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## Nonlinear parabolic problems on manifolds, and a
nonexistence result for the noncompact Yamabe problem

### Qi S. Zhang

**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.

*Copyright 1997 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**03** (1997), pp. 45-51
- Publisher Identifier: S 1079-6762(97)00022-X
- 1991
*Mathematics Subject Classification*. Primary 35K55; Secondary 58G03
- Received by the editors February 19, 1997
- Posted on May 20, 1997
- Communicated by Richard Schoen
- Comments (When Available)

**Qi S. Zhang**

Department of Mathematics, University of Missouri, Columbia, MO 65211

*E-mail address:* `sz@mumathnx6.math.missouri.edu`

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