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Large global solutions for nonlinear Schrödinger equations. II: Mass-supercritical, energy-subcritical cases. (English) Zbl 1462.35346

Summary: In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schrödinger equation, \[ i \partial_t u + \Delta u = |u|^p u, \quad (t, x) \in \mathbb{R}^{d+1}, \] with \(p \in (\frac{4}{d}, \frac{4}{d-2})\). We prove that under some restrictions on \(d, p\), any radial function in the rough space \(H^{s_0} (\mathbb{R}^d)\), for some \(s_0 < s_c\) with the support away from the origin, there exists an incoming/outgoing decomposition, such that the initial data in the outgoing part leads to the global well-posedness and scattering forward in time; while the initial data in the incoming part leads to the global well-posedness and scattering backward in time. The proof is based on Phase-Space analysis of the nonlinear dynamics.
For Part I, see [“Large global solutions for nonlinear Schrödinger equations. I: Mass-subcritical cases”, Preprint, arXiv:1809.09831].

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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