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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 113, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2005/113\hfil Diffusion equation with convection and absorption] 
{Vanishing of solutions of diffusion equation with
convection and absorption}
\author[A. Gladkov, S. Prokhozhy\hfil EJDE-2005/113\hfilneg]
{Alexander Gladkov, Sergey Prokhozhy}  % in alphabetical order

\address{Alexander Gladkov \hfill\break
Mathematics Department, Vitebsk State University, Moskovskii pr.
33, 210038 Vitebsk, Belarus}
\email{gladkov@vsu.by}

\address{Sergey Prokhozhy \hfill\break
Mathematics Department, Vitebsk State University, Moskovskii pr.
33, 210038 Vitebsk, Belarus}
\email{prokhozhy@vsu.by}


\date{}
\thanks{Submitted June 10, 2005. Published October 17, 2005.}
\subjclass[2000]{35K55, 35K65}
\keywords{Diffusion equation; vanishing of solutions}

\begin{abstract}
 We study the vanishing of solutions of the Cauchy problem for
  the equation
 $$
 u_t = \sum_{i,j=1}^N a_{ij}(u^m)_{x_ix_j} + \sum_{i=1}^N
 b_i(u^n)_{x_i} - cu^p, \quad (x,t)\in S = \mathbb{R}^N\times(0,+\infty).
 $$
 Obtained results depend on relations of parameters
 of the problem and growth of initial data at infinity.
\end{abstract}

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

\section{Introduction}\label{in}

We consider the Cauchy problem for the equation
\begin{equation} \label{1.1}
u_t = \sum_{i,j=1}^N a_{ij}(u^m)_{x_ix_j}
+ \sum_{i=1}^N b_i(u^n)_{x_i}
- cu^p, \quad (x,t)\in S = \mathbb{R}^N\times(0,+\infty)
\end{equation}
with initial data
\begin{equation}
\label{1.2} u(x,0) = u_0(x), \quad x  \in \mathbb{R}^N,
\end{equation}
where $m>1>p>0$, $n\ge 1$,\hspace{0.2cm} $a_{ij}$, $b_i$ $(i,j=1,\dots,N)$,
$c$ are real numbers, $a_{ij}=a_{ji}$, $\sum_{i,j=1}^N a_{ij}\xi_i\xi_j>0$
for $\sum_{i=1}^N\xi_i^2>0$ $(\xi_i\in\mathbb{R}, i=1,\dots,N)$,
$c>0$, $u_0(x)$ is a nonnegative continuous function which can be
increasing at infinity. Equation
\eqref{1.1} is encountered, for example, when simulating a process of
diffusion or heat propagation accompanied by convection and absorption. It is
parabolic for $u>0$ and degenerates into a first-order equation for $u=0$. Due
to degeneracy the Cauchy problem \eqref{1.1}, \eqref{1.2} can have not a
classical solution even when initial data are smooth.

Put $B_h=\{x\in\mathbb{R}^N:|x|<h\}$ $(0<h<+\infty)$. By
$\vec{\nu}=(\nu_1,\dots,\nu_N)$ we denote the outward unit
normal to the boundary of a considered domain.
\smallskip

\noindent\textbf{Definition.} The function $u(x,t)$ which is nonnegative and
continuous in $\overline{S}$ we call a generalized solution of the equation
\eqref{1.1} in $S$ if $u(x,t)$ satisfies the integral identity

\begin{equation} \label{1.3}
\begin{aligned}
&\int_{t_1}^{t_2}\int_{B_R}\{uf_t+u^m\sum_{i,j=1}^N a_{ij}f_{x_ix_j} -
u^n\sum_{i=1}^N b_i f_{x_i} - cu^pf\}dx\, dt
 - \int_{B_R} uf\big|_{t_1}^{t_2}dx \\
&- \int_{t_1}^{t_2} \int_{\partial
B_R}u^m\sum_{i,j=1}^Na_{ij}f_{x_j} \nu_i ds\, dt = 0
\end{aligned}
\end{equation}
for all $R>0$, $0<t_1<t_2<+\infty$ and any nonnegative function
$f(x,t)\in C_{x,t}^{2,1}(\overline{B}_R\times[t_1,t_2])$ such that
$f(x,t)=0$ for $|x|=R$, $t_1<t<t_2$.

If the equal sign in (\ref{1.3}) is replaced by $\ge$ $(\le)$ then we
obtain the
definition of a generalized subsolution (supersolution) of the equation
\eqref{1.1} in $S$.
\smallskip

\noindent\textbf{Definition.}
 The function $u(x,t)$ is called a generalized
solution of the Cauchy problem \eqref{1.1}, \eqref{1.2} if it is a generalized
solution of the equation \eqref{1.1} in $S$ and condition \eqref{1.2} is
satisfied.

In the present paper we investigate the conditions when the generalized
solution of the Cauchy problem \eqref{1.1}, \eqref{1.2} vanishes at every
point $x\in\mathbb{R}^N$ in a finite time $T_0(x)$ depending on $x$. For
$n>(m+p)/2$ one-dimensional equation \eqref{1.1} is considered.

Behavior for large values of the time of unbounded generalized solutions of
the Cauchy problem \eqref{1.1}, \eqref{1.2} with $a_{ij}=1$ for $i=j$,
$a_{ij}=0$ for $i\ne j$ and $b_i=0$ ($i,j=1,\dots,N$) has been studied in \cite{Herrero}
and \cite{Gladkov1} for $m=1$ and $m>1$ respectively. The case $n=(m+p)/2$
has been considered in \cite{Khramtsov} in terms of the control theory.

The distribution of the paper is as follows. In the next section we introduce
notations and give existence and uniqueness results which we need in the
following. The condition on the initial data for the vanishing of solutions of
the Cauchy problem \eqref{1.1}, \eqref{1.2} at every point of $\mathbb{R}^N$ in a
finite time in the case $n<(m+p)/2$ we point out in Section 3. The same
results for the one-dimensional equation \eqref{1.1} in the cases
$(m+p)/2<n<m$, $n=m$ and $n>m$ are established in Sections 4--6 respectively.

\section{Existence and uniqueness}\label{Eu}

We begin with an existence theorem which reduces the vanishing problem to the
construction of a suitable upper bound for the generalized solution. This
statement can be proved in a similar way as the corresponding theorems in
\cite{Kalashnikov} -- \cite{Kamin}.

\begin{theorem} \label{thm2.1}
Suppose that $\Phi(x,t)\ge 0$ is a generalized
supersolution of the equation \eqref{1.1} in $S$ and $u_0(x)\le\Phi(x,0)$.
Then there exists a generalized solution $u(x,t)$ of the Cauchy problem
\eqref{1.1}, \eqref{1.2} in $S$, which is minimal in the set of solutions of
this Cauchy problem, such that $0\le u(x,t)\le\Phi(x,t)$ in $S$.
\end{theorem}

To construct a generalized supersolution we shall use the following lemma
which is easily proved by integration by parts.

\begin{lemma} \label{lem2.1}
Let $v(x,t)$ be a continuous nonnegative function in
$\overline{S}$ that satisfies the inequality $$ -v_t+\sum_{i,j=1}^N a_{ij}
(v^m)_{x_ix_j} + \sum_{i=1}^N b_i (v^n)_{x_i} - cv^p \le
0 \quad(\ge 0) $$ and belongs to the space $C^{2,1}_{x,t}$ in $S$ outside a
set $G$ that consists for each fixed $t\in(0,+\infty)$ of finitely many
bounded closed hypersurfaces each of which is formed by finitely many
piecewise smooth surfaces. Furthermore, suppose that $\nabla(v^m)$ is
continuous on $G$. Then $v(x,t)$ is a generalized supersolution (subsolution)
of the equation \eqref{1.1}.
\end{lemma}

Part of existence and uniqueness classes of the Cauchy problem \eqref{1.1},
\eqref{1.2} have been established in \cite{Gladkov2,Prokhozhy1,Gladkov3}
and others can be obtained in a similar way. Let us formulate
that results in the part which is necessary for our aim.

\noindent(a) Consider the case $n<(m+1)/2$. It is well known that a
positive definite
quadratic form $\sum_{i,j=1}^N a_{ij}\xi_i\xi_j$ reduces to the shape
$\sum_{i=1}^N \eta_i^2$ by means of linear transformation
\begin{equation}
\label{2.1} \xi_i=\sum_{j=1}^Nc_{ij}\eta_j, \quad i=1,\dots,N,
\end{equation}
where $c_{ij}=c_{ji}$ $(i,j=1,\dots,N)$. Put for $x\in\mathbb{R}^N$
\begin{equation}
\label{2.2} \mathop{\rm dist}(x)
=\Big[\sum_{i=1}^N\Big(\sum_{j=1}^Nc_{ij}x_j\Big)^2\Big]^{1/2}.
\end{equation}
Obviously, $\mathop{\rm dist}(x)>0$ for $x\ne 0$. Denote
$r=\mathop{\rm dist}(x)$.

We define the class $\mathcal{K}_1$ of nonnegative functions $\varphi(x,t)$ and
$\varphi(x)$ which satisfy in arbitrary layer $S_T=\mathbb{R}^N\times[0,T]$ and
$\mathbb{R}^N$ respectively the following condition
\begin{equation}
\label{2.3} \varphi\le M_1(\gamma_1+r)^k, \quad 0\le k<2/(m-1).
\end{equation}
Here and below by $M_i$ and $\gamma_i$ $(i=1,2,\dots)$ we shall denote
positive and nonnegative constants respectively. Constants $k$, $M_1$ and
$\gamma_1$ in \eqref{2.3} can depend on $T$ and function $\varphi$.

\begin{theorem} \label{thm2.2}
Let $u_0(x)\in\mathcal{K}_1$. Then the Cauchy problem
\eqref{1.1}, \eqref{1.2} has a minimal generalized solution
$u(x,t)\in\mathcal{K}_1$ in $S$.
The generalized solution is unique in the class $\mathcal{K}_1$.
\end{theorem}

\noindent(b) Assume now that $n=(m+1)/2$. In contrast to the previous case now the
second term in the right hand side of \eqref{1.1} is essential for
existence and uniqueness. Let us consider the equation \eqref{1.1} for the
dimension $N=1$
\begin{equation}
\label{2.4}  \mathcal{L}_1(u)\equiv -u_t+a(u^m)_{xx}+b(u^n)_x-cu^p=0
\end{equation}
with the initial data \eqref{1.2}. For definiteness here and below in the
paper we shall suppose $b>0$. Else the case $b=0$ has been studied in
\cite{Gladkov1} and for $b<0$ space variable substitution $x$ to $(-x)$ leads
to \eqref{2.4} with $b>0$.

Define the class $\mathcal{K}_2$ of nonnegative functions $\varphi(x,t)$ and
$\varphi(x)$ satisfying in arbitrary strip $S_T=\mathbb{R}\times[0,T]$
and $\mathbb{R}$
respectively the following inequalities
\begin{gather}
\label{2.4,3} \varphi\le M_2(\gamma_2+x)^k \quad\mbox{for } x\ge 0, \quad
0\le k<2/(m-1),
\\
\label{2.4,6} \varphi\le\Big(
\frac{b(m-1)}{2am}(\gamma_3+|x|)\Big)^{2/(m-1)} \quad\mbox{for } x<0.
\end{gather}
Constants $k$, $M_2$, $\gamma_2$ and $\gamma_3$ in \eqref{2.4,3} and
(\ref{2.4,6}) can depend on $T$ and function $\varphi$.

\begin{theorem} \label{thm2.3}
Let $u_0(x)\in\mathcal{K}_2$. Then the Cauchy problem
\eqref{2.4}, \eqref{1.2} has a minimal generalized solution $u(x,t)$ in $S$.
The generalized solution is unique in the class $\mathcal{K}_2$.
\end{theorem}

\noindent(c) Consider the case $(m+1)/2<n<m$. Equation \eqref{2.4} has
a family of
stationary solutions $u_s(x)$ satisfying the ordinary differential equation
\begin{equation}
\label{2.5}
a(u^m_s)''+b(u^n_s)'-cu^p_s=0.
\end{equation}

By $o(h(s))$ we shall denote the functions with the following property
$$
\lim_{s\to +\infty}\frac{o(h(s))}{h(s)}=0.
$$
It is known (see \cite{Prokhozhy2}) that for $u_s(x)$ satisfying the
conditions
\begin{equation}
\label{2.6} u_s(x_0)=M_3, \quad u'_s(x_0)=0, \quad x_0\in\mathbb{R},
\end{equation}
the following equalities are true
\begin{gather}
\label{2.7} u_s(x)=c_+x^{1/(n-p)}+o(x^{1/(n-p)}) \quad\mbox{for } x\ge 0,
\\
\label{2.8} u_s(x)=c_-|x|^{1/(m-n)}+o(|x|^{1/(m-n)}) \quad\mbox{for } x<0,
\end{gather}
where
\begin{equation}
\label{2.9} c_+=(\frac{c(n-p)}{bn})^{1/(n-p)},\quad
c_-=(\frac{b(m-n)}{am})^{1/(m-n)}.
\end{equation}
Note that asymptotic formulas \eqref{2.7} and \eqref{2.8} are valid for any
$(m+p)/2< n<m$. Denote
\begin{equation}
\label{2.10} c_-^*=(\frac{bn(m-n)}{am(2m-n)})^{1/(m-n)}.
\end{equation}
Obviously, $c_-^*<c_-$ for $n<m$. Define the class $\mathcal{K}_3$ of nonnegative
functions $\varphi(x,t)$ and $\varphi(x)$ satisfying in arbitrary strip
$S_T=\mathbb{R}\times [0,T]$ and $\mathbb{R}$ respectively the following
inequalities
\begin{gather}
\label{2.11} \varphi\le M_4(\gamma_4+x)^k \quad\mbox{for } x\ge 0, \quad 0\le
k<1/(n-1),
\\
\label{2.12} \varphi\le M_5(\gamma_5+|x|)^{1/(m-n)} \quad\mbox{for } x<0,
\quad M_5<c_-^*.
\end{gather}
Constants $k$, $M_4$, $M_5$, $\gamma_4$, $\gamma_5$ in \eqref{2.11} and
\eqref{2.12} can depend on $T$ and function $\varphi$.

