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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 141, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/141\hfil Behavior of the maximal solution]
{Behavior of the maximal solution of the Cauchy problem for some
nonlinear pseudoparabolic equation as $|x|\to\infty$}

\author[T. Kavitova\hfil EJDE-2012/141\hfilneg]
{Tatiana Kavitova}

\address{Tatiana Kavitova \newline
Department of Mathematics, Vitebsk State University,
Moskovskii pr. 33, 210038 Vitebsk, Belarus}
\email{KavitovaTV@tut.by}

\thanks{Submitted March 22, 2012. Published August 20, 2012.}
\subjclass[2000]{35B40, 35B51, 35K70}
\keywords{Pseudoparabolic equation; comparison principle; stabilization}

\begin{abstract}
 We prove a comparison principle for solutions of the Cauchy problem of the
 nonlinear pseudoparabolic equation $u_t=\Delta u_t+ \Delta\varphi(u) +h(t,u)$
 with nonnegative bounded initial data.
 We show stabilization of a maximal solution to a maximal solution of
 the Cauchy problem for the corresponding ordinary differential
 equation $\vartheta'(t)=h(t,\vartheta)$ as $|x|\to\infty$ under certain
 conditions on an initial datum.
\end{abstract}

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

\section{Introduction}

In this article we consider the Cauchy problem for the  pseudoparabolic equation
\begin{equation}\label{in:eq}
    u_t=\Delta u_t+ \Delta\varphi(u) +h(t,u),\quad x\in\mathbb{R}^n,\;t>0,
\end{equation}
subject to the initial condition
\begin{equation}\label{in:id}
    u(x,0)=u_0(x),\quad x\in\mathbb{R}^n.
\end{equation}
Put $R_{+}=(0,+\infty)$ and $\Pi_T=\mathbb{R}^n\times[0,T]$, $n\geq1,\;T>0$.
Throughout this paper we suppose that the functions $\varphi$ and $h$ satisfy
the following conditions:
\begin{equation} \label{in:koof}
\parbox{10cm}{
\(\varphi(p)\) is defined for \(p\geq0\),  \(h(t,p)\) is defined for \(t\geq0\)
and  \(p\geq0\),
$\varphi(p)\in C^2(\overline R_{+})\cap C^3(R_{+})$,
$h(t,p)\in C^{0,\alpha}_{\rm loc}(\overline R_{+}\times\overline R_{+})
\cap C^{0,1+\alpha}_{\rm loc}(\overline R_{+}\times R_+)$,
$0<\alpha<1$, $h(t,0)=0$, $t\in \overline R_{+}$,
$\varphi(p)+h(t,p)$ does not decrease in \(p\) 
for all \(t\in \overline R_{+}\).}
\end{equation}
Assume that one of the following conditions is satisfied:
\begin{equation}\label{in:koof+h(x,p)}
h(t,p)\geq0,\quad t\in\overline R_{+},\;p\in\overline R_{+},
\end{equation}
or
\begin{equation}\label{in:koof-h(x,p)}
h(t,p) \text{ does not increase in \(p\) for all \(t\in \overline R_{+}\)}.
\end{equation}
Let the initial data have the following properties:
\begin{gather}
    u_0(x)\in C^2(\mathbb{R}^n), \quad 0\leq u_0(x)\leq M \,(M\geq0), \;
 x\in\mathbb{R}^n,\label{in:u0(x)}\\
    \lim_{|x|\to\infty}u_0(x)=M.\label{in:limu0(x)}
\end{gather}
Equations $u_t=\Delta u_t+ \Delta u^p+u^q$ and
$u_t=\Delta u_t+ \Delta (u^l+u^p)-u^p$, where $p,l\geq2,\;q>0$,
are typical examples of equation~\eqref{in:eq} satisfying~\eqref{in:koof}
under conditions~\eqref{in:koof+h(x,p)} and~\eqref{in:koof-h(x,p)} respectively.

If we suppose $u_0(x)\equiv M$ in~\eqref{in:id} then a solution of the
 Cauchy problem for the corresponding ordinary differential equation
\begin{equation}\label{in:odu}
    \vartheta'(t)=h(t,\vartheta), \quad \vartheta(0)=M
\end{equation}
will be a solution of~\eqref{in:eq}, \eqref{in:id}.

\begin{remark} \label{rmk1.1} \rm
    We note that problem~\eqref{in:odu} may have more than one solution.
 Indeed, we put $h(t,\vartheta)=\vartheta^p, \, 0<p<1$, and $M\equiv0$
then problem~\eqref{in:odu} has the solutions $\vartheta_1(t)\equiv0$
and $\vartheta_2(t)=(1-p)^{\frac{1}{1-p}}t^{\frac{1}{1-p}}$.
\end{remark}

\begin{definition} \label{def1.1} \rm
    A nonnegative solution $\vartheta(t)$ of~\eqref{in:odu} is called maximal
on $[0,T)$ if for any other nonnegative solution $f(t)$ of~\eqref{in:odu}
the inequality $f(t)\leq \vartheta(t)$ is satisfied for $0\leq t<T$.
\end{definition}

We suppose that the maximal nonnegative solution $\vartheta(t)$ of~\eqref{in:odu}
exists on $[0,T_0)$, $T_0\leq+\infty$. Similarly we define the maximal
solution of~\eqref{in:eq}, \eqref{in:id}.

Assume that~\eqref{in:koof} and \eqref{in:u0(x)} hold. Then there exists a
 nonnegative solution $u(x,t)\in C^{2,1}(\Pi_T)$ of~\eqref{in:eq}, \eqref{in:id}
(see~\cite{Kavitova}) satisfying for any $T<T_0$ the inequality
    \begin{gather*}
        0\leq u(x,t)\leq \vartheta(t),\quad (x,t)\in\Pi_T.
    \end{gather*}
The main result of this article is the following statement.

\begin{theorem}\label{in:behavior:thm}
    Let~\eqref{in:koof}, \eqref{in:u0(x)}, \eqref{in:limu0(x)} hold and
$u(x,t)$, $\vartheta(t)$ are maximal solutions of problems~\eqref{in:eq},
\eqref{in:id} and~\eqref{in:odu} respectively. Suppose that either
\eqref{in:koof+h(x,p)} or \eqref{in:koof-h(x,p)} is satisfied in addition.
 Then we have
    \begin{equation*}
        u(x,t)\to\vartheta(t)\quad \text{as }|x|\to\infty
    \end{equation*}
uniformly in $[0,T] \, (T<T_0)$.
\end{theorem}

Results similar to Theorem~\ref{in:behavior:thm}  were obtained
in~\cite{Gladkov1,Gladkov3} and~\cite{GU1,GU2,Igarashi_Umeda,S,SSU,ShimUm}
respectively in studying of an asymptotic behavior of solutions of parabolic
equations, systems and blow-up solutions of nonlinear heat equations and
reaction-diffusion systems at infinity. Pseudoparabolic equations has been
analyzed by many authors (see \cite{SAKP} and the references therein).

