  In this thesis the Cauchy problem and in particular the question of
singularity formation for co--rotational wave maps from 3+1 Minkowski space to
the three--sphere $S^3$ is studied. Numerics indicate that self--similar
solutions of this model play a crucial role in dynamical time evolution. In
particular, it is conjectured that a certain solution $f_0$ defines a universal
blow up pattern in the sense that the future development of a large set of
generic blow up initial data approaches $f_0$. Thus, singularity formation is
closely related to stability properties of self--similar solutions.
  In this work, the problem of linear stability is studied by functional
analytic methods. In particular, a complete spectral analysis of the
perturbation operators is given and well--posedness of the linearized Cauchy
problem is proved by means of semigroup theory and, alternatively, the
functional calculus for self--adjoint operators. These results lead to growth
estimates which provide information on the stability of self--similar wave
maps. Finally, convergence properties of $f_n$ for large $n$ and the spectra of
the corresponding perturbation operators are investigated.
  The thesis is intended to be self--contained as far as possible, i.e. all the
mathematical requirements are carefully introduced, including proofs for many
results which could be found elsewhere.
