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\begin{document}

\title[Nonlinear parabolic problems]{Nonlinear parabolic problems on 
manifolds, and a nonexistence result for the  
noncompact Yamabe problem}
\author{Qi S. Zhang}
\address{Department of Mathematics, University of Missouri, Columbia, MO 65211}
\email{sz@mumathnx6.math.missouri.edu}
\begin{abstract}We study the Cauchy problem for the  
semilinear parabolic equations $\Delta u - R u 
 + 
u^{p} - u_{t} =0$ on $\mathbf{M}^{n} \times (0, \infty )$  with initial 
value $u_{0} \ge 0$,
where $\mathbf{M}^{n}$ is a Riemannian manifold including the ones  with 
nonnegative Ricci curvature. 
 In the Euclidean case and when $R=0$, it is well known that $1+ 
\frac{2}{n}$ is the critical exponent, i.e., if 
 $p > 1 + \frac{2}{n}$ and $u_{0}$ 
is smaller than a small Gaussian, then 
the Cauchy problem has global positive solutions, and if $p<1+\frac{2}{n}$, 
then 
all positive solutions blow up in finite time.  
In this paper, we show that  on certain Riemannian  manifolds, the above 
equation with certain conditions on $R$ also has a critical exponent. More 
importantly, we reveal an explicit relation between the size of the critical 
exponent and geometric properties  
such as the growth rate of geodesic balls.   To achieve the results we 
introduce a new estimate 
for related heat kernels. 

As an application, we show that the well-known noncompact Yamabe problem 
(of prescribing constant positive scalar curvature) on a manifold with 
nonnegative Ricci curvature cannot be solved if the existing scalar 
curvature decays ``too fast'' and the volume of geodesic balls does not 
increase ``fast enough''. We also find some complete manifolds with positive 
scalar curvature, which are conformal to complete manifolds with positive 
constant and with zero scalar curvatures. This is a new phenomenon which 
does not happen in the compact case.  
\end{abstract}
\date{February 19, 1997}
\subjclass{Primary 35K55; Secondary 58G03}
\commby{Richard Schoen}
\maketitle

\section{Introduction }\label{sec:1}

We shall announce new results in the study of the global existence  and blow up 
of the  following semilinear parabolic Cauchy problem: 
\begin{equation}\label{eq:1.1}
\begin{cases}H u \equiv H_{0} u + u^{p} = \Delta u - R u- 
\partial _{t} u 
 +  u^{p}  = 0   \quad \text{in} \quad \mathbf{M}^{n}\times (0, \infty ),\\
u(x, 0) = u_{0}(x) \ge 0 \quad \text{in} \quad \mathbf{M}^{n},
\end{cases}
%\tag{1.1}
\end{equation}
where $\mathbf{M}^{n}$ with $n \ge 3$ is a noncompact 
complete Riemannian manifold,  $\Delta $ is the Laplace-Beltrami operator, 
and $R=R(x)$ is a bounded function.


Throughout the paper we make the following assumptions. 


(i) There are positive constants $b$,  $C$, and $K$   such
that
\begin{equation}\label{eq:1.2}
   |B(x,  r)| \le C   r^{n}; \quad |B(x, 2r)| \le C 2^{n} 
|B(x, r)|, \ r>0; 
\quad \text{Ricci} \ge -K.
%\tag{1.2}
\end{equation}

(ii) $G$, the  fundamental solution 
of the linear operator $H_{0}= \Delta - R - \partial _{t}  $ in \eqref{eq:1.1},  has 
global Gaussian upper bound, i.e., 
\begin{equation}\label{eq:1.3}
0 \le G(x,t;y,s) \le \frac{C}{|B(x, (t-s)^{1/2})|} e^{- b
\frac{d(x,y)^{2}}{t-s}},
%\tag{1.3}
\end{equation}
for 
all $x, y \in \mathbf{M}^{n}$ and  all $t>s$.