\begin{theorem} \label{thm2.4}
Assume that $ u_0(x)\le u_s(x)$ for some function  $u_s(x)$
satisfying \eqref{2.5}. Then the Cauchy problem \eqref{2.4},
\eqref{1.2} has a minimal generalized solution $u(x,t)$ in $S$. If
additionally $u_0(x)\in\mathcal{K}_3$ then $u(x,t)\in\mathcal{K}_3$ and the
generalized solution is unique in the class $\mathcal{K}_3$.
\end{theorem}

It isn't known if the constant $c_-^*$ is optimal with respect to the
uniqueness. One can find in \cite{Gladkov2} the example of nonuniqueness of
generalized solution of the Cauchy problem \eqref{2.4}, \eqref{1.2} with $c=0$
in the class of functions satisfying \eqref{2.11} and \eqref{2.12} with
$M_5=c_-$. But there isn't example of non-uniqueness for $c_-^*\le M_5<c_-$. In
\cite{Kamin}, \cite{McLeod} for the similar Cauchy problem for the equation
\eqref{2.4} with $b=0$ and $1<p<m$ the optimality of the uniqueness result is
not established too.

\noindent(d) Suppose that $n=m$. Then for the solutions of the
problem \eqref{2.5}, \eqref{2.6} asymptotic formulas \eqref{2.7} and
\begin{equation}
\label{2.14} u_s(x)=M_6\exp(-\frac{b}{am}x)+o(\exp(-\frac{b}{am}x))
\quad\mbox{for } x<0
\end{equation}
are true (see \cite{Prokhozhy2}) with some positive constant $M_6$ depending
on $\,x_0$, $\,M_3\,$ and parameters of the equation \eqref{2.5}. Also one can
verify that the functions $$ w_s(x)=[M_7\exp(-\frac bax)+\gamma_6]^{1/m} $$
with arbitrary $M_7$ and $\gamma_6$ are stationary classical supersolutions of
the equation \eqref{2.4}.

 Define the class $\mathcal{K}_4$ of nonnegative functions $\varphi(x,t)$ and $\varphi(x)$
satisfying in arbitrary strip $S_T=\mathbb{R}\times [0,T]$ and $\mathbb{R}$
respectively the inequalities \eqref{2.11} and
\begin{equation}
\label{2.15} \varphi\le\delta(x)\exp(-\frac{b}{am}x) \quad\mbox{for }x<0,
\end{equation}
where $\delta(x)\ge 0$ and $\lim_{x\to -\infty}\delta(x)=0$. Constants $M_4$,
$\gamma_4$, $k$ in \eqref{2.11} and function $\delta(x)$ in \eqref{2.15} can
depend on $T$ and function $\varphi$.

\begin{theorem} \label{thm2.5}
Assume that $u_0(x)$ satisfies for $x<0$ the inequality
$u_0(x)\le w_s(x)$ and \eqref{2.11} holds with $\varphi=u_0(x)$. Then the
Cauchy problem \eqref{2.4}, \eqref{1.2} has a minimal generalized solution
$u(x,t)$ in $S$. The generalized solution is unique in the class
$\mathcal{K}_4$.
\end{theorem}

One can find in \cite{Gladkov2} for the equation \eqref{2.4} with $c=0$ the
example which indicates the impossibility of replacement in \eqref{2.15}
$\delta(x)$ to any positive constant without loss of uniqueness for the Cauchy
problem.

\noindent(e) And finally we consider the case $n>m$. Define the class $\mathcal{K}_5$ of
nonnegative functions $\varphi(x,t)$ and $\varphi(x)$ satisfying in arbitrary
strip $S_T=\mathbb{R}\times [0,T]$ and $\mathbb{R}$ respectively the inequality
\eqref{2.11} where the constants $M_4$, $\gamma_4$ and $k$ can depend on $T$
and function $\varphi$.

The proof of the following theorem is very similar to the arguments from
 \cite[Theorems 1 and 3]{Gladkov3}.

\begin{theorem} \label{thm2.6}
Let $u_0(x)\in\mathcal{K}_5$. Then the Cauchy problem
\eqref{2.4}, \eqref{1.2} has a minimal generalized solution
$u(x,t)\in\mathcal{K}_5$ in $S$.
The generalized solution is unique in the class $\mathcal{K}_5$.
\end{theorem}

Note that no assumption has to be made on the behavior of $u_0(x)$ as
$x\to -\infty$ in Theorem \ref{thm2.6}.

\begin{remark} \label{rmk2.1} \rm
  If $u_0(x)$ satisfies \eqref{2.3} for $n<(m+1)/2$ or \eqref{2.4,3}
 for $n=(m+1)/2$ with $k=2/(m-1)$ or \eqref{2.11} for $n>(m+1)/2$
with $k=1/(n-1)$ then a minimal generalized
solution of the Cauchy problem \eqref{1.1}, \eqref{1.2} for $n<(m+1)/2$
and \eqref{2.4}, \eqref{1.2}
for $n \ge (m+1)/2$ may blow up in a finite time (see, for example,
 \cite{Gladkov2} for the equation
\eqref{2.4} with $c=0$).
\end{remark}

\section{The case $n<(m+p)/2$}\label{C1}

In this section we prove the vanishing of generalized solutions of the Cauchy
problem \eqref{1.1}, \eqref{1.2} with initial data having definite growth at
infinity. In the end of the section we show certain optimality of obtained
results. Put $c_N=\{\frac{\displaystyle c(m-p)^2}{\displaystyle
2m(2p+N(m-p))}\}^{1/(m-p)}$.

