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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 83, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/83\hfil Uniqueness of self-similar very singular solution]
{Uniqueness of self-similar very singular solution for non-Newtonian
polytropic filtration equations with gradient absorption}

\author[H. Ye, J. Yin \hfil EJDE-2015/83\hfilneg]
{Hailong Ye, Jingxue Yin}

\address{Hailong Ye \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou, Guangdong 510631, China}
\email{ye2006hailong@yeah.net}

\address{Jingxue Yin \newline
School of Mathematical Sciences, South China Normal University,
Guangzhou, Guangdong 510631, China}
\email{yjx@scnu.edu.cn}

\thanks{Submitted November 20, 2014. Published April 7, 2015.}
\subjclass[2000]{35K65, 35K92, 35K15}
\keywords{Polytropic filtration; gradient absorption; uniqueness; self-similar;
\hfill\break\indent very singular}

\begin{abstract}
 Uniqueness of self-similar very singular solutions with compact support are proved
 for the non-Newtonian polytropic filtration equation with gradient absorption
 $$
 \frac{\partial u}{\partial t}
 =\operatorname{div}(|\nabla u^m|^{p-2}\nabla u^m)
 -|\nabla u|^{q},\quad x\in \mathbb{R}^N,\quad t>0,
 $$
 where $m>0$, $p>1$, $m(p-1)>1$ and $q>1$.
\end{abstract}

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

\section{Introduction}

This article concerns the  non-Newtonian
polytropic filtration equation with gradient absorption
\begin{equation}\label{problem}
\frac{\partial u}{\partial t}
=\operatorname{div}(|\nabla u^m|^{p-2}\nabla u^m)
-|\nabla u|^{q},\qquad
x\in \mathbb{R}^N,\quad t>0,
\end{equation}
where $m>0, p>1$, $m(p-1)>1$ and $q>1$.


Such an equation, especially the case $m=1$ and $p=2$, appears as the
viscosity approximation to the well-known Hamilton-Jacobi equation,
in the stochastic control theory, as well as in a number
of interesting and different physical considerations. For more details,
see \cite{PBM,MKG,JKH} and the references therein.


In this article, we pay  attention to self-similar very singular
solutions of  \eqref{problem}.
Due to the possible degeneracy and singularity, it is necessary to
clarify the concept of weak solutions of \eqref{problem}.
A non-negative function $u$ is said to be a weak solution of \eqref{problem}, if
$u\in C_{\mathrm{loc}}(0,\infty; L^2(\mathbb{R}^N))$,
$u^m\in L_{\mathrm{loc}}^p(0,\infty; W_{loc}^{1,p}(\mathbb{R}^N))$,
$|\nabla u|\in L_{\mathrm{loc}}^q(\mathbb{R}^N\times(0,\infty))$ and $u$ satisfies
\eqref{problem} in the sense of distributions in $\mathbb{R}^N\times(0,\infty)$.
Further, by a very singular solution $u$, we mean a weak solution with
$u\in C(\mathbb{R}^N\times[0,\infty)\backslash\{(0,0)\})$ satisfying
\begin{equation}\label{singular}
\lim_{t\to0}\sup_{|x|>\varepsilon}u(x,t)=0
\end{equation}
and
\begin{equation}\label{verysingular}
\lim_{t\to0}\int_{|x|<\varepsilon}u(x,t)\,\mathrm{d}x=\infty
\end{equation}
for any $\varepsilon>0$.