(iii) When $d(x, y)^{2} \le t-s$, $G$ satisfies 
\begin{equation}
G(x, t; y, s) \ge \min \left\{ \frac{1}{C |B(x, (t-s)^{1/2})|}, 
\frac{1}{C |B(y, (t-s)^{1/2})|}\right\}.
\tag{1.3$'$}\label{eq:1.3prime}
\end{equation}


If one  assumes that the Ricci curvature of $\mathbf{M}^{n}$ is nonnegative 
and $R=0$, it is well known that that above properties hold (see \cite{LY}). 
More general conditions on $\mathbf{M}^{n}$ (without the nonnegativity of 
Ricci) that support \eqref{eq:1.2}, \eqref{eq:1.3}, and \eqref{eq:1.3prime} can be found in 
\cite{G}. When $R$ is 
not identically zero, \eqref{eq:1.3} and \eqref{eq:1.3prime} pose certain restrictions on $R$ as 
indicated in Corollary \ref{cor2.3} below.

Equation \eqref{eq:1.1} with $R=0$ contains the following important special cases. 
When the Riemannian metric  is just the Euclidean metric, \eqref{eq:1.1} becomes  the 
semilinear parabolic equation 
which 
has been  studied by many authors. In the paper \cite{Fu}, 
Fujita proved the following results:

(a) when  
$10$, 
problem \eqref{eq:1.1} possesses no global positive solutions;

(b) when 
  $p>1+\frac{2}{n}$ and $u_{0}$ is smaller than a small Gaussian, then \eqref{eq:1.1} 
has global positive solutions. So $1+\frac{2}{n}$ is the critical exponent. 

For a rich literature on the subsequent development of the 
topic, we refer the reader to the survey paper \cite{Le}.

In recent years many authors have undertaken 
the research on  
semilinear elliptic operators on manifolds, including the well-known Yamabe 
problem (see \cite{Sc} and \cite{Yau}).  In contrast, not much has been done 
for their semilinear parabolic counterpart. It looks more imperative to 
fill this gap when one takes into account the tremendous literature about 
the heat kernel of a complete Riemannian manifold, which is still a linear 
theory. The study of Ricci flows also leads to semilinear parabolic 
problems (see \cite{H} and \cite{Sh}).  We shall need some new techniques to 
study when blow up of solutions occur, and when 
global positive solutions exist for equation \eqref{eq:1.1}.  
 In fact the prevailing methods in treating semilinear problems are 
variational  and comparison method. The variational approach seems hard to 
apply when the manifold is noncompact. The comparison method also faces 
some difficulties on curved manifolds because the intrinsic operators 
are not of constant coefficients. 

The method we are using 
 is based on some new inequalities involving the heat kernels. We are able 
to reveal an explicit relation between the size of the critical exponent and 
geometric properties of the manifold such as the growth rate of geodesic 
balls (see Theorem \ref{thmb}). 

It is interesting to note that Theorem \ref{thmb} immediately leads to a 
nonexistence result for the well-known noncompact Yamabe problem of 
prescribing positive constant scalar curvatures, i.e., whether the following 
elliptic problem has positive solutions on $\mathbf{M}^{n}$ (see Theorem
\ref{thmc}):
\begin{equation}\label{eq:1.4}
\Delta u - \frac{n-2}{4(n-1)}R u
 +  u^{(n+2)/(n-2)}  = 0.
%\tag{1.4}
\end{equation}
This problem has been posed by J. Kazdan \cite{K} and S. T. 
Yau \cite{Yau}. The compact 
version of the problem was proposed by Yamabe \cite{Yam}, proved by 
Trudinger \cite{Tr} and Aubin \cite{Au} in some cases, and eventually 
proved by R. Schoen \cite{Sc} completely. In the noncompact case, Aviles 
and McOwen \cite{AM} obtained some existence result for the problem of 
prescribing constant negative scalar curvature. However, as far as we know, 
there has been no result on the problem of prescribing constant positive 
scalar curvature, which was termed more difficult in \cite{AM}. 

When $\mathbf{M}^{n}$ is $\mathbf{R}^{n}$ with the Euclidean metric, then 
problem \eqref{eq:1.4} becomes $\Delta u + u^{(n+2)/(n-2)}=0$, which does have 
positive solutions (see \cite{Ni})
\begin{equation*}u_{\lambda }(x) = \frac{[n(n-2) \lambda 
^{2}]^{(n-2)/4}}{(\lambda ^{2} + |x|^{2})^{(n-2)/2}}, \qquad \lambda >0.
\end{equation*}
So it is reasonable to expect that \eqref{eq:1.4} has a solution. 
However, Theorem \ref{thmc} below asserts that unlike the compact Yamabe problem, 
problem \eqref{eq:1.4} cannot be solved in general. In fact, if the existing scalar 
curvature decays ``too fast" and the volume of geodesic balls does not 
increase ``fast enough", then \eqref{eq:1.4} has no solution.


