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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 178, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/178\hfil Dynamical problems]
{Dynamical problems without initial conditions for
elliptic-parabolic equations in spatial unbounded domains}

\author[M. Bokalo\hfil EJDE-2010/178\hfilneg]
{Mykola Bokalo}

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

\thanks{Submitted October 28, 2010. Published December 22, 2010.}
\subjclass[2000]{34A09, 34G20, 35B15, 35K65, 47J35}
\keywords{Problems without initial conditions;
degenerate implicit equations;
\hfill\break\indent nonlinear evolution equations}

\begin{abstract}
 We consider a problem without initial conditions for degenerate
 nonlinear evolution equations with nonlinear dynamical boundary
 condition in spatial unbounded domains.
 We obtain sufficient conditions for the well-posedness
 of this problem without any restrictions at infinity.
\end{abstract}

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

\section{Introduction}\label{Sect1}

For $x=(x_1,\dots,x_n)\in \mathbb{R}^n$, with $n\geq 1$, consider
Euclidean norm $|x|:=(|x_1|^2+\dots+x_n|^2)^{1/2}$.
Let $\Omega$ be a domain in $\mathbb{R}^n $ with
 boundary $\partial\Omega$ which is  a $C^1$ manifold of
dimension $n-1$.
Let $\Gamma_0$ be the closure of an open set on
$\partial\Omega$ (in particular, $\Gamma_0=\partial\Omega$ or
$\Gamma_0$ is an empty set),
$\Gamma_1:=\partial\Omega\setminus{\Gamma_0} $.
Let $\nu=(\nu_1,\dots,\nu_n)$ be the unit vector of the outer normal
to $\partial\Omega$. Let $S$ be either  $(-\infty,0]$,
$(-\infty,+\infty)$ or $(0,1]$. Put $Q := \Omega\times S, $
$\Sigma_0 :=\Gamma_0 \times S$,  $\Sigma_1 := \Gamma_1\times S$.
We will use this notation hereafter.
Also we will assume that all quantities in this article
are real-valued.

Consider the  problem:
\begin{quote}
Find a function
$u:\overline\Omega\times \overline S\to\mathbb{R}$ such that
 \begin{equation}\label{1}
  \frac{\partial}{\partial t}(b_1(x)u)-\sum_{i=1}^{n}
  \frac{d}{dx_i}a_i(x,t,u,\nabla u)+ a_0(x,t,u,\nabla u)=
  f_1(x,t),\quad (x,t)\in Q   ,
   \end{equation}
 \begin{equation}\label{2}
  u(y,t)=0, \quad (y,t)\in \Sigma_0,
 \end{equation}
 \begin{equation} \label{3}
  \frac{\partial}{\partial t}(b_2(y) u) + \sum_{i=1}^{n}
  a_i(y,t,u,\nabla u) \nu_i(y)+c(y,t,u)= f_2(y,t), \quad  (y,t)\in
  \Sigma_1,
   \end{equation}
 and if $S=(0,1]$ then in addition
 \begin{equation}\label{4}
\begin{gathered}
 b_1(x)(u(x,t)-u_1(x))|_{t=0}=0,\quad x\in \Omega, \\
 b_2(y)(u(y,t)-u_2(y))|_{t=0}=0, \quad y\in \Gamma_1,
\end{gathered}
 \end{equation}
where $a_i$ $(i=\overline{0,n})$,
$c,f_1,f_2,u_1,u_2$, $b_1 \geq 0$, $b_2\geq 0$
are  given real-valued functions.
\end{quote}

We allow at least one of following two relations to hold:
$b_1=0$ and $b_2\ne 0$ on subsets of $\Omega$,
or $\Gamma_1$ is of nonzero measure.
Moreover we assume that the space
part of the differential expression in the left side of
\eqref{1} is nonlinear elliptic. Thus the partial differential
equation \eqref{1} is parabolic at those $x\in \Omega$ for which
$b_1(x)>0$ and elliptic where $b_1(x)=0$. Note that boundary
condition \eqref{3} is dynamical on the subset of $\Gamma_1$ where
$b_2>0$ and of the second type on the other part of $\Gamma_1$.

Further we will call this problem: problem
\eqref{1}--\eqref{4} if $S=(0,1]$, and  problem
 \eqref{1}--\eqref{3} provided that $S$ is either
$(-\infty,0]$ or $(-\infty,+\infty)$.

In the case when $S=(0,1]$ and $\Omega$ is bounded ,
\eqref{1}--\eqref{4}
  can be considered as the Cauchy problem
for an implicit evolution equation of the form
 \begin{equation}\label{FirstEq1}
 \bigl({\mathcal{B}} u(t)\bigr)'+{\mathcal{A}}\bigl(t, u(t)\bigr)=f(t),\quad t\in S,
 \end{equation}
where ${\mathcal{A}}(t,\cdot)$ and ${\mathcal{B}}$ are some operators
(see, e.g., \cite{Showalter69}). The well-posedness  of this problem
 has been studied extensively
by many authors \cite{Bahuguna, Gajewski, Kuttler, Kuttler_Shillor, Lions,
Showalter69, Showalter77, Showalter97}.
  Note if ${\mathcal{B}}$ is linear and ${\mathcal{A}}$ is
either linear or nonlinear, the monographs by  Showalter
\cite{Showalter77,Showalter97} give  sufficient
conditions for existence and uniqueness of solutions of the Cauchy
problem for equation \eqref{FirstEq1}.

If $S=(0,1]$ and $\Omega$ is unbounded then it is  known that for
linear and  some quasilinear  equations \eqref{1}
the existence and uniqueness  of solutions of
\eqref{1}--\eqref{4} need  the additional assumptions
on the growth of the data-in  and the behavior of a solution
as $|x| \to +\infty$ \cite{Oleinik_Iosifjan, Shishkov}. Though such
assumptions are not necessary for some  nonlinear equations
\cite{Bernis, Brezis, Gladkov_Guedda}.

In the case when $S$ is either $(-\infty,0]$ or
$(-\infty,+\infty)$ the initial conditions \eqref{4} are missed
and the problem \eqref{1}--\eqref{3} is called the problem without
initial conditions for evolutional equations. These problems arise
when describing different non-stationary processes in nature under
hypothesis  that we consider so distant the initial time that the
initial condition practically has no influence on present time,
while boundary conditions do affect it. Thus we can assume that
either $t=0$ or $\infty$ is the {\it final time}, while
$t=-\infty$ is the initial time. Sufficiently full survey of
results on the problem can be found in \cite{Bokalo_Lorenzi}. Here
we recall those results which are close to our investigation.

First of all recall that  when dealing with problems
 related to equations in the form \eqref{1} different approaches are needed
subject to whether the domain $\Omega$  is either bounded or not.
 In first case the problem without initial conditions
\eqref{1}--\eqref{3} may be written in the form \eqref{FirstEq1}
(recall that $S$ is either $(-\infty,0]$ or $(-\infty,+\infty)$
now). It is known that when ${\mathcal{B}}$ is linear and
${\mathcal{A}}$ is either linear or almost linear this problem is
well-posed if in addition some restrictions on behaviour of the
solution and growth  of data-in as $t$ goes to $-\infty$ are
imposed \cite{Bokalo90, Bokalo_Lorenzi, Hu, Pankov, Vafodorova}.
Same results were obtained in \cite{Showalter80} when
${\mathcal{A}}$ may be set-valued. But the papers \cite{Bokalo90,
Bokalo_Dmytryshyn} and others
 imply that for some nonlinear operators
 ${\mathcal{A}}$ equation \eqref{FirstEq1} (with linear ${\mathcal{B}}$)
 admits a unique solution without any restrictions on its behaviour at
$-\infty$ and the growth of the right-hand side as
$t\to-\infty$ (that is, in the class
of locally integrable functions on $S$).

When $\Omega$ is unbounded,  a solution of the
  problem \eqref{1} -- \eqref{3} cannot always be identified with
 the solution of the abstract equation \eqref{FirstEq1}. In addition
 a question about additional conditions on behaviour of the solution both
 as  $t\to-\infty$ and
 as $|x|\to +\infty$ can arise. The answer to this question can be both
 positive  and negative.
 The former one  is in case of linear and almost linear equations
  \cite{Bokalo94, Oleinik_Iosifjan} (in the case of equations in the forms \eqref{1} and
  \eqref{3} there are
  $b_1=1$, $b_2=0$ respectively). The letter  may be only
 for some nonlinear evolution  equations.  The results of such kind were obtained in
 \cite{Bokalo_Pauchok} for the
 equations in the form \eqref{1} provided $b_1=1$, and when $\Gamma_0=\partial\Omega$
 (that is,  condition \eqref{3} is missed).

In this paper, we generalize the results of \cite{Bokalo_Pauchok}
for the problem \eqref{1}-\eqref{3} when possible  $b_1=0$ on
nonzero measure subset of $\Omega$ and
 $b_2\ne 0$ on nonzero surface measure subset of $\Gamma_1$. We obtain
sufficient conditions for existence and uniqueness  of solutions of this problem
  without  an
additional assumptions on the behavior of the solutions and data-in
 both as $t\to-\infty$ and
as $|x|\to +\infty$. We also establish the continuous
dependence on
 data-in of  solutions of this problem.

Our paper is organized as follows.  In Section~\ref{Sect2} we
 state a problem and formulate the main results.
Section~\ref{Sect3} is devoted to some
 auxiliary statements needed in the sequel.
We prove our main results in Section~\ref{Sect4}.

\section{Statement of the problem and main results}\label{Sect2}

Let us assume hereafter that $\Omega$ is unbounded and $S$ is the
interval $(-\infty,0]$. Suppose that $0\in \Omega$ and, for every
$R>0$,  $\Omega_R$ is  the connected component of the set
$\Omega\cap\{x:|x|<R\}$ containing $0$. For arbitrary $R>0$ denote
$\Gamma_{0,R} := \overline{\partial\Omega_R\setminus\Gamma_1}$,
$\Gamma_{1,R} :=\partial\Omega_R\setminus\Gamma_{0,R}$;
$S_R:=(-R,0]$, $Q_R := \Omega_R\times S_R$,
 $\Sigma_{1,R} := \Gamma_{1,R}\times S_R$.

Let $p>2$, $q>2$ be real numbers which remain invariable
throughout the paper. Denote $p':=p/(p-1)$, $q':=q/(q-1)$.

Hereafter we use some linear locally convex spaces which are
introduced here.

Let $G$ is either a domain or a regular surface in $\mathbb{R}^k$
for either $k=n$ or $k=n+1$, and let $Bs(G)$ be the set consisting
of bounded measurable  subsets of $G$. For each $r\in[1,\infty]$
define
 $$
  L_{r,{\rm loc}}(\overline G) := \{v(z),z\in G  |  v\in L_r(G')
\text{ for all }  G'\in Bs(G)\}.
 $$
It is obvious that $L_{r,{\rm loc}}(\overline G)=L_{r}(G)$ when
$G$ is a bounded set.
 Suppose that on the space
$L_{r,{\rm loc}}(\overline G)$ there are introduced the standard
linear operations and the system of semi-norms
$\{\|\cdot\|_{L_r(G')} |  G' \in Bs(G)\}$. In particular, it
means that the sequence $\{v_k\}_{k=1}^\infty$ converges to $v$ in
$L_{r,{\rm loc}}(\overline G)$ provided the sequence
 $\{v_k|_{G'}\}_{k=1}^\infty$ converges to
$v|_{G'}$ in $L_r(G')$ for every $G' \in Bs(G)$. (Hereinafter for
the function $g$ defined on $G$ and a subset $G'$ of the set $G$
the notation $g|_{G'}$  means the restriction of $g$ on $G'.$)

Define $ L_{r,{\rm loc}}^{0,+}(\overline G)$ be the subset of $
L_{r,{\rm loc}}(\overline G)$ consisting of nonnegative functions,
and $ L_{r,{\rm loc}}^{+}(\overline G)$ be the subset of $
L_{r,{\rm loc}}^{0,+}(\overline G)$ whose each element $g$ is a
function such that $\mathop{\text{ess}\inf}_{z\in G'} g(z)>0$ for
all bounded  $G'\subset G$.

Let
 $$
  H^1_{\rm loc}(\overline \Omega) :=
  \{v\in L_{2,{\rm loc}}(\overline \Omega) |
  v|_{\Omega_R}\in H^1(\Omega_R)\quad\text {for all}\quad R>0\}
 $$
 with the system of semi-norms
$\{\|\cdot\|_{H^1(\Omega_R))} |  R>0\}$.
(Hereinafter $H^1(\widetilde{\Omega}):=\{v\in
L_2(\widetilde{\Omega}) |
   v_{x_i}\in L^2(\widetilde{\Omega}),
  i=\overline{1,n}\}$ is the Sobolev space with the norm
$\|v\|_{H^1(\widetilde{\Omega})}:=
(\|v\|_{L_2(\widetilde{\Omega})}^2+
\sum_{i=1}^n\|v_{x_i}\|_{L_2(\widetilde{\Omega})}^2)^{1/2}$
for any domain $\widetilde{\Omega}\subset \mathbb{R}^n$.)

Define
$$
L_{2,{\rm loc}}(S;H^1_{\rm loc}(\overline \Omega)):= \{v:S\to
H^1_{\rm loc}(\overline\Omega) |
  v\in L_2(S_R;H^1(\Omega_R))\quad\text{for all } R>0 \}
$$
with the system of semi-norms
 $\{\|\cdot\|_{L_2(S_R;H^1(\Omega_R))} |  R>0\}$.
Put
 $$
  C(S;L_{2,{\rm loc}}(\overline\Omega)) :=
  \{v:S\to L_{2,{\rm loc}}(\overline\Omega) |
  v\in C(S_R;L_2(\Omega_R))\quad\text{for all } R>0 \}
 $$
with the system of semi-norms
$\{\|\cdot\|_{C(S_R;L_2(\Omega_R))} |  R>0\}$,
 $$
  C(S;L_{2,{\rm loc}}(\overline{\Gamma_1})) :=
  \{w:S\to  L_{2,{\rm loc}}(\overline{\Gamma_1}) |
  v\in C(S_R;L_2(\Gamma_{1,R}))\quad\text{for all } R>0 \}
 $$
 with the system of semi-norms
$\{\|\cdot\|_{C(S_R;L_2(\Gamma_{1,R}))} |  R>0\}$.
Let
 $$
  \mathbb F_{\rm loc}:=\{(f_1,f_2) |  f_1\in
  L_{p',{\rm loc}}(\overline Q),
  f_2\in L_{q',{\rm loc}}(\overline {\Sigma_1}) \} \equiv  L_{p',{\rm loc}}
  (\overline Q)\times L_{q',{\rm loc}}(\overline {\Sigma_1})
 $$
with the topology generated by the Cartesian product of topological spaces.


The notation $\mathbb{V}_{\rm loc}$ means the linear locally
convex space obtained by the closure of the space $\{v\in
C^1(\overline \Omega): \operatorname{supp} v \in
Bs(\overline\Omega), \operatorname{dist}\{\operatorname{supp}v,
\Gamma_0\}>0 \}$ in the topology generated by the system of
semi-norms $\{\|\cdot\|_R
:=\|\cdot\|_{H^1(\Omega_R)}+\|\cdot\|_{L_p(\Omega_R)}+\quad$
$\|\cdot\|_{L_q(\Gamma_{1,R})} |  R>0 \}$. Note that
$\|\cdot\|_R$ is the norm for the space $H^1(\Omega_R)\cap
L_p(\Omega_R) \cap L _q(\Gamma_{1,R})$, where  $R>0$.

