论文标题
非线性schrödinger方程的衰减属性的注释
A note on decay property of nonlinear Schrödinger equations
论文作者
论文摘要
在本说明中，我们显示了一个特殊的解决方案$ u $，以换成$ 3D $的立方nls，该$ 3D $的存在，该$ 3D $的所有$ s> 0 $都以$ h^{s} $的形式生存，但以非常缓慢的方式散布到线性解决方案。我们证明了这个$ u $，对于所有$ε> 0 $，一个人都有$ \ sup_ {t> 0} t^ε\ | U（t）-e^{itΔ} u^{+^{+} \ | _ {\ dot {\ dot {请注意，如果一个人在$ l^{1} $中进一步构成$ u（0）$的初始数据，那么这种缓慢的渐近收敛是不可能的。我们期望其他NLS模型类似的结构。可以看出，缓慢的收敛性是由于初始数据中存在延迟的向后散射曲线而引起的，我们还说明了为什么$ l^{1} $初始数据的条件会摆脱这种现象。
In this note, we show the existence of a special solution $u$ to defocusing cubic NLS in $3d$, which lives in $H^{s}$ for all $s>0$, but scatters to a linear solution in a very slow way. We prove for this $u$, for all $ε>0$, one has $\sup_{t>0}t^ε\|u(t)-e^{itΔ}u^{+}\|_{\dot{H}^{1/2}}=\infty$. Note that such a slow asymptotic convergence is impossible if one further pose the initial data of $u(0)$ be in $L^{1}$. 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 $L^{1}$ condition of initial data will get rid of this phenomena.