Before stating the main results explicitly we list some notation.
  
\begin{definition1} A function $u=u(x, t)$ such that 
$u \in L^{2}_{loc}(\mathbf{M}^{n} \times (0, \infty ))$ is called a solution 
of 
\eqref{eq:1.1} if
\begin{equation*}u(x, t) = \int _{\mathbf{M}^{n}} G(x, t; y, 0) u_{0}(y)\, dy + 
\int ^{t}_{0}\int _{\mathbf{M}^{n}} G(x, t; y, s) u^{p}(y, s)\, dy\,ds
\end{equation*}
for all $(x, t) \in \mathbf{M}^{n} \times (0, \infty )$. 
\end{definition1}


$G=G(x, t; y, s)$ will denote the fundamental solution of the  
linear operator $H_{0}$ in \eqref{eq:1.1}.
For any $c>0$, we write   
\begin{equation*}G_{c}(x, t;y, s)=
\begin{cases}\frac{1}{|B(x, (t-s)^{1/2})|}\exp (-c \frac{d(x, y)^{2}}{t-s}), 
\qquad t>s,\\
0, \qquad t0$;  
we write
\begin{equation}\label{eq:1.5}
h_{a}(x, t) = \int _{\mathbf{M}^{n}}G_{a}(x,t;y,0) 
u_{0}(y)\,dy;
%\tag{1.5}
\end{equation}
\begin{equation}\label{eq:1.6}
h(x, t) = \int _{\mathbf{M}^{n}}G(x,t;y,0) u_{0}(y)\,dy.
%\tag{1.6}
\end{equation}
Given $V=V(x, t)$ and $c>0$, we introduce the notation 
\begin{equation}\label{eq:1.7}
\begin{split}
N_{c, \infty }(V)& \equiv \operatorname{sup}_{x \in \mathbf{M}^{n},t>0}\int ^{\infty 
}_{-\infty }\int _{\mathbf{M}^{n}}
|V(y,s)|
G_{c}(x,t;y,s)\, dy\,ds\\
 & \quad+  \text{sup}_{y\in \mathbf{M}^{n},s>0}\int ^{\infty }_{\infty }\int _{
\mathbf{M}^{n}}|V(x,t)|
G_{c}(x,t;y,s)\, dx\,dt.
\end{split}%\tag{1.7} 
\end{equation}
We note that $N_{c, \infty }(V)$ may be infinite for some 
$V$. However, the fact that $N_{c, \infty }(V)$ is finite for some specific 
functions will play a key role in the proof of the theorems. 


\section{Main results}\label{sec:2}

The main results of the paper are the next three theorems.    


\begin{theorem}\label{thma}
Suppose $R=0$ and \eqref{eq:1.2}, \eqref{eq:1.3}, and \eqref{eq:1.3prime}
hold; the critical exponent of 
\eqref{eq:1.1} is $p^{*}= 1+ \frac{1}{s^{*}}$, where
\begin{equation*}\begin{split}
s^{*}= \sup \{s \mid &\ \lim _{t \rightarrow \infty }\sup t^{s} \|W(\cdot, 
t)\|_{L^{\infty }} < \infty ,\\
 & \text{for a nonnegative and nontrivial } \ W  \ \text{such that} \ 
H_{0} W = 0 \}.
\end{split}\end{equation*}\end{theorem}


\begin{remark}\label{rem2.1} Theorem \ref{thma}, which establishes the existence of the critical 
exponent, can be proved by following the proof of Theorem 1 in 
\cite{Me}. However, this theorem does not provide any estimate of the size 
of the exponent. It does not even tell whether $p^{*}$ is closer to $1$ or 
$\infty $. Our main concern is therefore to find an explicit relation 
between the critical exponent and geometrical properties of the manifold. 
This is done in the next 
\end{remark}


\begin{theorem}\label{thmb} 
Let $\mathbf{M}^{n}$ be any Riemannian manifold and $R=R(x)$ any bounded
functions such that \eqref{eq:1.2}, \eqref{eq:1.3}, and \eqref{eq:1.3prime} 
hold; then the following
conclusions are true.

