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It is well known from the work of [2] that the Fujita phenomenon forreaction-diffusion evolution equations with power nonlinearities does not occuron the hyperbolic space $\mathbb{H}^N$, thus marking a striking difference withthe Euclidean situation. We show that, on classes of manifolds in which thebottom $\Lambda$ of the $L^2$ spectrum of $-\Delta$ is strictly positive (thehyperbolic space being thus included), a different version of the Fujitaphenomenon occurs for other kinds of nonlinearities, in which the role of thecritical Fujita exponent in the Euclidean case is taken by $\Lambda$. Suchnonlinearities are time-independent, in contrast to the ones studied in [2]. Asa consequence of our results we show that, on a class of manifolds much largerthan the case $M=\mathbb{H}^N$ considered in [2], solutions to (1.1) with powernonlinearity $f(u)=u^p$, $p>1$, and corresponding to sufficiently small data,are global in time.

Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds