  We consider reaction-diffusion equations driven by the $p$-Laplacian on
noncompact, infinite volume manifolds assumed to support the Sobolev inequality
and, in some cases, to have $L^2$ spectrum bounded away from zero, the main
example we have in mind being the hyperbolic space of any dimension. It is
shown that, under appropriate conditions on the parameters involved and
smallness conditions on the initial data, global in time solutions exist and
suitable smoothing effects, namely explicit bounds on the $L^\infty$ norm of
solutions at all positive times, in terms of $L^q$ norms of the data. The
geometric setting discussed here requires significant modifications w.r.t. the
Euclidean strategies.