(a) Suppose, for $t>s \ge 0$,
\begin{equation*}\sup _{x \in \mathbf{M}^{n}, t}\int ^{\infty }_{r_{0}}\!\int 
_{\mathbf{M}^{n}} \frac{G_{c}(x, t; y, s)}{|B(y, s^{1/2})|^{p-1}} \,dy\,ds + 
\sup _{y \in \mathbf{M}^{n}, s}\int ^{\infty }_{r_{0}}\!\int _{\mathbf{M}^{n}} 
\frac{G_{c}(x, t; y, s)}{|B(x, t^{1/2})|^{p-1}}\, dx\,dt < \infty 
\end{equation*}
for some $r_{0}>0$ and a suitable $c>0$; then \eqref{eq:1.1} has 
global positive solutions for  some $u_{0} \ge 0$. 

(b) Suppose for some $x_{0} \in \mathbf{M}^{n}$,  
\begin{equation*}\lim _{r \rightarrow \infty } \inf \frac{|B(x_{0}, 
r)|}{r^{\alpha }}
< \infty;\end{equation*}
then \eqref{eq:1.1} has no global positive solutions for any 
$p<1+ \frac{2}{\alpha }$ and any $u_{0} \ge 0$. \end{theorem}


To exemplify the conditions in Theorem \ref{thmb}, we provide two corollaries.


\begin{corollary}\label{cor2.1} Under the same assumptions as in 
Theorem \ref{thmb}, suppose
\begin{equation*}\int ^{\infty }_{r_{0}} \sup _{x \in \mathbf{M}^{n}} 
\frac{1}{|B(x, r^{1/2})|^{p-1}}\, dr< \infty \end{equation*}
for some $r_{0}>0$; then \eqref{eq:1.1} has global positive solutions for some  $u_{0}
\ge 0$. In particular, if, for an $\alpha >0$,   $\inf _{x \in \mathbf{M}^{n}} 
|B(x,
r)| \ge C r^{\alpha }$  when $r$ is sufficiently large, then  for  $p >1+
\frac{2}{\alpha }$, \eqref{eq:1.1} has global positive solutions for some 
$u_{0}$.\end{corollary}


In the next corollary, we show that if $\mathbf{M}^{n}$ has bounded geometry 
in the sense of E. B. Davies (see p. 172 in \cite{D}), which means the 
existence of a function $b(r)$ and $c>0$ such that
\begin{equation}\label{eq:2.1}
c^{-1} b(r) \le |B(x, r)| \le c b(r)
%\tag{2.1}
\end{equation}
for all $x \in \mathbf{M}^{n}$ and $r>0$, then the critical 
exponent of \eqref{eq:1.1}  can be explicitly determined.


\begin{corollary}\label{cor2.2} 
Suppose, in addition to the assumptions 
in Theorem \ref{thmb},  \eqref{eq:2.1} holds; then 
$p^{*}$, the critical exponent of \eqref{eq:1.1}, is given by $p^{*} = 1+ 
\frac{2}{\alpha ^{*}}$, where
\begin{equation*}\alpha ^{*} = \inf \left\{ \alpha >0 \Bigm| \lim _{r \rightarrow 
\infty } \inf \frac{|B(x, r)|}{r^{\alpha }}
< \infty , \ x \in \mathbf{M}^{n}\right\}.
\end{equation*}\end{corollary}


\begin{remark}\label{rem2.2} By \eqref{eq:2.1}, the above number $\alpha ^{*}$ is independent of 
the choice of $x \in \mathbf{M}^{n}$. 
\end{remark}


\begin{remark}\label{rem2.3} 
Under the assumptions of part (a) of Theorem \ref{thmb}, \eqref{eq:1.1} has global positive 
solutions if $u_{0} \ge 0$ satisfies
\begin{equation*}u_{0} \in C^{2}(\mathbf{M}^{n}), 
\quad \lim _{d(x, 0) \rightarrow \infty } u_{0}(x)=0, \quad \text{and} 
\quad \|u_{0}\|_{L^{\infty }(\mathbf{M}^{n})} + 
\|u_{0}\|_{L^{1}(\mathbf{M}^{n})} \le b_{0},
\end{equation*}
where $b_{0}$ is a sufficiently small number and $0$ is a 
point in $\mathbf{M}^{n}$. This result is new even in the Euclidean case.
\end{remark}