In 1986, Brezis et al \cite{HBL} investigated the  semilinear heat
equation with concentration absorption
\begin{equation*}
\frac{\partial u}{\partial t}=\Delta u-u^q;
\end{equation*}
they proved the existence and uniqueness of self-similar very singular solutions when
$1<q<1+2/N$. Since that time the self-similar very singular solutions of diffusion
equations with concentration absorption have been studied extensively, see for example,
\cite{YQM1,XCYQ,GL,LAJ,SKL}. Recently, the equations with gradient absorption have
attracted much attention. In 2001, Qi et al \cite{YQM} and Benachour et al
\cite{SBP,SBH} independently obtained the existence and uniqueness of self-similar
very singular solutions of viscous Hamilton-Jacobi equation
\begin{equation}\label{HJequation}
\frac{\partial u}{\partial t}=\Delta u-|\nabla u|^q
\end{equation}
by two different methods. Afterwards, these previous results on \eqref{HJequation}
were extended to the $p$-Laplacian with gradient absorption
\begin{equation}\label{p-Laplace}
\frac{\partial u}{\partial t}=\operatorname{div}(|\nabla u|^{p-2}\nabla u)-|\nabla u|^q.
\end{equation}
For $1<p<2$, i.e., fast diffusion case,  Shi \cite{PS} proved the existence and uniqueness
of self-similar very singular solutions of \eqref{p-Laplace} when $1<q<p-\frac{N}{N+1}$.
After that, Iagara et al \cite{RGIP,RGIP1} generalized the corresponding results restricted
to $q>1$. More precisely, they proved that there exists a unique self-similar very singular
solution of \eqref{p-Laplace} when $\frac{2N}{N+1}<p<2$ and $\frac{p}{2}<q<p-\frac{N}{N+1}$.
On the other hand, for slow diffusion case, i.e., $p>2$,  Shi \cite{PS0} obtained existence
of self-similar very singular solutions with compact support of \eqref{p-Laplace} when
$p-1<q<p-\frac{N}{N+1}$. And soon, the corresponding existence result in \cite{PS0} was
extended to the equation \eqref{problem} in \cite{PSM}.


As far as we know, however, the uniqueness of self-similar very singular solutions
of \eqref{problem} has not been obtained. In the present paper, we shall show the
uniqueness of self-similar very singular solutions of \eqref{problem}, which not
only extends the corresponding results in \cite{SBP,SBH,YQM}, but completes the
investigations in \cite{PSM}. Our main result is the following:
\begin{theorem}\label{uniqueness}
The equation \eqref{problem} has at most one
self-similar very singular solution with compact support.
\end{theorem}


\begin{remark}\label{remark0} \rm
According to the main result in \cite{PSM}, there exists a (forward) self-similar
very singular solution with compact support of \eqref{problem} if and only if
$m(p-1)<q<\frac{p+Nm(p-1)}{N+1}$, and so, under which the uniqueness will be
discussed in what follows.
\end{remark}

This article is organized as follows. In Section 2, we derive some properties
of self-similar very singular solutions of \eqref{problem}. In particular,
we prove the monotonicity of self-similar solutions with
respect to initial data in the sense that two positive orbits do not intersect
each other. Finally, the proof of Theorem \ref{uniqueness} is given in Section 3.


\section{Preliminaries}

In this section, we derive some properties of self-similar very singular solutions
of \eqref{problem} which are important for the proof of Theorem \ref{uniqueness}.
Owing to the homogeneity of \eqref{problem}, we actually look for a (forward)
self-similar very singular solution $u$ to \eqref{problem} of the form
\begin{equation}\label{self-similarsolution}
u(x,t)=(\frac{\alpha}{t})^{\alpha}f(r)
\end{equation}
where $r=|x|(\frac{\alpha}{t})^{\alpha\beta}$, for some profile
$f$ and exponents $\alpha$ and $\beta$ to be determined. Inserting this setting
in \eqref{problem} gives the vales of $\alpha$ and $\beta$
$$
\alpha=\frac{p-q}{p(q-1)-q(m(p-1)-1)}>0, \quad \beta=\frac{q-m(p-1)}{p-q}>0,
$$
and implies that the profile $f$ is a solution of the ordinary differential equation
\begin{equation}\label{ODE}
\big(|(f^m)'|^{p-2}(f^m)'\big)'+\frac{n-1}{r}|(f^m)'|^{p-2}(f^m)'+\beta rf'+f-|f'|^q=0, \ r>0
\end{equation}
with
\begin{equation}\label{zeroderivative}
(f^m)'(0)=0, \quad f(0)=a,
\end{equation}
where $a$ is a positive constant to be determined. Note that condition \eqref{singular}
is equivalent to, if $u$ is given by \eqref{self-similarsolution},
\begin{equation}\label{condition}
\lim_{r\to\infty}r^{1/{\beta}}f(r)=0.
\end{equation}
In addition, it is easy to see that if $N\beta<1$ (i.e. $q<\frac{p+Nm(p-1)}{N+1}$)
and the solution $f$ of \eqref{ODE} satisfies \eqref{zeroderivative}--\eqref{condition},
then $u$ given explicitly by \eqref{self-similarsolution} satisfies \eqref{verysingular}
automatically. According to \cite[Lemma 3.1]{PSM}, however, the condition \eqref{condition}
does not hold if $N\beta\ge1$, that is, there is no self-similar singular solution.



