The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

Abstract

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]).

Locations

• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF