  In this paper, blow-up solutions of autonomous ordinary differential
equations (ODEs) which are unstable under perturbations of initial conditions
are studied. Combining dynamical systems machinery (e.g. phase space
compactifications, time-scale desingularizations of vector fields) with tools
from computer-assisted proofs (e.g. rigorous integrators, parameterization
method for invariant manifolds), these unstable blow-up solutions are obtained
as trajectories on stable manifolds of hyperbolic (saddle) equilibria at
infinity. In this process, important features are obtained: smooth dependence
of blow-up times on initial points near blow-up, level set distribution of
blow-up times, singular behavior of blow-up times on unstable blow-up
solutions, organization of the phase space via separatrices (stable manifolds).
In particular, we show that unstable blow-up solutions themselves, and
solutions defined globally in time connected by those blow-up solutions can
separate initial conditions into two regions where solution trajectories are
either globally bounded or blow-up, no matter how the large initial points are.
