Abstract
In this note, we show the existence of a special solution u to defocusing cubic NLS in 3d, which lives in Hs for all s > 0, but scatters to a linear solution in a very slow way. We prove for this u, for all > 0, one has supt>0 tu(t) − eitΔu+H 1/2 = ∞. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u(0) be in L1. We expect that similar construction hold the for other NLS models. It can been seen the slow convergence is caused by the fact that there are delayed backward scattering profile in the initial data, we also illustrate why L1 condition of initial data will get rid of this phenomena.
Original language | English |
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Pages (from-to) | 2527-2542 |
Number of pages | 16 |
Journal | Proceedings of the American Mathematical Society |
Volume | 151 |
Issue number | 6 |
DOIs | |
Publication status | Published - Jun 2023 |
Keywords
- Nonlinear Schrödinger equation
- convergence rate
- decay estimate
- scattering
- scattering rate
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Fan, C., & Zhao, Z. (2023). A NOTE ON DECAY PROPERTY OF NONLINEAR SCHRÖDINGER EQUATIONS. Proceedings of the American Mathematical Society, 151(6), 2527-2542. https://doi.org/10.1090/proc/16296