Now remark that since $\partial \Omega \in C^1$ then for every
element of $H^1_{\rm loc}(\overline \Omega)$ there exists its
(uniquely) defined trace on $\partial\Omega$, which is the element
of $ L_{2,{\rm loc}}(\overline{\partial\Omega}) $ and for every
smooth function on $\overline \Omega $ it coincides with
restriction of this function on the  $\partial\Omega$. Therefore,
taking into account the definition of the family of
 semi-norms on $\mathbb{V}_{\rm loc}$ (in particular, it follows
that $\mathbb{V}_{\rm loc}\subset H^1_{\rm loc}(\overline
\Omega)$),  we can conclude the proper
 definiteness, linearity and continuity  of  the operator $\gamma:\mathbb{V}_{\rm loc}\to
L_{q,{\rm loc}}(\overline{\Gamma_1})$ which is the restriction of
standard trace operator on the space $H^1_{\rm loc}(\overline
\Omega)$ to $\mathbb{V}_{\rm loc}$.
Put
 $$
 \mathbb{V}_c :=\{v\in \mathbb{V}_{\rm loc} | \operatorname{supp}
  v   \text{ is a bounded set}\}.
$$

Let us agree for every linear locally convex space  $W$ and
interval $I\subset \mathbb{R}$ to understand the $(I \to W)$ as
the linear space that is the factorization of the linear space of
mappings of the set $I$ to $W$ by  such equivalence relation that
two mappings are equivalent if their values coincide for almost
every value of the argument.

We will also need the space
\begin{align*}
  \mathbb{U}_{\rm loc}
:= \{& u\in (S\to\mathbb{V}_{\rm loc}) |   u\in
  L_{2,{\rm loc}}(S;H^1_{\rm loc}(\overline\Omega))\cap L_{p,{\rm loc}}
  (\overline Q),   b_1^{1/2}u \in  C(S;L_{2,{\rm loc}}
(\overline\Omega)),  \\
& \gamma u \in L_{q,{\rm loc}}(\overline{\Sigma_1}),
b_2^{1/2}\gamma u
  \in C(S;L_{2,{\rm loc}}(\overline {\Gamma_1})) \}
\end{align*}
with the topology generated by the system of semi-norms
\begin{align*}
\big\{\|u\|_R^* &=\|u\|_{L_2(S_R;H^1(\Omega_R))}+\|u\|_{L_p(Q_R)}+
\sup_{t\in S_R} \|b_1^{1/2}(\cdot)u(\cdot,t)\|_{L_2(\Omega_R)}\\
&\quad  +\|\gamma u\|_{L_q(\Sigma_{1,R})}+\sup_{t\in
S_R}\|b_2^{1/2}(\cdot) \gamma
u(\cdot,t)\|_{L_2(\Gamma_{1,R})} |   R>0\big\}.
\end{align*}


 Let $\mathbb B$ be the set of pairs $b=(b_1,b_2)$ of the functions
satisfying  the condition
\begin{itemize}
\item[(B)]
 $ b_1\in L_{p^*,{\rm loc}}(\overline\Omega)$,
 $b_1\ge 0$ on $\Omega$;
 $b_2\in L_{q^*,{\rm loc}}(\overline{\Gamma_1})$,
$b_2\ge 0$ on $\Gamma_1$,
 where $p^*=p/(p-2)$, $q^*=q/(q-2)$.
\end{itemize}
Consider the set whose any element  is an array
$(a_0,a_1,\dots,a_n)$ of
 $n+1$ real-valued functions  satisfying  the following conditions:
\begin{itemize}
\item[(A1)] for each  $i\in \{0,1,\dots,n\}$ 
the function $Q\times \mathbb{R} \times \mathbb{R}^n\ni (x,t,s,\xi)\to
a_i(x,t,s,\xi)$ is a Caratheodory; i.e., for all
$(s,\xi)\in \mathbb{R} \times \mathbb{R}^n \equiv
\mathbb{R}^{1+n}$ the function $a_i(\cdot,\cdot,s,\xi):Q\to
\mathbb{R}$ is Lebesgue measurable and for a.e. $(x,t)\in Q$ the
function $a_i(x,t,\cdot,\cdot): \mathbb{R}^{1+n}\to \mathbb{R}$ is
continuous;

\item[(A1')] $ a_i(x,t,0,0)=0 $ for a.e.
 $(x,t)\in Q$ and all $i\in \{0,1,\dots,n\}$;

\item[(A2)] for a.e. $(x,t)\in Q$ and every $(s,\xi)\in
\mathbb{R}^{1+n}$,
 $$
  |a_0(x,t,s,\xi)|\le h_{0,1}(x,t)(|s|^{p-1}+|\xi|^{2/p'})+
  h_{0,2}(x,t)
 $$
 where $h_{0,1}\in L_{\infty,{\rm loc}}^{+}(\overline Q),
h_{0,2}\in L_{p',{\rm loc}}^{0,+}(\overline Q)$;

\item[(A3)] for a.e. $(x,t)\in Q$ and every $
(s,\xi),(r,\eta)\in  \mathbb{R}^{1+n} $,
 $$
 \sum_{i=1}^n|a_i(x,t,s,\xi)-a_i(x,t,r,\eta)|\le d_1(x,t) |\xi-\eta|  +
    d_2(x,t)  |s-r|,
 $$
where $d_1\in L_{\infty,{\rm loc}}^{+}(\overline
Q) ,
 d_2 \in L_{\infty,{\rm loc}}^{0,+}(\overline Q)$ are arbitrary
 functions;

\item[(A4)] for a.e. $(x,t)\in Q$ and every
$(s,\xi),(r,\eta)\in \mathbb{R}^{1+n}$,
\begin{align*}
&\sum_{j=1}^{n}
  (a_i(x,t,s,\xi)-a_i(x,t,r,\eta))(\xi_i-\eta_i) +
  (a_0(x,t,s,\xi)-a_0(x,t,r,\eta))(s-r)\\
&\ge  \rho_1(x,t)  |\xi-\eta|^2 + \rho_2(x,t) |s-r|^p,
\end{align*}
 where $\rho_1,\rho_2\in L_{\infty,{\rm loc}}^+(\overline Q)$.
\end{itemize}

 On this set,  define
    an equivalence relation such that the element $(a_0,a_1,\dots,a_n)$
     is equivalent to the element $(\widetilde a_0,
\widetilde a_1,\dots, \widetilde a_n)$ if for every $i\in
\{0,1,\dots,n\}$ the equality $a_i(x,t,s,\xi)=\widetilde
a_i(x,t,s,\xi)$ holds for every $(s,\xi)\in
\mathbb{R}^{1+n}$ and a.e. $(x,t)\in Q$ . We denote by $\mathbb{A}$ 
the quotient-space obtained
 by this equivalence relation. We will not distinguish the notations of
 the elements of the space $\mathbb{A}$ (that are the classes of equivalent
 functions arrays) and their representatives. On the set $\mathbb{A}$
 introduce the notion of convergence in such a way that a sequence
  $\{(a_0^k,a_1^k,\dots,a_n^k)\}_{k=1}^{\infty}$  is convergent to
  $(a_0,a_1,\dots,a_n)$ in $\mathbb{A}$, provided
  \begin{equation}\label{zba}
\begin{aligned}
&\lim_{k\to\infty} \mathop{\rm ess\,sup}
  _{(x,t)\in Q'}\sup
  _{(s,\xi)\in \mathbb{R}^{1+n}}
  \Bigl[\sum_{i=1}^{n}  |a_i^k(x,t,s,\xi) -
  a_i(x,t,s,\xi)|/(1+|s|+|\xi|)\\
&\quad +|a_0^k(x,t,s,\xi)-a_0(x,t,s,\xi)|/(1+|s|^{p-1}
+|\xi|^{2/p'})\Bigr]=0 \end{aligned}
  \end{equation} for every
bounded domain $Q'\subset Q$.


\begin{remark} \label{rmk2.1} \rm
It is easy to show that if
$a_0(x,t,s,\xi)=\bar{a}_0(x,t)|s|^{p-2}s$,
$a_i(x,t,s,\xi)=\bar{a}_i(x,t)\xi_i$  $(i=\overline{1,n})$, where
$\bar{a}_j\in L_{\infty,{\rm loc}}^{+}(\overline Q)$
$(j=\overline{0,n})$, then the array $(a_0,a_1,\dots,a_n)$ is an
element of $\mathbb{A}$. Also note that for
$a_0^k(x,t,s,\xi)=\bar{a}_0^k(x,t)|s|^{p-2}s$,
$a_i^k(x,t,s,\xi)=\bar{a}_i^k(x,t)\xi_i$ $(i=\overline{1,n})$,
where $k\in\mathbb{N}$, $\bar{a}_j^k\in L_{\infty,{\rm
loc}}^{+}(\overline Q)$ $(j=\overline{0,n})$, the sequence
  $\{(a_0^k,a_1^k,\dots,a_n^k)\}_{k=1}^{\infty}$  is convergent to
  $(a_0,a_1,\dots,a_n)$ in $\mathbb{A}$ if and only if
 $\bar{a}_j^k\to\bar{a}_j$ in
$L_{\infty,{\rm loc}}(\overline Q)$ $(j=\overline{0,n})$.
\end{remark}

Consider the set of real-valued functions $c(y,t,s)$,
$(y,t,s)\in \Sigma_1\times \mathbb{R}$, satisfying the conditions:
\begin{itemize}

\item[(C1)] $c$ is a Caratheodory function, that is for every
$s\in \mathbb{R} $ the function
$c(\cdot,\cdot,s): \Sigma_1\to \mathbb{R}$
is Lebesgue measurable and for a.e. (in the sense of surface measure)
 $(y,t)\in\Sigma_1$ the function
$c(y,t,\cdot):\mathbb{R}\to\mathbb{R}$ is continuous;


\item[(C1')] $c(y,t,0)=0$ for a.e. $(y,t)\in \Sigma_1$;

\item[(C2)] for a.e.  $(y,t)\in \Sigma_1 $ and every
$s\in \mathbb{R} $,
$$
  |c(y,t,s)|\le g_1(y,t)  |s|^{q-1}+g_2(y,t),
$$
 where $g_1\in L_{\infty,{\rm loc}}^{+}(\overline{\Sigma_1}), g_2\in
 L_{q',{\rm loc}}^{0,+}(\overline{\Sigma_1})$;

\item[(C3)]  for a.e. $(y,t)\in \Sigma_1$
 and every $s,r\in \mathbb{R}$ the inequality
 $$
  (c(y,t,s)-c(y,t,r))(s-r)\ge \rho_3(y,t)|s-r|^q
 $$
is satisfied, where $\rho_3\in L_{\infty,{\rm loc}}^+
(\overline\Sigma_1)$ is some function.
\end{itemize}

On this set, introduce
    an equivalence relation such that two functions $c$ and $\widetilde c$ of the
     given set
    are equivalent if $c(y,t,s)=\widetilde c(y,t,s)$ for all $s\in \mathbb{R}$ 
    and for a.e.
     $(x,t)\in \Sigma_1$. We denote by
     $\mathbb C$ the obtained quotient-set. Define the convergence
     notation of the sequences of the elements of the set $\mathbb C$
      such that the sequence
$\{c^k\}_{k=1}^{\infty}$ is convergent to $c$ in $\mathbb C$ if
 \begin{equation} \label{zbb}
  \lim_{k\to\infty} \operatorname{ess\,sup}
  _{(y,t)\in \Sigma'} \sup
  _{s\in \mathbb{R}}   |c^k(y,t,s) - c(y,t,s)|/(1+|s|^{q-1})=0
 \end{equation}
for every bounded subset $\Sigma'\subset \Sigma_1$.

\begin{remark} \label{rmk2.2} \rm
It is easy to show that $c(x,t,s)=\bar{c}(x,t)|s|^{q-2}s$ is an
element of $\mathbb C$, when
$\bar{c}\in L_{\infty,{\rm loc}}^+(\overline \Sigma_1)$. Also note that for
$c^k(x,t,s)=\bar{c}^k(x,t)|s|^{q-2}s$, where $k\in\mathbb{N},
\bar{c}^k\in L_{\infty,{\rm loc}}^+(\overline Q)$, the sequence
  $\{c^k\}_{k=1}^{\infty}$  is convergent to
  $c$ in $\mathbb C$ if and only if
 $\bar{c}^k\to\bar{c}$ in
 $L_{\infty,{\rm loc}}(\overline \Sigma_1) $.
\end{remark}

\begin{remark}\label{remk1} \rm
Conditions {\rm (A1')} and {\rm (C1')} are not essential.
 Indeed, let any of them or both do not hold. Then it is sufficient
 to suppose that for every $i\in\{1,\ldots,n\}$
 the function $t\to a_i(\cdot,t,0,0)$ belongs to the space
  $L_{2,{\rm loc}}(S;H^1_{\rm loc}(\overline\Omega))$ and in equations \eqref{1}
   and \eqref{3} make the substitution $a_i(x,t,u,\nabla u)$
   for $\tilde a_i(x,t,u,\nabla u):=
  a_i(x,t,u,\nabla u)-a_i(x,t,0,0)  (i\in\{0,1,\ldots,n\})$,
  $f_1(x,t)$ for $\tilde f_1(x,t):=f_1(x,t)-a_0(x,t,0,0)+
  \sum_{i=1}^n\frac{\partial}{\partial x_i}a_i(x,t,0,0),  $
  $c(y,t,u)$ for $\tilde c(y,t,u):=c(y,t,u)-c(y,t,0)$,
  $f_2(y,t)$ for $\tilde f_2(y,t):=f_2(y,t)-c(y,t,0)-
  \sum_{i=1}^n a_i(y,t,0,0)\nu_i$. It is obvious that the functions
 $(\tilde a_0,\ldots,\tilde a_n), \tilde c$, $(\tilde f_1,\tilde f_2)$
 satisfy all above mentioned conditions for
  $(a_0,\ldots,a_n), c$, $(f_1,f_2)$ respectively.
 \end{remark}


\begin{definition} \label{def2.4} \rm
Let $(b_1,b_2)\in \mathbb B $, $(a_0,a_1,\dots,a_n)\in \mathbb{A},
 c\in \mathbb C, (f_1,f_2)\in \mathbb{F}_{\rm loc}. $
We say that the function $u\in \mathbb{U}_{\rm loc} $ is
 generalized solution of
the problem \eqref{1}--\eqref{3} if
  \begin{equation}\label{inttot}
\begin{aligned}
&\iint_Q\Bigl\{\sum_{i=1}^n a_i(x,t,u,\nabla u)
   \psi_{x_i} \varphi + a_0(x,t,u,\nabla u)
  \psi  \varphi - b_1(x) u \psi  \varphi'\Bigr\}\,dx\,dt\\
& +\iint_{\Sigma_1}\{c(y,t,\gamma u) \gamma\psi
 \varphi- b_2(y)  \gamma u  \gamma \psi  \varphi'\}\,d\Gamma_y\,dt\\
&= \iint_Q  f_1 \psi \varphi\,dx\,dt+
 \iint_{\Sigma_1}f_2 \gamma \psi \varphi\,d\Gamma_y\,dt
\end{aligned}
  \end{equation}
for every
$\psi\in \mathbb{V}_c, \varphi\in C^1_0(-\infty,0)$.
\end{definition}

Hereinafter denote by $C^1_0(I)$, where $I$ is an interval
of the number axis,
the linear space of finite continuous-differentiable functions on $I$.