\begin{theorem} \label{thm3.1}
Assume that $n<(m+p)/2$ and $u_0(x)$ satisfies the
inequality
\begin{equation}
\label{3.1} u_0(x)\le Ar^{2/(m-p)}+o(r^{2/(m-p)}),
\end{equation}
where $r=\mathop{\rm dist}(x)$ is defined in (\ref{2.2}) and $0\le A<c_N$. Then the
generalized solution of the Cauchy problem \eqref{1.1}, \eqref{1.2} from the
class $\mathcal{K}_1$ vanishes at every point $y\in\mathbb{R}^N$ in a finite time
$T_0(y)$.
\end{theorem}

\begin{proof}
Fix arbitrary $y\in\mathbb{R}^N$ and construct a generalized
supersolution $W(x,t)$ of the equation \eqref{1.1} in $S$ satisfying the
conditions
\begin{equation}
\label{3.2} u_0(x)\le W(x,0)
\end{equation}
and $W(y,t)=0$ for $t\ge T_0$ where $T_0$ is finite and depends on $y$. We
shall seek a function $W(x,t)$ in the form $W(x,t)=w(r,t)$. Using (\ref{2.2}),
Lemma \ref{lem2.1} and the inequality $|\sum_{j=1}^Nc_{ij}x_j|\le r$ ($i=1,\dots,N$) we
conclude that $W(x,t)$ is a generalized supersolution of the equation
\eqref{1.1} if for $w(r,t)$ outside of finitely many curves of the form
$r=\zeta(t)$ the following inequality holds
\begin{equation}
\label{3.3} -w_t+(w^m)_{rr}+\frac{N-1}{r}(w^m)_r+d|(w^n)_r|-cw^p\le 0, \quad
r\ge 0,
\end{equation}
with $d=\sum_{i,j=1}^N |c_{ij}b_i|$ and condition
$(w^m)_r(0,t)=0$ takes place. At the points where \eqref{3.3} is not valid we
suppose that the derivative $(w^m)_r$ is continuous. Choose
$\varepsilon$ in the following way
\begin{equation}
\label{3.4} 0<\varepsilon<1-(A/c_N)^m.
\end{equation}
Set
\begin{equation}
\label{3.5} w(r,t)=\{\varepsilon g^m(t)+(1-\varepsilon)z^m(r)\}^{1/m},
\end{equation}
where
\begin{equation}
\label{3.6} g(t)=[K-\varepsilon^{(m-1)/m}c(1-p)t]_+^{1/(1-p)},
\end{equation}
and a positive constant $K$ and nonnegative nondecreasing function $z(r)$ will be
defined below. In (\ref{3.6}) the notation $s_+=\max\{s,0\}$ is used. Using the
convexity of the function $h(s)=s^{p/m}$ we obtain
\begin{equation}
\label{3.7}
w^p\ge\varepsilon g^p+(1-\varepsilon)z^p.
\end{equation}
It is not difficult to show the validity of the following relations:
\begin{gather}
\label{3.7,5}
-w_t\le c\varepsilon g^p,
\\
\label{3.8} |(w^n)_r|=(1-\varepsilon)(z^n)'D(r,t),
\end{gather}
where
\begin{equation}
\label{3.9} D(r,t)=\frac{z^{m-n}(r)}{\{\varepsilon
g^m(t)+(1-\varepsilon)z^m(r)\}^{(m-n)/m}}.
\end{equation}
Since $z(r)$ and $g(t)$ are nonnegative functions we have $0\le D(r,t)\le
(1-\varepsilon)^{-(m-n)/m}$ for $r\ge 0$ and $t\ge 0$. Therefore,
\begin{equation}
\label{3.10} d|(w^n)_r|\le B(1-\varepsilon)(z^n)',
\end{equation}
where $B=d(1-\varepsilon)^{-(m-n)/m}$. Thus from \eqref{3.5}--\eqref{3.10}
it follows that \eqref{3.3} holds when
\begin{equation}
\label{3.11}
\mathcal{L}_2(z)\equiv (z^m)''+\frac{N-1}{r}(z^m)'+B(z^n)'-cz^p\le 0.
\end{equation}
Moreover, we suppose that $z'(0)=0$ and at the points where (\ref{3.11}) is
not true the derivative $(z^m)'$ is continuous. Let us verify that the
following function
\begin{equation}
\label{3.12} z(r)=c_N(r^l-M_8)^{2/[l(m-p)]}_+
\end{equation}
with
\begin{equation}
\label{3.13} 0<l\le\min\big\{\frac{2(n-p)}{m-p},\frac{m+p-2n}{m-p}\big\}
\end{equation}
and sufficiently large $M_8$ satisfies the above conditions. Remark that sum
of the positive numbers in right hand side of (\ref{3.13}) is equal to one.
Hence we have $l\le 1/2$. Obviously, $\mathcal{L}_2(z)=0$ for $r < M_8^{1/l}.$
For $r>M_8^{1/l}$ elementary calculations give us
\begin{equation}\label{3.14}
\begin{aligned}
\mathcal{L}_2(z)
&=cc_N^p(r^l-M_8)^{2p/[l(m-p)]}\Big\{\frac{2m-l(m-p)}
{2p+N(m-p)}(1-M_8r^{-l})^{-2+2/l} \\
&\quad +\frac{(m-p)(l+N-2)}{2p+N(m-p)} (1-M_8r^{-l})^{-1+2/l}\\
&\quad + \frac{2Bnc_N^{n-p}}{c(m-p)}(r^l-M_8)^ {-1+2(n-p)/[l(m-p)]}r^{l-1} - 1
\Big\}.
\end{aligned}
\end{equation}
For $N\ge 2$ we apply the inequality
\begin{equation}
\label{3.15} s^{\alpha}<s \quad\mbox{for } 0<s<1 \mbox{ and }\alpha>1
\end{equation}
to the first and second terms in the braces of (\ref{3.14}) and conclude that
$\mathcal{L}_2(z)\le 0$ if
\begin{equation}
\label{3.16} -\delta M_8 + \frac{2Bnc_N^{n-p}}{c(m-p)}(r^l-M_8)^
{-1+2(n-p)/[l(m-p)]}r^{2l-1}\le 0,
\end{equation}
where $\delta=1$. For $N=1$ transforming the second term in the braces of
(\ref{3.14}) and using (\ref{3.15}) we get that $\mathcal{L}_2(z)\le 0$ if
(\ref{3.16}) holds with $\delta=[2p+l(m-p)]/(m+p)>0$. Choosing sufficiently
large $M_8$ and using (\ref{3.13}) we obtain that (\ref{3.16}) is correct for
$r>M_8^{1/l}$. Moreover, $(z^m)'$ is continuous at the point $r=M_8^{1/l}$ and
$z'(0)=0$. Thus due to Lemma \ref{lem2.1} constructed function $W(x,t)$ is the generalized
supersolution of the equation \eqref{1.1}.

