\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 $h(t,\vartheta)=\vartheta^p, \, 0
0,\;(x,t)\in \Pi_T$.
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 $\sigma^+(x,t)\in C^{0,1}(Q_T)$,
$-\infty0$.
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, $k 0$ 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 $T_\varepsilon0$.
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 $T_\varepsilon0$ and $00$ such that
for any $\varepsilon<\varepsilon_1$ the inequality $T0$ 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