Let $z=f^m, a^m=b$, then the problem \eqref{ODE}--\eqref{zeroderivative}
is replaced by the following problem with respect to $z$,
\begin{equation}\label{problem3}
\begin{gathered}
(|z'|^{p-2}z')'+\frac{n-1}{r}|z'|^{p-2}z'+\beta r(z^{1/m})'+z^{1/m}-|(z^{1/m})'|^q=0,  \quad r>0,\\
z(0)=b>0,\quad z'(0)=0
\end{gathered}
\end{equation}
and the condition \eqref{condition} is replaced by
\begin{equation}\label{conditionz}
\lim_{r\to\infty}r^{1/{\beta}}z^{1/m}(r)=0.
\end{equation}
By the standard theory of ordinary differential equations, the local existence
and uniqueness of solution for \eqref{problem3} is easy to be obtained.
Let $z(\cdot;b)$ be the solution of \eqref{problem3} and define
$$
R(b):=\sup\{r_0>0:z(r;b)>0,\quad r\in[0,r_0)\}.
$$
In the sequel, where there is no confusion, we will omit $b$ and let $z=z(\cdot;b)$.

Before going further, we present some basic properties of $z$ which
have already been proved in \cite{PSM}.

\begin{lemma}\label{lemma}
Assume that $\alpha>0$, $\beta>0$ and $b>0$. Let $z$ be a solution to
\eqref{problem3} with support $[0,R(b))$. Then
\begin{itemize}
\item[(i)] $z'(r)<0$ in $(0,R(b))$;
\item[(ii)]  $\lim_{r\to R(b)^-}z(r)=0$;
\item[(iii)] $\lim_{r\to R(b)^-}z'(r)=0$ when $R(b)=\infty$.
\end{itemize}
\end{lemma}


Next, we prove the monotonicity of solutions of \eqref{problem3} with
respect to $b$ in the sense that two positive orbits do not intersect each other.

\begin{lemma}\label{monotonicity}
Assume that  $\alpha, \beta>0$, $z_i$ are solutions of \eqref{problem3} on
$[0,R_i)$ with initial data $z_i(0)=b_i, i=1,2$ and $\min\{R_1, R_2\}<\infty$, where $[0,R_i)$ denotes the maximal
existence interval of $z_i$ and the $R_i>0$ are possibly infinity. If $b_1<b_2$, then
$$
z_1(r)<z_2(r), \quad\text{for all} 0\le r\le R:=\min\{R_1, R_2\}.
$$
\end{lemma}

\begin{proof}
Suppose contrarily that there exists $R_0\in[0,R]$ such that $z_1(r)<z_2(r)$
for $r\in[0,R_0)$ and $z_1(R_0)=z_2(R_0)$. We define
$$
g_k(r):=k^{-mp/(m(p-1)-1)}z_1(kr), \quad r\in[0,R_1/k)
$$
for $k>0$ and then $g_k(r)$ solves
\begin{equation}\label{1}
\begin{aligned}
&(|g'_k|^{p-2}g'_k)'+\frac{N-1}{r}|g'_k|^{p-2}g'_k+\beta r(g_k^{1/m})'\\
&+g_k^{1/m}-k^{\frac{q(p+1-m(p-1))-p}{m(p-1)-1}}|(g_k^{1/m})'|^q=0
\end{aligned}
\end{equation}
Note that $g_k$ is strictly decreasing with respect to $k$, and $\lim_{k\to0}g_k(r)=+\infty$
for any $r\in[0,R]$, then there exists a small $k_0>0$ such that
$$
z_2(r)<g_k(r)\quad\text{ for any $r\in[0,R]$ and $k\in[0,k_0]$}.
$$
Define
$$
\tau:=\sup\big\{k_0>0;z_2(r)<g_k(r)\text{ for $r\in[0,R_0]$ and  
$k\in[0,k_0]$}\big\},
$$
we see that $\tau<1$, $g_\tau(r)\ge z_2(r)$ and there exists $r_0\in[0,R_0]$ such that
$g_\tau(r_0)=z_2(r_0)$.