Our main research tool is a comparison principle.

\begin{theorem}\label{CP:pr-comp:th}
    Let \eqref{in:koof} hold and $u_1(x,t)$, $u_2(x,t)$ be nonnegative
bounded solutions of \eqref{in:eq} in $\Pi_T$ and one of them is
not less some positive constant. Suppose that the corresponding initial data
$u_{01}(x)$  and $u_{02}(x)$ satisfying~\eqref{in:u0(x)} and the  inequality
    \begin{equation*}
        u_{01}(x)\leq u_{02}(x),\quad x\in\mathbb{R}^n.
    \end{equation*}
    Then
    \begin{equation*}
        u_1(x,t)\leq u_2(x,t),\quad (x,t)\in\Pi_T.
    \end{equation*}
\end{theorem}

For problem~\eqref{in:eq}, \eqref{in:id} with $\varphi(u)=u^2$ and $h(t,u)=0$
the comparison  principle was established in~\cite{Furaev}. For an initial--boundary
value problem for equation~\eqref{in:eq} with $h(t,u)=h(u)$ it was proved
in~\cite{Kozhanov}.

This paper is organized as follows. In the next section we prove
Theorem~\ref{CP:pr-comp:th}. Some auxiliary statements used for description a
behavior of the maximal solution of~\eqref{in:eq}, \eqref{in:id} at infinity
are established in Section~\ref{Beh}. Theorem~\ref{in:behavior:thm} is proved
in Section~\ref{Inf}.


\section{Proof of Theorem~\ref{CP:pr-comp:th}}\label{CP}

 Without loss of generality we may assume that
$u_2(x,t)\geq\varepsilon,\;\varepsilon>0,\;(x,t)\in \Pi_T$.
Obviously, the function $w(x,t)=u_2(x,t)-u_1(x,t)$ satisfies the problem
\begin{gather}\label{CP:urav1}
     w_t=\Delta{w_t}+\Delta{(aw)}+bw, \quad (x,t)\in \mathbb{R}^n \times (0,T),\\
\label{CP:nach1}
    w(x,0)=u_{02}(x)-u_{01}(x),\quad x\in \mathbb{R}^n.
\end{gather}
Here
\begin{equation*}
    a(x,t)=\int_0^1{\varphi'(z(\theta))\,d\theta},\quad
 b(x,t) =\int_0^1{h_{z(\theta)}(t,z(\theta))\,d\theta},
\end{equation*}
where $z(\theta)=\theta u_2(x,t)+(1-\theta)u_1(x,t)$.
By~\eqref{in:koof} the functions $a(x,t)$ and $b(x,t)$ have the following properties:
\begin{equation}\label{CP:ab}
    \begin{gathered}
    a(x,t)\in C^{2,0}(\Pi_T ),\quad b(x,t)\in C^{\alpha,0}_{\rm loc}(\Pi_T ),\\
    a(x,t)+b(x,t)\geq0,\quad |a(x,t)|+|b(x,t)|\leq m,\quad (x,t)\in\Pi_T,
    \end{gathered}
\end{equation}
where $m$ is some positive constant.

\begin{lemma}\label{CP:uniq:th}
    Let $a(x,t)$ and $b(x,t)$ be  functions such that conditions~\eqref{CP:ab}
 are satisfied. Then a solution of~\eqref{CP:urav1}, \eqref{CP:nach1} is unique.
\end{lemma}

The proof of the above lemma is analogous to the proof the same statement
for problem~\eqref{in:eq}, \eqref{in:id} with $h(t,p)=0$ in~\cite{Gladkov2}.

Let \(Q\) be a bounded domain in \(\mathbb{R}^n\) for $n\geq1$ with a smooth
boundary $\partial Q$. We denote $Q_{T}=Q\times(0,T)$ and
$S_{T}=\partial Q\times(0,T)$.
Let us consider the equation
\begin{equation}\label{CP:Fur_eq}
u_t=\Phi(x,t,u)+F(u(\cdot,t)),\quad (x,t)\in Q_T,
\end{equation}
subject to the initial data
\begin{equation}\label{CP:Fur_in_datum}
u(x,0)=u_0(x),\quad x\in Q,
\end{equation}
where the function $\Phi(x,t,\xi)$ is defined on the set
$\overline Q\times[0,T]\times R$ and $F(u(\cdot,t))$ is a nonlinear integral operator.

\begin{definition} \label{CP:sub-super}  \rm
We shall say that a function $\sigma^+(x,t)\in C^{0,1}(Q_T)$,
 $-\infty<m_T\leq\sigma^+(x,t)\leq M_T<+\infty$, $(x,t)\in Q_T$,
is a supersolution  of~\eqref{CP:Fur_eq}, \eqref{CP:Fur_in_datum} in $Q_T$ if
\begin{equation}\label{CP:sigma}
\begin{gathered}
  \sigma^+_t(x,t)\geq \Phi(x,t,\sigma^+)+F(\sigma ^+(\cdot,t)),\quad
 (x,t)\in Q_T,\\
  \sigma^+(x,0)\geq u_0(x),\;x\in\overline Q,
\end{gathered}
\end{equation}
where $m_T$, $M_T$ are constants depending on $T$.
\end{definition}

Analogously we say that $\sigma^-(x,t)\in C^{0,1}(Q_T)$,
 $m_T\leq\sigma^-(x,t)\leq M_T$, $(x,t)\in Q_T$, is a subsolution of~\eqref{CP:Fur_eq}, \eqref{CP:Fur_in_datum} in $Q_T$ if it satisfies inequalities~\eqref{CP:sigma} in the reverse order. Under the assumption $\sigma^-(x,t)\leq\sigma^+(x,t),\;(x,t)\in Q_T$, we introduce the set $O(\sigma^-,\sigma^+)=\{u\in C(\overline Q_T)|\,\sigma^-\leq u \leq \sigma^+,\;(x,t)\in Q_T\}$  and
make the following assumptions on data of~\eqref{CP:Fur_eq}, \eqref{CP:Fur_in_datum}:
\begin{equation}\label{CP:condition1}
\parbox{10cm}{
There exist a supersolution $\sigma^+(x,t)$ and a subsolution $\sigma^-(x,t)$
of \eqref{CP:Fur_eq}, \eqref{CP:Fur_in_datum} in $Q_T$ such that
$\sigma^-(x,t)\leq\sigma^+(x,t),\;(x,t)\in Q_T$.}
\end{equation}
\begin{equation}\label{CP:condition2}
 \text{$\Phi(x,t,\xi)$ and $\Phi_\xi(x,t,\xi)$ are continuous functions on the set
$\overline Q\times[0,T]\times R$.}
\end{equation}
\begin{equation}\label{CP:condition3}
\parbox{10cm}{
The operator $F(u(\cdot,t))$, on $C(\overline Q_T)$ into $C(\overline Q_T)$,
is completely continuous and monotone on $O(\sigma^-,\sigma^+)$.}
\end{equation}
\begin{equation}\label{CP:condition4}
   u_0(x)\in C(\overline Q).
\end{equation}
The following existence theorem has been proved in \cite{Furaev}.

\begin{theorem}\label{CP:Fur:th_}
Assume that~\eqref{CP:condition1}-\eqref{CP:condition4} hold. Then there
exists a solution of problem~\eqref{CP:Fur_eq}, \eqref{CP:Fur_in_datum}
in $Q_T$ such that
\begin{equation*}
\sigma^-(x,t)\leq u(x,t)\leq\sigma^+(x,t),\quad (x,t)\in Q_T.
\end{equation*}
\end{theorem}