Now let us choose $K$ from (\ref{3.6}) to satisfy the inequality (\ref{3.2}).
By virtue of (\ref{3.1}), (\ref{3.4}), (\ref{3.5}) and (\ref{3.12}) there
exists the maximal root $R$ of the equation
\begin{equation}
\label{3.18} (1-\varepsilon)^{1/m}z(s)=\max_{\mathop{\rm dist}(x)\le s}u_0(x).
\end{equation}
From (\ref{3.1}), (\ref{3.5}) and (\ref{3.18}) we conclude that (\ref{3.2}) is
correct for $\mathop{\rm dist}(x)\ge R$. To satisfy (\ref{3.2}) for $\mathop{\rm dist}(x)<R$
it is sufficient that $\varepsilon^{1/m}g(0)=\max_{\mathop{\rm dist}(x)\le R}u_0(x)$.
Hence we can set $$ K=[\varepsilon^{-1/m}\max_{\mathop{\rm dist}(x)\le
R}u_0(x)]^{1-p}. $$ Now Theorems \ref{thm2.1} and \ref{thm2.2}
 applied to the generalized
solution $u(x,t)$ from the class $\mathcal{K}_1$ and to the generalized
supersolution $W(x,t)$ give us the estimate
\begin{equation}
\label{3.19} u(x,t)\le W(x,t) \quad\mbox{in } S.
\end{equation}
Setting $M_8\ge [\mathop{\rm dist}(y)]^l$ we have
\begin{equation} \label{3.17}
z(\mathop{\rm dist}(y))=0.
\end{equation}
As a consequence of (\ref{3.5}), (\ref{3.6}) and (\ref{3.17}) we get $W(y,t)=0$
for $t\ge T_0=K/[(1-p) c \varepsilon^{(m-1)/m}]$. This completes the proof.
\end{proof}

\begin{remark} \label{rmk3.1}\rm
 Let us show that Theorem \ref{thm3.1} is optimal in a certain sense.
Indeed, the equation \eqref{1.1} with $a_{ij}=1$ for $i=j$, $a_{ij}=0$ for
$i\ne j$ and $b_i=0$ ($i,j=1,\dots,N$) has explicit stationary solution from
the class $\mathcal{K}_1$ $$ u_s(r)=c_Nr^{2/(m-p)}. $$ It satisfies (\ref{3.1})
with $A=c_N$ and does not vanish at every point of $\mathbb{R}^N$.
\end{remark}

\section{The case $(m+p)/2<n<m$}\label{C2}

For the rest of the paper we shall consider one-dimensional equation
\eqref{2.4}. Let $w_i(\xi)$ $(i=1,2)$ be nonnegative functions satisfying for
$\xi\ge 0$ the differential inequalities
\begin{equation}
\label{4.1}  \mathcal{T}_i(w_i)\equiv w'_i+a(w^m_i)''+d_i(w^n_i)'-cw^p_i\le 0
\end{equation}
with constants $d_1=b$ and $d_2=-b$ and the conditions
\begin{equation}
\label{4.2}  w_i(0)=M_9, \quad w'_i(0)=0.
\end{equation}
For each of functions $w_i$ the inequality (\ref{4.1}) can be not fulfilled at
finitely many points $\xi^{(i)}_{j_i}$ $(j_i=1,\dots,k_i;\; i=1,2)$ where the
derivative $(w_i^m)'$ is continuous. Note that Cauchy problems for the
equations $\mathcal{T}_i(w_i)=0$ with initial conditions \eqref{4.2} have unique
solutions with the following asymptotic behavior
\begin{equation}
\label{4.2,5} w_1(\xi)=c_+\xi^{1/(n-p)}+o(\xi^{1/(n-p)}),\quad
w_2(\xi)=c_-\xi^{1/(m-n)}+o(\xi^{1/(m-n)}),
\end{equation}
where the constants $c_+$ and $c_-$ were defined in \eqref{2.9} (see
\cite{Prokhozhy2} for similar problems).

 For $\overline x\in\mathbb{R}$ define the auxiliary function
\begin{equation}
\label{4.3} w_{\overline x}(x)=
\begin{cases}
w_1(x-\overline x)&\mbox{for } x\ge \overline x,\\
 w_2(\overline x-x)&\mbox{for }x< \overline x.\\
\end{cases}
\end{equation}

Now we can formulate the main result of this section.

\begin{theorem} \label{thm4.1}
 Assume that $(m+p)/2<n<m$ and $u_0(x)$ satisfies the
inequality
\begin{equation}
\label{4.4} u_0(x)\le w_{\overline x}(x)
\end{equation}
for some $\overline x \in \mathbb{R}$ and some function $w_{\overline x}(x)$
constructed in \eqref{4.3}. Suppose also that $u_0(x)\in\mathcal{K}_i$
$(i=1,2,3)$ for $n<(m+1)/2$, $n=(m+1)/2$ and $n>(m+1)/2$ respectively. Then
the generalized solutions of the Cauchy problem \eqref{2.4}, \eqref{1.2} from
the classes $\mathcal{K}_i$ $(i=1,2,3)$ for $n<(m+1)/2$, $n=(m+1)/2$
and $n>(m+1)/2$  respectively vanish at every point $y\in\mathbb{R}$
in a finite time $T_0(y)$.
\end{theorem}

\begin{proof}
 We prove the theorem in two steps. At first we show that the
generalized solution of the Cauchy problem \eqref{2.4}, \eqref{1.2} is bounded
in $\Omega\times [0,+\infty)$ where $\Omega$ is any bounded domain in $\mathbb{R}$.
Then we establish the vanishing of the generalized solution at every
point of $\mathbb{R}$ in a finite time.