For every $(b_1,b_2)\in \mathbb B $,
$\rho_1,\rho_2$ in
 $ L_{\infty,{\rm loc}}^{+}(\overline Q)$,
$\rho_3\in   L_{\infty,{\rm loc}}^{+}(\overline {\Sigma_1})$,
 $ d_1,d_2\in L_{\infty,{\rm loc}}^{0,+}(\overline Q)$
 put
    \begin{equation}\label{2.24}
\begin{aligned}
&\Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R)\\
&:=R^{-2p/(p-2)} \iint_{Q_{R}}
 \rho_1^{-p/(p-2)}\rho_2^{-2/(p-2)}
    d_1^{2p/(p-2)}\,dx\,dt\\
&\quad +R^{-p/(p-2)}\iint_{Q_{R}}
    \rho_2^{-2/(p-2)}b_1^{p/(p-2)}\,dx\,dt\\
&\quad + R^{-p/(p-2)}\iint_{Q_{R}}
           \rho_2^{-2/(p-2)}d_2^{p/(p-2)}\,dx\,dt\\
&\quad +R^{-q/(q-2)}\iint_{\sum_{1,R}}
  \rho_3^{-2/(q-2)}b_2^{q/(q-2)}
  d\Gamma_{y}\,dt,\quad R>0.
\end{aligned}
\end{equation}

Denote by $\mathbb{B}\mathbb{A}\mathbb{C}$
the subset of the Cartesian product
$\mathbb{B}\times\mathbb{A}\times\mathbb{C}$
 whose any element $((b_1,b_2),(a_0,a_1,\dots,a_n),c)$
 such that $(a_0,a_1,\dots,a_n)$
satisfies  conditions (A3), (A4), and $c$
satisfies condition (C3) with
$\rho_1,\rho_2,\rho_3,d_1,d_2$ satisfying
\begin{equation}\label{psiton}
\Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R)\to 0
 \quad\text{as } R\to+\infty.
\end{equation}

\begin{theorem}\label{t1}
Let $((b_1,b_2),(a_0,a_1,\dots,a_n), c)\in
\mathbb{B}\mathbb{A}\mathbb{C}$, $(f_1,f_2)\in \mathbb{F}_{\rm loc}$. Then the problem \eqref{1}--\eqref{3} has a unique
generalized solution. Moreover, for every
 $R_0,R$, $0<R_0<R$, this solution satisfies the estimation
\begin{equation}\label{ozinka}
\begin{aligned}
&\sup_{t\in[-R_0,0]}\Bigl[
\int_{\Omega_{R_0}}b_1(x) |u(x,t)|^2\,dx +
\int_{\Gamma_{1,R_0}}b_2(y)|\gamma u(y,t)|^2
   d\Gamma_y\Bigr]\\
&+\iint_{Q_{R_0}}\Bigl[ \rho_1|\nabla u|^2+\rho_2|u|^p \Bigr]\,dx\,dt+
  \iint_{\Sigma_{1,R_0}} \rho_3|\gamma u|^q d\Gamma_y dt \\
& \le   C\Bigl(R/(R-R_0)\Bigr)^\sigma
   \Bigl[\Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R)\\
&\quad   +\iint_{Q_R}\rho_2^{-1/(p-1)}|f_1(x,t)|^{p'}\,dx\,dt+
  \iint_{\Sigma_{1,R}}\rho_3^{-1/(q-1)}|f_2(y,t)|^{q'} \,
 d\Gamma_y \,dt\Bigr],
\end{aligned}
  \end{equation}
where $\sigma:=\max\big\{p/(p-2),q/(q-2)\big\}+2p/(p-2)$, $C>0$ is
a constant  depending only on $p,q$.

In addition, for any sequences $\{(a_0^k,a_1^k,\dots,a_n^k)\},
\{c^k\}$ and $\{(f_1^k,f_2^k)\}$
such that \quad$((b_1,b_2),(a_0^k,a_1^k,\dots,a_n^k), c^k)\in
\mathbb{B}\mathbb{A}\mathbb{C}$ and
 $(a_0^k,a_1^k,\dots,a_n^k)\to (a_0,a_1,\dots,a_n)$
 in $\mathbb{A}$,  $c^k\to c $
in $\mathbb C$, $(f_1^k,f_2^k) \to (f_1,f_2)$
 in $ \mathbb{F}_{\rm loc}$ as $k\to \infty$
we have $u^k \to u$  in $\mathbb{U}_{\rm loc}$ as $k\to \infty$,
where for every $k\in \mathbb{N}$
 $u^k$ is a generalized solution of the problem differing from the
 problem ( \ref{1})--( \ref{3}) only by having functions
  $a_0^k,a_1^k,\dots,a_n^k,  c^k,  f_1^k,f_2^k$ instead of
   $ a_0,a_1,\dots,a_n,  c,  f_1,f_2 $ respectively.
\end{theorem}


\begin{remark} \label{rmk2.6} \rm
If the functions $b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2$ are
constant (positive), then  condition \eqref{psiton} is equivalent
to the condition
\begin{equation}\label{psiton1}
 \operatorname{meas} \Omega_R\cdot R^{-2/(p-2)} +  \operatorname{meas}
 \Gamma_{1,R}\cdot R^{-2/(q-2)}\to 0
\quad\text{as } R\to+\infty.
\end{equation}
\end{remark}


\section{Auxiliary statements}\label{Sect3}

 Now we state some technical results needed later.

\begin{lemma} \label{l1}
Let $R>0$,   $\tau_1<\tau_2$ be any
numbers, $(b_1,b_2)\in \mathbb B$. Suppose that a function
$ v\in ((\tau_1,\tau_2)\to\mathbb{V}_{\rm loc})\cap
L_2(\tau_1,\tau_2;H^1_{\rm loc}(\overline\Omega))\cap
L_{p,{\rm loc}}\big( \overline{\Omega\times(\tau_1,\tau_2)} \big)$,
$\gamma v \in L_{q,{\rm loc}}\big(\overline{\Gamma_1
\times(\tau_1,\tau_2)} \big)$,
 and $ g_0\in L_{p',{\rm loc}}\big( \overline{\Omega\times
(\tau_1,\tau_2)} \big)$,
$g_i\in L_{2,{\rm loc}}\big( \overline{\Omega\times(\tau_1,\tau_2)}
\big)$, $(i=\overline{1,n})$,
$h\in L_{q',{\rm loc}}\big( \overline{\Gamma_1\times(\tau_1,\tau_2)}
\big)$  satisfy
\begin{equation} \label{totlem}
\begin{aligned}
&\int_{\tau_1}^{\tau_2}\int_{\Omega_R}\Bigl\{\sum_{i=1}^n
  g_i \psi_{x_i} \varphi + g_0 \psi  \varphi -
  b_1 v \psi \varphi'\Bigr\}\,dx\,dt\\
&+ \int_{\tau_1}^{\tau_2}\int_{\Gamma_{1,R}}\{h \gamma\psi \varphi -
  b_2 \gamma v \gamma\psi  \varphi'\}\,d\Gamma_y dt=0
\end{aligned}
\end{equation}
for  $\varphi\in C^1_0(\tau_1,\tau_2)$,
$\psi\in \mathbb{V}_c$,
$\operatorname{supp}  \psi\subset \overline{\Omega_R}$.
Then $b_1^{1/2}v\in C([\tau_1,\tau_2];L_2(\Omega_{R^*}))$,
$b_2^{1/2}\gamma v\in C([\tau_1,\tau_2];L_2(\Gamma_{1,R^*}))$,
for every $R^*\in (0,R)$. In addition, for every functions
$\theta\in C^1([\tau_1,\tau_2])$,
$w\in C^1(\overline{\Omega})$,
$\operatorname{supp}w \subset \overline{\Omega_R}$,  $w\geq 0$
and arbitrary numbers $t_1,t_2 $ such that $\tau_1\le t_1<t_2\le
\tau_2$, the equality holds
 \begin{equation}\label{ozhlem}
\begin{aligned}
&\theta(t)\Bigl(\int_{\Omega_R}b_1(x)|v(x,t)|^2w(x)  dx +
  \int_{\Gamma_{1,R}}b_2(y) |\gamma v(y,t)|^2w(y)
  d\Gamma_y\Bigr)\Big|_{t=t_1}^{t=t_2}\\
& -\int_{t_1}^{t_2}
  \Bigl( \int_{\Omega_R}b_1(x) |v(x,t)|^2w(x)\,dx +
  \int_{\Gamma_{1,R}}b_2(y) |\gamma v(y,t)|^2w(y)
  d\Gamma_y  \Bigr)\theta'(t)\,dt \\
&+ 2\int_{t_1}^{t_2} \Bigl(  \int_{\Omega_R}
  \Bigl\{
  \sum_{i=1}^n g_i(vw)_{x_i} +
  g_0 v  w\Bigr\}\,dx + \int_{\Gamma_{1,R}}h
  \gamma v w\,d\Gamma_y \Bigr)\theta\,dt=0.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
We assume without loss of  generality that  $\tau_1=0, \tau_2=T $,
where $T>0$ is any number.
We will use some ideas with proof  of
\cite[Proposition~1.2, p.~106]{Showalter97}.
First of all construct the extension of the functions
$\widehat v,\widehat g_i (i=\overline{0,n})$
for functions $v,g_i (i=\overline{0,n})$ respectively
onto the cylinder $\Omega\times
(-T,2T)$ by putting for a.e. $x\in \Omega$,
 $$
  \widehat v(x,t):=\begin{cases}
  v(x,-t),& -T<t<0,\\
  v(x,t), & 0 \leq t  \leq T,\\
  v(x,2T-t), & T < t < 2T,
   \end{cases} \;
  \widehat g_i(x,t):=\begin{cases}
  - g_i(x,-t),& -T<t<0,\\  g_i(x,t), & 0 \leq t \leq T,\\
  - g_i(x,2T-t),& T < t < 2T.
   \end{cases}
 $$
Construct also the extension  $\widehat h$ of the
function $h$ onto the surface
$\Gamma_1\times(-T,2T)$:
$$
  \widehat h(y,t):= \begin{cases}
    -h(y,-t),& -T<t<0,\\
   h(y,t),& 0 \leq t  \leq T,\\
  - h(y,2T-t),& T < t < 2T
   \end{cases}
 $$
for a.e. $y\in \Gamma_1$.
It is to verify that
   \begin{equation}\label{hattot}
\begin{aligned}
&\int_{-T}^{2T}\int_{\Omega_R}\Bigl\{ \sum_{i=1}^n
     \widehat g_i \psi_{x_i}\varphi +
  \widehat g_0 \psi \varphi - b_1  \widehat v  \psi  \varphi'\Bigr\}
\,dx\,dt \\
&+  \int_{-T}^{2T}\int_{\Gamma_{1,R}}\{ \widehat h \gamma\psi
  \varphi - b_2 \gamma \widehat v
  \gamma\psi  \varphi' \}\,d\Gamma_y dt=0
\end{aligned}
 \end{equation}
is fulfilled for every $\varphi\in C^1_0(-T,2T)$,
$\psi\in \mathbb{V}_c, \operatorname{supp} \psi
\subset \overline{\Omega_R}$.

Indeed, it is easy to ascertain that  \eqref{hattot} holds
for every $\psi\in \mathbb{V}_{\rm loc}$ and $\varphi\in
C^1_0(-T,2T)$, provided
 $\operatorname{supp} \varphi\subset(-T,0)\cup(0,T)\cup(T,2T)$
  (it is enough to make the corresponding substitution of
the variable  $t$ in to identity \eqref{inttot}). It remains to
consider the case when
$\operatorname{supp}\varphi\cap\{0,T\}\neq \emptyset$. For the
simplicity we will assume without loss of generality that
$\operatorname{supp}\varphi\subset(-T,T)$. Then for every
$m\in\mathbb{N}$   choose the function $\chi_m\in C^1(\mathbb{R})$
such that $|\chi_m(t)|\leq 1$,
   $|\chi'_m(t)|\leq 2m$ and $\chi_m(-t)=\chi_m(t)$  when
$t\in\mathbb{R}$,   $\chi_m(t)=1$ as
$t\in(-\infty,-2/m)\cup(2/m,+\infty)$ and $\chi_m(t)=0$,
   when $t\in(-1/m,1/m)$.

It is obvious that for every $t\in\mathbb{R}\backslash\{0\}$ we have
 $\chi_m(t)\to 1$ as $m\to +\infty$. The above mentioned yields
 that \eqref{hattot} is fulfilled provided
 $\psi\in \mathbb{V}_{\rm loc}$ and
 with $\varphi$ instead of $\chi_m \varphi$, where $m\in\mathbb{N}$.
  After the simple transformations we obtain
\begin{equation}\label{g61}
\begin{aligned}
&\int_{-T}^{2T}\int_{\Omega_R}
    \Big\{\sum_{i=1}^n\widehat{g}_i \psi_{x_i} \varphi+
    \widehat{g}_0 \psi \varphi-b_1\widehat{v} \psi
    \varphi'\Big\}\chi_m\,dx\,dt
\\
&+\int_{-T}^{2T}\int_{\Gamma_{1,R}}\Big\{\widehat{h}
     \gamma \psi \varphi-
   b_2 \gamma\widehat{v} \gamma\psi \varphi'\Big\}\chi_m
   d\Gamma_y\,dt
\\
& -\int_{-2/m}^{2/m}\int_{\Omega}b_1\widehat{v} \psi
    \varphi \chi'_m\,dx\,dt-
    \int_{2/m}^{-2/m}\int_{\Gamma_{1,R}}
   b_2 \gamma\widehat{v} \gamma\psi \varphi \chi'_m
   d\Gamma_y\,dt=0,
\end{aligned}
\end{equation}
   where $m\in\mathbb{N}$, $\varphi\in C^1_0(-T,2T)$, $\operatorname{supp}\varphi\subset(-T,T)$,
    $\psi\in\mathbb{V}_{\text{c}},   \operatorname{supp} \psi\subset \overline{\Omega_R}$.

Change the third and fourth terms of the left side part of
\eqref{g61}. We obtain
    \begin{equation}\label{g62}
\begin{aligned}
&\int_{-2/m}^{2/m}\int_{\Omega_R}b_1\widehat{v} \psi
    \varphi \chi'_m\,dx\,dt\\
&= \int_{1/m}^{2/m}\int_{\Omega_R}b_1(x)\widehat{v}(x,t)
    \psi(x) \varphi(t) \chi'_m(t)\,dx\,dt\\
&\quad+  \int_{-2/m}^{-1/m}\int_{\Omega_R}b_1(x)\widehat{v}(x,t)
    \psi(x) \varphi(t) \chi'_m(t)\,dx\,dt \\
&= \int_{1/m}^{2/m}\int_{\Omega_R}b_1(x){v}(x,t) \psi(x)
    \varphi(t) \chi'_m(t)\,dx\,dt\\
&\quad + \int_{1/m}^{2/m}\int_{\Omega_R}b_1(x){v}(x,t) \psi(x)
    \varphi(-t) \chi'_m(-t)\,dx\,dt\\
&= \int_{1/m}^{2/m}\int_{\Omega_R}b_1(x){v}(x,t) \psi(x)
    (\varphi(t)-\varphi(-t)) \chi'_m(t)\,dx\,dt\\
&= 2\int_{1/m}^{2/m}\int_{\Omega_R}t \varphi'(\xi(t))
    \chi'_m(t)b_1(x){v}(x,t) \psi(x)  \,dx\,dt,
\end{aligned}
\end{equation}
where  $\xi(t)$ is some number between $-t$ and $t$.
    Here we made the replacement of $t$
    by $-t$ in one of terms, used the definition of $\widehat{v}$
    and the Lagrange Theorem of finite decrements:
     $\varphi(t)-\varphi(-t)=\varphi'(\xi(t)) t$, $t>0$.
Note that $|t \chi'_m(t)|\leq
    (2/m)  2m=4$ for every $t\in[1/m,2/m]$ and
\begin{equation}\label{g63}
    \operatorname{meas}_{n+1}\{(x,t)| x\in\operatorname{supp}\psi,  t\in(1/m,2/m)\}\to 0
    \quad{as } m\to +\infty.
\end{equation}
It is obvious that
\begin{equation}\label{g64}
   |t \varphi'(\xi(t))
   \chi'_m(t) b_1(x){v}(x,t) \psi(x) |
   \leq K|b_1(x){v}(x,t) \psi(x)|,\quad
   (x,t)\in\Omega\times(-T,2T),
\end{equation}
where $K>0$ is some constant not depending on $m$. Since the right
side of the inequality  \eqref{g64} belongs to
$L_1(\operatorname{supp}\psi\times(-T,2T))$, thus the left side
belongs to $L_1(\operatorname{supp}\psi\times(-T,2T))$.

