Scattering theory for a nonlinear system of wave equations with critical growth
Volume 106 / 2006
Colloquium Mathematicum 106 (2006), 69-81
MSC: 35L05, 35L15, 35P25.
DOI: 10.4064/cm106-1-6
Abstract
We consider scattering properties of the critical nonlinear system of wave equations with Hamilton structure $$ \cases{ u_{tt}-{\mit \Delta } u=-F_1(|u|^2,|v|^2)u,&\cr v_{tt}-{\mit \Delta } v=-F_2(|u|^2, |v|^2)v, \cr }$$ for which there exists a function $F(\lambda , \mu )$ such that $$ {\partial F(\lambda ,\mu )\over \partial \lambda }=F_1(\lambda ,\mu ),\hskip 1em {\partial F(\lambda ,\mu )\over \partial \mu }=F_2(\lambda ,\mu ). $$ By using the energy-conservation law over the exterior of a truncated forward light cone and a dilation identity, we get a decay estimate for the potential energy. The resulting global-in-time estimates imply immediately the existence of the wave operators and the scattering operator.
