论文标题
急剧散射，用于将批判性NLS聚焦于高维波指导歧管上
Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds
论文作者
论文摘要
我们研究焦点中的中间nls \ begin {align} \ label {Abstract_nls} i \ pt_t u+δ_{x，x，y} u = - | 5 $和$α\ in（\ frac {4} {d}，\ frac {4} {d-1}）$。在$ d \ leq 4 $的情况下，借助于半毒品消失的理论\ cite {luo_inter}，作者能够构建一个尖锐的阈值，它以基础状态解决方案为特征，以基础状态解决方案的独特特征，从而迅速确定了全球散射和有限的时间爆破解决方案的依赖性依赖性依从性的依从性功能的依赖性。由于非线性电势的导数不再是$ d \ geq 5 $中的lipschitz，并且基础域具有各向异性性质，因此使用浓度紧凑的原则，无法将其扩展到更高的尺寸模型。在本文中，我们利用了对莫拉维兹 - 多德森 - 穆尔菲（IMDM）估计的量化良好的适应性，该估计仅适用于欧几里得空间中，以证明在\ cite {luo_inter}以{luo_inter}中的形式表明，以{luo_inter}持续conly for lu $ d $ d $ d $ d $ d $ d。 Together with Tzvetkov-Visciglia \cite{TzvetkovVisciglia2016} and the author \cite{Luo_inter}, we thus give a complete characterization of the large data scattering for \eqref{abstract_nls} in both defocusing and focusing case and in arbitrary dimension.
We study the focusing intercritical NLS \begin{align}\label{abstract_nls} i\pt_t u+Δ_{x,y}u=-|u|^αu\tag{NLS} \end{align} on the semiperiodic waveguide manifold $\R^d_x\times \T_y$ with $d\geq 5$ and $α\in(\frac{4}{d},\frac{4}{d-1})$. In the case $d\leq 4$, with the aid of the semivirial vanishing theory \cite{Luo_inter}, the author was able to construct a sharp threshold, which being uniquely characterized by the ground state solutions, that sharply determines the bifurcation of global scattering and finite time blow-up solutions in dependence of the sign of the semivirial functional. As the derivative of the nonlinear potential is no longer Lipschitz in $d\geq 5$ and the underlying domain possesses an anisotropic nature, the proof in \cite{Luo_inter}, which makes use of the concentration compactness principle, can not be extended to higher dimensional models. In this paper, we exploit a well-tailored adaptation of the interaction Morawetz-Dodson-Murphy (IMDM) estimates, which were only known to be applicable on Euclidean spaces, into the waveguide setting, in order to prove that the large data scattering result formulated in \cite{Luo_inter} continues to hold for all $d\geq 5$. Together with Tzvetkov-Visciglia \cite{TzvetkovVisciglia2016} and the author \cite{Luo_inter}, we thus give a complete characterization of the large data scattering for \eqref{abstract_nls} in both defocusing and focusing case and in arbitrary dimension.
