  We consider reaction-diffusion equations either posed on Riemannian manifolds
or in the Euclidean weighted setting, with pow\-er-type nonlinearity and slow
diffusion of porous medium time. We consider the particularly delicate case
$p<m$ in problem (1.1), a case largely left open in [21] even when the initial
datum is smooth and compactly supported. We prove global existence for L$^m$
data, and a smoothing effect for the evolution, i.e. that solutions
corresponding to such data are bounded at all positive times with a
quantitative bound on their L$^\infty$ norm. As a consequence of this fact and
of a result of [21], it follows that on Cartan-Hadamard manifolds with
curvature pinched between two strictly negative constants, solutions
corresponding to sufficiently large L$^m$ data give rise to solutions that blow
up pointwise everywhere in infinite time, a fact that has no Euclidean
analogue. The methods of proof of the smoothing effect are functional analytic
in character, as they depend solely on the validity of the Sobolev inequality
and on the fact that the L$^2$ spectrum of $\Delta$ on $M$ is bounded away from
zero (namely on the validity of a Poincar\'{e} inequality on $M$). As such,
they are applicable to different situations, among which we single out the case
of (mass) weighted reaction-diffusion equation in the Euclidean setting. In
this latter setting, a modification of the methods of [37] allows to deal also,
with stronger results for large times, with the case of globally integrable
weights.