Let $G_n(x,\xi)$ be the Green function of the boundary value problem for
the operator $L=I-\Delta$ in $Q$. It is known that
\begin{equation*}
G_n(x,\xi)=\mathcal{E}_n(x-\xi)+g_n(x,\xi),\quad (x,\xi)\in Q\times Q,
\end{equation*}
where $\mathcal{E}_n(x)$ is the fundamental solution of the operator $L$
of $\mathbb{R}^n$ tending to zero as $|x|\to\infty$ and for any fixed
 $\xi\in Q$ the function $g_n\in C^2(Q)\cap C(\overline Q)$ satisfies the equation
\begin{equation*}
L_xg_n=0,\quad x\in Q,
\end{equation*}
and the boundary condition
\begin{equation*}
g_n(x,\xi)|_{x\in\partial Q}=-\mathcal{E}_n(x-\xi)|_{x\in\partial Q},\quad \xi\in Q.
\end{equation*}
It is well known that
\begin{equation}
\mathcal{E}_n(x)=c_n|x|^{(2-n)/2}K_{(2-n)/2}(|x|),
\end{equation}
where $K_\mu(|x|)$ is the $\mu$th order Macdonald function and $c_n$ is the
normalizing multiplier such that
$\int_{\mathbb{R}^n}{\mathcal{E}_n(x)}d\,x=1$.

We note some properties of the Green function (see \cite{Gladkov}):
\begin{equation}\label{ÑP:Green}
\begin{gathered}
0<G_n(x,\xi)<\mathcal{E}_n(x-\xi),\quad (x,\xi)\in Q \times Q, \\
\frac{\partial G_n(x,\xi)}{\partial \nu_\xi}\leq 0,\quad
 \xi\in\partial Q,\;x\in Q,\\
\int_{Q}{ G_n(x,\xi)\,d\xi}=1+\int_{\partial Q}{\frac{\partial G_n(x,\xi)}
{\partial \nu_\xi} \,dS},\quad x\in Q,\\
\min _{y\in \partial Q}(-\mathcal{E}_n(x-y))<g_n(x,\xi)<0,\quad (x,\xi)\in Q\times Q,
\end{gathered}
\end{equation}
where $\nu_\xi$ is the outward normal derivative on $\partial Q$ in variables
of $\xi$.

We consider the integro--differential equation, in $Q_T$,
\begin{equation}\label{CP:int-diff-equat}
    w_t(x,t)=-a(x,t)w(x,t)+\int_{Q}{G_n(x,\xi)[a(\xi,t)+b(\xi,t)]w(\xi,t)\,d\xi}
\end{equation}
subject to the  initial condition, in $Q$,
\begin{equation}\label{CP:nach12}
    w(x,0)=u_{02}(x)-u_{01}(x).
    \end{equation}
Let $u_{02}(x)-u_{01}(x)\leq M_1$, $x\in\mathbb{R}^n,\;M_1\in \overline R_{+}$.

\begin{lemma}\label{CP:exist:lemma}
Let conditions~\eqref{CP:ab} hold. Then there exists a solution of~\eqref{CP:int-diff-equat}, \eqref{CP:nach12} in $Q_T$ such that
    \begin{gather}\label{CP:ner}
        0\leq w(x,t)\leq  M_1e^{2mt},\quad (x,t)\in  Q_T.
    \end{gather}
\end{lemma}

\begin{proof}
We use the following functions
\begin{equation*}
\Phi(x,t,w)=-a(x,t)w(x,t),\quad
F(w(\cdot,t))=\int_{Q}{G_n(x,\xi)[a(\xi,t)+b(\xi,t)]w(\xi,t)\,d\xi}
\end{equation*}
and show that the conditions of Theorem~\ref{CP:Fur:th_} are valid.
It is obvious, that $\sigma^-(x,t)\equiv0$ is the subsolution
of~\eqref{CP:int-diff-equat}, \eqref{CP:nach12}.
We shall show that $\sigma^+(x,t)=M_1e^{2mt}$ is the supersolution
of~\eqref{CP:int-diff-equat}, \eqref{CP:nach12}.
Indeed,
\begin{gather*}
\begin{aligned}
\Phi(x,t,\sigma^+)+F(\sigma^+)
&=-a(x,t)M_1e^{2mt}+\int_{Q}{G_n(x,\xi)[a(\xi,t)+b(\xi,t)]M_1e^{2mt}\,d\xi} \\
&\leq m M_1e^{2mt}+mM_1e^{2mt} \leq 2mM_1e^{2mt}\\
&=\sigma^{+}_t(x,t),\;(x,t)\in Q_T,
\end{aligned}\\
\sigma^+(x,0)=M_1\geq w_0(x),\quad x\in\overline Q.
\end{gather*}
Condition~\eqref{CP:condition2} of  Theorem~\ref{CP:Fur:th_} is satisfied by
virtue of~\eqref{CP:ab}.
As $a(x,t)+b(x,t)\geq0$ then the operator $F$ is monotone on $O(\sigma^-,\sigma^+)$.
We shall prove that the operator $F$ is completely continuous on
$O(\sigma^-,\sigma^+)$. Let $w\in O(\sigma^-,\sigma^+)$ then
\begin{equation*}
|F(w(\cdot,t))|=\Big|\int_{Q}{ G_n(x,\xi) [a(\xi,t)+b(\xi,t)]w(\xi,t)}\,d\xi \Big|
\leq mM_1e^{2mT}.
\end{equation*}
Hence, the operator $F$ is bounded. Suppose $x,y\in Q$ and
$w\in O(\sigma^-,\sigma^+)$. Then
\begin{align*}
|F(w(x,t))-F(w(y,t))|
&=\Big|\int_{Q}{[G_n(x,\xi)-G_n(y,\xi)](a(\xi,t)+b(\xi,t))w(\xi,t)}\,d\xi \Big|\\
&\leq m M_1e^{2mT}\int_{Q}{|G_n(x,\xi)-G_n(y,\xi)|\,d\xi},
\end{align*}
that implies the validity of~\eqref{CP:condition3}.
Relations~\eqref{in:u0(x)} for the initial data $u_{01}(x)$ and $u_{02}(x)$
are valid then all conditions of Theorem~\ref{CP:Fur:th_} are satisfied.
Hence, there exists a solution $w(x,t)$ of~\eqref{CP:int-diff-equat},
\eqref{CP:nach12} in $Q_T $ for which inequality \eqref{CP:ner} holds.
\end{proof}

\begin{lemma}\label{CP:exist:th2}
If conditions~\eqref{CP:ab} are satisfied then there exists a nonnegative
solution of~\eqref{CP:urav1}, \eqref{CP:nach1} in $\Pi_T$.
\end{lemma}

\begin{proof}
Let $G_n(x,\xi,l)$ be the Green function of the boundary value problem for
the operator $L=I-\Delta$ in $Q_l=\{x\in\mathbb{R}^n:|x|<l\},\;l>0$.
Let the  functions of the sequence $w_l(x,t)\;(l=1,2,\dots)$ satisfy
equation~\eqref{CP:int-diff-equat} in $Q_{l,T}=Q_l\times(0,T)$ and initial
data~\eqref{CP:nach12} in $Q_l$. According to Lemma~\ref{CP:exist:lemma}
there exists a solution $w_l(x,t)$ of~\eqref{CP:int-diff-equat}, \eqref{CP:nach12}
in $Q_{l,T}$ such that
\begin{equation}\label{CP:ocenkaWl}
    0\leq w_l(x,t)\leq M_1e^{2mt},\quad (x,t)\in Q_{l,T}.
\end{equation}
Differentiating \eqref{CP:int-diff-equat} with respect 
to $x_i$ $(i=1,\dots,n)$
 we obtain
\begin{align*}
 w_{lt{x_i}}(x,t)
&=-a_{x_i}(x,t)w_l(x,t)-a(x,t)w_{lx_i}(x,t)\\
&\quad +\int_{Q_l}G_{nx_i}(x,\xi,l)[a(\xi,t)+b(\xi,t)]w_l(\xi,t)\,d\xi,\quad
(x,t)\in Q_{l,T},
\end{align*}
from which we find that
\begin{equation}\label{CP:dif}
    w_{lx_i}(x,t)=e^{-\int_0^t{a(x,\tau)}\,d\tau}
\Big[u_{02}(x)-u_{01}(x)+\int_0^t{p_l(x,\tau)e^{\int_0^\tau{a(x,\tau_1)\;
    d\tau_1}}\,d\tau}\Big],
\end{equation}
where
$$
p_l(x,t)=-a_{x_i}(x,t)w_l(x,t)+\int_{Q_l}{G_{nx_i}(x,\xi,l)[a(\xi,t)
+b(\xi,t)]w_l(\xi,t)\,d\xi}.
$$
It follows from~\eqref{ÑP:Green}, \eqref{CP:int-diff-equat}, \eqref{CP:ocenkaWl}
and \eqref{CP:dif} that absolute values of functions
$w_l,\;w_{lt},\;w_{l{x_i}}\;(i=1,2,\dots,n)$ are  uniformly  bounded with respect
to $l$ on each set $\overline Q_{k,T}$, where $k$ is an arbitrary fixed
natural number, $k <l$. According to the Arzela--Ascoli theorem  the
sequence $w_l (x, t)$ is compact in $\overline Q_{k,T}$. By applying diagonal
process we can extract from the sequence $w_l(x,t)$ a subsequence $w_{l_s}(x,t)$
such that
\begin{equation} \label{CP:shod}
 w_{l_s}(x,t)\to w(x,t)\quad\text{uniformly in } \overline Q_{k,T}.
\end{equation}
Without loss of generality we assume that~\eqref{CP:shod} is valid for the sequence
 $w_l(x,t)$. Integrating equation~\eqref{CP:int-diff-equat} with respect to $t$
 we obtain
\begin{equation}\label{CP:eq}
\begin{aligned}
w_l(x,t)&=u_{02}(x)-u_{01}(x)-\int^t_0{a(x,\tau)w_l(x,\tau)\,d\tau}\\
&\quad +\int^t_0\int_{Q_l}{G_n(x,\xi,l)[a(\xi,\tau)+b(\xi,\tau)]w_l(\xi,\tau)\,d\xi\,
 d\tau},\quad (x,t)\in Q_{l,T}.
\end{aligned}
\end{equation}
Let $(x,t)$ be an arbitrary point of $\Pi_T$ and let $k$ be such that
$(x,t)\in\overline Q_{k,T},\;k<l$. By virtue of~\eqref{ÑP:Green}, \eqref{CP:ocenkaWl}
and \eqref{CP:shod} we obtain
\begin{equation}\label{CP:vs_equil}
\begin{aligned}
&\lim_{l\to\infty}{\int_0^t{\int_{Q_l}{G_n(x,\xi,l)[a(\xi,\tau)
 +b(\xi,\tau)] w_l(\xi,\tau)\,d\xi\,d\tau }}}\\
&=\int_0^t{\int_{\mathbb R^n}{\mathcal{E}_n(x-\xi)[a(\xi,\tau)
 +b(\xi,\tau)] w(\xi,\tau)\,d\xi\,d\tau}}.
\end{aligned}
\end{equation}
Letting $l\to\infty$ in~\eqref{CP:eq} and using~\eqref{CP:shod} and
\eqref{CP:vs_equil} we conclude that
\begin{equation} \label{CP:fineq}
\begin{aligned}
w(x,t)&=u_{02}(x)-u_{01}(x)-\int^t_0{a(x,\tau)w(x,\tau)\,d\tau}\\
&\quad +\int^t_0\int_{\mathbb R^n}{\mathcal{E}_n(x- \xi)[a(\xi,\tau)
 +b(\xi,\tau)]w(\xi,\tau)\,d\xi\, d\tau},\quad (x,t)\in \Pi_T.
\end{aligned}
\end{equation}
By \eqref{CP:ab} the solution $w(x,t)$ of~\eqref{CP:fineq} belongs to the
 class $C^{2,1}(\Pi_T)$
and
\begin{gather*}
\begin{aligned}
&\Delta \left (w_t(x,t)+a(x,t)w(x,t)\right)\\
&=\Delta\int_{\mathbb R^n}{\mathcal{E}_n(x-\xi)[a(\xi,t)+b(\xi,t)]w(\xi,t)\,d\xi}\\
&=-[a(x,t)+b(x,t)]w(x,t)+\int_{\mathbb R^n}{\mathcal{E}_n(x-\xi)[a(\xi,t)
 +b(\xi,t)]w(\xi,t)\,d\xi}\\
&=w_t(x,t)-b(x,t)w(x,t),\;(x,t)\in \mathbb{R}^n \times (0,T),
\end{aligned}\\
w(x,0)=u_{02}(x)-u_{01}(x),\quad x\in\mathbb{R}^n.
\end{gather*}
\end{proof}

According to Lemmas~\ref{CP:uniq:th} and \ref{CP:exist:th2} we have
$$
u_2(x,t)\geq u_1(x,t),\quad (x,t)\in \Pi_T.
$$

\begin{remark} \label{rmk2.5} \rm
    The comparison principle is valid without the condition that one of the
 solution is not less some positive constant if we assume that
$h(t,p)\in C^{0,1+\alpha}_{\rm loc}(\overline R_{+}\times\overline R_{+})$,
$0<\alpha<1$.
\end{remark}

\begin{remark}  \rm\label{CP:rem}
   If the inequality $u_0(x)\geq m>0$ and~\eqref{in:koof+h(x,p)} hold then
problem~\eqref{in:eq}, \eqref{in:id} has an unique solution.
\end{remark}

Indeed, in the same way as it was done in~\cite{Kavitova} we can show the
existence of the solution $u(x,t)$ of problem~\eqref{in:eq}, \eqref{in:id}
such that $u(x,t)\geq m>0$.

\section{Auxiliary statements}\label{Beh}

Let condition~\eqref{in:koof+h(x,p)} hold. We consider the Cauchy problem
for equation~\eqref{in:eq} subject to the initial condition
\begin{equation}\label{Beh:id}
    u(x,0)=u_0(x)+\varepsilon,\quad x\in\mathbb{R}^n.
\end{equation}
If we suppose $u_0(x)\equiv M$ in~\eqref{Beh:id} then a solution of the
Cauchy problem for the corresponding ordinary differential equation
\begin{equation}\label{Beh:zCodu}
    \vartheta'(t)=h(t,\vartheta),\quad \vartheta(0)=M+\varepsilon
\end{equation}
will be a solution of~\eqref{in:eq}, \eqref{Beh:id}.

