q_m} f_m\psi\varphi\,dx\,dt
\end{aligned}
\end{equation}
for each $\psi \in\mathbb{V}_p$, $\varphi \in C_0^1(-m,+\infty)$,
where $f_m(x,t):=f(x,t)$ if  $(x,t) \in Q_m$,
and $f_m(x,t):=0$ if  $(x,t) \in Q \setminus Q_m$.
The existence and uniqueness of the
 function $u_m$ follows from a well-known fact (see, for example,
\cite{fu-pan-2010}).

We extend $u_m$ on $Q$ by zero and this extension is denoted  by $u_m$ again.
Further we prove that the sequence $\{ u_m \}$ converges in  
$\mathbb{U}_{p,\rm loc}^b$ 
to a weak solution of problem \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}.
Indeed, note that for
each $m\in \mathbb{N}$ the fuction $u_m$ is a weak solution of the
problem which  differs
 from problem \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}
 in $f_m$ instead of    $f$.
 Using Lemma \ref{lem2} for each natural numbers $m$ and  $k$ we have
\begin{equation} \label{eq19}
\begin{aligned}
&\max_{t\in [t_0,t_0-R_0]}\int_{\Omega}b(x)|u_m(x,t) - u_k(x,t)|^2 dx
\\
&+\int_{t_0-R_0}^{t_0}\int_{\Omega}\Big[ \sum_{i=1}^n |u_{m,x_i}-u_{k,x_i}|^{p_i(x)} +
|u_m - u_k|^{p_0(x)} \Big]\,dx\,dt
\\
&\leq C_4\Big\{ R^{-2/(p_0^+-2)}+
\int_{t_0-R}^{t_0}\int_{\Omega} |f_m-f_k|^{p_0(x)} \,dx\,dt \Big\},
\end{aligned}
\end{equation}
where $R,R_0,t_0$ are arbitrary numbers such
that $t_0\in R,\ R\geq 1,\ 0<R_0<R/2$.

We show that for fixed  $t_0$ and $R_0$
 the left side of inequality \eqref{eq19} converges to zero
when  $m,k\to +\infty$. Actually,
let $\varepsilon>0$ be an arbitrary  small number.
We choose $R\geq \max\{1,2R_0\}$ to be big enough such
that the following inequality holds
\begin{equation}\label{eq20}
C_4 R^{-2/(p_0^+-2)}<\varepsilon.
\end{equation}
This is possible as $p_0^+-2>0$.
Under
 \eqref{eq20}  for arbitrary $m,k\in\mathbb{N}$ such that $\max\{ -m,-k \}\leq t_0 -R$
 (then $f_m=f_k$ almost everywhere on $\Omega\times (t_0-R,t_0)$)
the right side of inequality \eqref{eq19} is
 less than  $\varepsilon$. From this it follows
 that the restriction of the terms of the sequence  $\{u_m\}$ on
 $Q_{t_0-R_0,t_0}$ is a Cauchy sequence
 in   $\widetilde{W}^{1,0}_{p(\cdot)}(Q_{t_0-R_0,t_0})\cap
 C([t_0-R_0,t_0];\widetilde{H}^b (\Omega))$. 
   Therefore, since $t_0$ and  $R_0$ are arbitrary,
    it follows that there exists a function
    $u\in \mathbb{U}_{p,\rm loc}^b$ such that 
$u_m\to u$ in $\mathbb{U}_{p,\rm loc}^b$.
Assuming that in \eqref{eq18} the
integration on $Q_m$ can be replaced by integration on $Q$,
we take the limit of this equality for $m\to \infty$.
As a result we obtain \eqref{EJDEtotozh}
for all $\psi \in\mathbb{V}_p$ and
 $\varphi \in C_0^1(\mathbb{R})$.
It means that the function $u$ is a weak solution of  problem
 \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}.
  Estimate \eqref{EJDEotsin_rozv} directly
 follows from   Lemma \ref{lem2} putting
 $u_1=u$, $u_2=0$, $f_{0,1}=f$, $f_{i,1}=0$ $(i=1,\ldots,n)$, $f_{j,2}=0$
 $(j=0,\ldots,n)$. Continuous dependence of  a weak solution of problem
    \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}
    on input data is easily proved using  Lemma \ref{lem2}
  with $u_k$ and $f_k$ instead of   $u_1$ and
  $f_{0,1}$ respectively, and also $u$
  and $f$ instead of  $u_2$ and $f_{0,2}$ respectively, putting
    $f_{i,1}=f_{i,2}=0\, (i=1,\ldots,n)$. 
\end{proof}

