Blowup and scattering problems for the nonlinear Schrödinger equations (Q371226)

In [\textit{J. Holmer} et al., Commun. Math. Phys. 282, No. 2, 435--467 (2008; Zbl 1155.35094)], scattering (asymptotic linear behavior) is proved for radial \(H^1\) solutions to the 3D focusing cubic nonlinear Schrödinger equation (\(\dot H^{\frac 12}\)-critical equation) satisfying \(M[u]E[u]<M[Q]E[Q]\) (\(M\) is the mass and \(E\) is the energy) and an initial mass-gradient bound \(\|u_0\|_{L^2}\|\nabla u_0\|_{L^2}<\|Q\|_{L^2}\|\nabla Q\|_{L^2}\), where Q is the ground state. This result was extended by \textit{T. Duyckaerts} et al. [Math. Res. Lett. 15, No. 5--6, 1233--1250 (2008; Zbl 1171.35472)] to the nonradial setting.NEWLINENEWLINEIn the present paper, the authors extend these results to any \(L^2\) supercritical and energy subcritical power nonlinear Schrödinger equations in any space dimension, using a suitable generalization of the criteria above.NEWLINENEWLINEThese works follow the concentration-compactness/rigidity theory developed in the energy critical case by \textit{C. E. Kenig} and \textit{F. Merle} [Acta Math. 201, No. 2, 147--212 (2008; Zbl 1183.35202)] and [Invent. Math. 166, No. 3, 645--675 (2006; Zbl 1115.35125)].