算法
注释
计算机科学
类型(生物学)
人工智能
地质学
古生物学
作者
Chenjie Fan,Zehua Zhao
出处
期刊:Proceedings of the American Mathematical Society
[American Mathematical Society]
日期:2022-10-13
摘要
In this note, we show the existence of a special solution u u to defocusing cubic NLS in 3 d 3d , which lives in H s H^{s} for all s > 0 s>0 , but scatters to a linear solution in a very slow way. We prove for this u u , for all ϵ > 0 \epsilon >0 , one has sup t > 0 t ϵ ‖ u ( t ) − e i t Δ u + ‖ H ˙ 1 / 2 = ∞ \sup _{t>0}t^{\epsilon }\|u(t)-e^{it\Delta }u^{+}\|_{\dot {H}^{1/2}}=\infty . Note that such a slow asymptotic convergence is impossible if one further pose the initial data of u ( 0 ) u(0) be in L 1 L^{1} . 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 L 1 L^{1} condition of initial data will get rid of this phenomena.
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