We start with construction in $S$ a traveling-wave generalized supersolution
of the equation \eqref{2.4}. Put
\begin{equation}
\label{4.7} W(x,t)=\begin{cases}
w_1(x-\overline x-t)&\mbox{for } x\ge\overline x+t,\\
M_9&\mbox{for }\overline x-t<x< \overline x+t,\\
w_2(\overline x-x-t)&\mbox{for }x\le\overline x-t.
\end{cases}
\end{equation}
In view of (\ref{4.1}) we have $\mathcal{L}_1(W)\le 0$ everywhere except the
lines $x\pm t=\overline x$, $x-t=\overline x+\xi^{(1)}_{j_1}$ and
$x+t=\overline x-\xi^{(2)}_{j_2}$ $(j_i=1,\dots,k_i;\; i=1,2)$ where the
derivative $(W^m)_x$ is continuous. Moreover, from (\ref{4.4}) we obtain that
(\ref{3.2}) holds since $W(x,0)=w_{\overline x}(x)$.
Applying Theorem \ref{thm2.1} and
one of Theorems \ref{thm2.2}--\ref{thm2.4} for $n<(m+1)/2$, $n=(m+1)/2$ and $n>(m+1)/2$
respectively, we obtain the estimate (\ref{3.19}). From (\ref{3.19}) and
(\ref{4.7}) we conclude that for all $y\in\mathbb{R}$
\begin{equation}
\label{4.8} u(y,t)\le M_9 \quad\mbox{for }t\ge |y-\overline x|.
\end{equation}
Let us consider the function
\begin{equation}
\label{4.9} w(x,t)=\{\varepsilon
g^m(t-t_0)+(1-\varepsilon)z^m(\sigma)\}^{1/m}, \quad \sigma = |x-y|,
\end{equation}
where $g(t)$ was defined in (\ref{3.6}), $t_0>0$ and nonnegative nondecreasing
function $z(\sigma)$ will be defined below, $\varepsilon$ is arbitrary number
from the interval $(0,1)$. It is easy to see
that relations (\ref{3.7})--(\ref{3.10}) with $d=b$ remain true after
replacement $r$ to $\sigma$. Thus to satisfy the inequality $\mathcal{L}_1(w)\le
0$ it is sufficient to require that
\begin{equation}
\label{4.9,5}
a(z^m)''+B(z^n)'-cz^p \le 0.
\end{equation}
We define now $z(\sigma)$ as follows
\begin{equation}
\label{4.10} z(\sigma)=M_{10}(\sigma^l-1)^{1/[l(n-p)]}_+,
\end{equation}
where $l<(m-p)/[2(n-p)]$ and $M_{10}$ is small enough. Then the function $z(\sigma)$
satisfies (\ref{4.9,5}) and the condition $z'(0)=0.$ Obviously, the equation
$z(\sigma)=M_9(1-\varepsilon)^{-1/m}$ has a unique root
\begin{equation}
\label{4.11} \sigma_0=[(M_9(1-\varepsilon)^{-1/m}/M_{10})^{l(n-p)}+1]^{1/l}.
\end{equation}
Fix arbitrary $y\in\mathbb{R}$. Choose in (\ref{4.9}) $t_0$ in the following way:
\begin{equation}
\label{4.12} t_0=|y-\overline x|+\sigma_0.
\end{equation}
The relations (\ref{4.9}), (\ref{4.10}) -- (\ref{4.12}) yield
\begin{equation}
\label{4.13} w(y\pm \sigma_0,t)\ge M_9, \quad t\ge t_0.
\end{equation}
Setting in (\ref{3.6}) $K=(\varepsilon^{-1/m}M_9)^{1-p}$ we obtain
\begin{equation}
\label{4.14} w(x,t_0)\ge M_9, \quad y-\sigma_0\le x\le y+\sigma_0.
\end{equation}
From (\ref{4.8}), (\ref{4.13}), (\ref{4.14}) we conclude that on the
parabolic boundary of the domain $Q=[y-\sigma_0,y+\sigma_0]\times [t_0,+\infty)$
the inequality
\begin{equation}
\label{4.15} u(x,t)\le w(x,t)
\end{equation}
holds. Moreover, $\mathcal{L}_1(w)\le 0$ in $Q$.
Applying the comparison theorem (see, for example, \cite{Diaz}) we obtain the
estimate (\ref{4.15}) in Q. But by virtue of (\ref{4.9}), (\ref{4.10}) and
(\ref{3.6}) $w(y,t)=0$ for $t\ge T_0=t_0+K/[(1-p)c\varepsilon^{(m-1)/m}]$.
Theorem is proved.
\end{proof}

\begin{remark} \label{rmk4.1} \rm
 Without assumptions $u(x,t)\in\mathcal{K}_i$ $(i=1,2,3)$ in
Theorem \ref{thm4.1} we can conclude only the vanishing of the
{\it minimal} generalized solution of the Cauchy problem
\eqref{2.4}, \eqref{1.2} since we know nothing about its
uniqueness (see item (c) of Section 2).
\end{remark}

\begin{remark} \label{rmk4.2}\rm
 Let us show a certain optimality of Theorem \ref{thm4.1}. Passing to the
integral equation and using Schauder-Tychonoff theorem one can show that there
exists a stationary solution $u_1(x)$ of the equation \eqref{2.4} such that
$u_1(x)=0$ for $x\le 0$ and $u_1(x)>0$ for $x>0$. Arguing as in
\cite{Prokhozhy2} one can obtain asymptotic formula \eqref{2.7} for this
solution. Therefore, $u_1(x)$ belongs to the class $\mathcal{K}_1$ for
$n<(m+1)/2$, to the class $\mathcal{K}_2$ for $n=(m+1)/2$ and to the class $\mathcal{K}_3$
for $n>(m+1)/2$ but it doesn't vanish at every point of $\mathbb{R}$ in a
finite time. Note that $u_1(x)$ has the same first term of asymptotic
behavior as $x\to +\infty$ as $w_{\overline x}(x)$.

In a similar way we can construct a stationary solution $u_2(x)$ of the
equation \eqref{2.4} such that $u_2(x)=0$ for $x\ge 0$ and $u_2(x)>0$ for
$x<0$. This solution has the same first term of asymptotic behavior as $x\to
-\infty$ as $w_{\overline x}(x)$ and belongs to the class $\mathcal{K}_1$ for
$n<(m+1)/2$ and to the class $\mathcal{K}_2$ for $n=(m+1)/2$. Note that for
$n>(m+1)/2$ the function $w_{\overline x}(x)$ grows as $x\to -\infty$
faster than any
function from the class $\mathcal{K}_3$.
\end{remark}

\section{The case $n=m$}\label{C3}

The main result of this section is the following.

\begin{theorem} \label{thm5.1}
Let $n=m$ and initial data satisfy the inequalities
\eqref{2.15} with $\varphi=u_0(x)$ and
\begin{equation}
\label{5.0,5} u_0(x)\le A_+x^{1/(n-p)}+o(x^{1/(n-p)})
\quad\mbox{for } x\ge 0,
\end{equation}
where $0\le A_+<c_+$ and the constant $c_+$ was defined in \eqref{2.9}. Then
the generalized solution of the Cauchy problem \eqref{2.4}, \eqref{1.2} from
the class $\mathcal{K}_4$ vanishes at every point $y\in\mathbb{R}$ in a
finite time $T_0(y)$.
\end{theorem}

