2008/05/11 by Terence Tao, Tao, Terence · 1 citation
Mathematics · Physics and Astronomy · #35Q55 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.0805.1544
openalex publication_date 2008/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the asymptotic behavior of large data solutions in the energy space H := H1(\Rd) in very high dimension d ≥ 11 to defocusing Schrödinger equations i ut + Δu = |u|p-1 u + Vu in \Rd, where V ∈ C^∞0(\Rd) is a real potential (which could contain bound states), and 1+(4)/(d) < p < 1+(4)/(d-2) is 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.