If $r_0=R_0$, then
$$
g_\tau(R_0)=\tau^{-mp/(m(p-1)-1)}z_1(\tau R_0)=z_2(R_0).
$$
Since $z_1(R_0)=z_2(R_0)$ and $g_\tau$ is strictly decreasing with respect
to $\tau$, we conclude that $\tau=1$ and this contradicts to the hypothesis;
while if $r_0\in(0,R_0)$,  we have
$$
g_\tau(r_0)=z_2(r_0), \quad g'_\tau(r_0)=z'_2(r_0), \quad
g''_\tau(r_0)\ge z''_2(r_0).
$$
Since that $\alpha>0$, that is,
$p>q>\frac{p}{p+1-m(p-1)}$,  we deduce from \eqref{problem3} that
\begin{align*}
\lefteqn{(|g'_\tau|^{p-2}g'_\tau)'(r_0)-(|z_2'|^{p-2}z_2')'(r_0)}\\
&  =   (\tau^{\frac{q(p+1-m(p-1))-p}{m(p-1)-1}}-1)|(z_2^{1/m})'|^q\\
  <   0
\end{align*}
which contradicts $g''_\tau(r_0)\ge z''_2(r_0)$.

Thus, $r_0=0$ and $g_\tau(r)>z_2(r)$ for $r\in(0,R_0]$. Then we have
$$
g_\tau(0)=z_2(0),
$$
$$
\lim_{r\to0^+}g_\tau'(r)=\lim_{r\to0^+}z_2'(r)=0,
$$
and
$$
\lim_{r\to0^+}(|g'_\tau|^{p-2}g'_\tau)'(r)=\lim_{r\to0^+}(|z_2'|^{p-2}z_2')'(r)=-\frac{b_2^{1/m}}{N}<0.
$$
By continuity there exists $\varepsilon>0$ such that
$$
g_\tau(r)>z_2(r)>0\quad\text{and}\quad  0>g'_\tau(r)>z'_2(r)
$$
for $r\in(0,\varepsilon)$. Further, we can choose $\varepsilon>0$ small enough
such that the following inequalities hold for $r\in(0,\varepsilon)$,
\begin{gather*}
(|g'_\tau|^{p-2}g'_\tau)'(r)-(|z_2'|^{p-2}z_2')'(r) >0,\\
(g^{1/m}_\tau)'(r)-(z^{1/m}_2)'(r) >0,\\
|(z_2^{1/m})'|^q-\tau^{\frac{q(p+1-m(p-1))-p}{m(p-1)-1}}|(g_{\tau}^{1/m})'|^q >0.
\end{gather*}
Thus, we obtain that
\begin{align*}
0 &=\Big((|g'_\tau|^{p-2}g'_\tau)'-(|z_2'|^{p-2}z_2')'\Big)(r)+\frac{N-1}{r}
         \Big(|g'_\tau|^{p-2}g'_\tau-|z_2'|^{p-2}z_2'\Big)(r)  \\
&\quad  +\beta r\Big((g^{1/m}_\tau)'-(z^{1/m}_2)'\Big)(r)+\Big((g^{1/m}_\tau)-(z^{1/m}_2)\Big)(r)&
    \\
&\quad +\Big(|(z_2^{1/m})'|^q-\tau^{\frac{q(p+1-m(p-1))-p}{m(p-1)-1}}|(g_{\tau}^{1/m})'|^q\Big)(r)>0&
    \\
\end{align*}
for $r\in(0,\varepsilon)$, which is impossible.
Summing up, we completed the proof of Lemma \ref{monotonicity}.
\end{proof}


According to Lemma \ref{monotonicity}, we can define three sets for every $b>0$,
\begin{gather*}
\mathcal{A}=\{b>0;R(b)<\infty\ {\rm and}\ z'(R(b))<0\},\\
\mathcal{B}=\{b>0;R(b)<\infty\ {\rm and}\ z'(R(b))=0\},\\
\mathcal{C}=\{b>0;R(b)=\infty\ {\rm and}\ z(r)>0, r\ge0\}.
\end{gather*}
Obviously, these sets are disjoint and 
$\mathcal{A}\cup\mathcal{B}\cup\mathcal{C}=(0,\infty)$.
From \cite[Theorem 1.1]{PSM}, we have the following lemma.