Next we turn our attention to  the noncompact Yamabe problem of prescribing 
positive scalar curvatures. 

  
\begin{theorem}\label{thmc} Suppose \eqref{eq:1.2}, \eqref{eq:1.3}, and
\eqref{eq:1.3prime} hold and 
\begin{equation}\label{eq:2.2}
\lim _{r \rightarrow \infty } \inf \frac{|B(x_{0}, 
r)|}{r^{\alpha }}< \infty %\tag{2.2}
\end{equation}
for some $x_{0}$ and $\alpha < (n-2)/2$; then the Yamabe 
problem \eqref{eq:1.4} has 
no solutions. \end{theorem}


It is important to know what conditions should be imposed on the Ricci 
curvature and the scalar curvature $R$ so that the conditions of Theorem
\ref{thmc} be met. To this end we have


\begin{corollary}\label{cor2.3} (a) Suppose the Ricci curvature of
$\mathbf{M}^{n}$ is nonnegative, and for a suitable $c>0$, $N_{c, \infty 
}(R)$ is
sufficiently small; then \eqref{eq:1.2}, \eqref{eq:1.3}, 
and \eqref{eq:1.3prime} hold. Hence the Yamabe 
problem
has no solutions if \eqref{eq:2.2} holds. 

(b) In particular, suppose $\mathbf{M}^{n}$ is a manifold with nonnegative 
Ricci
curvature and  $|B(x, r)| \sim r^{\alpha }$ for $2<\alpha <(n-2)/2$ and large
$r>0$. Then if  
\begin{equation*}0 \le R(x) \le \epsilon / (1+ d^{2+\delta 
}(x, 0)) \end{equation*}
for a
sufficiently small $\epsilon $ and a $\delta >0$, then the Yamabe problem 
\eqref{eq:1.4}
has no solution. Here $0$ is a point in $\mathbf{M}^{n}$. An example is 
$\mathbf{M}^{9}= \mathbf{R}^{3} \times S^{1} \times \cdots \times S^{1}$ with the
metric tensor being the direct product of those usual ones on $\mathbf{R}^{3}$
and $S^{1}$.\end{corollary}


\begin{remark}\label{rem2.4} In the above example, we can just change the metric in
$\mathbf{R}^{3}$ suitably so that the scalar curvature of $\mathbf{M}^{9}$ be
positive but \eqref{eq:1.4} still have no solution. Indeed, by part (b) of the 
corollary,
any metric $g'$ on $\mathbf{R}^{3}$ satisfying the next three conditions can 
be a
choice: (i) $|B(x', r)| \sim r^{3}$, $x' \in \mathbf{R}^{3}$; (ii) Ricci is
nonnegative; (iii) the scalar curvature $R' \le \epsilon /(1 +
d^{2+\delta }(x', 0))$ for $x' \in \mathbf{R}^{3}$, $\delta >0$, and a small
$\epsilon >0$. \end{remark}


\begin{remark}\label{rem2.5} 
We would like to point out another fundamental difference 
between the noncompact Yamabe problem and the compact one. In the compact 
case, no manifold with positive scalar curvature is conformal to a 
manifold with zero scalar curvature. This is reflected from the fact that 
the equation $\Delta u -R u = 0$ with $R>0$ has no positive solution on 
compact manifolds. However, this is not the case for noncompact manifolds. 
The following is a  complete noncompact manifold which is conformal to a 
complete manifold of constant positive scalar curvature, and to a complete 
manifold with zero scalar curvature.  Let $\mathbf{M}=S^{3} \times 
\mathbf{R}^{1}$ with the metric being the direct product of the usual ones 
on $S^{3}$ and $\mathbf{R}^{1}$. Then $R=6$, $n=4$ and hence equation \eqref{eq:1.4} 
becomes
\begin{equation*}\Delta u - u + u^{3} = 0,
\end{equation*}
which has  a solution $u=1$.  At the same time the equation
\begin{equation*}\Delta u - u=0
\end{equation*}
has a positive solution 
\begin{equation*}u(x_{1}, x_{2}) = e^{x_{2}} + 
e^{-x_{2}} \ge 1,
\end{equation*}
where $x_{1} \in S^{3}$ and $x_{2} \in \mathbf{R}^{1}$. Clearly the above 
solution generates a complete metric of zero scalar curvature since it is
bounded away from zero. Similar phenomenon can be shown for $\mathbf{M} 
\times \mathbf{R}^{k}$ where $\mathbf{M}$ is any compact manifold with 
positive scalar
curvature (see \cite{Zhang5}). 
\end{remark}


