  In this paper the semilinear equation of the spin-$\frac{1}{2}$ fields in the
de Sitter space is investigated. We prove the existence of the global in time
small data solution in the expanding de Sitter universe. Then, under the
Lochak-Majorana condition, we prove the existence of the global in time
solution with large data. The sufficient conditions for the solutions to blow
up in finite time are given for large data in the expanding and contracting de
Sitter spacetimes. The influence of the Hubble constant on the lifespan is
estimated.
