In the framework of the PDE's algebraic topology, previously introduced by A. Pr\'astaro, are considered { \em exotic differential equations}, i.e., differential equations admitting Cauchy manifolds $N$ identifiable with exotic spheres, or such that their boundaries $\partial N$ are exotic spheres. For such equations are obtained local and global existence theorems and stability theorems. In particular the smooth ($4$-dimensional) Poincar\'e conjecture is proved. This allows to complete the previous Theorem 4.59 in \cite{ PRA17} also for the case $n=4$.
Categories: math.GM