Let us briefly discuss the method we are going to adopt. 
We will use the Schauder 
fixed point
theorem to achieve existence. This requires some new     
estimates involving the heat kernel on $\mathbf{M}^{n}$, such as, for 
$0s \ge 0$,
\begin{equation*} \int ^{t}_{s}\int _{\mathbf{M}^{n}}G_{a}(x, t; z, \tau ) \ 
|V(z, \tau )| \ 
G_{b}(z, \tau ; y, s)\, dz\, d\tau \le C_{a, b}  
 N_{c, \infty }(V) G_{a}(x, t; y, s);
\end{equation*}
\begin{equation*}  \int ^{t}_{s}\int _{\mathbf{M}^{n}}G_{b}(x, t; z, \tau ) 
\ |V(z, \tau )| \
G_{a}(z, \tau ; y, s)\, dz\, d\tau \le C_{a, b} 
 N_{c, \infty }(V) G_{a}(x, t; y, s).
\end{equation*}

 To prove Theorem \ref{thmc},  the key idea is to obtain  global  bounds \eqref{eq:1.3} and 
\eqref{eq:1.3prime}  for the fundamental solution of $H_{0}$, which involves certain 
conditions on the scalar curvature. Then we show that the parabolic problem 
\eqref{eq:1.1} with $p=\frac{n+2}{n-2}$ has no global positive solutions under the 
assumptions of Theorem \ref{thmc}. Since every positive solution of the Yamabe 
equation \eqref{eq:1.4} is a global positive solution of the parabolic problem \eqref{eq:1.1}, 
the former cannot exist either.

Details of the proof can be found in the paper \cite{Zhang5}.

\section*{Acknowledgment}I should 
 thank Professor J. Beem for helpful discussions,  and Professors N. Garofalo, 
H. A. Levine, and  Z. Zhao for their interest and support.
 