  From \eqref{g62} by  \eqref{g63} and \eqref{g64} we
deduce
\begin{equation}\label{g65}
   \int_{-2/m}^{2/m}\int_{\Omega_R}b_1\widehat{v}
   \psi \varphi \chi'_m\,dx\,dt\to 0\quad \text{as }
    m\to \infty.
\end{equation}
Arguing the same way, we derive
\begin{equation}\label{g66}
   \int_{-2/m}^{2/m}\int_{\Gamma_{1,R}}b_2
   \gamma\widehat{v} \gamma\psi \varphi
   \chi'_m\,d \Gamma_y\,dt\to 0\quad \text{as } m\to \infty.
\end{equation}
Passing to the limit in \eqref{g61} as $m\to +\infty$, taking into
account \eqref{g65}, \eqref{g66} and the Lebesgue Theorem of
 boundary transition under the integral sign.
It is obvious that as a result we obtain \eqref{hattot}, which is
required.


Let $\{\omega_{\rho}|  \rho>0\}$ be the mollifier kernels, that is
 $\omega_{\rho}\in C^{\infty}(\mathbb{R})$,
$\omega_{\rho}$ is an even function,
$\operatorname{supp}\omega_{\rho}\subset [-\rho,\rho]$,
$\int_\mathbb{R}\omega_{\rho}(s)\,ds=1$ for every $\rho>0$.
Choose a number $k_0\in \mathbb{N}$ such that $1/k_0<T/2$,
and for every  $k\geq k_0$ put
\begin{gather*}
  \widehat v_k(x,\tau):=(\widehat v\ast\omega_{1/k})(x,\tau)\equiv\int
  _\mathbb{R}\widehat v(x,t) \omega_{1/k}(t-\tau)\,dt,
  \quad (x,\tau)\in \Omega\times(-T/2,3T/2),
\\
 \widehat g_{i,k}(x,\tau):=(\widehat g_i\ast\omega_{1/k})(x,\tau)
  \equiv\int_\mathbb{R}\widehat g_i(x,t)
  \omega_{1/k}(t-\tau)\,dt,   (x,\tau)\in
  \Omega\times(-T/2,3T/2),
\\
 i\in\{0,\ldots,n\},
\\
  \widehat h(y,\tau):=(\widehat h\ast\omega_{1/k})(y,\tau)
  \equiv\int_\mathbb{R}\widehat h(y,t) \omega_{1/k}(t-\tau)\,dt,
  \quad (y,\tau)\in \Gamma_1\times(-T/2,3T/2).
\end{gather*}
It is easy to ascertain the fact
$$
  (\gamma\widehat v\ast\omega_{1/k})(y,\tau)
=\gamma\widehat v_k(y,\tau),
  \quad (y,\tau)\in \Gamma_1\times(-T/2,3T/2).
 $$
  From well-known facts of homogenization theory we conclude
\begin{gather}
  \widehat v_k\mathop{\longrightarrow}_{k\to\infty}\widehat v
  \quad\text{in }
   L_{p,\mathrm{loc}}\big( \overline{\Omega\times(-T/2, 3T/2)} \big)
  \cap
  L_2(-T/2, 3T/2; H^1_{\mathrm{loc}}(\overline{\Omega})),
\notag \\
  \gamma\widehat v_k\mathop{\longrightarrow}_{k\to\infty}
  \gamma\widehat v \quad\text{in }
  L_{q,\mathrm{loc}}\big( \overline{\Gamma_1\times(-T/2, 3T/2)} \big),
\notag \\
  \widehat g_{i,k}\mathop{\longrightarrow}_{k\to\infty}
  \widehat g_i \quad\text{in} \quad
  L_{2,\mathrm{loc}}\big( \overline{\Omega\times(-T/2, 3T/2)} \big)
  \quad(i=\overline{1,n}),
\notag \\
  \widehat g_{0,k}\mathop{\longrightarrow}_{k\to\infty}
  \widehat g_0 \quad\text{in} \quad
  L_{p',\mathrm{loc}}\big(\overline{\Omega\times(-T/2, 3T/2)}\big),
\notag \\
 \widehat h_k\mathop{\longrightarrow}_{k\to\infty}\widehat h
  \quad\text{in} \quad
  L_{q',\mathrm{loc}}\big(\overline{\Gamma_1\times(-T/2, 3T/2)}\big).
\label{eq11z}
\end{gather}
In \eqref{hattot} set $\varphi(t)=\omega_{1/k}(t-\tau)$,
$t\in(-T, 2T)$, where $\tau\in[-T/2, T]$, $k\geq k_0$ are some
numbers. After simple transformations we obtain
  \begin{equation}\label{eq12z}
\begin{aligned}
&\int_{\Omega_R}\big\{b_1(x)
   \dfrac{d}{d\tau}\widehat v_k(x,\tau) \psi(x)
  +\sum_{i=1}^n
  \widehat g_{i,k}(x,\tau) \psi_{x_i}(x)+
  \widehat g_{0,k}(x,\tau) \psi(x)\big\}dx
\\
&+\int_{\Gamma_{1,R}}\big\{b_2(y) \dfrac{d}{d\tau}
  \gamma\widehat v_k(y, \tau) \gamma\psi(y)
  +\widehat h_k(y, \tau) \gamma\psi(y)\big\}d\Gamma_y=0
\end{aligned}
\end{equation}
for every $\tau\in[-T/2, T]$, $\psi\in\mathbb{V}_c,
\operatorname{supp}\psi\subset\overline{\Omega_R}$.

Let $k$, $l$ be arbitrary natural numbers bigger than $k_0$.
Subtracting from \eqref{eq12z} the same equality with $k=l$ and
putting $\widehat v_{kl}:=\widehat v_k-\widehat v_l$, $\widehat
g_{i,kl}:=\widehat g_{i,k}-\widehat g_{i,l}$ $(i=\overline{0,n})$,
$\widehat h_{kl}:=\widehat h_k-\widehat h_l$, we deduce
\begin{equation}\label{eq13z}
\begin{aligned}
&\int_{\Omega_R}\big\{b_1(x)
  \dfrac{d}{d\tau}\widehat v_{kl}(x,\tau) \psi(x)
  +\sum_{i=1}^n
  \widehat g_{i,kl}(x,\tau) \psi_{x_i}(x)+
  \widehat g_{0,kl}(x,\tau) \psi(x)\big\}dx
\\
&+\int_{\Gamma_{1,R}}\big\{b_2(y) \dfrac{d}{d\tau}
  \gamma\widehat v_{kl}(y, \tau) \gamma\psi(y)
  +\widehat h_{kl}(y, \tau) \gamma\psi(y)\big\}d\Gamma_y=0
\end{aligned}
 \end{equation}
for every $\psi\in\mathbb{V}_c$,
$\operatorname{supp}\psi\subset\overline{\Omega_R}$,
 $\tau\in[-T/2, T]$, $k,l\geq k_0$.

Let $w\in C^1(\overline{\Omega})$ be any function such that
$\mathrm{supp} w\subset\overline{\Omega_R}$, $  w\geq 0$, and
let $\theta\in C^1(\mathbb{R})$ be an arbitrary function. In
\eqref{eq13z} let for every $\tau\in[-T/2, T]$
$\psi(x)=\widehat v_{kl}(x,\tau) w(x) \theta(\tau)$,
 $ x\in\Omega$.
Integrate the obtained equality over $\tau$ between
$t_1$ and $t_2$ $(-T/2\leq t_1<t_2\leq
T)$, keeping in mind that for every $\tau\in [-T/2,T]$
\begin{gather*}
  \frac{d}{d\tau}\widehat v_{kl}(x,\tau)\cdot
  \widehat v_{kl}(x,\tau)\cdot\theta(\tau)
  =\frac{1}{2}\frac{d}{d\tau}
  \bigl(|\widehat v_{kl}(x,\tau)|^2\theta(\tau)\bigr)
  -\frac{1}{2}
  |\widehat v_{kl}(x,\tau)|^2\frac{d}{d\tau}\theta(\tau),
\\
  \frac{d}{d\tau}\gamma\widehat v_{kl}(x,\tau)\cdot
  \gamma\widehat v_{kl}(x,\tau)\cdot\theta(\tau)
  =\frac{1}{2}\frac{d}{d\tau}
  \bigl(|\gamma\widehat v_{kl}(x,\tau)|^2\theta(\tau)\bigr)
  -\frac{1}{2}
  |\gamma\widehat v_{kl}(x,\tau)|^2\frac{d}{d\tau}\theta(\tau).
\end{gather*}
As a result,
\begin{align}
&\frac{1}{2}\int_{\Omega_R}\big(b_1(x)|\widehat v_{kl}(x,\tau)|^2
  w(x)\theta(\tau)\big)\Big |_{\tau=t_1}^{\tau=t_2}dx \notag \\
&+  \frac{1}{2}\int_{\Gamma_{1,R}}\big(b_2(y)|\gamma\widehat v_{kl}(y,\tau)|^2
  w(y)\theta(\tau)\big)\Big |_{\tau=t_1}^{\tau=t_2}d\Gamma_y \notag\\
&-\frac{1}{2}\int_{t_1}^{t_2}
  \Big (\int_{\Omega_R}b_1(x)|\widehat v_{kl}(x,\tau)|^2
  w(x)dx
  +\int_{\Gamma_{1,R}}b_2(y)|\gamma\widehat v_{kl}(y,\tau)|^2
  w(y)d\Gamma_y\Big )\theta'\,d\tau \notag \\
&+\int_{t_1}^{t_2}\Big ( \int_{\Omega_R}
  \Big\{ \sum_{i=1}^n \widehat g_{i,kl}(x,\tau)
  (\widehat v_{kl}(x,\tau)w(x))_{x_i}+
  \widehat g_{0,kl}(x,\tau)\widehat v_{kl}(x,\tau)w(x)
  \Big\}\,dx \notag \\
&+\int_{\Gamma_{1,R}}\widehat h_{kl}(y,\tau)
  \gamma\widehat v_{kl}(y,\tau) w(y)\,d\Gamma_y\Big)
  \theta(\tau)\,d\tau=0. \label{eq14z}
\end{align}
Now, we impose additional conditions on the function $\theta$:
\begin{gather*}
  0\leq\theta(\tau)\leq 1\quad\text{when
  $\tau\in\mathbb{R}$},\quad
  \theta(\tau)=0\quad\text{when $\tau\leq -T/2$},
\\
  \theta(\tau)=1\quad\text{when $\tau\geq 0$},\quad
  |\theta^{\prime}(\tau)|\leq 4/T\quad\text{when $\tau\in  [-T/2, 0]$}.
\end{gather*}
 Then from \eqref{eq14z}, having chosen $t_1=-T/2$ and $t_2$ be any
number from the interval $[0,T]$, we derive
\begin{align*}
&\max_{\tau\in [0,T]}\Big ( \int_{\Omega_R}b_1(x)|\widehat v_{kl}(x,\tau)|^2
  w(x)dx+\int_{\Gamma_{1,R}}b_2(y)|\gamma\widehat v_{kl}(y,\tau)|^2
  w(y)d\Gamma_y \Big)
\\
& \leq\frac{4}{T}\int_{-T/2}^{0}
  \Big ( \int_{\Omega_R}b_1(x)|\widehat v_{kl}(x,\tau)|^2
  w(x)dx+\int_{\Gamma_{1,R}}b_2(y)|\gamma\widehat v_{kl}(y,\tau)|^2
  w(y)d\Gamma_y \Big)d\tau
\\
&\quad  +2\int_{-T/2}^{T}\Big ( \int_{\Omega_R}
  \Big\{ \sum_{i=1}^n |\widehat g_{i,kl}(x,\tau)|
  |(\widehat v_{kl}(x,\tau)w(x))_{x_i}|\\
&\quad +|\widehat g_{0,kl}(x,\tau)|
  |\widehat v_{kl}(x,\tau)|w(x) \Big\}dx
 +\int_{\Gamma_{1,R}}|\widehat h_{kl}(y,\tau)|
  |\gamma\widehat v_{kl}(y,\tau)|w(y)d\Gamma_y\Big)d\tau.
\end{align*}  %\label{eq15z}
 From above inequality by  \eqref{eq11z} this implies that as
$k, l\to +\infty$,
\begin{gather*}
  (w b_1)^{1/2} \widehat v_{k,l}\to  0 \quad\text{in}
  \quad\;C([0,T];L_2(\Omega_{R})),\\
  (w b_2)^{1/2} \gamma\widehat v_{k,l}\to 0 \quad\text{in}
  \quad\;C([0,T];L_2(\Gamma_{1,R})).
\end{gather*}
 This means that the sequences
$\{(w b_1)^{1/2}\widehat v_k\}_{k=1}^\infty$,
$\{(w b_2)^{1/2}\gamma\widehat v_k\}_{k=1}^\infty$
are fundamental in the spaces
$C([0,T];L_2(\Omega_{R}))$,
$C([0,T];L_2(\Gamma_{1,R}))$ respectively and
\begin{equation}\label{zbizh}
\begin{gathered}
  (w b_1)^{1/2}\widehat v_{k}\mathop{\longrightarrow}_{k\to+\infty}
    (w b_1)^{1/2}\widehat v\quad\text{in }
  C([0,T];L_2(\Omega_{R})),
\\
(w b_2)^{1/2}\gamma\widehat v_{k}
   \mathop{\longrightarrow}_{k\to+\infty}
   (w b_2)^{1/2}\gamma\widehat v\quad\text{in }
  \;C([0,T];L_2(\Gamma_{1,R})).
\end{gathered}
\end{equation}
 Thus we conclude that
 $$
  b_1^{1/2}v\in\;C([0,T];L_2(\Omega_{R^*})),\quad
  b_2^{1/2}\gamma v\in\;C([0,T];L_2(\Gamma_{1,R^*}))
 $$
 for every $R^*\in(0,R)$.

