The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

We undertake a comprehensive study of the nonlinear Schrödinger equation where u (t, x) is a complex-valued function in spacetime, λ1 and λ2 are nonzero real constants, and. 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 -critical, respectively -critical NLS, that is, λ1, λ2 > 0 and,. The results at the endpoint are conditional on a conjectured global existence and spacetime estimate for the -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 for solutions to the nonlinear Schrödinger equation with, which was first obtained by Ginibre and Velo (1985 Ginibre, J. , Velo, G. (1985). Scattering theory in the energy space for a class of nonlinear Schrödinger equations. J. Math. Pure. Appl. 64: 363 – 401. Web of Science ®, Google Scholar).