The Proofs of Corollaries \ref{coro1}--\ref{coro3}
follow from estimate  \eqref{EJDEotsin_rozv}. 

\begin{proof}[Proof of Theorem \ref{thm2}]
 Let $u$ denote a weak solution of
 problem \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}.
Put $u^{(\mu)}(x,t):=u(x,t+\mu)$, $f^{(\mu)}(x,t):=f(x,t+\mu)$,
$a_j^{(\mu)}(x,t):=a_j(x,t+\mu)$, $(x,t) \in Q$, where  $\mu \in \mathbb{R}$.
Replace variable  $t$ by  $t+\mu$ ($\mu \in \mathbb{R}$
 is arbitrary at present) in \eqref{EJDEtotozh}.
 As a result we obtain an identity which we will write in the form
\begin{equation} \label{eq21}
\begin{aligned}
&\iint_Q\Big\{ \big( \sum_{i=1}^n a_i^{(0)}
|u_{x_i}^{(\mu)}|^{p_i(x)-2} u_{x_i}^{(\mu)}\psi_{x_i} +
 a_0^{(0)} |u^{(\mu)}|^{p_0(x)-2} u^{(\mu)}\psi \big)\varphi \\
&\quad - bu^{(\mu)}\psi\varphi' \Big\}\,dx\,dt \\
&=\iint_Q
\Big( \sum_{i=1}^n (a_i^{(0)}-a_i^{(\mu)})
|u_{x_i}^{(\mu)}|^{p_i(x)-2} u_{x_i}^{(\mu)}\psi_{x_i} \\
&\quad +(a_0^{(0)}-a_0^{(\mu)}) |u^{(\mu)}|^{p_0(x)-2} u^{(\mu)}\psi \Big)
\varphi \,dx\,dt
+\iint_Q f^{(\mu)}\psi\varphi\,dx\,dt
\end{aligned}
\end{equation}
for all $\psi \in\mathbb{V}_p, \ \varphi \in C_0^1(\mathbb{R})$.
From this, putting $\mu=\sigma$ and using periodicity
of the functions $a_j\,(j=0,\ldots,n)$ and $f$,
we obtain that the function $u^{(\sigma)}$ is a weak solution of problem
 \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6}. Taking this into consideration
 and the fact of uniqueness of
 a weak solution of the
 problem \eqref{EJDEp.4.2.5}, \eqref{EJDEp.4.2.6},
  we get  $u^{(0)}=u^{(\sigma)}$
 almost everywhere on $Q$.
 Therefore the statement of Theorem \ref{thm2} is proved.
\end{proof}


\begin{proof}[Proof of  Theorem \ref{thm3}]
 Similarly as in the proof of Theorem \ref{thm2} we arrive to  equality
\eqref{eq21}.
Let $\delta_*:=\min \{ 1;K_1/2\}$ and $ \sigma \in F_{\delta_*}$,
where $F_\varepsilon$ is defined in the formulation of given theorem.
 We consider the identity \eqref{eq21} at first
 for  $\mu =0$ and afterwards for  $\mu=\sigma$.
Then using Lemma \ref{lem2} with  $u_1 = u^{(0)}$,
$u_2 = u^{(\sigma)}$, $a_j = a_j^{(0)}$ $(j=0,\ldots,n)$,
$f_{0,1} = f^{(0)}$,
$f_{0,2}=(a_0^{(0)} - a_0^{(\sigma)}) | u^{(\sigma)} | ^{p_0(x)-2} u^{(\sigma)} 
+ f^{(\sigma)}$,
$f_{i,1} = 0 $, $f_{i,2}=(a_i^{(0)} - a_i^{(\sigma)})
 | u_{x_i}^{(\sigma)} | ^{p_i(x)-2} u_{x_i}^{(\sigma)}$, 
$(i=1,\ldots,n)$,
 $t_0 =\tau \in \mathbb{R}$,
 $ R_0=1$, $R=l \in \mathbb{N}$  $(l\geq 2)$, we obtain
\begin{equation} \label{eq22}
\begin{aligned}
&\max_{t\in [\tau-1,\tau]}\int_{\Omega}b(x)|u^{(\sigma)}(x,t) - u^{(0)}(x,t)|^2 dx\\
& + \int_{\tau-1}^{\tau}\int_{\Omega}
 \Big[ \sum_{i=1}^n |u_{x_i}^{(\sigma)}-u_{x_i}^{(0)}|^{p_i(x)}
 + |u^{(\sigma)} - u^{(0)}|^{p_0(x)} \Big]\,dx\,dt\\
&\leq C_4 \Big(l^{-2/(p_0^+-2)}
+ \int_{\tau-l}^{\tau}\int_{\Omega} \Big\{ \big(|f^{(\sigma)}-f^{(0)}|
+|a_0^{(\sigma)}-a_0^{(0)}| |u^{(\sigma)}|^{p_0(x)-1}\big)^{{p_0}'(x)}\\
&\quad + \sum_{i=1}^n | a_i^{(\sigma)}-a_i^{(0)}
 |^{{p_i}'(x)}\cdot |u_{x_i}^{(\sigma)}|^{p_i(x)}\Big\}\,dx\,dt\Big).
\end{aligned}
\end{equation}
From the inequality
$
(a+c)^{q}\leq 2^{q-1}(a^{q}+c^{q})$, $a\geq 0$, $c\geq 0$, $q\geq 1$,
we have
\begin{equation} \label{eq23}
\begin{aligned}
&\int_{\tau-l}^{\tau}\int_{\Omega}  \big(|f^{(\sigma)}-f^{(0)}|+
|a_0^{(\sigma)}-a_0^{(0)}| |u^{(\sigma)}|^{p_0(x)-1}\big)^{{p_0}'(x)}\,dx\,dt
\\
&\leq 2^{1/(p_0^- -1)}\int_{\tau-l}^{\tau}\int_{\Omega}\big(|f^{(\sigma)}-f^{(0)}|^{{p_0}'(x)}
+|a_0^{(\sigma)}-a_0^{(0)}|^{{p_0}'(x)} |u^{(\sigma)}|^{p_0(x)}\big)\,dx\,dt
\\
&\leq 2^{1/(p_0^- -1)}\int_{\tau-l}^{\tau}\int_{\Omega}
|f^{(\sigma)}-f^{(0)}|^{{p_0}'(x)}\,dx\,dt
\\
&\quad +\Big(\sup_{t\in \mathbb{R}}\,
\|a_0^{(\sigma)}(\cdot,t) - a_0^{(0)}(\cdot,t)\|_{L_\infty(\Omega)} \Big)^{(p_0^+)'}
\int_{\tau-l}^{\tau}\int_{\Omega} |u^{(\sigma)}|^{p_0(x)}\,dx\,dt,
\end{aligned}
\end{equation}
\begin{equation} \label{eq24}
\begin{aligned}
&\int_{\tau-l}^{\tau}
\int_{\Omega}\Big( \sum_{i=1}^n |a_i^{(\sigma)}-a_i^{(0)}|^{{p_i}'(x)}\cdot
|u_{x_i}^{(\sigma)}|^{p_i(x)} \Big)\,dx\,dt
\\
&\leq \max _{i\in \{ 1,\dots ,n \}}
\Big(\sup_{t\in \mathbb{R}}\,
\|a_i^{(\sigma)}(\cdot,t) - a_i^{(0)}(\cdot,t)\|_{L_\infty(\Omega)} \Big)^{(p_i^+)'}
\int_{\tau-l}^{\tau}\int_{\Omega} \sum_{i=n}^n |u^{(\sigma)}_{x_i}|^{p_i(x)}\,dx\,dt,
\end{aligned}
\end{equation}
where $(p^{+}_j)':= p^{+}_j/(p^{+}_j -1)$ $(j=0,\ldots,n) $.


