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

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.