\begin{lemma}\label{threesets}
Assume that $N\beta<1$, then
\begin{itemize}
\item[(i)] set $\mathcal{A}$ is nonempty and open;
\item[(ii)] set $\mathcal{B}$ is nonempty and closed, and the interface relation
\begin{equation}\label{interface}
\lim_{r\to R(b)^-}\Big(z^{\frac{m(p-1)-1}{m(p-1)}}\Big)'(r;b)
=-\frac{m(p-1)-1}{m(p-1)}(\beta R(b))^{1/(p-1)}
\end{equation}
holds if $b\in\mathcal{B}$;
\item[(iii)] set $\mathcal{C}$ is nonempty and open, and $\lim_{r\to\infty}r^{1/{\beta}}z^{1/m}(r;b)>0$ if $b\in\mathcal{C}$.
\end{itemize}
\end{lemma}


\begin{remark}\label{remark} \rm
By Lemma \ref{threesets}, it is easy to see that the solution $z(\cdot;b)$
of the problem \eqref{problem3} satisfies \eqref{conditionz} if and only if
$b\in\mathcal{B}$. That is to say, to obtain the uniqueness of self-similar
very singular solution of \eqref{problem}, it is suffice to show that the
set $\mathcal{B}$ consists only one element.
\end{remark}

\section{Proof of the Theorem \ref{uniqueness}}

We need an auxiliary lemma.
Let $z(\cdot;b)$ be a solution of \eqref{problem3}
satisfying $b\in\mathcal{B}$, then $R(b)<\infty$ and \eqref{conditionz} holds.
Denote $\xi_0=R(b)$ and define
$$
U(x,t)=k^{1/m}(\frac{\alpha}{t})^{\alpha}z^{1/m}(\xi),
$$
where $\xi=k^{-\gamma}|x|(\frac{\alpha}{t})^{\alpha\beta}$ and 
$\gamma=\frac{m(p-1)-1}{mp}$,
then
$$
\operatorname{supp}U=\big\{(x,t)\in\mathbb{R}^N\times(0,\infty);
 |x|\le\xi_0k^{\gamma}(\frac{\alpha}{t})^{-\alpha\beta}\big\}.
$$

\begin{lemma}\label{U}
For $t>0$ fixed and $\delta>0$ small enough there exists 
$\theta=\theta(\delta)\in(0,1)$
such that
$U(x,t)<U(x,t+\delta)$
for
$$
\theta \xi_0\le k^{-\gamma}|x|(\frac{\alpha}{t})^{\alpha\beta}\le \xi_0.
$$
Moreover, we have
$$
\lim_{\delta\to0}\theta(\delta)=\theta_0\in(0,1).
$$
\end{lemma}

\begin{proof}
It suffices to prove the existence of $\xi_1\in(0,\xi_0)$ such that
\begin{equation}\label{2}
\big(\frac{\alpha}{t}\big)^{\alpha}z^{1/m}(\xi)
<\big(\frac{\alpha}{t+\delta}\big)^{\alpha}
z^{1/m}\Big(\xi\big(1+\frac{\delta}{t}\big)^{-\alpha\beta}\Big),\quad 
\xi_1\le\xi\le\xi_0.
\end{equation}
That is,
$$
z^\lambda(\xi)<\big(1+\frac{\delta}{t}\big)^{-\alpha m\lambda}z^\lambda
\Big(\xi\big(1+\frac{\delta}{t}\big)^{-\alpha\beta}\Big),\quad \xi_1\le\xi\le\xi_0,
$$
where $\lambda=\frac{m(p-1)-1}{m(p-1)}$.
Denote $\varepsilon=\delta/t$, then we need prove that there exists the smallest
$\xi_1\le\xi_0$ such that
\begin{equation}\label{3}
z^\lambda(\xi)<(1+\varepsilon)^{-\alpha m\lambda}z^\lambda
\big(\xi(1+\varepsilon)^{-\alpha\beta}\big)
\end{equation}
holds on $[\xi_1, \xi_0]$. Note that
\begin{align*}
& (1+\varepsilon)^{-\alpha m\lambda}
z^\lambda(\xi(1+\varepsilon)^{-\alpha\beta})\\
&  =  z^\lambda(\xi)-\alpha\lambda m\varepsilon z^\lambda(\xi)
-\alpha\beta\varepsilon\xi(z^\lambda)'(\xi)+O(\varepsilon^2),
\end{align*}
so \eqref{3} reads 
\begin{equation}\label{4}
z^\lambda(\xi)<-\frac{\beta\xi}{m\lambda}(z^\lambda)'(\xi)+O(\varepsilon).
\end{equation}
Recalling \eqref{conditionz}, the set of $\eta\in(0,\xi_0)$ such that
$$
z^\lambda(\eta)=-\frac{\beta\xi}{m\lambda}(z^\lambda)'(\eta)
$$
is not empty. Let $\tilde{\xi}$ be the least upper bound of this set, 
then $0<\tilde{\xi}<\xi_0$
and
$$
z^\lambda(\xi)=-\frac{\beta\xi}{m\lambda}(z^\lambda)'(\xi)
$$
for $\tilde{\xi}<\xi<\xi_0$.
For $\varepsilon>0$ small enough we can deduce the existence of 
$\xi_1\in(\tilde{\xi}, \xi_0)$
such that \eqref{4} hold on $[\xi_1, \xi_0]$.
 Denote $\theta=\theta(\delta)=\xi_1/\xi_0$
and $\theta_0=\tilde{\xi}/\xi_0$, it is obvious that
$$
\lim_{\delta\to0}\xi_1=\tilde{\xi} \quad\text{and}\quad
 \lim_{\delta\to0}\theta(\delta)=\theta_0\in(0,1).
$$
The proof is complete.
\end{proof}