Now for every  $\tau\in [0,T]$, in \eqref{eq12z},  put
$\psi(x)=\widehat v_k(x,\tau)w(x)\theta(\tau)$, $x\in\Omega$,
where $w\in C^1(\overline{\Omega})$, $\text{supp}
w\subset\overline{\Omega}_R$,    $w\geq 0$,
$\theta\in C^1([0,T])$, and integrate over $\tau$ between $t_1$ and
$t_2$. The transformations similar to those we made above to obtain
 \eqref{eq14z} yield
\begin{align*}
&\frac{1}{2}\int_{\Omega_R}\big(b_1(x)|\widehat v_{k}(x,\tau)|^2
  w(x)\theta(\tau)\big)\Big |_{\tau=t_1}^{\tau=t_2}dx\\
&+  \frac{1}{2}\int_{\Gamma_{1,R}}\big(b_2(y)|\gamma\widehat v_{k}(y,\tau)|^2
  w(y)\theta(\tau)\big)\Big |_{\tau=t_1}^{\tau=t_2}d\Gamma_y
\\
&-\frac{1}{2}\int_{t_1}^{t_2}
  \Big (\int_{\Omega_R}b_1(x)|\widehat v_{k}(x,\tau)|^2
  w(x)dx
  +\int_{\Gamma_{1,R}}b_2(y)|\gamma\widehat v_{k}(y,\tau)|^2
  w(y)d\Gamma_y\Big )\theta'\,d\tau
\\
&+\int_{t_1}^{t_2}
  \Big ( \int_{\Omega_R}\Big\{ \sum_{i=1}^n \widehat g_{i,k}(x,\tau)
  (\widehat v_{k}(x,\tau)w(x))_{x_i}+\widehat g_{0,k}(x,\tau)\widehat v_{k}(x,\tau)w(x)
  \Big\}\,dx
\\
& +\int_{\Gamma_{1,R}}\widehat h_{k}(y,\tau)
  \gamma\widehat v_{k}(y,\tau)w(y)\,d\Gamma_y\Big)
  \theta(\tau)\,d\tau=0.
 \end{align*} %\label{eq16z}
Passing to the limit as $k\to +\infty$, by
 \eqref{eq11z}, \eqref{zbizh} we obtain \eqref{ozhlem}.
\end{proof}

\begin{lemma}\label{l2}
 Let $(b_1,b_2)\in \mathbb B $, $(a_0,a_1,\dots,a_n)\in \mathbb{A}$,
$c\in \mathbb C $.
 Suppose that, for every $k\in\{1,2\}$,
$u_k\in \mathbb{U}_{\rm loc}$,
$(f_{1,k}, f_{2,k})\in \mathbb{F}_{\rm loc}$,
$\bar{f}_{i,k}\in L_{2,{\rm loc}}(\overline Q)$
$(i=\overline{1,n})$ and
\begin{equation}\label{inttotozh}
\begin{aligned}
&  \iint_Q\Bigl\{\sum_{i=1}^n a_i(x,t,u_k,\nabla u_k)
   \psi_{x_i} \varphi + a_0(x,t,u_k,\nabla u_k)
  \psi  \varphi - b_1(x) u_k \psi  \varphi'\Bigr\}\,dx\,dt
\\
&  +\iint_{\Sigma_1}\{c(y,t,\gamma u_k) \gamma\psi
 \varphi- b_2(y)  \gamma u_k  \gamma \psi  \varphi'\}\,d\Gamma_y dt
\\
&= \iint_Q \Big\{ \sum_{i=1}^n\bar{f}_{i,k} \psi_{x_i}\varphi +
  f_{1,k} \psi \varphi\Big\}\,dx\,dt+ \iint_{\Sigma_1}f_{2,k}
  \gamma \psi \varphi\,d\Gamma_y  dt
\end{aligned}
\end{equation}
 holds for every $\psi\in \mathbb{V}_c,  \operatorname{supp}
 \psi\subset \overline{\Omega_R}$,
  $\varphi\in C_0^1(-\infty,0),\
  \operatorname{supp}\varphi\subset S_R$,  where  $R\geq 1$
is some number.
Then for arbitrary $R_0\in(0,R)$ we have
\begin{align}
& \max_{t\in[-R_0,0]}
   \int_{\Omega_{R_0}}b_1(x)|u_1(x,t)-u_2(x,t)|^2 dx\notag \\
&+ \max_{t\in[-R_0,0]}\int_{\Gamma_{1,R_0}}b_2(y)|
   \gamma u_1(y,t)-\gamma u_2(y,t)|^2d\Gamma_y \notag\\
&+\iint_{Q_{R_0}}\big\{ \rho_1|\nabla u_1-\nabla u_2|^2
   +\rho_2|u_1-u_2|^p  \big\}\,dx\,dt +
   \iint_{\Sigma_{1,R_0}}
   \rho_3|\gamma u_1-\gamma u_2|^q d\Gamma_y dt \notag\\
&\leq    C   \big(R/(R-R_0) \big)^\sigma
    \Big[\Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R) \notag\\
&\quad +  \iint_{Q_{R}}\Big\{\big(\rho_1^{-1}+n\rho_1d_1^{-2}\big)
    \Big(\sum_{i=1}^n|\bar{f}_{i,1}-\bar{f}_{i,2}|^2\Big)
  +\rho_2^{-1/(p-1)}  |f_{1,1}-f_{1,2}|^{p^{\prime}}\Big\}  \,dx\,dt
  \notag \\
&\quad +\iint_{\Sigma_{1,R}}\rho_3^{-1/(q-1)}
   |f_{2,1}-f_{2,2}|^{q^{\prime}} d\Gamma_y dt \Big], \label{eq18z}
\end{align}
 where $C,\sigma, \Psi$ are the same as in Theorem \ref{t1}.
\end{lemma}

\begin{proof}
Introduce two  ``cutting''   functions (see \cite{Bernis}):
 $$
\zeta(x)=   \begin{cases}
            (R^2-|x|^2)/R,& |x|<R,\\
            0,&|x|\geq R,
        \end {cases} \quad
\chi(t)= \begin{cases}
            t+R,& -R\leq t\leq 0,\\
            0,& t< -R.
        \end{cases}
$$
 For given $\psi\in\mathbb{V}_c,
  \varphi\in C^1_0(-\infty,0)  $ such that
  $\operatorname{supp} \psi
  \subset\overline{\Omega_R},  \operatorname{supp}\varphi \subset S_R$,
  consider  \eqref{inttotozh} when $k=1$
  and the same equality when $k=2$. Subtract these equalities. Put
\begin{gather*}
  u_{12}(x,t):=u_1(x,t)-u_2(x,t),\quad f_{1,12}(x,t):= f_{1,1}(x,t)-
  f_{1,2}(x,t),
\\
  a_{i,12}(x,t):=a_i(x,t,u_1(x,t),
  \nabla u_1(x,t))-a_i(x,t,u_1(x,t),  \nabla u_1(x,t)),
\\
   \bar{f}_{i,12}(x,t):= \bar{f}_{i,1}(x,t)-  \bar{f}_{i,2}(x,t),
   \quad (i=\overline{1,n}),
\\
  (x,t)\in Q, \quad i=\overline{0,n},
\\
  \gamma u_{12}(y,t):=\gamma u_1(y,t)-\gamma u_2(y,t),\quad f_{2,12}(y,t)=
  f_{2,1}(y,t)-f_{2,2}(y,t),
\\
  c_{12}(y,t):=c(y,t,\gamma u_1(y,t))-c(y,t,\gamma u_2(y,t)),\quad
  (y,t)\in\Sigma_1.
\end{gather*}
 Apply Lemma \ref{l1} to the obtained equality with
$g_0:=a_{0,12}-f_{1,12}$,
$g_i=a_{i,12}-\bar{f}_{i,12}$ ($i=\overline{1,n}$),
 $h=c_{12}-f_{2,12}$, $w=\zeta^s$,
 $\theta=\chi^r$,
where $r=\max\big\{p/(p-2),q/(q-2)\big\}$, $s=2p/(p-2)$,
$t_1= -R$, $t_2=\tau\in( -R,0]$. After simple transformation we
obtain
\begin{align}
&\eta^r(\tau)\Big(\int_{\Omega_R}b_1(x)
  |u_{12}(x,\tau)|^2\zeta^s(x)dx+
  \int_{\Gamma_{1,R}}b_2(y)
  |\gamma u_{12}(y,\tau)|^2\zeta^s(y)d\Gamma_y\Big)\notag \\
&+2\iint_{Q_R^\tau}\Big\{\sum_{i=1}^n  a_{i,12}(u_{12})_{x_i}
  +a_{0,12}u_{12}\Big\}\zeta^s\eta^r\,dx\,dt+
  2\iint_{\Sigma_{1,R}^\tau} b_{12} \gamma u_{12} \zeta^s
  \eta^r\,d\Gamma_y\,dt  \notag \\
& =r\iint_{Q_R^\tau} b_1|u_{12}|^2\zeta^s\eta^{r-1}  \,dx\,dt+
  r\iint_{\Sigma_{1,R}^\tau}
   b_2|\gamma u_{12}|^2\zeta^s\eta^{r-1} d\Gamma_y\,dt \notag\\
&\quad   -2s\iint_{Q_R^\tau}\Big(\sum_{i=1}^n a_{i,12} \zeta_{x_i}\Big)
  u_{12} \zeta^{s-1}\eta^r
  \,dx\,dt+
  2\iint_{Q_{R}^{\tau}}\sum_{i=1}^n\bar{f}_{i,12}
    (u_{12})_{x_i}\zeta^s\eta^r\,dx\,dt  \notag \\
&\quad +2s\iint_{Q_{R}^{\tau}}
    \Big(\sum_{i=1}^n\bar{f}_{i,12}\zeta_{x_i}\Big)
    u_{12}\zeta^{s-1}\eta^r\,dx\,dt+
  2\iint_{Q_R^\tau} f_{1,12} u_{12} \zeta^s\chi^r  \,dx\,dt \notag\\
&\quad +2\iint_{\Sigma_{1,R}^\tau}
    f_{2,12} \gamma u_{12} \zeta^s\chi^r d\Gamma_y\,dt, \label{eq19z}
\end{align}
 where $Q_R^\tau:= \Omega_R\times(-R,\tau),
 \Sigma_{1,R}^\tau := \Gamma_{1,R} \times(-R,\tau)$
  when $ \tau\in (-R,0]$.

Making the appropriate estimation of the integrals of equality
\eqref{eq19z}.
From conditions (A4) and (B3) we have respectively
\begin{gather}
\begin{aligned}
&\iint_{Q_{R}^{\tau}}
\bigg\{\sum_{i=1}^{n}a_{i,12}(u_{12})_{x_{i}}+
a_{0,12}u_{12}\bigg\}\zeta^{s}\eta^{r}\,dx\,dt\\
&\geq \iint_{Q_{R}^{\tau}}\bigg\{\rho_1|\nabla
u_{12}|^{2}+\rho_2|u_{12}|^{p}\bigg\} \zeta^{s}\eta^{r}\,dx\,dt,
\end{aligned} \label{2.12}
\\
\iint_{Q_{R}^{\tau}}b_{12} \gamma u_{12}
\zeta^{s}\eta^{r}d\Gamma_{y}dt\geq
\iint_{\sum_{R}^{\tau}}\rho_3|\gamma
u_{12}|^q \zeta^{s}\eta^{r}d\Gamma_{y}dt.
\label{2.13}
\end{gather}
Hereinafter we will use the Young inequality:
For every $a\geq 0$, $b\geq0$, $\varepsilon>0$, $\nu>1$, we have
\begin{equation}\label{2.14}
ab\leq\varepsilon a^{\nu}+M(\nu,\varepsilon)b^{\nu'},
\end{equation}
where $1/\nu+1/\nu'=1$, $M(\nu,\varepsilon)>0$ is the
constant depending only on $\nu$ and $\varepsilon$.
Choose $\nu=p/2$
  $a=\rho_2^{1/\nu}|u_{12}|^{2}\zeta^{s/\nu}\eta^{r/\nu}$,
  $b=\rho_2^{-1/\nu}b_1\zeta^{s/\nu'}\eta^{r/\nu'-1}$,
  $\varepsilon=\varepsilon_1>0$ $(\nu'=p/(p-2))$.
  By  \eqref{2.14} we deduce
\begin{equation}\label{2.15}
\begin{aligned}
&\iint_{Q_{R}^{\tau}}b_1|u_{12}|^{2}\zeta^{s}\eta^{r-1}\,dx\,dt\\
&\leq  \varepsilon_1\iint_{Q_{R}^{\tau}}
  \rho_2|u_{12}|^{p}\zeta^{s}\eta^{r}\,dx\,dt\\
&\quad +  M(p/2,\varepsilon_1)\iint_{Q_{R}^{\tau}}
  \rho_2^{-{2}/{(p-2)}}b_1^{{p}/{(p-2)}}
  \zeta^{s}\eta^{r-p/(p-2)}\,dx\,dt,
\end{aligned}
\end{equation}
where $\varepsilon_1>0$ is an arbitrary number.
In the same way we obtain the inequality
   \begin{equation}\label{2.16}
\begin{aligned}
& \iint_{\sum_{1,R}^{\tau}}b_2|\gamma u_{12}|^{2}\zeta^{s}\eta^{r-1}
   d\Gamma_{y}dt\\
&\leq  \varepsilon_2\iint_{\sum_{1,R}^{\tau}}\rho_3
|\gamma u_{12}|^q  \zeta^{s}\eta^{r}d\Gamma_{y}dt\\
&\quad +   M(q/2,\varepsilon_2)
   \iint_{\sum_{1,R}^{\tau}}\rho_3^{-{2}/{(q-2)}}
   b_2^{q/(q-2)}\zeta^{s}\eta^{r-q/(q-2)}\,d\Gamma_{y}dt,
\end{aligned}
\end{equation}
where $\varepsilon_2>0$ is any number.
On the basis of condition (A3), taking into account that
    $|\zeta_{x_{i}}|\leq2$    $(i=\overline{1,n})$, we have
   \begin{equation}\label{2.17}
\begin{aligned}
& \Big|\iint_{Q_{R}^{\tau}}
   \bigg(\sum_{i=1}^{n} a_{i,12}\zeta_{x_{i}}\bigg)
   u_{12}\zeta^{s-1}\eta^{r}\,dx\,dt\Big|\\
&\leq    2\iint_{Q_{R}^{\tau}}
   \bigg(\sum_{i=1}^{n}|a_{i,12}|\bigg)|u_{12}|
   \zeta^{s-1}\eta^{r}\,dx\,dt \\
&\leq 2\iint_{Q_{R}^{\tau}}
   d_1|\nabla u_{12}\|u_{12}|\zeta^{s-1}\eta^{r}\,dx\,dt+
   2\iint_{Q_{R}^{r}}d_2|u_{12}|^{2}\zeta^{s-1}\eta^{r}\,dx\,dt.
\end{aligned}
\end{equation}
  Choose  $\nu=2$, $a=\rho_1^{1/2}|\nabla u_{12}|\zeta^{s/2}\eta^{r/2}$,
$b=\rho_1^{-1/2}d_1|u_{12}|\zeta^{s/2-1}\eta^{r/2}$,
   $\varepsilon=\varepsilon_3>0$.
    From  Young's inequality  \eqref{2.14}, we derive
\begin{equation}\label{2.18}
\begin{aligned}
&\iint_{Q_{R}^{\tau}}d_1|\nabla u_{12}| |u_{12}|
   \zeta^{s-1}\eta^{r}\,dx\,dt\\
&\leq    \varepsilon_3\iint_{Q_{R}^{\tau}}\rho_1|\nabla
   u_{12}|^{2}\zeta^{s}\eta^{r}\,dx\,dt
 +   M(2,\varepsilon_3)
   \iint_{Q_{R}^{\tau}}\rho_1^{-1}
   d_1^{2}|u_{12}|^{2} \zeta^{s-2}\eta^{r}\,dx\,dt,
\end{aligned}
\end{equation}
   where $\varepsilon_3>0$ is an arbitrary number.

