  This work is concerned with the derivation of an a posteriori error estimator
for Galerkin approximations to nonlinear initial value problems with an
emphasis on finite-time existence in the context of blow-up. The stucture of
the derived estimator leads naturally to the development of both h and hp
versions of an adaptive algorithm designed to approximate the blow-up time. The
adaptive algorithms are then applied in a series of numerical experiments, and
the rate of convergence to the blow-up time is investigated.
