  In many situations, the notion of function is not sufficient and it needs to
be extended. A classical way to do this is to introduce the notion of weak
solution; another approach is to use generalized functions. Ultrafunctions are
a particular class of generalized functions that has been previously introduced
and used to define generalized solutions of stationary problems in
[4,7,9,11,12]. In this paper we generalize this notion in order to study also
evolution problems. In particular, we introduce the notion of Generalized
Ultrafunction Solution (GUS) for a large family of PDE's, and we confront it
with classical strong and weak solutions. Moreover, we prove an existence and
uniqueness result of GUS's for a large family of PDE's, including the nonlinear
Schroedinger equation and the nonlinear wave equation. Finally, we study in
detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's
of this equation provide a description of the phenomenon at microscopic level.
