Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 118, pp. 1-28.

Title: Stability of energy-critical nonlinear Schrodinger 
       equations in high dimensions
       
Authors: Terence Tao  (Univ. of California, Los Angeles, CA, USA)
         Monica Visan (Univ. of California, Los Angeles, CA, USA)

Abstract: 
 We develop the existence, uniqueness, continuity, stability,
 and scattering theory for energy-critical nonlinear Schrodinger
 equations in dimensions $n \geq 3$, for solutions which have large,
 but finite, energy and large, but finite, Strichartz norms.
 For dimensions $n \leq 6$, this theory is a standard extension
 of the small data well-posedness theory based on iteration in
 Strichartz spaces. However, in dimensions $n > 6$ there is an
 obstruction to this approach because of the subquadratic nature
 of the nonlinearity (which makes the derivative of the nonlinearity
 non-Lipschitz). We resolve this by iterating in exotic Strichartz
 spaces instead. The theory developed here will be applied in a
 subsequent paper of the second author, [21],
 to establish global well-posedness and scattering for the
 defocusing energy-critical equation for large energy data.
  
Submitted July 2, 2005. Published October 26, 2005.
Math Subject Classifications: 35J10.
Key Words: Local well-posedness; uniform well-posedness; scattering theory;
           Strichartz estimates