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ISSN 1079-6762



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

Author: Qi S. Zhang
Journal: Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 45-51
MSC (1991): Primary 35K55; Secondary 58G03
Published electronically: May 20, 1997
MathSciNet review: 1447067
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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.

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Qi S. Zhang
Affiliation: Department of Mathematics, University of Missouri, Columbia, MO 65211

Received by editor(s): February 19, 1997
Published electronically: May 20, 1997
Communicated by: Richard Schoen
Article copyright: © Copyright 1997 American Mathematical Society