Institute of Mathematics with Computer Center at the Bulgarian Academy of Sciences
We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see ). We also study decay properties of the spinor field.
With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris  and H. Takamura . They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.