We discuss relations between the nonlinear Klein-Gordon equation and the nonlinear Schrödinger equation in view of the global wellposedness in the energy space and L^2. In some critical cases, we show that the global wellposedness for the former equation with some uniform bounds implies that for the latter.
"Transfer of global wellposedness from nonlinear Klein-Gordon equation to nonlinear Schrödinger equation." Hokkaido Math. J. 37 (4) 749 - 771, November 2008. https://doi.org/10.14492/hokmj/1249046367