Suppose that the solution $\vartheta_\varepsilon(t)$ of~\eqref{Beh:zCodu}
exists on $[0,T_{0,\varepsilon})$, $T_{0,\varepsilon}\leq+\infty$.
It is easy to show (see~\cite{Kavitova}) that a solution $u_\varepsilon(x,t)$
of the integral equation
\begin{equation} \label{Beh:Ue}
\begin{aligned}
u_\varepsilon(x,t)
&=u_0(x)+\varepsilon-\int_0^t{\varphi(u_\varepsilon(x,\tau))}\,d\tau\\
&\quad +\int_0^t\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi)
[\varphi(u_\varepsilon(\xi,\tau))+h(\tau,u_\varepsilon(\xi,\tau))]}\,d\xi\,d\tau
\end{aligned}
\end{equation}
for any $T_\varepsilon<T_{0,\varepsilon}$ solves in $\Pi_{T_\varepsilon}$
problem~\eqref{in:eq}, \eqref{Beh:id} and satisfies the  inequality
\begin{equation}\label{Beh:ue}
    \varepsilon\leq u_\varepsilon(x,t)\leq \vartheta_\varepsilon(t),\quad
(x,t)\in\Pi_{T_\varepsilon}.
 \end{equation}
We note that problem~\eqref{Beh:zCodu} is equivalent to the integral equation
\begin{equation}\label{Beh:ve1}
    \vartheta_\varepsilon(t)=M+\varepsilon
+\int_0^t{h(\tau,\vartheta_\varepsilon(\tau))\,d\tau},\quad t\in[0,T_{0,\varepsilon}).
\end{equation}

\begin{lemma}\label{Beh:lemm}
Let~\eqref{in:koof}, \eqref{in:koof+h(x,p)}, \eqref{in:u0(x)} and \eqref{in:limu0(x)}
 hold. Then for some $T_{*,\varepsilon}<T_{0,\varepsilon}$ we have
    \begin{gather*}
        u_\varepsilon(x,t)\to\vartheta_\varepsilon(t)\quad \text{as }|x|\to\infty
    \end{gather*}
uniformly in $[0,T_{*,\varepsilon}]$.
\end{lemma}

\begin{proof}
Put $u_{0,\varepsilon}(x,t)\equiv\vartheta_{\varepsilon}(t)$.
We define a sequence of functions $u_{k,\varepsilon}(x,t)$ $(k=1,2,\dots)$
in the following way
\begin{equation} \label{Beh:seqU}
\begin{split}
 u_{k,\varepsilon}(x,t)
&=u_0(x)+\varepsilon-\int_0^t{\varphi(u_{k-1,\varepsilon}(x,\tau))}\,d\tau\\
&\quad +\int_0^t\!\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi)
 [\varphi(u_{k-1,\varepsilon}(\xi,\tau))+h(\tau,u_{k-1,\varepsilon}(\xi,\tau))]}
 \,d\xi\,d\tau.
\end{split}
\end{equation}
Fix any $T_\varepsilon$ such that $T_\varepsilon<T_{0,\varepsilon}$ and show that
the sequence $u_{k,\varepsilon}(x,t)$ converges to the solution
 $u_\varepsilon(x,t)$ of~\eqref{in:eq}, \eqref{Beh:id} as $k\to\infty$ uniformly
in some layer $\Pi_{T_{*,\varepsilon}}\;(T_{*,\varepsilon}\leq T_\varepsilon)$.

At first we show that the sequence $u_{k,\varepsilon}(x,t)$ is uniformly bounded
in some layer $\Pi_{T_{*,\varepsilon}}$. Using the method of mathematical
induction we prove the inequality
    \begin{equation}\label{Beh:rav_ogr}
        \frac{\varepsilon}{2}\leq u_{k,\varepsilon}(x,t)
\leq M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon}),\quad
(x,t)\in \Pi_{T_{*,\varepsilon}},\;k=0,1,\dots\,.
    \end{equation}
It is obviously that~\eqref{Beh:rav_ogr} is true for $k=0$. We assume
that~\eqref{Beh:rav_ogr} holds for $k=k_0$ and we shall prove the inequality
for $k=k_0+1$. Using the property of function $\varphi+h$ and the mean value
theorem we obtain
\begin{equation} \label{Beh:ner1}
\begin{aligned}
u_{k_0+1,\varepsilon}(x,t)
&=u_0(x)+\varepsilon-\int_0^t{\varphi(u_{k_0,\varepsilon}(x,\tau))}\,d\tau\\
&\quad +\int_0^t\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi)
 [\varphi(u_{k_0,\varepsilon}(\xi,\tau))+h(\tau,u_{k_0,\varepsilon}(\xi,\tau))
 ]}\,d\xi\,d\tau
\\
&\leq M+\varepsilon +\int_0^t\!\int_{\mathbb{R}^n}\mathcal{E}_n(x-\xi)
\Big[\varphi(M+\frac{3\varepsilon}{2} +\vartheta_{\varepsilon}(T_{\varepsilon})) \\
&\quad  +h(\tau,M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon}))
\Big]\,d\xi\,d\tau
 -\int_0^t{\varphi(u_{k_0,\varepsilon}(x,\tau))}\,d\tau
\\
&\leq M+\varepsilon +\int_0^t\Big\{\varphi(M+\frac{3\varepsilon}{2}
 +\vartheta_{\varepsilon}(T_{\varepsilon}))-\varphi(u_{k_0,\varepsilon}(x,\tau))\\
&\quad +h(\tau,M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon}))\Big\}
d\tau\\
&\leq M+\varepsilon+T_{*,\varepsilon}(M+\varepsilon
  +\vartheta_{\varepsilon}(T_{\varepsilon}))
\max_{\frac{\varepsilon}{2}\leq\theta\leq M+\frac{3\varepsilon}{2}
+\vartheta_{\varepsilon}(T_{\varepsilon})}|\varphi'(\theta)|\\
&\quad +T_{*,\varepsilon}\max_{0\leq t\leq T_{\varepsilon}}
 h(t,M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon}))
\end{aligned}
\end{equation}
    and
\begin{equation}\label{Beh:ner2}
\begin{aligned}
&u_{k_0+1,\varepsilon}(x,t)\\
&\geq\varepsilon+\int_0^t\left\{\varphi(\frac{\varepsilon}{2})
-\varphi(u_{k_0,\varepsilon}(x,\tau))+h(\tau,\frac{\varepsilon}{2})\right\}\,d\tau\\
&\geq\varepsilon-T_{*,\varepsilon}
\Big((M+\varepsilon+\vartheta_{\varepsilon}(T_{\varepsilon}))\max_{\frac{\varepsilon}{2}\leq\theta\leq M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon})}|\varphi'(\theta)|
        +\max_{0\leq t\leq T_{\varepsilon}}h(t,\frac{\varepsilon}{2})\Big).
\end{aligned}
\end{equation}
From~\eqref{Beh:ner1} and~\eqref{Beh:ner2} we conclude that
inequality~\eqref{Beh:rav_ogr} is valid for $k=k_0+1$ provided
\begin{equation}\label{T}
        T_{*,\varepsilon}\leq \min
\big\{T_\varepsilon,\;\frac{\varepsilon/2}{(M+\varepsilon
+\vartheta_{\varepsilon}(T_{\varepsilon}))\lambda+\mu}\big\},
\end{equation}
where 
$$
\lambda=\max_{\frac{\varepsilon}{2}\leq\theta
\leq M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon})}|
\varphi'(\theta)|,\quad
\mu=\max_{0\leq t\leq T_{\varepsilon},\,
\frac{\varepsilon}{2}\leq\theta\leq M+\frac{3\varepsilon}{2}
+\vartheta_{\varepsilon}(T_{\varepsilon})}h(t,\theta).
$$
   Using the method of mathematical induction it is easy to show the
validity in $\Pi_{T_{*,\varepsilon}}$ the  estimate
\begin{equation}\label{Beh:raznost}
 |u_{k,\varepsilon}(x,t)-u_{k-1,\varepsilon}(x,t)|
\leq M(2\lambda+\nu)^{k-1}\frac{t^{k-1}}{(k-1)!},
\end{equation}
where
$$
\nu=\max_{0\leq t\leq T_{\varepsilon},\,
\frac{\varepsilon}{2}\leq\theta\leq M+\frac{3\varepsilon}{2}
+\vartheta_{\varepsilon}(T_{\varepsilon})}|h_\theta(t,\theta)|.
$$
For $k=1$ we have
  \begin{equation*}
         |u_{1,\varepsilon}(x,t)-u_{0,\varepsilon}(x,t)|
=\vartheta_\varepsilon(t)-u_0(x)-\varepsilon
-\int_0^t h(\tau,\vartheta_\varepsilon(\tau))\,d\tau\leq M.
   \end{equation*}
We assume that~\eqref{Beh:raznost} holds for $k=k_0$ and we shall prove the
 inequality for $k=k_0+1$. By~\eqref{Beh:raznost} and the mean value theorem
we have
\begin{align*}
& |u_{k_0+1,\varepsilon}(x,t)-u_{k_0,\varepsilon}(x,t)|\\
&=|\int_0^t{\varphi'(\theta_1(x,\tau))[u_{k_0,\varepsilon}(x,\tau)
 -u_{k_0-1,\varepsilon}(x,\tau)]}\,d\tau|\\
&\quad +|\int_0^t\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi)
 \varphi'(\theta_2(\xi,\tau))[u_{k_0,\varepsilon}(\xi,\tau)
 -u_{k_0-1,\varepsilon}(\xi,\tau)]}\,d\xi\,d\tau|\\
&\quad +|\int_0^t\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi)h_{\theta_3}
 (\tau,\theta_3(\xi,\tau))[u_{k_0,\varepsilon}(\xi,\tau)
 -u_{k_0-1,\varepsilon}(\xi,\tau)]}\,d\xi\,d\tau|\\
&\leq M(2\lambda+\nu)^{k_0}\int_0^t{\frac{\tau^{k_0-1}}{(k_0-1)!}}\,d\tau\\
&\leq M(2\lambda+\nu)^{k_0}\frac{t^{k_0}}{k_0!},
 \end{align*}
where $\frac{\varepsilon}{2} \leq\theta_i\leq M+\frac{3\varepsilon}{2}
+\vartheta_\varepsilon(T_{*,\varepsilon})$, $i=1,2,3$.

   To show that the sequence $u_{k,\varepsilon}(x,t)$ converges uniformly in
$\Pi_{T_{*,\varepsilon}}$ we consider the series
   \begin{equation}\label{Beh:rjad}
u_{0,\varepsilon}(x,t)+\sum_{n=1}^{\infty}(u_{n,\varepsilon}(x,t)
-u_{n-1,\varepsilon}(x,t)).
   \end{equation}
Then $u_{k,\varepsilon}(x,t)$ is the $(k+1)$th partial sum of~\eqref{Beh:rjad}.
By~\eqref{Beh:raznost} every term of series~\eqref{Beh:rjad} for all
 $(x,t)\in\Pi_{T_{*,\varepsilon}}$ is not greater than the absolute value of
the corresponding term of the following convergent series
   \begin{equation*}
\vartheta_\varepsilon(t)+M\sum_{n=0}^{\infty}(2\lambda+\nu)^n
\frac{{T_{*,\varepsilon}^{n}}}{n!}.
   \end{equation*}
Hence, series~\eqref{Beh:rjad} as well as the sequence $u_{k,\varepsilon}(x,t)$
converge uniformly in $\Pi_{T_{*,\varepsilon}}$.
Let
$$
u_\varepsilon(x,t)=\lim_{k\to\infty}u_{k,\varepsilon}(x,t).
$$
Passing to the limit as $k\to\infty$ in~\eqref{Beh:seqU} and using the Lebesgue
theorem we obtain that the function $u_\varepsilon(x,t)$ satisfies~\eqref{Beh:Ue}.
Hence, $u_\varepsilon(x,t)$ solves problem~\eqref{in:eq}, \eqref{Beh:id} in
$\Pi_{T_{*,\varepsilon}}$.

Using the method of mathematical induction we shall prove that
    \begin{gather}\label{Beh:shod uu_{k,e}}
        u_{k,\varepsilon}(x,t)\to\vartheta_\varepsilon(t)\quad
\text{ as }|x|\to\infty,\;k=0,1,\dots
    \end{gather}
uniformly in $[0,T_{*,\varepsilon}]$.

It is obviously that~\eqref{Beh:shod uu_{k,e}} is true for $k=0$.
We assume that~\eqref{Beh:shod uu_{k,e}} holds for $k=k_0$ and we shall
 prove~\eqref{Beh:shod uu_{k,e}} for $k=k_0+1$. Fix an arbitrary $\delta>0$.
   By the induction assumption for any $\delta_0>0$ there exists a constant
$A_0=A_0(\delta_0,\varepsilon,T_{*,\varepsilon},k_0)$ such that if $|x|>A_0$
 and $0\leq t\leq T_{*,\varepsilon}$ then
   \begin{equation*}
    |u_{k_0,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|<\delta_0.
   \end{equation*}
From \eqref{Beh:ve1} and \eqref{Beh:seqU} we have
  \begin{align*}
&|u_{k_0+1,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|\\
&=|u_0(x)+\varepsilon-\int_0^t{\varphi(u_{k_0,\varepsilon}(x,\tau))}\,d\tau
  +\int_0^t\int_{\mathbb{R}^n} \mathcal{E}_n(x-\xi)
 \Big[\varphi(u_{k_0,\varepsilon}(\xi,\tau))\\
&\quad +h(\tau,u_{k_0,\varepsilon}(\xi,\tau)) \Big]\,d\xi\,d\tau
 -M-\varepsilon-\int_0^t{h(\tau,\vartheta_{\varepsilon}(\tau))}\,d\tau| \\
&\leq|u_0(x)-M|+\int_0^t{|\varphi'(\theta_1(x,\tau))|\cdot|u_{k_0,\varepsilon}
 (x,\tau)-\vartheta_\varepsilon(\tau)|}\,d\tau\\
&\quad +\int_0^t\int_{|\xi|\leq A_0}{\mathcal{E}_n(x-\xi)
 (|\varphi'(\theta_2(\xi,\tau))|+|h_{\theta_3}(\tau,\theta_3(\xi,\tau))|)
 |u_{k_0,\varepsilon}(\xi,\tau)-\vartheta_\varepsilon(\tau)|}\,d\xi\,d\tau\\
&\quad +\int_0^t\int_{|\xi|>A_0}{\mathcal{E}_n(x-\xi)
 (|\varphi'(\theta_2(\xi,\tau))|+|h_{\theta_3}(\tau,\theta_3(\xi,\tau))|)
   |u_{k_0,\varepsilon}(\xi,\tau)-\vartheta_\varepsilon(\tau)|}\,d\xi\,d\tau,
\end{align*}
where $\frac{\varepsilon}{2}\leq\theta_i\leq M+\frac{3\varepsilon}{2}
+\vartheta_\varepsilon(T_{*,\varepsilon})$, $i=1,2,3$.
 By \eqref{in:limu0(x)} for any $\delta_1>0$ there exists a constant
$A_1=A_1(\delta_1)$ such that $|u_0(x)-M|<\delta_1$ if $|x|>A_1$.
   Using the property of the fundamental solution $\mathcal{E}_n$ and
 \eqref{Beh:rav_ogr} we obtain that for any $\delta_2>0$ there exists
a constant $A_2=A_2(\delta_2,\varepsilon)$ such that if $|x|>A_2$ then
   \begin{align*}
&\int_0^t\int_{|\xi|\leq A_0}{\mathcal{E}_n(x-\xi)(|\varphi'
(\theta_2(\xi,\tau))|+|h_{\theta_3}(\tau,\theta_3(\xi,\tau))|)
|u_{k_0,\varepsilon}(\xi,\tau)-\vartheta_\varepsilon(\tau)|}\,d\xi\,d\tau\\
&<\delta_2.
\end{align*}
Hence, we obtain
\begin{equation*}
 |u_{k_0+1,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|
<\delta_1+\delta_2+T_{*,\varepsilon}(2\lambda+\nu)\delta_0,
\end{equation*}
where
$$
\lambda=\max_{\frac{\varepsilon}{2}\leq\theta\leq M+\frac{3\varepsilon}{2}
+\vartheta_{\varepsilon}(T_{\varepsilon})}|\varphi'(\theta)|,  \quad
\nu=\max_{0\leq t\leq T_{\varepsilon},\,\frac{\varepsilon}{2}\leq\theta
\leq M+\frac{3\varepsilon}{2}+\vartheta_{\varepsilon}(T_{\varepsilon})}
|h_\theta(t,\theta)|.
$$
 Let $\delta_0=\frac{\delta}{3T_{*,\varepsilon}(2\lambda+\nu)}$,
$\delta_1=\frac{\delta}{3},\;\delta_2=\frac{\delta}{3}$ and
$A=\max(A_0,A_1,A_2)$ then
 $$
|u_{k_0+1,\varepsilon}(x,t)-\vartheta_{\varepsilon}(t)|<\delta
$$
 if $0\leq t\leq T_{*,\varepsilon}$ and $|x|>A$. It follows that for any
 $\delta>0$ by suitable choosing $k$ and $A$ we obtain
    \begin{align*}
 |u_\varepsilon(x,t)-\vartheta_\varepsilon(t)|
&=|u_\varepsilon(x,t)-u_{k,\varepsilon}(x,t)+u_{k,\varepsilon}(x,t)
 -\vartheta_\varepsilon(t)|\\
&\leq|u_\varepsilon(x,t)-u_{k,\varepsilon}(x,t)|
 +|u_{k,\varepsilon}(x,t)-\vartheta_\varepsilon(t)|< \delta
\end{align*}
for $0\leq t\leq T_{*,\varepsilon}$ and $|x|>A$.
\end{proof}

\begin{lemma}\label{Beh:ue-ve:lemm}
    Let~\eqref{in:koof}, \eqref{in:koof+h(x,p)}, \eqref{in:u0(x)} and
 \eqref{in:limu0(x)}  hold. Then for any $T_\varepsilon<T_{0,\varepsilon}$ we have
     \begin{equation*}
        u_\varepsilon(x,t)\to\vartheta_\varepsilon(t)\quad \text{as }|x|\to\infty
    \end{equation*}
uniformly in $[0,T_\varepsilon]$.
\end{lemma}

\begin{proof}
Fix any $T_\varepsilon$ such that $T_\varepsilon<T_{0,\varepsilon}$.
 We recall that for any $T_\varepsilon<T_{0,\varepsilon}$ the solution
$u_\varepsilon(x,t)$ exists in $\Pi_{T_\varepsilon}$ and satisfies
inequality~\eqref{Beh:ue}. Note that the solution $u_\varepsilon(x,t)$
of~\eqref{in:eq}, \eqref{Beh:id} is unique by Remark~\ref{CP:rem}.

By  Lemma~\ref{Beh:lemm} there exists $T_{*,\varepsilon}\leq T_\varepsilon$
such that $ u_\varepsilon(x,t)\to\vartheta_\varepsilon(t)$
as $|x|\to\infty$ uniformly in $[0,T_{*,\varepsilon}]$.
If $T_{*,\varepsilon}< T_\varepsilon$ then we construct for
 $t\geq T_{*,\varepsilon}$ new sequence $u_{k,\varepsilon}(x,t)$
in the following way:
\begin{gather*}
u_{0,\varepsilon}(x,t)\equiv\vartheta_{\varepsilon}(t), \\
\begin{aligned}
    u_{k,\varepsilon}(x,t)
&=u_\varepsilon(x,T_{*,\varepsilon})-\int_{T_{*,\varepsilon}}^t
{\varphi(u_{k_0-1,\varepsilon}(x,\tau))}\,d\tau\\
&\quad   +\int_{T_{*,\varepsilon}}^t\int_{\mathbb{R}^n}{\mathcal{E}_n(x-\xi)
[\varphi(u_{k_0-1,\varepsilon}(\xi,\tau))+h(\tau,u_{k_0-1,\varepsilon}(\xi,\tau))
]}\,d\xi\,d\tau,
\end{aligned}
\end{gather*}
for $k=1,2,\dots$.
By the similar arguments to Lemma~\ref{Beh:lemm} we can prove that the
sequence $u_{k,\varepsilon}(x,t)$ converges to the solution $u_\varepsilon(x,t)$
of~\eqref{in:eq}, \eqref{Beh:id} as $k\to\infty$ uniformly in the layer
$\mathbb{R}^n\times[T_{*,\varepsilon},T_{*,\varepsilon}+\Delta T_\varepsilon]$
provided $\Delta T_\varepsilon$ satisfies condition~\eqref{T} with
$T_{*,\varepsilon}=\Delta T_\varepsilon$ and the inequality
 $T_{*,\varepsilon}+\Delta T_\varepsilon\leq T_\varepsilon$. It follows that
\begin{equation*}
        u_\varepsilon(x,t)\to\vartheta_\varepsilon(t)\quad \text{as }|x|\to\infty
\end{equation*}
uniformly in $[T_{*,\varepsilon},T_{*,\varepsilon}+\Delta T_\varepsilon]$.
Repeating this procedure we obtain the conclusion of the theorem.
\end{proof}

