A large data critical problem

Abstract

Throughout the remainder of these notes we restrict attention to the defocusing energy-critical NLS

$$ i\partial _t u\, + \,\Delta u\, = \,\,\left| u \right|^{\frac{4} {{d - 2}}} u\,\,\,\,{\rm with}\,\,\,\,u\left( 0 \right)\, = \,u_0 \, \in \,\dot H_x^1 . $$

For arguments and further references in the focusing case, see [22].