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

DOI:
https://doi.org/10.1090/S1079-6762-97-00022-X

Published electronically:
May 20, 1997

MathSciNet review:
1447067

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the Cauchy problem for the semilinear parabolic equations on with initial value , where is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when , it is well known that is the critical exponent, i.e., if and is smaller than a small Gaussian, then the Cauchy problem has global positive solutions, and if , 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 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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Additional Information

**Qi S. Zhang**

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

Email:
sz@mumathnx6.math.missouri.edu

DOI:
https://doi.org/10.1090/S1079-6762-97-00022-X

Received by editor(s):
February 19, 1997

Published electronically:
May 20, 1997

Communicated by:
Richard Schoen

Article copyright:
© Copyright 1997
American Mathematical Society