Now we give the proof of the main result.

\begin{proof}[Proof of Theorem \ref{uniqueness}]
By Remark \ref{remark}, it is suffice to show that the set 
$\mathcal{B}$ consists only one element.
 We give the proof by contradiction. Without loss of generality, assume that $z$ and
$Z$ are two solutions of \eqref{problem3} satisfying $z(0),Z(0)\in\mathcal{B}$ 
and $z(0)<Z(0)$.
Denote
$$
R_1:=\inf\{r\ge0:z(r)=0\},\quad 
R_2:=\inf\{r\ge0:Z(r)=0\}.
$$
By Lemma \ref{monotonicity}, we
obtain $R_1<R_2$ and $z(r)<Z(r)$ for $r\in[0,R_1]$. We define
$$
z_k(r)=kz(k^{-\gamma}r),
$$
where $\gamma=\frac{m(p-1)-1}{mp}$. Then $z_k$ will be larger than $Z$ on $[0,R_2]$
for sufficiently large $k$. We now define
$$
\tau=\inf\left\{k\ge1; z_k(r)\ge Z(r), r\in[0,R_2]\right\}.
$$
Obviously, if $\tau\le1$, then $z(r)=z_1(r)\ge Z(r)$ for $r\in[0,R_2]$, which
contradicts the hypothesis. Thus, we suppose that $\tau>1$ in the following proof.
By the definition of $\tau$, $z_{\tau}(r)$ must touch $Z(r)$ at $r_0\in[0,R_2]$
from the above, so we divide the next proof into two cases: $r_0\in[0,R_2)$ 
and $r_0=R_2$.
\smallskip

\noindent\textbf{Case (i).} 
If $z_{\tau}(r)$ touch $Z(r)$ at $r_0\in[0,R_2)$, by the
similar proof to that of Proposition \ref{monotonicity}, we will derive a
contradiction, so $z_{\tau}(r)$ can not touch $Z(r)$ at $r_0\in[0,R_2)$.
\smallskip

\noindent\textbf{Case (ii).}
 We firstly define the functions $u, U_{\tau}$
corresponding to $Z$ and $z_\tau$ by
\begin{gather*}
u(x,t):=(\frac{\alpha}{t})^{\alpha}Z^{1/m}(r), \\
U_\tau(x,t):=(\frac{\alpha}{t})^{\alpha}z_\tau^{1/m}(r)=\tau^{1/m}
(\frac{\alpha}{t})^{\alpha}z^{1/m}(\tau^{-\gamma}r).
\end{gather*}
Then $u$ is a solution of \eqref{problem} and $U_\tau$ is a supersolution. Indeed,
a straightforward computation shows that
$$
\frac{\partial U_\tau}{\partial t}
-\operatorname{div}(|\nabla U_\tau^m|^{p-2}\nabla U_\tau^m)
+|\nabla U_\tau|^{q}=(1-\tau^{\frac{q(m(p-1)-p-1)+p}{mp}}|\nabla U_\tau|^{q})\ge0.
$$
By Lemma \ref{U}, for sufficiently small $\delta>0$, there exist
$\theta_0, \theta(\delta)\in(0,1)$ such that
$$
U_\tau(x,1)<U_\tau(x,1+\delta)
$$
for $\theta(\delta)R_2\tau^{\gamma}\le|x|<R_2\tau^{\gamma}(1+\delta)^{\beta}$ and
$$
\lim_{\delta\to0}\theta(\delta)=\theta_0.
$$
Combining this with $z_{\tau}(r)\ge Z(r)$ for $r\in[0,R_2]$, we obtain that
\begin{equation}\label{onlyone1}
u(x,1)<U_\tau(x,1+\delta)
\end{equation}
for $\theta(\delta)R_2\tau^{\gamma}\le|x|<R_2\tau^{\gamma}(1+\delta)^{\beta}$.