  
\bibliographystyle{amsalpha}
\begin{thebibliography}{xxxxxxx}


\bibitem[A]{A}
D.G. Aronson, {\em Nonnegative solutions of linear parabolic equations}, 
Annali della Scuola Norm. Sup. Pisa {\bf 22} (1968), 607--694.
\MR{55:8553}
\bibitem[Au]{Au}
T. Aubin, {\em The scalar curvature}, Differential geometry and relativity, 
edited by Cahen and Flato, Reidel, Dordrecht, 1976, pp. 5--18.
\MR{55:6476}
\bibitem[AM]{AM}
P. Aviles and R. C. McOwen, {\em Conformal deformation to constant negative 
scalar curvature on noncompact Riemannian manifolds}, J. Diff. Geometry 
{\bf 27} (1988), 225--239.
\MR{89b:58225}
\bibitem[D]{D}
E. B. Davies, {\em Heat kernels and spectral theory}, Cambridge Univ. Press, 
Cambridge and New York, 1989. 
\MR{90e:35123}
\bibitem[Fu]{Fu}
H. Fujita, {\em On the blowing up of solutions of the Cauchy problem for $u_{t}= 
\Delta u + u^{1 + \alpha }$}, J. Fac. Sci. Univ. Tokyo, Sect. I {\bf 13} 
(1966), 109--124.
 \MR{35:5761}
\bibitem[G]{G}
A. A. Grigoryan, {\em The heat equation on noncompact Riemannian manifolds}, 
Mat. Sb. {\bf 182} (1991), 55--87; English transl., Math. USSR Sb.
{\bf 72} (1992), 47--77.
 \MR{92h:58189}
\bibitem[H]{H}
R. Hamilton, {\em Three-manifolds with positive Ricci curvature}, J. Diff. 
Geom. {\bf 17} (1982), 255--306.
\MR{84a:53050}
\bibitem[K]{K}
J. Kazdan, {\em Prescribing the curvature of a Riemannian manifold}, Amer. 
Math. Soc., Providence, RI, 1985.
\MR{86h:53001}
\bibitem[Le]{Le}
H. A. Levine, {\em The role of critical exponents in blowup theorems}, SIAM 
Review {\bf 32} (1990), 269--288.
 \MR{91j:35135}
\bibitem[LN]{LN}
Tzong-Yow Lee and Wei-Ming  Ni, {\em Global existence, large time behavior 
and life span of solutions of a semilinear parabolic Cauchy problem}, 
Transactions AMS {\bf 333} (1992), 365--378.
\MR{92k:35134}
\bibitem[LSU]{LSU}
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, {\em Linear and 
quasilinear equations of parabolic type}, Transl. Math. Monographs, vol. 23,
Amer. Math. Soc., Providence, RI, 1968.
 \MR{39:3159b}
\bibitem[LY]{LY}
P. Li and S. T. Yau, {\em On the parabolic kernel of the Schr\"{o}dinger 
operator}, Acta Math. {\bf 156} (1986), 153--201.
\MR{87f:58156}
\bibitem[Me]{Me}
P. Meier, {\em On the critical exponent for reaction-diffusion equations}, 
Arch. Rat. Mech. Anal. {\bf 109} (1990), 63--71.
\MR{90k:35038}
\bibitem[Mo]{Mo}
J. Moser, {\em A Harnack inequality for parabolic differential equations}, 
Comm. Pure and Appl. Math. {\bf 17} (1964), 101--134.
\MR{28:2357}
\bibitem[Ni]{Ni}
W. M. Ni, {\em On the elliptic equation $\Delta u + K(x) u^{(n+2)/(n-2)} = 
0$, its generalizations and applications in geometry}, Indiana Univ. Math. 
J. {\bf 31} (1982), 493--529.
\MR{84e:35049}
\bibitem[Sa]{Sa}
L. Saloff-Coste, {\em A note on Poincar\'{e}, Sobolev and Harnack inequality}, 
IMRN, Duke Math. J. {\bf 2} (1992), 27--38.
 \MR{93d:58158}
\bibitem[Sc]{Sc}
R. Schoen, {\em Conformal deformation of a Riemannian metric to constant 
scalar curvature}, J. Diff. Geometry {\bf 20} (1984), 479--495.
\MR{86i:58137}
\bibitem[Sh]{Sh}
W.-X. Shi, {\em Deforming the metric on complete Riemannian manifolds}, J. 
Diff. Geometry {\bf 30} (1989), 225--301.
\MR{90i:58202}
\bibitem[Tr]{Tr}
N. Trudinger, {\em Remarks concerning the conformal deformation of 
Riemannian structures on compact manifolds}, Ann. Scuola Norm. Sup. Pisa 
Cl. Sci. (4) {\bf 22} (1968), 265--274.
\MR{39:2093} 
\bibitem[Yam]{Yam}
H. Yamabe, {\em On a deformation of Riemannian structures on compact 
manifolds}, Osaka Math. J. {\bf 12} (1960), 21--37.
\MR{23:A2847}
\bibitem[Yau]{Yau}
S. T. Yau, {\em Seminar on differential geometry}, Annals of Math. Studies, 
vol. 102, Princeton Univ. Press, Princeton, NJ 1982.
\MR{83a:53002}
\bibitem[Zhao]{Zhao}
Z. Zhao, {\em On the existence of positive solutions of nonlinear elliptic 
        equations---A probabilistic potential theory approach}, Duke Math. J. 
{\bf 69} (1993), 247--258.
\MR{94c:35090}
\bibitem[Zhang1]{Zhang1}
Qi Zhang, {\em On a parabolic equation with a singular lower order term}, 
Transactions of AMS {\bf 348} (1996), 2811--2844.
\MR{96k:35073}
\bibitem[Zhang2]{Zhang2}
Qi Zhang, {\em On a parabolic equation with a singular lower order term, 
Part II}, Indiana U. Math. J., to appear.
\bibitem[Zhang3]{Zhang3}
Qi Zhang, {\em Global existence and local continuity of solutions for 
semilinear 
parabolic equations}, Comm. PDE, to appear.
\bibitem[Zhang4]{Zhang4}
Qi Zhang, {\em The critical exponent of a reaction-diffusion equation on 
some Lie groups}, Math. Z., to appear.
\bibitem[Zhang5]{Zhang5}
Qi Zhang, {\em Semilinear parabolic problems on manifolds and applications 
to the noncompact Yamabe problem}, preprint.
\end{thebibliography}

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