\begin{proof}
 Fix an arbitrary $y\in\mathbb{R}$. We construct a supersolution
$w(x,t)$ of the equation \eqref{2.4} in the form (\ref{3.5}), (\ref{3.6}),
where $r=x$,
\begin{equation}
\label{5.1} 0<\varepsilon<1-(A_+/c_+)^m
\end{equation}
and function $z(x)$ will be determined below. Since the inequalities
(\ref{3.7}) and (\ref{3.7,5}) remain true and $(w^n)_x=(1-\varepsilon)(z^m)'$
we conclude that $\mathcal{L}_1(w)\le 0$ when
\begin{equation}
\label{5.2} a(z^m)''+b(z^m)'-cz^p \le 0.
\end{equation}
At the points where (\ref{5.2}) is not valid we suppose that the derivative
$(z^m)'$ is continuous. For $x\ge y$ these conditions are fulfilled for the following function
\begin{equation}
\label{5.3} z(x)=c_+[(x-y)^l-M_{11}]^{1/[l(m-p)]}_+,
\end{equation}
where $l<1/2$ and $M_{11}$ is large enough. Clearly,
\begin{equation}
\label{5.4} z(y)=0.
\end{equation}
For $x\in [y-\delta,y)$ ($\delta>0$) put $z(x)=(y-x)^{\gamma}$,
$\gamma>2/(m-p)$, and note that (\ref{5.2}) holds here if $\delta$ is small
enough. Further, for $x<y-\delta$ set
\begin{equation}
\label{5.5} z(x)=\big(M_{12}\exp(-\frac{b}{a}x)+\beta\big)^{1/m},
\end{equation}
where the constants $M_{12}>0$ and $\beta$ are determined from the
corresponding continuity conditions of $z(x)$ and $(z^m)'(x)$ at the point
$x=y-\delta$. It is not difficult to check that the function (\ref{5.5})
satisfies (\ref{5.2}). Applying Lemma \ref{lem2.1} we have that $w(x,t)$ is the
generalized supersolution of the equation \eqref{2.4} in $S$.

Taking into account hypotheses of Theorem \ref{thm5.1}, (\ref{5.1}), (\ref{5.3}) and
(\ref{5.5}) we conclude that there exists the minimal $R_-$ and the maximal
$R_+$ roots of the equation
\begin{equation}
\label{5.6} (1-\varepsilon)^{1/m}z(x)=u_0(x).
\end{equation}
Thus from (\ref{3.5}) and (\ref{5.6}) we have
\begin{equation}
\label{5.7} w(x,0)\ge u_0(x)
\end{equation}
for $x\ge R_+$ and $x\le R_-$. To satisfy (\ref{5.7}) for $R_-<x<R_+$ we put
in (\ref{3.6}) $$ K=[\varepsilon^{-1/m}\max_{R_-\le x\le R_+}u_0(x)]^{1-p}.$$
Applying Theorems \ref{thm2.1} and \ref{thm2.5}, (\ref{3.5}), (\ref{3.6}) and (\ref{5.4}) we
complete the proof.
\end{proof}

\begin{remark} \label{rmk5.1} \rm
 Slightly modifying the proof of Theorem \ref{thm5.1} we can construct
a generalized supersolution of the Cauchy problem \eqref{2.4}, \eqref{1.2}
vanishing at every point of $\mathbb{R}$ in a finite time as well for initial
data satisfying \eqref{5.0,5} and the inequality
\begin{equation}
\label{5.8}  u_0(x)\le M_{13}\exp(-\frac b{am}x)+o(\exp(-\frac b{am}x))
\quad\mbox{for }x<0
\end{equation}
with arbitrary positive constant $M_{13}$. Now we can conclude
only that the {\it  minimal } generalized solution of the Cauchy
problem \eqref{2.4}, \eqref{1.2} vanishes at every point of
$\mathbb{R}$ in a finite time since this solution may be
non-unique (see item (d) of Section 2). Note that solutions
$u_s(x)$ of the problem \eqref{2.5}, \eqref{2.6} have asymptotic
representation for $x<0$ as right hand side of (\ref{5.8}) where
$M_{13}$ depends on $M_3$.
\end{remark}

\begin{remark} \label{rmk5.2} \rm
From \eqref{2.7} it follows that for the stationary solution
$u_1(x)$ of the equation \eqref{2.4} constructed as in Remark \ref{rmk4.2} the
inequality \eqref{5.0,5} with $A_+=c_+$ holds. This fact demonstrates a certain
optimality of Theorem \ref{thm5.1}. Note that stationary solution $u_2(x)$ of the
equation \eqref{2.4} constructed as in Remark \ref{rmk4.2} grows as $x\to -\infty$
faster than any function from the class $\mathcal{K}_4$.
\end{remark}

\section{The case $n>m$}\label{C4}


We show here that some generalized solutions of the Cauchy problem
\eqref{2.4}, \eqref{1.2} with any growing as $x\to -\infty$ initial function
vanish at every point of $\mathbb{R}$ in a finite time.

\begin{theorem} \label{thm6.1}
Assume that $n>m$ and $u_0(x)$ satisfies the inequality
\eqref{5.0,5}. Then the generalized solution of the Cauchy problem
\eqref{2.4}, \eqref{1.2} from the class $\mathcal{K}_5$ vanishes at every point
$y\in\mathbb{R}$ in a finite time $T_0(y)$.
\end{theorem}

\begin{proof}
Let $W(x,t)$ be a travelling-wave generalized supersolution of the
equation \eqref{2.4} of the form (\ref{4.7}) where $\overline x=0$ and
functions $w_i(\xi)$ $(i=1,2)$ satisfy (\ref{4.1}) and \eqref{4.2}. It isn't
difficult to check that the function $$ w_1(\xi)=B_+(M_9+\xi^2)^{1/[2(n-p)]},
\quad \xi\ge 0,$$ where $A_+<B_+<c_+$,  satisfy the above
requirements and (\ref{3.2}) holds for $x\ge 0$ if $M_9$ is large enough.

Let us construct $w_2(\xi)$. Suppose that
\begin{equation} \label{6.2,5} M_9\ge(\frac{2}{bn})^{1/(n-1)}.
\end{equation}
At first we prove the following auxiliary result.
\end{proof}

\begin{lemma} \label{lem6.1}
The solution $g(\xi)$ of the equation
\begin{equation}
\label{6.3} \mathcal{T}_2(g)=0
\end{equation}
satisfying the conditions \eqref{4.2} and \eqref{6.2,5} exists only in a
finite half-interval $[0,\xi_0)$ and $\xi_0 \to 0$ as $M_9\to +\infty$.
Moreover,
\begin{equation}
\label{6.11} \lim_{\xi\to \xi_0-0}g'(\xi)[g(\xi)]^{-(n-m+1)}=\frac{b}{am}.
\end{equation}
\end{lemma}

\begin{proof} It is easy to see that
\begin{equation}
\label{6.4} g(\xi)>M_9 \quad\mbox{and }\quad g'(\xi)>0 \quad\mbox{for }\xi>0.
\end{equation}
Using \eqref{6.2,5} and (\ref{6.4}) we have
\begin{equation}
\label{6.6} (g^m)''\ge\frac{b}{2a}(g^n)'+\frac caM_9^p.
\end{equation}
Integrating (\ref{6.6}) over $(0,\xi)$ and taking into account \eqref{4.2} we
get
\begin{equation}
\label{6.7} (g^m)'\ge\frac{b}{2a}g^n-\frac{b}{2a}M_9^n+\frac caM_9^p\xi.
\end{equation}
Putting $v=g^m-M_9^m$ and using the inequality $$
r^{\alpha}-s^{\alpha}>(r-s)^{\alpha}, \quad 0<s<r, \quad \alpha>1, $$ with
$\alpha=n/m$ we obtain for $\xi>0$
\begin{equation}
\label{6.8} v'\ge\frac{b}{2a}v^{n/m}.
\end{equation}
As a consequence of (\ref{6.4}) and (\ref{6.7}) we have
\begin{equation}
\label{6.9} v\ge\frac c{2a}M_9^p\xi^2.
\end{equation}
Fixing arbitrary $\varepsilon_1>0$, integrating (\ref{6.8}) over
$(\varepsilon_1,\xi)$ and using (\ref{6.9}) we deduce the inequality
\begin{equation}
\label{6.10} v(\xi)>\Big[(\frac c{2a}M_9^p\varepsilon_1^2)^{-(n-m)/m} -\frac
{b(n-m)}{2am}(\xi-\varepsilon_1)\Big]^{-m/(n-m)}.
\end{equation}
The first part of Lemma \ref{lem6.1} follows from (\ref{6.10}) by virtue of
arbitrariness of $\varepsilon_1$. Integrating (\ref{6.3}) over $(0,\xi)$,
$\xi<\xi_0$, it is easy to obtain (\ref{6.11}). Lemma is proved.
\end{proof}

Pass in (\ref{4.1}) to new unknown function $f(\xi)=[w_2(\xi)]^{-(n-m)}$. If
we multiply obtained inequality for $f(\xi)$ by $(n-m)f^{(2n-m)/(n-m)}/m$ the
relations (\ref{4.1}), \eqref{4.2} can be written in the form
\begin{gather}
\label{6.12} \begin{aligned}
\mathcal{L}_3(f)&\equiv-aff''+\frac{an}{n-m}(f')^2+\frac{bn}{m}f'-\frac
1mf^{(n-1)/(n-m)}f'\\
&\quad  -\frac{c(n-m)}{m}f^{(2n-m-p)/(n-m)}\le 0,
\end{aligned} \\
\label{6.13} f(0)=M_9^{-(n-m)}, \quad f'(0)=0.
\end{gather}
Using Lemma \ref{lem6.1} and (\ref{6.4}) it is not difficult to verify that solutions of
the Cauchy problem for the equation
\begin{equation}
\label{6.14} \mathcal{L}_3(f)=0
\end{equation}
with initial conditions (\ref{6.13}) have the following properties:
\begin{equation}
\label{6.15}
\begin{gathered}
f(\xi)>0, \quad -b(n-m)/(am)<f'(\xi)<0, \quad f''(\xi)<0 \quad\mbox{ for }
0<\xi<\xi_0, \\
\lim_{\xi\to \xi_0-0}f(\xi)=0, \quad
\lim_{\xi\to \xi_0-0}f'(\xi)=-b(n-m)/(am), \\
\xi_0\to 0 \mbox{ as } M_9^{-(n-m)}\to 0.
\end{gathered}
\end{equation}
Put $\overline{u}_0(\xi)=[u_0(-\xi)+1]^{-(n-m)}$, $\xi\ge 0$. Obviously, for
$x\le 0$ (\ref{3.2}) follows from the inequality
\begin{equation}
\label{6.16} f(\xi)\le \overline{u}_0(\xi).
\end{equation}
Let a constant $M_{14}$ be so large that $\xi_0<1/2$ when $M_9\ge M_{14}$.
Choose $M_9$ as follows
\begin{equation}
\label{6.17} M_9\ge\Big[\min\{M_{14}^{-(n-m)},\frac{b(n-m)}{4am},
\overline{u}_0(2)\}\Big]^{-1/(n-m)}.
\end{equation}
In the interval $[0,\xi_*]$ we put the function $f(\xi)$ equal to the solution
of the Cauchy problem (\ref{6.14}), (\ref{6.13}) where the point $\xi_*<\xi_0$ is
defined in such a way that the function $q(\xi)$, linear for $\xi\le 1$, which
passes through the points $(\xi_*,f(\xi_*))$ and
$(1,\min\{f(0)/3,\overline{u}_0(3)\})$,
satisfies the equality $q'(\xi_*)=f'(\xi_*)$. We define the sequence
$\{q(k)\}$ by the recurrence relation
\begin{equation}
\label{6.18} q(k)=\min\{q(k-1)/3,\overline{u}_0(k+2)\}, \quad k=2,3,\ldots.
\end{equation}
Let function $q(\xi)$ be piecewise-linear for $\xi\ge 1$ and have a graphical
representation obtained by joining points $(k,q(k))$, $k=1,2,\ldots$. Denote
the mollification of $q(\xi)$ by $q_h(\xi)$. Assume that $h<1/2.$ Using the
definition of $q(\xi)$ and the properties of mollifiers we have
\begin{equation} \label{6.19}
\begin{gathered}
q_h(\xi)\in C^{\infty}(\xi_*,\infty), \quad q_h(\xi)
=q(\xi)\mbox{ for } \xi_*<\xi<1-h, \\
  k-1+h<\xi<k-h, \quad k=2,3,\dots,\\
q''_h(\xi)\ge 0,\quad
q'_h(\xi)\le 0,\quad q(\xi)\le q_h(\xi)\le\overline{u}_0(\xi)
\quad \mbox{ for }\xi>\xi_*.
\end{gathered}
\end{equation}
 For $\xi\ge \xi_*$ we put $f(\xi)=q_h(\xi)$. Due to
\eqref{6.2,5}, (\ref{6.17}) -- (\ref{6.19}) inequality (\ref{6.12}) is valid
for $\xi\neq \xi_*$. Nevertheless $f'(\xi)$ is continuous at the point
$\xi=\xi_*$. The function $w_2(\xi)$ is constructed.

The rest of the proof completely repeats the same arguments as in the proof of
Theorem \ref{thm4.1} except for the inequality (\ref{4.9,5}). Instead of it we require
that
\begin{equation}
\label{6.20} a(z^m)''+\frac{bn}{m}\{\varepsilon
K^{m/(1-p)}+(1-\varepsilon)z^m(r)\}^{(n-m)/m}(z^m)'-cz^p \le 0,
\end{equation}
but the same function $z(\sigma)$, which is defined by (\ref{4.10}), satisfies
(\ref{6.20}).

\begin{remark} \label{rmk6.1} \rm
Let us show a certain optimality of Theorem \ref{thm6.1}. Let $u_1(x)$
be the stationary solution of the equation \eqref{2.4} constructed as in
Remark \ref{rmk4.2}. This solution belongs to the class $\mathcal{K}_5$ and satisfies
\eqref{5.0,5} with $A_+=c_+$ but it doesn't vanish at every point of
 $\mathbb{R}$ in a finite time.
\end{remark}

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\end{document}