\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 $$\label{1.1} u_t = \sum_{i,j=1}^N a_{ij}(u^m)_{x_ix_j} + \sum_{i=1}^N b_i(u^n)_{x_i} - cu^p, \quad (x,t)\in S = \mathbb{R}^N\times(0,+\infty)$$ with initial data $$\label{1.2} u(x,0) = u_0(x), \quad x \in \mathbb{R}^N,$$ 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 $B_h=\{x\in\mathbb{R}^N:|x|0$, $0(m+p)/2$ 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 $(m+p)/2m$ 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 $$\label{2.1} \xi_i=\sum_{j=1}^Nc_{ij}\eta_j, \quad i=1,\dots,N,$$ where $c_{ij}=c_{ji}$ $(i,j=1,\dots,N)$. Put for $x\in\mathbb{R}^N$ $$\label{2.2} \mathop{\rm dist}(x) =\Big[\sum_{i=1}^N\Big(\sum_{j=1}^Nc_{ij}x_j\Big)^2\Big]^{1/2}.$$ 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 $$\label{2.3} \varphi\le M_1(\gamma_1+r)^k, \quad 0\le k<2/(m-1).$$ 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 $N=1$ $$\label{2.4} \mathcal{L}_1(u)\equiv -u_t+a(u^m)_{xx}+b(u^n)_x-cu^p=0$$ 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 $(m+1)/2m$. 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 $$\label{3.1} u_0(x)\le Ar^{2/(m-p)}+o(r^{2/(m-p)}),$$ where $r=\mathop{\rm dist}(x)$ is defined in (\ref{2.2}) and $0\le AM_8^{1/l}$ elementary calculations give us \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} For $N\ge 2$ we apply the inequality \label{3.15} s^{\alpha}1 to the first and second terms in the braces of (\ref{3.14}) and conclude that $\mathcal{L}_2(z)\le 0$ if $$\label{3.16} -\delta M_8 + \frac{2Bnc_N^{n-p}}{c(m-p)}(r^l-M_8)^ {-1+2(n-p)/[l(m-p)]}r^{2l-1}\le 0,$$ 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 $$\label{3.18} (1-\varepsilon)^{1/m}z(s)=\max_{\mathop{\rm dist}(x)\le s}u_0(x).$$ 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 $\mathop{\rm dist}(x)(m+1)/2$ 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 \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(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}$$$\label{4.8} u(y,t)\le M_9 \quad\mbox{for }t\ge |y-\overline x|.$$ Let us consider the function $$\label{4.9} w(x,t)=\{\varepsilon g^m(t-t_0)+(1-\varepsilon)z^m(\sigma)\}^{1/m}, \quad \sigma = |x-y|,$$ 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 $$\label{4.9,5} a(z^m)''+B(z^n)'-cz^p \le 0.$$ We define now$z(\sigma)$as follows $$\label{4.10} z(\sigma)=M_{10}(\sigma^l-1)^{1/[l(n-p)]}_+,$$ 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 $$\label{4.11} \sigma_0=[(M_9(1-\varepsilon)^{-1/m}/M_{10})^{l(n-p)}+1]^{1/l}.$$ Fix arbitrary$y\in\mathbb{R}$. Choose in (\ref{4.9})$t_0$in the following way: $$\label{4.12} t_0=|y-\overline x|+\sigma_0.$$ The relations (\ref{4.9}), (\ref{4.10}) -- (\ref{4.12}) yield $$\label{4.13} w(y\pm \sigma_0,t)\ge M_9, \quad t\ge t_0.$$ Setting in (\ref{3.6})$K=(\varepsilon^{-1/m}M_9)^{1-p}$we obtain $$\label{4.14} w(x,t_0)\ge M_9, \quad y-\sigma_0\le x\le y+\sigma_0.$$ 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 $$\label{4.15} u(x,t)\le w(x,t)$$ 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 $$\label{5.0,5} u_0(x)\le A_+x^{1/(n-p)}+o(x^{1/(n-p)}) \quad\mbox{for } x\ge 0,$$ where$0\le A_+0$) 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$x0$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 $$\label{5.6} (1-\varepsilon)^{1/m}z(x)=u_0(x).$$ Thus from (\ref{3.5}) and (\ref{5.6}) we have $$\label{5.7} w(x,0)\ge u_0(x)$$ for$x\ge R_+$and$x\le R_-$. To satisfy (\ref{5.7}) for$R_-m$}\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_+M_9 \quad\mbox{and }\quad g'(\xi)>0 \quad\mbox{for }\xi>0. Using \eqref{6.2,5} and (\ref{6.4}) we have $$\label{6.6} (g^m)''\ge\frac{b}{2a}(g^n)'+\frac caM_9^p.$$ Integrating (\ref{6.6}) over $(0,\xi)$ and taking into account \eqref{4.2} we get $$\label{6.7} (g^m)'\ge\frac{b}{2a}g^n-\frac{b}{2a}M_9^n+\frac caM_9^p\xi.$$ Putting $v=g^m-M_9^m$ and using the inequality $$r^{\alpha}-s^{\alpha}>(r-s)^{\alpha}, \quad 01,$$ with $\alpha=n/m$ we obtain for $\xi>0$ $$\label{6.8} v'\ge\frac{b}{2a}v^{n/m}.$$ As a consequence of (\ref{6.4}) and (\ref{6.7}) we have $$\label{6.9} v\ge\frac c{2a}M_9^p\xi^2.$$ Fixing arbitrary $\varepsilon_1>0$, integrating (\ref{6.8}) over $(\varepsilon_1,\xi)$ and using (\ref{6.9}) we deduce the inequality $$\label{6.10} v(\xi)>\Big[(\frac c{2a}M_9^p\varepsilon_1^2)^{-(n-m)/m} -\frac {b(n-m)}{2am}(\xi-\varepsilon_1)\Big]^{-m/(n-m)}.$$ 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 $$\label{6.14} \mathcal{L}_3(f)=0$$ with initial conditions (\ref{6.13}) have the following properties: \label{6.15} \begin{gathered} f(\xi)>0, \quad -b(n-m)/(am)\xi_*. \end{gathered} 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 $$\label{6.20} a(z^m)''+\frac{bn}{m}\{\varepsilon K^{m/(1-p)}+(1-\varepsilon)z^m(r)\}^{(n-m)/m}(z^m)'-cz^p \le 0,$$ 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}. 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