\section{Behavior of maximal solution at infinity}\label{Inf}

\begin{proof}[Proof of Theorem~\ref{in:behavior:thm}]
Let~\eqref{in:koof+h(x,p)} hold and $u_\varepsilon(x,t)$,
$\vartheta_\varepsilon(t)$ be solutions of problems~\eqref{in:eq}, \eqref{Beh:id}
 and~\eqref{Beh:zCodu} respectively.
Using Theorem~\ref{CP:pr-comp:th} for $\varepsilon_1\geq\varepsilon_2$ we obtain:
\begin{gather*}
    u(x,t)\leq u_{\varepsilon_2}(x,t)\leq u_{\varepsilon_1}(x,t),\quad
(x,t)\in\Pi_{T_{\varepsilon_1}},\\
    \vartheta(t)\leq\vartheta_{\varepsilon_2}(t)\leq\vartheta_{\varepsilon_1}(t),\quad
 t\in[0,T_{\varepsilon_1}].
\end{gather*}
According to  Dini's theorem the sequences~$u_\varepsilon(x,t)$ and
$\vartheta_\varepsilon(t)$ convergence  to some solutions $u(x,t)$ and
$\vartheta(t)$ of problems~\eqref{in:eq}, \eqref{in:id} and~\eqref{in:odu} as
$\varepsilon\to0$   uniformly respectively in $\Pi_T$ and $[0,T]$, where
$T<T_0$. It is easy to see that $u(x,t)$ and $\vartheta(t)$ are maximal solutions
of problems~\eqref{in:eq}, \eqref{in:id} and~\eqref{in:odu} respectively.

We fix an arbitrary $\delta>0$ and $0<T<T_0$. Choose $\varepsilon_1>0$ such that
for any $\varepsilon<\varepsilon_1$ the inequality $T<T_{0,\varepsilon}$ holds.
By the uniform convergence functions $u_\varepsilon(x,t)$ to $u(x,t)$  in
$\Pi_T$ and $\vartheta_\varepsilon(t)$ to $\vartheta(t)$ in $[0,T],\;(T<T_0)$ as
 $\varepsilon\to0$ we can take $\varepsilon_2>0$ such that for any
$\varepsilon<\varepsilon_2$,
\begin{gather}\label{vspom1}
    |u_{\varepsilon}(x,t)-u(x,t)|<\frac{\delta}{3},\quad (x,t)\in\Pi_T, \\
\label{vspom2}
   |\vartheta_{\varepsilon}(t)-\vartheta(t)|<\frac{\delta}{3},\quad t\in[0,T].
   \end{gather}
Put $\varepsilon_0=\min(\varepsilon_1,\varepsilon_2)$.
 From  Lemma~\ref{Beh:ue-ve:lemm} there exists the constant
$A_0=A_0(\delta,\varepsilon_0,T)$ such that for any $|x|>A_0$ we obtain
    \begin{equation}\label{vspom3}
|u_{\varepsilon_0}(x,t)-\vartheta_{\varepsilon_0}(t)|
<\frac{\delta}{3},\quad (x,t)\in\Pi_T.
\end{equation}
By \eqref{vspom1}--\eqref{vspom3} we conclude that by suitable choosing
$\varepsilon=\varepsilon_0$ and $A=A_0$,
\begin{align*}
 |u(x,t)-\vartheta(t)|
&=|u(x,t)-u_\varepsilon(x,t)+u_\varepsilon(x,t)+\vartheta_\varepsilon(t)
 -\vartheta_\varepsilon(t)-\vartheta(t)|\\
&\leq|u_\varepsilon(x,t)-u(x,t)|+|u_\varepsilon(x,t)-\vartheta_\varepsilon(t)|
+|\vartheta_\varepsilon(t)-\vartheta(t)|<\delta
\end{align*}
for $0\leq t\leq T$ and $|x|>A$.

Let~\eqref{in:koof-h(x,p)} hold. Consider the Cauchy problems
    \begin{equation}\label{in:eqvsp}
    \begin{gathered}
    \omega_t=\Delta \omega_t+ \Delta\varphi(\omega) +h(t,\omega)
-h(t,\varepsilon),\quad x\in\mathbb{R}^n,\;t>0,\\
    \omega(x,0)=u_0(x)+\varepsilon,\quad x\in\mathbb{R}^n,
    \end{gathered}
    \end{equation}
and
    \begin{equation}\label{Beh:zCoduvsp}
    g'(t)=h(t,g)-h(t,\varepsilon),\quad g(0)=M+\varepsilon.
    \end{equation}
We suppose that the maximal nonnegative solution $g_\varepsilon(t)$
of~\eqref{Beh:zCoduvsp} exists on $[0,T_{0,\varepsilon})$,
$T_{0,\varepsilon}\leq+\infty$.
It is easy to show (see~\cite{Kavitova}) that for any
$T_\varepsilon<T_{0,\varepsilon}$ there exists  in $\Pi_{T_\varepsilon}$
a solution $\omega_\varepsilon(x,t)$ of~\eqref{in:eqvsp} satisfying the inequality
\begin{equation*}
    \varepsilon\leq \omega_\varepsilon(x,t)
\leq g_\varepsilon(t),\quad (x,t)\in\Pi_{T_\varepsilon}.
\end{equation*}
Applying Theorem~\ref{CP:pr-comp:th} we conclude that the solution
$\omega_\varepsilon(x,t)$ of~\eqref{in:eqvsp} is unique.
Let $\varepsilon_1\geq\varepsilon_2$ and $\omega_{\varepsilon_1}(x,t)$,
$\omega_{\varepsilon_2}(x,t)$ are nonnegative bounded solutions
of~\eqref{in:eqvsp} with $\varepsilon=\varepsilon_1$ and
 $\varepsilon=\varepsilon_2$ respectively. Then
\begin{equation*}
    \omega_{\varepsilon_1}(x,t)\geq \omega_{\varepsilon_2}(x,t),\quad
(x,t)\in\Pi_{T_{\varepsilon_1}}.
\end{equation*}
The proof of this statement is analogous to the proof of
Theorem~\ref{CP:pr-comp:th}.
Then we consider the sequence $\omega_{k,\varepsilon}(x,t)\;(k=0,1,\dots)$:
\begin{gather*}
\omega_{0,\varepsilon}(x,t)\equiv g_\varepsilon(t), \\
\begin{aligned}
    \omega_{k,\varepsilon}(x,t)
&=u_0(x)+\varepsilon-\int_0^t{\varphi(\omega_{k-1,\varepsilon}(x,\tau))}\,d\tau
    +\int_0^t\!\int_{\mathbb{R}^n} \mathcal{E}_n(x-\xi)
\Big[\varphi(\omega_{k-1,\varepsilon}(\xi,\tau))\\
&\quad +h(\tau,\omega_{k-1,\varepsilon}(\xi,\tau))-h(\tau,\varepsilon
    )\Big] \,d\xi\,d\tau,\quad k=1,2,\dots\,.
\end{aligned}
\end{gather*}
Analogous to the arguments in Section~\ref{Beh} can be shown that for any
$T_\varepsilon<T_{0,\varepsilon}$
     \begin{equation*}
        \omega_\varepsilon(x,t)\to g_\varepsilon(t)\quad \text{as }|x|\to\infty
    \end{equation*}
uniformly in $[0,T_\varepsilon]$.
Further arguments are similar to reasoning in the proof of this theorem
 with condition~\eqref{in:koof+h(x,p)}.
\end{proof}


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\end{document}