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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