The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher

Rowan Killip, Monica Visan
2010 American Journal of Mathematics  
Publications Goldberg, M., and Visan, M. (2005) A counterexample to dispersive estimates for Schrödinger operators in higher dimensions. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4 . Abstract of the Dissertation We consider the initial value problem for the defocusing energy-critical nonlinear where the initial data u 0 is chosen to belong to the energy-spaceḢ 1 (R n ), and u(t, x) is a complex-valued function in spacetime R t
more » ... n in spacetime R t × R n x . We use the induction on energy technique introduced by J. Bourgain and extend the frequency-localized interaction Morawetz inequality introduced by J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao in dimension n = 3 to higher dimensions. The results obtained are global well-posedness, scattering, and global L 2(n+2) n−2 t,x spacetime bounds for solutions to the initial value problem ( * ). x
doi:10.1353/ajm.0.0107 fatcat:qm3z2jbi2vh63do6qmj4edgnf4