A global compact attractor for high-dimensional defocusing non-linear Schrödinger equations with potential

Abstract

We study the asymptotic behavior of large data solutions in the energy space) is a real potential (which could contain bound states), and 1 + 4is an exponent which is energy-subcritical and mass-supercritical.In the spherically symmetric case, we show that as t → +∞, these solutions split into a radiation term that evolves according to the linear Schrödinger equation, and a remainder which converges in H to a compact attractor K, which consists of the union of spherically symmetric almost periodic orbits of the NLS flow in H.The main novelty of this result is that K is a global attractor, being independent of the initial energy of the initial data; in particular, no matter how large the initial data is, all but a bounded amount of energy is radiated away in the limit.Contents 1. Introduction 102 2. Reduction to a quasi-Liouville theorem 105 3. Polynomial spatial decay 107 4. Virial inequalities 111 5. Remarks and possible generalisations 114

Locations

• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF
• Dynamics of Partial Differential Equations - View - PDF
• arXiv (Cornell University) - View - PDF
• arXiv (Cornell University) - PDF