  We investigate the well-posedness of the fast diffusion equation (FDE) in a
wide class of noncompact Riemannian manifolds. Existence and uniqueness of
solutions for globally integrable initial data was established in [5]. However,
in the Euclidean space, it is known from Herrero and Pierre [20] that the
Cauchy problem associated with the FDE is well posed for initial data that are
merely in $ L^1_{\mathrm{loc}} $. We establish here that such data still give
rise to global solutions on general Riemannian manifolds. If, in addition, the
radial Ricci curvature satisfies a suitable pointwise bound from below
(possibly diverging to $-\infty$ at spatial infinity), we prove that also
uniqueness holds, for the same type of data, in the class of strong solutions.
Besides, under the further assumption that the initial datum is in
$L^2_{\mathrm{loc}}$ and nonnegative, a minimal solution is shown to exist, and
we are able to establish uniqueness of purely (nonnegative) distributional
solutions, which to our knowledge was not known before even in the Euclidean
space. The required curvature bound is in fact sharp, since on model manifolds
it turns out to be equivalent to stochastic completeness, and it was shown in
[13] that uniqueness for the FDE fails even in the class of bounded solutions
on manifolds that are not stochastically complete. Qualitatively this amounts
to asking that the curvature diverges at most quadratically at infinity. A
crucial ingredient of the uniqueness result is the proof of nonexistence of
distributional subsolutions to certain semilinear elliptic equations with power
nonlinearities, of independent interest.