Since $\sigma\in F_{\delta_*}$
and $f$ is Stepanov almost periodic, it follows that
$a^{(\sigma)}(x,t)\geq K_1/2$ $(j=0,\ldots,n)$  for a. e.  $(x,t) \in Q$  and
$\sup_{s\in \mathbb{R}}
\int_{s-1}^{s}\int_{\Omega} |f^{(\sigma)}(x,t)|^{p_0(x)}\,dx\,dt\leq C_6$,
where $C_6>0$ is a constant independent on $\sigma$.
From this under Corollary \ref{coro2} we have
\begin{equation}\label{eq25}
\sup_{s\in \mathbb{R}} \int_{s-1}^{s}\int_{\Omega} \Big[ |u^{(\sigma)}|^{p_0(x)} +
 \sum_{i=1}^{n}|u^{(\sigma)}_{x_i}|^{p_i(x)} \Big]\,dx\,dt \leq C_7,
\end{equation}
where  $C_7>0$ is a constant independent of $\sigma$.
Thus, from \eqref{eq22} using \eqref{eq23} and \eqref{eq24}, we obtain
\begin{equation} \label{eq26}
\begin{aligned}
&\int_{\Omega}b(x)|u^{(\sigma)}(x,\tau) - u^{(0)}(x,\tau)|^2 dx \\
&+ \int_{\tau-1}^{\tau}\int_{\Omega}\Big[ \sum_{i=1}^n |u_{x_i}^{(\sigma)}-
u_{x_i}^{(0)}|^{p_i(x)} + |u^{(\sigma)} - u^{(0)}|^{p_0(x)} \Big]\,dx\,dt
\\
&\leq C_8 \Big\{ l^{-2/(p_0^+-2)} +
\sum_{k=1}^l \int_{\tau-k}^{\tau-k+1}\int_{\Omega}
|f^{(\sigma)} - f^{(0)}|^{{p_0}'(x)}\,dx\,dt
\\
&\quad + \max_{j\in \{ 0,\dots,n \}}
\Big( \sup_{t\in \mathbb{R}} || a_j^{(\sigma)} (\cdot,t)
- a_j^{(0)} (\cdot,t) ||_{L_{\infty}(\Omega)} \Big)^{(p_j^+)'}
\sum_{k=1}^l \int_{\tau-k}^{\tau-k+1}
\int_{\Omega}\Big[ |u^{(\sigma)}|^{p_0(x)}
\\
&\quad +\sum_{i=1}^n |u_{x_i}^{(\sigma)}|^{p_i(x)} \Big]\,dx\,dt \Big\},
\end{aligned}
\end{equation}
where $C_8$ is a  constant independent of $\tau,\sigma$ and $l$.

Let $\varepsilon>0$ be an arbitrary small fixed number.
 We show that the set
\begin{align*}
U_{\varepsilon}:= \Big\{& \sigma\in \mathbb{R}:
 \sup_{t\in \mathbb{R}} \int_{\Omega}
b(x)|u(x,t+\sigma) - u(x,t)|^2\,dx \leq \varepsilon,
\\
&\sup_{\tau\in\mathbb{R}} \int_{\tau-1}^{\tau}\int_{\Omega}
\Big[ \sum_{i=1}^n |u_{x_i}(x,t+\sigma) - u_{x_i}(x,t)|^{p_i(x)} \\
&+ |u(x,t+\sigma) - u(x,t)|^{p_0(x)}| \Big]\,dx\,dt\leq\varepsilon
\Big\}
\end{align*}
contains a set $F_{\delta}$ for
some $ \delta \in (0,\delta_*]$ implying the relative density of the 
set $U_{\varepsilon}$.
Indeed, choose big enough
$l\in \mathbb{N} \,(l\geq 2)$ satisfying the  inequality
\begin{equation}\label{eq27}
C_8 l^{-2/(p_0^+-2)} \leq \varepsilon/2,
\end{equation}
and fix this value $l$.
Then take $\delta \in (0,\delta_*]$
such that the following inequality remains true
\begin{equation}\label{eq28}
C_8\Big(\delta+\max_{j\in \{ 0,\dots,n \}}\delta^{(p_j^+)'}C_7 \Big)l
\leq \varepsilon/2.
\end{equation}
Therefore, if  $\delta \in F_{\delta}$,  then the right side of the
inequality \eqref{eq26} is less than or equal to $\varepsilon$.
This implies that
$F_{\delta}\subset U_{\varepsilon}$, that is the fact we had to prove.
\end{proof}

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\end{document}