  We consider the semilinear heat equation, to which we add a nonlinear
gradient term, with a critical power. We construct a solution which blows up in
finite time. We also give a sharp description of its blow-up profile. The proof
relies on the reduction of the problem to a finite dimensional one, and uses
the index theory to conclude.
  Thanks to the interpretation of the parameters of the finite-dimensional
problem in terms of the blow-up time and point, we also show the stability of
the constructed solution with respect to initial data. This note presents the
results and the main arguments. For the details, we refer to our paper
\cite{TZ15}.