To estimate the integral in the second term of the right side
of \eqref{2.18} use again the Young's inequality \eqref{2.14}, taking
 $\nu=p/2 $, $a=\rho_2^{1/\nu}|u_{12}|^{2}\zeta^{s/\nu}
   \eta^{r/\nu}$,
   $b=\rho_1^{-1}\rho_2^{-1/\nu}d_1^{2}\zeta^{s/\nu'-2}\eta^{r/\nu'}$,
   $\varepsilon=\varepsilon_4>0     (\nu'=p/(p-2))$.
   As a result we conclude
    \begin{equation}\label{2.19}
\begin{aligned}
&\iint_{Q_{R}^{\tau}}\rho_1^{-1}d_1^{2}|u_{12}|^{2}
    \zeta^{s-2}\eta^{r}\,dx\,dt\\
&\leq    \varepsilon_4\iint_{Q_{R}^{\tau}}
    \rho_2|u_{12}|^{p}\zeta^{s}\eta^{r}\,dx\,dt\\
&\quad +
    M(p/2,\varepsilon_4)\iint_{Q_{R}^{\tau}}\rho_1^{-p/(p-2)}
    \rho_2^{-2/(p-2)}d_1^{2p/(p-2)}\zeta^{s-2p/(p-2)}\eta^{r}\,dx\,dt,
\end{aligned}
\end{equation}
 where $\varepsilon_4>0$ is an arbitrary number.
 In the same way, we obtain
   \begin{equation}\label{2.20}
\begin{aligned}
&\iint_{Q_{R}^{\tau}}d_2|u_{12}|^{2}\zeta^{s-1}\eta^{r}\,dx\,dt\\
&\leq   \varepsilon_5
   \iint_{Q_{R}^{\tau}}\rho_2|u_{12}|^{p}\zeta^{s}\eta^{r}\,dx\,dt\\
&\quad +   M(p/2,\varepsilon_5)\iint_{Q_{R}^{\tau}}
   \rho_2^{-2/(p-2)}d_2^{p/(p-2)}
   \zeta^{s-p/(p-2)}\eta^{r}\,dx\,dt,
\end{aligned}
\end{equation}
where $\varepsilon_5>0$ is an arbitrary number.
Using the Cauchy inequality we have
    \begin{equation}\label{2.201}
\begin{aligned}
&\Big|\iint_{Q_{R}^{\tau}}\sum_{i=1}^n\bar{f}_{i,12}
    (u_{12})_{x_i}\zeta^s\eta^r\,dx\,dt\Big|\\
&\leq \varepsilon_6
    \iint_{Q_{R}^{\tau}}\rho_1|\nabla u_{12}|^2\zeta^s\eta^r\,dx\,dt
 + \frac{1}{4\varepsilon_6}   \iint_{Q_{R}^{\tau}}\rho_1^{-1}
    \Big(\sum_{i=1}^n|\bar{f}_{i,12}|^2\Big)\zeta^s\eta^r\,dx\,dt,
\end{aligned}
\end{equation}
and
   \begin{equation}\label{2.202}
\begin{aligned}
&\Big|\iint_{Q_{R}^{\tau}}
    \Big(\sum_{i=1}^n\bar{f}_{i,12}\zeta_{x_i}\Big)
    u_{12}\zeta^{s-1}\eta^r\,dx\,dt\Big|\\
&\leq    2\iint_{Q_{R}^{\tau}}\Big(\sum_{i=1}^n|\bar{f}_{i,12}|\Big)
    |u_{12}|\zeta^{s-1}\eta^r\,dx\,dt\\
&\leq \iint_{Q_{R}^{\tau}}\rho_1^{-1}d_1^{2}|u_{12}|^{2}
    \zeta^{s-2}\eta^{r}\,dx\,dt +
    n\iint_{Q_{R}^{\tau}}\rho_1d_1^{-2}
    \Big(\sum_{i=1}^n|\bar{f}_{i,12}|^2\Big)\zeta^s\eta^r\,dx\,dt.
\end{aligned}
\end{equation}
Also on the basis of Young's inequality, we obtain
    \begin{equation}\label{2.21}
\begin{aligned}
& \Big|\iint_{Q_{R}^{\tau}}
    f_{1,12}u_{12}\zeta^{s}\eta^{r}\,dx\,dt\Big|\leq
    \iint_{Q_{R}^{\tau}}|f_{1,12}|
    |u_{12}|\zeta^{s}\eta^{r}\,dx\,dt\\
& \leq \varepsilon_7\iint_{Q_{R}^{\tau}}\rho_2|u_{12}|^{p}
    \zeta^{s}\eta^{r}\,dx\,dt+
    M(p,\varepsilon_7)\iint_{Q_{R}^{\tau}}
    \rho_2^{-1/(p-1)}|f_{1,12}|^{p'}
    \zeta^{s}\eta^{r}\,dx\,dt,
\end{aligned}
 \end{equation}
and
\begin{equation}\label{2.22}
\begin{aligned}
 &\Big|\iint_{\sum_{1,R}^{\tau}}f_{2,12}
    \gamma u_{12}  \zeta^{s}\eta^{r}
    d\Gamma_{y}dt\Big|\\
&\leq \varepsilon_8     \iint_{\sum_{1,R}^{\tau}}
    \rho_3|\gamma     u_{12}|^q \zeta^{s}\eta^{r}d\Gamma_{y}dt
 + M(p,\varepsilon_{7})\iint_{\sum_{1,R}^{\tau}}
    \rho_3^{-1/(q-1)}|f_{2,12}|^{q'}
    \zeta^{s}\eta^{r}d\Gamma_{y}dt,
\end{aligned}
 \end{equation}
where $\varepsilon_{7}>0$, $\varepsilon_8>0$ are arbitrary numbers.
Then from \eqref{eq19z}, using  \eqref{2.12}, \eqref{2.13},
   \eqref{2.15}--\eqref{2.22} and taking
   $\varepsilon_1,\dots,\varepsilon_8$ be small enough,
we deduce the estimate
\begin{equation} \label{2.23}
\begin{aligned}
&\eta^{r}(\tau)\Big(\int_{\Omega_{R}}b_1(x)|u_{12}(x,\tau)|^{2}
   \zeta^{s}(x)dx+
   \int_{\Gamma_{1,R}}b_2(y)|\gamma u_{12}(y,\tau)|^{2}
   \zeta^{s}(y)d\Gamma_{y}\Big) \\
& +\iint_{Q_{R}^{\tau}}\big\{\rho_1|\nabla u_{12}|^{2}+
   \rho_2|u|^{p}\big\}\zeta^{s}\eta^{r}\,dx\,dt+
   \iint_{\sum_{1,R}^{\tau}}\rho_3
   |\gamma u_{12}|^q \zeta^{s}\eta^{r}
   d\Gamma_{y}dt \\
&\leq C_1\Big(\iint_{Q_{R}^{\tau}}\rho_2^{-2/(p-2)}
   b_1^{p/(p-2)}\zeta^{s}\eta^{r-p/(p-2)}\,dx\,dt\\
&\quad +   \iint_{\sum_{1,R}^{\tau}}\rho_3^{-2/(q-2)}
   b_2^{q/(q-2)}\zeta^{s}\eta^{r-q/(q-2)}d\Gamma_{y}dt\\
&\quad +\iint_{Q_{R}^{\tau}}\rho_1^{-p/(p-2)}\rho_2^{-2/(p-2)}
   d_1^{2p/(p-2)}\zeta^{s-2p/(p-2)}\eta^{r}\,dx\,dt\\
&\quad +   \iint_{Q_{R}^{\tau}}\rho_2^{-2/(p-2)}d_2^{p/(p-2)}
   \zeta^{s-p/(p-2)}\eta^{r}\,dx\,dt\Big)\\
&\quad +   C_2\Big(\iint_{Q_{R}^{\tau}}\rho_2^{-1/(p-1)}|f_{1,12}|^{p'}
   \zeta^{s}\eta^{r}\,dx\,dt\\
&\quad + \iint_{Q_{R}^{\tau}}\big(\rho_1^{-1}+n\rho_1d_1^{-2}\big)
    \Big(\sum_{i=1}^n|\bar{f}_{i,12}|^2\Big)\zeta^s\eta^r\,dx\,dt\\
&\quad +  \iint_{\sum_{1,R}^{\tau}}\rho_2^{-1/(q-1)}|f_{2,12}|^{q'}
   \zeta^{s}\eta^{r}\,d\Gamma_{y}dt \Big),
\end{aligned}
\end{equation}
    where $C_1>0,C_2>0$ are  constants depending only on $p,q$.
Let $R_{0}\in(0,R)$ be any number.
    Since $0\leq \zeta(x)\leq R$
    for every $x\in \mathbb{R}^{n}$, $\zeta(x)\geq R-R_{0}$
when $|x|\leq R_{0}$,
     and $0\leq\eta(t)\leq R$ for all $t\in \mathbb{R}$,
$\eta(t)\geq R-R_{0}$ when $t\geq - R_{0}$, from
    \eqref{2.23} we obtain
\begin{align*}
&\max_{\tau\in[-R_{0},0]}\Big(\int_{\Omega_{R_{0}}}b_1(x)
    |u_{12}(x,\tau)|^{2}dx+
    \int_{\Gamma_{1,R_{0}}}b_2(y)
    |\gamma u_{12}(y,\tau)|^{2}d\Gamma_{y}\Big)\\
&+\iint_{Q_{R_{0}}}\big\{\rho_1|\nabla u_{12}|^{2}+
    \rho_2|u_{12}|^{p}\big\}\,dx\,dt+\iint_{\sum_{1,R_{0}}}\rho_3
    |\gamma u_{12}|^q d\Gamma_{y}dt\\
&\leq (R/(R-R_{0}))^{s+r}\Big[C_3\Big(R^{-p/(p-2)}\iint_{Q_{R}}
    \rho_2^{-2/(p-2)}b_1^{p/(p-2)}\,dx\,dt \\
&\quad + R^{-2p/(p-2)}\iint_{Q_{R}}\rho_1^{-p/(p-2)}\rho_2^{-2/(p-2)}
    d_1^{2p/(p-2)}\,dx\,dt\\
&\quad+R^{-p/(p-2)}\iint_{Q_{R}}
    \rho_2^{-2/(p-2)}d_2^{p/(p-2)}\,dx\,dt\\
&\quad +    R^{-q/(q-2)}\iint_{\sum_{1,R}}\rho_3^{-2/(q-2)}
    b_2^{q/(q-2)}  d\Gamma_{y}\,dt\Big) \\
&\quad + C_4\Big( \iint_{Q_{R}}\big(\rho_1^{-1}+n\rho_1d_1^{-2}\big)
    \Big(\sum_{i=1}^n|\bar{f}_{i,12}|^2\Big)\,dx\,dt\\
&\quad +\iint_{Q_{R}}\rho_2^{-1/(p-1)}|f_{1,12}|^{p'}\,dx\,dt+
    \iint_{\sum_{1,R}}
    \rho_3^{-1/(q-1)}|f_{2,12}|^{q'}\,d\Gamma_{y}\,dt\Big)\Big],
\end{align*}
    where $C_3>0,\;C_4>0$ are  constants depending only on $p,q$.
This yields \eqref{eq18z}.
    \end{proof}

\begin{remark}\label{reml1} \rm
If in addition to the condition of Lemma \ref{l1} we assume that
$\operatorname{supp}  v\subset \overline{\Omega_R\times
(\tau_1,\tau_2)}$,
 then the assertion of Lemma \ref{l1}  is also true  when
$R_*=R$ and $w=1$.
\end{remark}

 \begin{corollary} \label{cor}
 Let $(b_1,b_2)\in \mathbb B $, $(a_0,a_1,\dots,a_n)\in \mathbb{A},
 c\in \mathbb C, (f_1,f_2)\in \mathbb{F}_{\rm loc}$. Suppose that
 for some $R>0$ there exist constants
 $\alpha_j>0,  \beta_j>0  (j=1,2), \alpha_3\geq 0,
 \beta_3\geq 0, \mu_1>0, \mu_2\geq 0$ such that
 for a.e. $(x,t)\in Q_R$ and every
 $(s,\xi)\in  \mathbb{R}^{1+n}$ we have
  \begin{gather}\label{est1}
 \sum_{i=1}^n|a_i(x,t,s,\xi)|\leq \alpha_1|\xi|  +
    \alpha_2|s|+\alpha_3, \\
\label{est2}
 \sum_{j=1}^{n}  a_i(x,t,s,\xi) \xi_i +  a_0(x,t,s,\xi)s \geq
  \beta_1 |\xi|^2 + \beta_2|s|^p - \beta_3, \\
\label{est3}
 c(x,t,s) s \geq
  \mu_1 |s|^q - \mu_2,
\end{gather}
 Then for any  generalized solution  $u$ of \eqref{1}--\eqref{3}
and for every $R^*\in (0,R)$ the estimate
 \begin{equation}\label{est4}
 \|u\|_{L_2(S_{R^*};H^1(\Omega_{R^*}))}+\|u\|_{L_p(Q_{R^*})}+
 \|\gamma u\|_{L_q(\Sigma_{1,R^*})}\leq C_5(R,R^*)
 \end{equation}
 takes place, where $C_5(R,R^*)>0$ is the constant depending only on
 $R, R^*, f_1|_{Q_R},\quad$ $ f_2|_{\Sigma_{1,R}}, \alpha_k,
 \beta_k   (k=1,2,3), \mu_j (j=1,2)$.
\end{corollary}

The statement of this Corollary  can be  obtained similarly as it
made for \eqref{eq18z}.

\section{Proof of main results}\label{Sect4}

  \begin{proof}[Proof of Theorem \ref{t1}]
\textbf{Step 1.} For every
  $k\in \mathbb{N}$ take the subdomain
 $\Omega^k$ of the domain $\Omega$ such that
$\partial \Omega^k\in C^1$,
 $\Omega_k\subset \Omega^k, \Omega^k\subset \Omega^{k+1}$.
 Put $Q^k=\Omega^k\times S_k,   \Gamma_0^k:=
 \overline{\partial \Omega^k\setminus\Gamma_1}$,
 $ \Gamma_1^k=\partial \Omega^k\setminus\Gamma_0^k$,
$\Sigma^k_0:=\Gamma^k_0\times S_k$,
 $\Sigma^k_1:=\Gamma^k_1\times S_k$.

