Geometry of super PDE's.

Superspaces and supermanifolds are introduced by using the concept of weak differentiability as usually given for locally convex spaces. This allows us to consider in algebraic way superdual spaces and superderivative spaces. In this way we obtain a good generalization of just known superstructures general enough to develop a formal theory for super PDE's that directly extends previous ones for standard manifolds of finite dimension. In particular, we give a Goldschmidt-type criterion of formal superintegrability for super PDE's, and show that a geometric theory of singular supersolutions, with singularities of Thom-Boardman type, can be formulated in the framework of super PDE's too. Conservation superlaws associated to super PDE's are considered and related with some spectral sequences and wholly cohomological character of these equations.