  We consider a class of abstract nonlinear evolution equations in
supermanifolds (smf's) modelled over Z_2-graded locally convex spaces. We show
uniqueness, local existence, smoothness, and an abstract version of causal
propagation of the solutions. If an a-priori estimate prevents the solutions
from blowing-up then an infinite-dimensional smf of "all" solutions can be
constructed.
  We apply our results to a class of systems of nonlinear field equations with
anticommuting fields which arise in classical field models used for realistic
quantum field theoretic models. In particular, we show that under suitable
conditions, the smf of smooth Cauchy data with compact support is isomorphic
with an smf of corresponding classical solutions of the model.