For every $k\in\mathbb{N}$, let $\mathbb{V}^k$ be the Banach space
obtained by closure of
 the space  $\{v\in C^1(\overline{\Omega^k}):\operatorname{dist}
 \{\operatorname{supp}v, \Gamma_0^k\}>0\}$ by the norm
 $\|v\|_{\mathbb{V}^k}:=\|v\|_{H^1(\Omega^k)}+\|v\|_{L_p(\Omega^k)}
 +\|v\|_{L_q(\Gamma_1^k)}$.
Note that for every $k\in\mathbb{N}$ the extensions
by zero on $\Omega$  of functions from $\mathbb{V}^k$ generate
the subspace of the
 space $\mathbb{V}_c\subset \mathbb{V}_{\rm loc}$.
   Thus we can consider the operator $\gamma^k:\mathbb{V}^k\to L_q(\Gamma_1^k)$ as
the contraction of the operator
$\gamma:\mathbb{V}_{\rm loc}\to L_{q,{\rm loc}}
 (\overline{\Gamma_1})$. So further we will write
$\gamma$ instead of $\gamma^k$.
Define
\begin{gather*}
 \mathbb{U}^k:=\{w\in(S_k\to \mathbb{V}^k):
w\in L_2(S_k;H^1(\Omega^k))\cap L_p(Q^k),\\
 b_1^{1/2}w\in C(\overline{S_k};L_2(\Omega^k)),
 \gamma w\in L_q(\Sigma_1^k), b_2^{1/2}\gamma w\in
 C(\overline{S_k};L_2(\Gamma_1^k))\}
\end{gather*}
  be the Banach space with the norm
\begin{align*}
 \|w\|_{\mathbb{U}^k}
&:=\|w\|_{L^2(S_k;H^1(\Omega^k))}+\|w\|_{L_p(Q^k)}
 +\max_{t\in S^k}\|b_1^{1/2}(\cdot)w(\cdot,t)\|_{L_2(\Omega^k)}\\
&\quad  +\|\gamma w\|_{L_q(\Sigma_1^k)}
 +\max_{t\in S^k}\|b_2^{1/2}(\cdot)
 \gamma w(\cdot,t)\|_{L_2(\Gamma_1^k)}.
\end{align*}
 Consider the family of mixed problems
 \begin{gather}
\begin{gathered}
 \frac{\partial}{\partial t}(b_1(x)u^k)-\sum_{i=1}^n
 \frac{d}{d x_i}
 a_i(x,t,u^k,\nabla u^k)+a_0(x,t,u^k,\nabla u^k)= f_1(x,t),  \\
(x,t)\in Q_k,
\end{gathered}\label{35} \\
\label{36}
 u^k(y,t)=0,\quad (y,t)\in\Sigma_0^k,
\end{gather}
and
\begin{gather}\label{38}
 \frac{\partial}{\partial t}(b_2(y)u^k)-\sum_{i=1}^n
 a_i(y,t,u^k,\nabla u^k)\nu_i(y)+c(y,t,u^k)=f_2(y,t),
 \quad (y,t)\in\Sigma_1^k, \\
\label{39}
 b_1^{1/2}u^k|_{t=-k}=0,\quad x\in\Omega^k,\quad
  b_2^{1/2}u^k|_{t=-k}=0,\quad y\in\Gamma_1^k.
\end{gather}


 \begin{definition} \label{def4.1} \rm
A function $u^k\in\mathbb{U}^k$ is called a generalized solution of
 the problem \eqref{35}-\eqref{39} (for arbitrary $k\in\mathbb{N}$)
if it  satisfies the initial data   \eqref{39}
 and the integral equality
\begin{equation}\label{40}
\begin{aligned}
&\iint_{Q^k}\Big\{\sum_{i=1}^n
    a_i(x,t,u^k,\nabla u^k) \psi_{x_i} \varphi+
    a_0(x,t,u^k,\nabla u^k) \psi \varphi-
    b_1(x) u^k \psi \varphi'\Big\}\,dx\,dt
\\
&+\iint_{\Sigma^k_1}\Big\{ c(y,t,\gamma u^k) \gamma
    \psi \varphi-b_2(y) \gamma u^k \gamma\psi
    \varphi'\Big\}\,d\Gamma_y\,dt
\\
&=\iint_{Q^k}f^k_1 \psi \varphi\,dx\,dt+
    \iint_{\Sigma^k_1}f^k_2 \gamma\psi
    \varphi \,d\Gamma_y\,dt
\end{aligned}
\end{equation}
for every $\psi\in\mathbb{V}^k$,
 $\varphi\in C^1([-k,0])$, $\varphi(0)=0$.
\end{definition}

The existence and uniqueness of the generalized solution of
 \eqref{35} -- \eqref{39} (for every  $k\in\mathbb{N}$)
 can be easily proved using research technique
from  \cite{Showalter97}.

\textbf{Step 2.} For every  $k\in\mathbb{N}$
extend $u^k$ by zero to $Q$ and
keep the notation $u^k$ to this extension.
 It easy to verify that $u^k\in\mathbb{U}_{\rm loc}$ for all
 $k\in\mathbb{N}$. Consider the sequence
   $\{u^k\}_{k=1}^\infty$ and show that it contains the subsequence
    converging to the generalized solution of
    problem  \eqref{1} -- \eqref{3} in some
    sense.

First we show that for every $R_0>0$ the sequences
$\{u^k|_{Q_{R_0}}\}_{k=1}^\infty$,
$\{\gamma u^k|_{\Sigma_{1,R_0}}\}_{k=1}^\infty$,
$\{b_1^{1/2} u^k|_{Q_{R_0}}\}_{k=1}^\infty$, and
$\{b_2^{1/2} \gamma u^k|_{\Sigma_{1,R_0}}\}_{k=1}^\infty$
are respectively fundamental in spaces
$L_2(S_{R_0};H^1(\Omega_{R_0}))\cap L_p(Q_{R_0})$,
$L_q(\Sigma_{1,R_0})$,
$C(\overline{S_{R_0}};L_2(\Omega_{R_0}))$ and
$C(\overline{S_{R_0}};L_2(\Gamma_{1,R_0}))$.
For this purpose use Lemma \ref{l2}, choosing $R>2R_0$ be any number,
 $u_1=u^k$, $u_2=u^l$ for arbitrary $k,l>R$.  From its assertion
 and inequality
 \begin{equation}\label{rro}
 {R}/(R-R_0)=1+{R_0}/(R-R_0)\leq 2\quad
\text{when } R\geq 2R_0
\end{equation}
 we obtain
\begin{equation}\label{41}
\begin{aligned}
&\max_{t\in \overline{S_{R_0}}}\|b_1^{1/2}(\cdot)
  (u^k(\cdot,t)-u^l(\cdot,t))\|^2_{L_2(\Omega_{R_0})}\\
&\quad+ \max_{t\in \overline{S_{R_0}}}\|b_2^{1/2}(\cdot)
  (\gamma u^k(\cdot,t)-\gamma u^l(\cdot,t))\|^2_{L_2(\Gamma_{1,R_0})}\\
&\quad +\|u^k-u^l\|^2_{L_2(S_{R_0};H^1(\Omega_{R_0}))}+
    \|u^k-u^l\|^p_{L_p(Q_{R_0})}+
    \|\gamma u^k-\gamma u^l\|^q_{L_q(\Sigma_{1,R_0})}\\
&\leq C_6(R_0)
    \Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R),
\end{aligned}
\end{equation}
where $C_6(R_0)>0$ is a constant depending on $R_0$,
 but not depending on  $R$.

 From  \eqref{psiton} and  \ref{rro} we
conclude that the right side of  \eqref{41} tends
to zero as $R\to +\infty$. Thus for arbitrarily small value
$\varepsilon>0$ there exists $k^*\in\mathbb{N}$ such that for every
$k,l\geq k^*$ the left side of
 \eqref{41} is less then $\varepsilon$. It yields the
  fundamentality  of the sequences $\{u^k|_{Q_{R_0}}\}$ in
  $L_2(S_{R_0};H^1(\Omega_{R_0}))\cap L_p(Q_{R_0})$,
  $\{\gamma u^k|_{\Sigma_{1,R_0}}\}$
  in  $L_q(\Sigma_{1,R_0})$,
   $\{b_1^{1/2} u^k|_{Q_{R_0}}\}$
   in $C(\overline{S_{R_0}};L_2(\Omega_{R_0}))$,
    $\{b_2^{1/2} \gamma u^k|_{\Sigma_{1,R_0}}\}$ in
     $C(\overline{S_{R_0}};L_2(\Gamma_{1,R_0}))$. Since $R_0$ is
      an arbitrary number, the above stated yields the existence of
       functions $u\in(S\to\mathbb{V}_{\rm loc})\cap
     L_{2,{\rm loc}}(S;H^1_{\rm loc}(\overline{\Omega}))
     \cap L_{p,{\rm loc}}(\overline{Q})$,
     $\gamma u\in L_{q,{\rm loc}}(\overline{\Sigma_1})$,
$\widehat{u}\in C(S;L_{2,{\rm loc}}(\overline{\Omega}))$,
$\widehat{u}\in C(S;L_{2,{\rm loc}}(\overline{\Gamma_1}))$ such
that
\begin{gather}\label{42}
    u^k\mathop{\longrightarrow}_{k\to\infty}u\quad\text{ in }
     L_{2,{\rm loc}}(S;H^1_{\rm loc}(\overline{\Omega}))\cap
    L_{p,{\rm loc}}(\overline{Q}), \\
\label{43}
    \gamma u^k\mathop{\longrightarrow}_{k\to\infty}\gamma
    u\quad\text{ in } L_{q,{\rm loc}}(\overline{\Sigma_1}),\\
\label{44}
    b_1^{1/2} u^k\mathop{\longrightarrow}_{k\to\infty}
    \widehat{u}\quad\text{in }
    C(S;L_{2,{\rm loc}}(\overline{\Omega})),\\
\label{45}
    b_2^{1/2} \gamma u^k\mathop{\longrightarrow}
    _{k\to\infty}\widehat{u}\quad\text{in }
     C(S;L_{2,{\rm loc}}(\overline{\Gamma_1})).
\end{gather}
It remains to show that
\begin{equation}\label{46}
    \widehat{u}=b_1^{1/2} u,\quad
     \widehat{u}=b_2^{1/2} \gamma u.
\end{equation}
Indeed, from \eqref{42}--\eqref{45} it follows that there exists
a subsequence $\{u^{k_j}\}_{j=1}^\infty$ such that
\begin{gather}\label{47}
    u^{k_j}\mathop{\longrightarrow}
    _{j\to\infty}u,\quad\partial u^{k_j}/\partial x_i
    \mathop{\longrightarrow}_{j\to\infty}
    \partial u/\partial x_i\quad\text{a.e. in } Q, \\
\label{48}
    \gamma u^{k_j}\mathop{\longrightarrow}
    _{j\to\infty}\gamma u \quad\text{a.e. on}\quad \Sigma_1,\\
\label{49}
    b_1^{1/2} u^{k_j}\mathop{\longrightarrow}
    _{j\to\infty}\widehat{u} \quad\text{a.e. in } Q,\\
\label{50}
    b_2^{1/2} \gamma u^{k_j}\mathop{\longrightarrow}
    _{j\to\infty}\widehat{u} \quad\text{a.e. on}\quad \Sigma_1.
\end{gather}
 From this it easily follows \eqref{46}. On the basis of
\eqref{44}-- \eqref{46} we conclude that $b_1^{1/2} u\in
C(S;L_{2,{\rm loc}} (\overline{\Omega}))$,
$b_2^{1/2} \gamma u\in  C(S;L_{2,{\rm loc}}(\overline{\Gamma_1}))$,
and therefore  $u\in \mathbb{U}_{\rm loc}$.

 Now show that $u$ is  generalized solution of \eqref{1}--\eqref{3}.
First of all note that under condition (A3) and
 \eqref{42}, we have
\begin{equation}\label{51}
    a_i(\cdot,\cdot,u^k(\cdot,\cdot),\nabla u^k(\cdot,\cdot))
    \mathop{\longrightarrow}_{k\to\infty}
    a_i(\cdot,\cdot,u(\cdot,\cdot),\nabla u(\cdot,\cdot))
    \quad\text{in } L_{2,{\rm loc}}(\overline{Q}),\;
     i=\overline{1,n}.
\end{equation}
Now prove the existence of a subsequence
$\{u^{k_i}\}_{i=1}^\infty$ of the sequence $\{u^{k}\}$
such that for arbitrary fixed $R>0$
\begin{gather}\label{52}
    a_0(\cdot,\cdot,u^{k_i}(\cdot,\cdot),\nabla u^{k_i}
    (\cdot,\cdot))\mathop{\longrightarrow}
    _{i\to\infty}a_0(\cdot,\cdot,u(\cdot,\cdot),
    \nabla u(\cdot,\cdot)) \quad\text{weakly in } L_{p'}(Q_R),\\
\label{53}
    c(\cdot,\cdot,\gamma u^{k_i}(\cdot,\cdot))\mathop{\longrightarrow}
    _{i\to\infty}c(\cdot,\cdot,\gamma u(\cdot,\cdot))
    \quad\text{weakly in } L_{q'}(\Sigma_{1,R}).
\end{gather}
Indeed,  \eqref{42} and  \eqref{43} yield the estimates
\begin{gather}\label{54}
    \|u^k\|_{L_2(S_R;H^1(\Omega_R))\cap L_{p}(Q_R)}\leq C_7(R),
    \quad k\in \mathbb{N}, \\
\label{55}
    \|\gamma u^k\|_{L_{q}(\Sigma_{1,R})}\leq C_8(R),
    \quad k\in \mathbb{N},
\end{gather}
where $C_7(R),C_8(R)>0$ are constants probably depending on $R$,
but not depending on $k$.

By (A2) and \eqref{54} we have
\begin{equation}\label{56}
    \|a_0(\cdot,\cdot,u^{k}(\cdot,\cdot),
    \nabla u^{k}(\cdot,\cdot))\|_{L_{p'}(Q_R)}\leq C_9(R),
    \quad k\in\mathbb{N},
\end{equation}
and on the basis of condition (B2) and
\eqref{55}, we have
\begin{equation}\label{57}
    \|c(\cdot,\cdot,\gamma u^{k}(\cdot,\cdot))\|_{L_{q'}
(\Gamma_{1,R})}\leq
    C_{10}(R),\quad k\in\mathbb{N},
\end{equation}
where $C_9(R),C_{10}(R)>0$ are constants probably depending on
$R$, but not depending on $k$. Using the
\cite[Proposition~3.4,p.~51]{Showalter97},
from $\mathbf{A}_1$, \eqref{47}
 and \eqref{56} we obtain \eqref{52}, and from (B1),
 \eqref{48} and \eqref{57}
 we obtain \eqref{53}.

Let $\psi\in\mathbb{V}_{\rm c}$, $\varphi\in C^1_0(-\infty,0)$ be
  arbitrary functions and $l$ be a natural number  such that
 $\operatorname{supp}\psi\subset \overline{\Omega^l}$,
  $\operatorname{supp}\varphi\subset (-l,0)$. Then for every
   $k>l$ $(k\in\mathbb{N})$ the equality \eqref{40} is fulfilled.
In fact the integrals in this equality are taken over
    $Q^l$ instead of $Q^k$
   and $\Sigma^l_1$  instead of  $\Sigma^k_1$.
    Put $k=k_i (i\in\mathbb{N})$ in \eqref{40} and pass to the limit
       as $i\to\infty$, taking into account
   \eqref{42},\eqref{43},\eqref{51}--\eqref{53}.
   As a result we obtain \eqref{inttot};
   i.e., exactly what is needed.

To obtain  \eqref{ozinka} let us use Lemma \ref{l2}.
 Let $u$ be the generalized solution of the
 problem \eqref{1}--\eqref{3} with given
 $(b_1,b_2)\in\mathbb{B}$, $(a_0,a_1,\ldots,a_n)\in\mathbb{A}$,
 $c\in\mathbb{C}$, $(f_1,f_2)\in\mathbb{F}_{\rm loc}$.
Note that on the basis of conditions
 $\mathbf{A'_1}, \mathbf{C'_1}$ the function
 $u=0$ is the solution of  \eqref{1}--\eqref{3} with the same
 $(b_1,b_2)$, $(a_0,a_1,\ldots,a_n)$,
 $b$, but with $(f_1,f_2) = (0,0)$.
 Let $R\geq 0$ be an arbitrary number, $u_1=u$, $u_2=0$,
 $f_{1,1}=f_1$, $f_{2,1}=f_2$, $f_{1,2}=0$, $f_{2,2}=0,
 \overline{f}_{i,1}=\overline{f}_{i,2} (i=\overline{1,n})$.
  Then from Lemma \ref{l2} (see \eqref{eq18z})
  it is easy to get \eqref{ozinka}.

