  In this paper, we propose a certified reduced basis (RB) method for
quasilinear parabolic problems. The method is based on a space-time variational
formulation. We provide a residual-based a-posteriori error bound on a
space-time level and the corresponding efficiently computable estimator for the
certification of the method. We use the Empirical Interpolation method (EIM) to
guarantee the efficient offline-online computational procedure. The error of
the EIM method is then rigorously incorporated into the certification
procedure. The Petrov-Galerkin finite element discretization allows to benefit
from the Crank-Nicolson interpretation of the discrete problem and to use a
POD-Greedy approach to construct the reduced-basis spaces of small dimensions.
It computes the reduced basis solution in a time-marching framework while the
RB approximation error in a space-time norm is controlled by the estimator.
Therefore the proposed method incorporates a POD-Greedy approximation into a
space-time certification.
