Global existence for the quadratic Dirac equation in two and three space dimensions

Abstract

In this paper we study global existence and pointwise decay estimates for the nonlinear Dirac equation with quadratic nonlinearity. We consider four cases depending on the spatial dimension n, the mass parameter m, and the initial data ψ0: i) (n,m)=(2,0) and ψ0 is compactly supported; ii)(n,m)=(3,0) and ψ0 is compactly supported; iii)(n,m)=(3,0) and ψ0 is not necessarily compactly supported; iv)n=3, m≥0 and ψ0 is compactly supported. In each of the cases i)-iii), we prove a small data global existence result, a sharp pointwise decay estimate and a scattering result for the global solution. In the case iv) we prove a uniform (in the mass parameter m) global existence result, a unified pointwise decay estimate, and a scattering result.