\textbf{Step 3.} Now prove
the continuous dependence on data-in  of generalized solutions
 of  \eqref{1}--\eqref{3}.
Let  $\{(a_0^k,a_1^k,\dots,a_n^k)\},
\{c^k\}$ and $\{(f_1^k,f_2^k)\}$ be the sequences
such that
$((b_1,b_2),(a_0^k,a_1^k,\dots,a_n^k), c^k)\in
 \mathbb{B}\mathbb{A}\mathbb{C}$ and
 \begin{gather}\label{581}
 (a_0^k,a_1^k,\dots,a_n^k)\mathop{\longrightarrow}_{k\to\infty}
 (a_0,a_1,\dots,a_n)\quad\text{in } \mathbb{A},
  \quad c^k\mathop{\longrightarrow}_{k\to\infty}  c
\quad\text{in } \mathbb C, \\
\label{58}
   (f^k_1,f^k_2)\mathop{\longrightarrow}_{k\to\infty}
   (f_1,f_2)\quad\text{in } \mathbb{F}_{\rm loc}.
\end{gather}
Take any number $k\in\mathbb{N}$. Reformulate the integral identity,
which define the function $u^k$ as a generalized solution
corresponding problem (similar to \eqref{inttot}), in the form
\begin{align}
&\iint_Q\Bigl\{\sum_{i=1}^n a_i(x,t,u^k,\nabla u^k)
   \psi_{x_i} \varphi + a_0(x,t,u^k,\nabla u^k)
  \psi  \varphi - b_1(x) u^k \psi  \varphi'\Bigr\}\,dx\,dt \notag\\
& +\iint_{\Sigma_1}\{c(y,t,\gamma u^k) \gamma\psi
 \varphi- b_2(y)  \gamma u^k  \gamma \psi  \varphi'\}\,d\Gamma_y dt
 \notag\\
& =\iint_Q\Bigl\{\sum_{i=1}^n
  \big(a_i(x,t,u^k,\nabla u^k)-a_i^k(x,t,u^k,\nabla u^k)\big)
   \psi_{x_i}\varphi \notag \\
&\quad +  \big(a_0(x,t,u^k,\nabla u^k)-a_0^k(x,t,u^k,\nabla u^k)+
  f_1^k \big) \psi \varphi
  \Bigr\}\,dx\,dt \notag \\
&\quad  +\iint_{\Sigma_1}
  \{c(y,t,\gamma u^k)-c^k(y,t,\gamma u^k)+f_2^k\} \gamma\psi
 \varphi\,d\Gamma_y dt \label{inttotk}
\end{align}
for every $\psi\in \mathbb{V}_c, \varphi\in C^1_0(-\infty,0)$.

 From \eqref{inttotk} and \eqref{inttot} on the basis of Lemma
\ref{l2},  putting $u_1=u^k$, $u_2=u$ and
$\bar{f}_{i,1}=a_i(\cdot,\cdot,u^k,\nabla u^k)- a_i^k(\cdot,
\cdot,u^k,\nabla u^k)$  $(i=\overline{1,n})$,
$f_{1,1}= a_0(\cdot,\cdot,u^k,\nabla u^k)
-a_0^k(\cdot,\cdot,u^k,\nabla u^k)+ f_1^k,$,
$f_{2,1}=c(\cdot,\cdot,\gamma u^k)-c^k(\cdot,\cdot,\gamma u^k)+f_2^k$,
 $\bar{f}_{i,2}=0$ $(i=\overline{1,n})$,
 $f_{1,2}=f_1$, $f_{2,2}=f_2$ and choosing $R>0$,
 $R_0\in (0,R)$ to be arbitrary, we obtain
\begin{align}
&\max_{t\in[-R_0,0]}
    \int_{\Omega_{R_0}}b_1(x)|u^k(x,t)-u(x,t)|^2\,dx \notag\\
&+    \max_{t\in[-R_0,0]}\int_{\Gamma_{1,R_0}}b_2(y)
    |\gamma  u^k(y,t)-\gamma u(y,t)|^2\,d\Gamma_y \notag\\
& +\iint_{Q_{R_0}}\Big\{\rho_1|\nabla u^k-\nabla u|^2+
    \rho_2|u^k-u|^p\Big\}\,dx\,dt+
    \iint_{\Sigma_{1,R_0}}\rho_3
    |\gamma u^k-\gamma u|^q\,d\Gamma_y\,dt \notag \\
&\leq C_{11}(R/(R-R_0))^\sigma
   \Big[\Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R) \notag\\
&\quad +   \iint_{Q_{R}}
   \rho_2^{-1/(p-1)}|f_1^k-f_1|^{p'}\,dx\,dt+
   \iint_{\Sigma_{1,R}}
   (\rho_3)^{-1/(q-1)}|f_2^k-f_2|^{q'}\,d\Gamma_y\,dt\notag \\
&\quad + \iint_{Q_{R}}\big(\rho_1^{-1}+n\rho_1d_1^{-2}\big)
    \sum_{i=1}^n |a_i(x,t,u^k,\nabla u^k)-a_i^k(x,t,u^k,
   \nabla u^k)|^2\,dx\,dt \notag \\
&\quad +  \iint_{Q_{R}}\rho_2^{-1/(p-1)}
  |a_0(x,t,u^k,\nabla u^k)-  a_0^k(x,t,u^k,\nabla u^k)|^{p'}\,dx\,dt
  \notag \\
&\quad   + \iint_{\Sigma_{1,R}}
   \rho_3^{-1/(q-1)}|c(y,t,\gamma u^k)-c^k(x,t,\gamma u^k)|^{q'}\,dx\,dt
   \Big], \label{59}
\end{align}
 where $\sigma,\Psi$ are the same
  as in the statement of the Theorem \ref{t1}, $C_{11}>0$
 is a constant  depending only on $p,q$.

Let $\varepsilon>0$ be an arbitrary small number and $R_0>0$ be
any number. By virtue of  \eqref{psiton}  we can take $R\geq 2R_0$
such that
\begin{equation}\label{61}
   \Psi(b_1,b_2,\rho_1,\rho_2,\rho_3,d_1,d_2;R)<
   \varepsilon/(5 C_{11} 2^\sigma )\,.
\end{equation}
Fix such $R$. On the basis of \eqref{58} there exists
$k_1\in\mathbb{N}$ such that
 \begin{equation}\label{62}
\begin{aligned}
&\iint_{Q_{R}}
   \rho_2^{-1/(p-1)}|f_1^k-f_1|^{p'}\,dx\,dt
 +\iint_{\Sigma_{1,R}}
   \rho_3^{-1/(q-1)}|f_2^k-f_2|^{q'}\,d \Gamma_y\,dt\\
&<   \varepsilon/(5 C_{11} 2^\sigma )
\end{aligned}
\end{equation}
for every $k\geq k_1$.
 Now we  show the existence of constants
 $k_2\geq k_1$  $(k_2\in\mathbb{N})$,  $ C_{12}>0$ such that
\begin{equation}\label{126}
\iint_{Q_R}\big(|u^k|^2+|u^k|^p+|\nabla u^k|^2\big)\,dx\,dt +
\iint_{\Sigma_{1,R}}|\gamma u^k|^q \leq C_{12}
\end{equation}
for all $k\geq k_2$.
To this effect we use Corollary \ref{cor}.
 Let $k$ be any natural number.
Put
 $\bar{\rho}_{j}:=\operatorname{ess\, inf}
 _{(x,t)\in Q_{R+1}}\rho_{j}(x,t)>0$
$(j=1,2)$,
$\bar{\rho}_3:=\operatorname{ess\,inf}
 _{(y,t)\in \Sigma_{1,R+1}}\rho_3(y,t)>0$,
 $\bar{d}_1:=\operatorname{ess\, sup}
 _{(x,t)\in Q_{R+1}}d_1(x,t)$,
 $\bar{d}_2:=\operatorname{ess sup} _{(x,t)\in Q_{R+1}}d_2(x,t)$.

 From conditions (A1'), (A3), (A4), (C1'), (C3)
and simple consideration for a.e. $(x,t)\in Q_{R+1}$ it follows
that
\begin{equation}\label{ocin2}
\begin{aligned}
&\sum_{i=1}^n|a_i^k(x,t,s,\xi)|\\
&\leq \sum_{i=1}^n|a_i(x,t,s,\xi)| +
\sum_{i=1}^n|a_i^k(x,t,s,\xi)-a_i(x,t,s,\xi)|\\
&\leq \bar{d}_1|\xi|+\bar{d}_2|s|\\
&\quad + \Big(\mathop{\rm ess\,sup}
_{(x,t)\in Q_{R+1}} \sup _{(s,\xi)\in
\mathbb{R}^{1+n}}\sum_{i=1}^{n}
 \frac{|a_i^k(x,t,s,\xi) -  a_i(x,t,s,\xi)|}{1+|s|+|\xi|}\Big)
 (1+|s|+|\xi|),
\end{aligned}
\end{equation}
and
\begin{align*} %\label{koer1}
& \sum_{i=1}^n a_i^k(x,t,s,\xi)\xi_i+a_0^k(x,t,s,\xi)s\\
&= \sum_{i=1}^n a_i(x,t,s,\xi)\xi_i+a_0(x,t,s,\xi)s\\
&\quad  + \sum_{i=1}^n \big(a_i^k(x,t,s,\xi)-a_i(x,t,s,\xi)\big)\xi_i+
 \big(a_0^k(x,t,s,\xi)-a_0(x,t,s,\xi)\big)s\\
&\geq \rho_1(x,t)|\xi|^2 + \rho_2(x,t)|s|^p -
 \Big(\sum_{i=1}^n \big|a_i^k(x,t,s,\xi)-a_i(x,t,s,\xi)\big\|\xi|\\
&\quad + \big|a_0^k(x,t,s,\xi)-a_0(x,t,s,\xi)\big\|s|\Big)\\
&\geq \bar{\rho}_1|\xi|^2+\bar{\rho}_2|s|^p\\
&\quad - \Big[\Big(\mathop{\rm ess\,sup} _{(x,t)\in Q_{R+1}}
\sup _{(s,\xi)\in\mathbb{R}^{1+n}}\sum_{i=1}^{n}
 \frac{|a_i^k(x,t,s,\xi) -  a_i(x,t,s,\xi)|}{(1+|s|+|\xi|)}\Big)
 (|\xi|+|s\|\xi|+|\xi|^2)\\
&\quad +\Big(\mathop{\rm ess\,sup} _{(x,t)\in Q_{R+1}}
\sup_{(s,\xi)\in\mathbb{R}^{1+n}}
\frac{|a_0^k(x,t,s,\xi)-a_0(x,t,s,\xi)|}{(1+|s|^{p-1}+|\xi|^{2/p'})}\Big)
(|s|+|s|^{p}+|s\|\xi|^{2/p'})\Big],
\end{align*}
and
\begin{equation}\label{coer2}
\begin{aligned}
c^k(y,t,s)s&= c(y,t,s)s - |c^k(y,t,s) - c(y,t,s)\|s|\\
&\geq \bar{\rho_3}|s|^p - \Big(\mathop{\rm ess\,sup} _{(x,t)\in
\Sigma_{1,R+1}} \sup_{s\in \mathbb{R}} \frac{|c^k(y,t,s)
- c(y,t,s)|}{1+|s|^{p-1}}\Big)\big(|s|+|s|^p\big).
\end{aligned}
\end{equation}
Now note that on the basis of  Young's inequalities
we obtain
\begin{gather*}
|\xi|\leq |\xi|^2/2+1/2, \quad
|s\|\xi|\leq |\xi|^2/2 + |s|^p/p + (p-2)/(2p), \\
|s|\leq |s|^p/p+1/p', \quad
|s\|\xi|^{2/p'}\leq |\xi|^2/p'+|s|^p/p.
\end{gather*}
  From this, \eqref{ocin2}-\eqref{coer2},
and \eqref{581} it follows that there exists natural number
$k_2\geq k_1$ such that for each $k\geq k_2$  we obtain
assumptions  of Corollary \ref{cor} (in particular,
\eqref{est1}-\eqref{est3})
 with $a_0^k,a_1^k,\dots,a_n^k,c^k$ instead of
$a_0,a_1,\dots,a_n,c$ respectively). Note that in the given case
constants $\alpha_j,\beta_j,\mu_l$ are independent of $k$. Taking
into consideration \eqref{58} from the statement of Corollary
\ref{cor} \eqref{126}  follows.


We resume to estimate the terms in the right side of \eqref{59}.
After easy transformations we obtain the inequalities
\begin{equation}  \label{610}
\begin{aligned}
&\iint_{Q_{R}}\big(\rho_1^{-1}+n \rho_1d_1^{-2}\big)
    \sum_{i=1}^n |a_i(x,t,u^k,\nabla u^k)-a_i^k(x,t,u^k,
\nabla u^k)|^2\,dx\,dt \\
&\leq 3\mathop{\rm ess\,sup} _{(x,t)\in Q_R}
\big(\rho_1^{-1}(x,t)+n\rho_1(x,t)d_1^{-2}(x,t)\big)\\
&\quad \times\mathop{\rm ess\,sup} _{(x,t)\in Q_R} \sup
_{(s,\xi)\in\mathbb{R}^{1+n}} \Big(\sum_{i=1}^{n}
|a_i^k(x,t,s,\xi) - a_i(x,t,s,\xi)|/(1+|s|+|\xi|)\Big)^2 \\
&\quad \times\iint_{Q_{R}}\big(1+|u^k|^2+|\nabla u^k|^2\big)\,dx\,dt,
\end{aligned}
\end{equation}
\begin{align*} % \label{611}
&\iint_{Q_{R}}\rho_2^{-1/(p-1)} |a_0(x,t,u^k,\nabla u^k)-
  a_0^k(x,t,u^k,\nabla u^k)|^{p'}\,dx\,dt\\
&\leq C_{13}(p)\mathop{\rm ess\,sup} _{(x,t)\in Q_R}
\rho_2^{-1/(p-1)}(x,t)\\
&\quad \times\mathop{\rm ess\,sup} _{(x,t)\in Q_R} \sup
_{(s,\xi)\in\mathbb{R}^{1+n}}
\big[|a_0^k(x,t,s,\xi)-a_0(x,t,s,\xi)|/(1+|s|^{p-1}
 +|\xi|^{2/p'})\big]^{p'} \\
&\quad  \times\iint_{Q_{R}}\big(1+|u^k|^p+|\nabla u^k|^2\big)\,dx\,dt,
\end{align*}
\begin{equation}
\begin{aligned}  \label{612}
&\iint_{\Sigma_{1,R}}    \rho_3^{-1/(q-1)}|
c(y,t,\gamma u^k)-c^k(x,t,\gamma u^k)|^{q'}\,dx\,dt \\
& \leq C_{14}(p)\mathop{\rm ess\,sup} _{(y,t)\in \Sigma_{1,R}}
\rho_3^{-1/(q-1)}(y,t)
 \Big(\mathop{\rm ess\,sup} _{(y,t)\in \Sigma_{1,R}}
\sup _{s\in\mathbb{R}}
|c^k(y,t,s)\\
&\quad -c(y,t,s)|/(1+|s|^{q-1})\Big)^{q'}
 \iint_{\Sigma_{1,R}}\big(1+|\gamma u^k|^q\big)\,d\Gamma_y\,dt.
\end{aligned}
\end{equation}
 From  \eqref{126} on the basis of  \eqref{581} it follows that
there exists natural number $ k_3\geq k_2$ such that the right
side of each inequalities \eqref{610}-\eqref{612} is less than
$\varepsilon/(5 2^\sigma C_{11})$ for all $k\geq k_3$.  From this
and \eqref{rro}, \eqref{61}, \eqref{62} it follows  that the right
side of the inequality \eqref{59} with $R$ and $k_3$ being chosen
above is less then $\varepsilon$ for every $k\geq k_3$. Therefore
$u^{k}\mathop{\longrightarrow}_{k\to\infty} u$ in
 $\mathbb{U}_{\rm loc}$.
\end{proof}

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\end{document}