@article{22047,
  abstract     = {We undertake a comprehensive study of the nonlinear Schrödinger equation (mathematical formular) where u(t, x) is a complex-valued function in spacetime R, xRn/x, λ1 and λ2 are nonzero real constants, and (mathematical formular). We address questions related to local and global well-posedness, finite time blowup, and asymptotic behaviour. Scattering is considered both in the energy space H^1(ℝ n ) and in the pseudoconformal space Σ := {f ∈ H^1(ℝ^n); xf ∈ L^2(ℝ^n)}. Of particular interest is the case when both nonlinearities are defocusing and correspond to the L2/x-critical, respectively H1/x-critical NLS, that is, λ1, λ2 > 0 and (mathematical formular) . The results at the endpoint p1= 4/n are conditional on a conjectured global existence and spacetime estimate for the L2/x-critical nonlinear Schrödinger equation, which has been verified in dimensions n ≥ 2 for radial data in Tao et al. (Tao et al. to appear a,b) and Killip et al. (preprint).
As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H1/x for solutions to the nonlinear Schrödinger equation (mathematical formular) with 4/n < p < 4/n-2, which was first obtained by Ginibre and Velo (Citation1985).},
  author       = {Tao, Terence and Visan, Monica and Zhang, Xiaoyi},
  issn         = {1532-4133},
  journal      = {Communications in Partial Differential Equations},
  keywords     = {Energy-critical, Mass-critical, Nonlinear Schrödinger equation, Wellposedness},
  number       = {8},
  pages        = {1281--1343},
  publisher    = {Informa UK Limited},
  title        = {{The nonlinear Schrödinger equation with combined power-type nonlinearities}},
  doi          = {10.1080/03605300701588805},
  volume       = {32},
  year         = {2007},
}