The Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

Abstract

Abstract. We undertake a comprehensive study of the nonlinear Schrödinger equationwhere u(t, x) is a complex-valued function in spacetime Rt × R n x , λ 1 and λ 2 are nonzero real constants, and 0 < p 1 < p 2 ≤ 4 n−2. 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 L 2 x -critical, respectivelyḢ 1 x -critical NLS, that is, λ 1 , λ 2 > 0 and. The results at the endpoint p 1 = 4 n are conditional on a conjectured global existence and spacetime estimate for the L 2x -critical nonlinear Schrödinger equation.As an off-shoot of our analysis, we also obtain a new, simpler proof of scattering in H 1 x for solutions to the nonlinear Schrödinger equation iut + ∆u = |u| p u,

The cubic nonlinear Schrödinger equation in two dimensions with radial data

“…By Lemma 1.5, it also implies (quantitative) continuous dependence upon initial data and stabillity under external forcing. To this one may add stability of well-posedness under perturbations of the equation; see [71].…”

Stability Solution of the Nonlinear Schr&#246;dinger Equation

“…Even though stability is a local question, it plays an important role in all existing treatments of the global well-posedness problem for nonlinear Schrödinger equation at critical case, for more see [7]. It has also proved useful in the treatment of local and global questions for more exotic nonlinearities [8,9]. In this section, we will only discus the stability theory for the mass-critical NLS.…”

Global well-posedness for the $L^2$ critical nonlinear Schrödinger equation in higher dimensions

“…The second ingredient in our proof is an a priori interaction Morawetz-type estimate for the solution u to (1.1) (see [12], [13], [18]). With the help of this estimate, some harmonic analysis and interpolation we obtain for any compact interval…”