On global regularity for systems of nonlinear wave equations with the null-condition (Q500071)

In this note the authors investigate the problem of global regularity for systems of semilinear wave equations on Minkowski space \(\mathbb R^{3+1}\), satisfying null condition. It has been well known that the problem admit global solutions provided the initial data is compactly supported and sufficiently small and smooth. The main result in this note is the pointwise decay for solutions of the form \(| u|\leq C (1+| t-| x\|)^{-1}(1+t+| x|)^{-1}\), by assuming small bound on \(\| (1+| x|)^{2s-1}(u(0),\partial u(0))\|_{H^s\times H^{s-1}}\) (\(s>3/2\) which is almost optimal concerning on the regularity) for compactly supported initial data.