Extended crystal PDE's stability.I: The general theory

This work, divided in two parts, follows some our previous works devoted to the algebraic topological characterization of PDE's. In this first part, the stability of PDE's is studied in details in the framework of the geometric theory of PDE's, and bordism groups theory of PDE's. In particular we identify criteria to recognize PDE's that are stable (in extended Ulam sense) and in their regular smooth solutions do not occur finite time unstabilities, ({\em stable extended crystal PDE's}). Applications to some important PDE's are considered in some details. (In the second part a stable extended crystal PDE encoding anisotropic incompressible magnetohydrodynamics is obtained.)