  We study the long-time behaviour of nonnegative solutions of the Porous
Medium Equation posed on Cartan-Hadamard manifolds having very large negative
curvature, more precisely when the sectional or Ricci curvatures diverge at
infinity more than quadratically in terms of the geodesic distance to the pole.
We find an unexpected separate-variable behaviour that reminds one of Dirichlet
problems on bounded Euclidean domains. As a crucial step, we prove existence of
solutions to a related sublinear elliptic problem, a result of independent
interest. Uniqueness of solutions vanishing at infinity is also shown, along
with comparison principles, both in the parabolic and in the elliptic case.
  Our results complete previous analyses of the Porous Medium Equation flow on
negatively curved Riemannian manifolds, which were carried out first for the
hyperbolic space and then for general Cartan-Hadamard manifolds with a negative
curvature having at most quadratic growth. We point out that no similar
analysis seems to exist for the linear heat flow.
  We also translate such results into some weighted Porous Medium Equations in
the Euclidean space having special weights.

Analysis of PDEs;Differential Geometry