Large global solutions for nonlinear Schrödinger equations II, mass-supercritical, energy-subcritical cases

Abstract

In this paper, we consider the defocusing mass-supercritical, energy-subcritical nonlinear Schrödinger equation, $$i{\partial }_{t}u+\Delta u=|u{|}^{p}u,(t,x)\in {\mathbb{R}}^{d+1},$$ 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 $H^{s_0}({\mathbb{R}}^d),\text{for some }{s}_0<s_c$ 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.

arXiv comment61 pages, last version, published in Comm. Math. Phys

References in corpus (8)

• Almost sure well-posedness of the cubic nonlinear Schrödinger equation below L^2(T)
• Ill-posedness for nonlinear Schrodinger and wave equations
• Global and local regularity of Fourier integral operators on weighted and unweighted spaces
• On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities
• Almost sure scattering for the energy-critical NLS with radial data below $H^1({\mathbb{R}}^4)$
• Monotonic Local Decay Estimates
• Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on ${\mathbb{R}}^3$
• Large Initial Data Global Well-Posedness for a Supercritical Wave Equation