On the other hand, as previously proved, $z_{\tau}(r)$ can not touch $Z(r)$ 
at $r_0\in[0,R_2)$, which implies for any fixed $\varepsilon_1>0$, 
there exists $\kappa\in(0,1)$ such that
$$
Z(|x|)<\kappa z_\tau(|x|),\quad |x|<(1-\varepsilon_1)R_2\tau^{\gamma};
$$
that is,
\begin{equation}\label{onlyone2}
u(x,1)<\kappa U_\tau(x,1),\quad |x|<(1-\varepsilon_1)R_2\tau^{\gamma}.
\end{equation}
Now we choose sufficiently small $\varepsilon_1>0$ and $\delta_0>0$ such that
$$
\theta(\delta)<1-\varepsilon_1
$$
for $\delta\in(0,\delta_0)$ and
$$
\theta_0<1-\varepsilon_1.
$$
So we obtain that
\begin{equation}\label{onlyone5}
\theta(\delta)R_2\tau^{\gamma}<(1-\varepsilon_1)R_2\tau^{\gamma},
\quad \delta\in[0,\delta_0).
\end{equation}
By continuity of $U_\tau$, there exists $\delta_1\in(0,\delta_0)$
such that
$$
\kappa U_\tau(x,1)\le U_\tau(x,1+\delta)
$$
for $\delta\in(0,\delta_1)$ and $|x|<(1-\varepsilon_1)R_2\tau^{\gamma}$.
Combining with \eqref{onlyone2}, we have
\begin{equation}\label{onlyone3}
u(x,1)<U_\tau(x,1+\delta)
\end{equation}
for $\delta\in(0,\delta_1)$ and $|x|<(1-\varepsilon_1)R_2\tau^{\gamma}$.
Thus, combining \eqref{onlyone1}, \eqref{onlyone5} and \eqref{onlyone3},
for any $x\in\mathbb{R}^N$ we have
$$
u(x,1)<U_\tau(x,1+\delta)=\tau^{1/m}(\frac{\alpha}{t+\delta})^{\alpha}
z^{1/m}(\tau^{-\gamma}|x|(\frac{\alpha}{t+\delta})^{\alpha\beta})
$$
Furthermore, from the continuity with respect to $\tau$, there exists 
$\tau_1\in(0,\tau)$ such that
$$
u(x,1)\le U_{\tau_1}(x,1+\delta)
$$
for $x\in\mathbb{R}^N$. By comparison we obtain $u(x,t)\le U_{\tau_1}(x,t+\delta)$;
 that is,
\begin{equation}\label{onlyone4}
(\frac{\alpha}{t})^{\alpha}Z^{1/m}
\Big(|x|(\frac{\alpha}{t})^{\alpha\beta}\Big)
\leq \tau_1^{1/m}\big(\frac{\alpha}{t+\delta}\Big)^{\alpha}z^{1/m}
Big(\tau_1^{-\gamma}|x|
(\frac{\alpha}{t+\delta})^{\alpha\beta}\Big)
\end{equation}
for any $(x,t)\in\mathbb{R}^N\times[1,\infty)$. 
Rewriting \eqref{onlyone4} in the form
$$
Z^{1/m}(r)\le\tau_1^{1/m}\big(\frac{t}{t+\delta}\big)^{\alpha}
z^{1/m}\Big(\tau_1^{-\gamma}r\big(\frac{t}{t+\delta}\big)^{\alpha\beta}\Big)
$$
and letting $t\to\infty$, we have
$$
Z(r)\le\tau_1z(\tau_1^{-\gamma}r)=z_{\tau_1}(r)
$$
which contradicts the fact that $\tau$ is the smallest constant with that property.
Thus, $z_\tau$ does not reach $Z$ at $r_0=R_0$ and we may conclude that $\tau\le1$
but it is impossible. Summing up, we completed the proof of Theorem \ref{uniqueness}.
\end{proof}



\subsection*{Acknowledgments}
This work is Supported by NNSFC (No. 11071099), and
by the Scientific Research Foundation of Graduate School of
South China Normal University.


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\end{document}