论文标题
通过无半几何形状将nls聚焦于$ \ mathbb {r}^d \ times \ mathbb {t} $的归一化基态和阈值散射。
Normalized ground states and threshold scattering for focusing NLS on $\mathbb{R}^d\times\mathbb{T}$ via semivirial-free geometry
论文作者
论文摘要
我们研究焦点nls \ begin {align} \ label {nls_abstract} i \ partial_t u+Δ_{x，x，y} intrical策略$α\ in（\ frac {4} {d}，\ frac {4} {d-1}）$。通过假设\ eqref {nls_abstract}独立于$ y $，它可以减少到$ \ mathbb {r}^d $上的焦点中的nls，该nls具有站立波浪和有限的时间爆破解决方案。自然，我们问是否存在具有非平凡$ y $依赖性的这些特殊解决方案。在本文中，我们对这个问题给出了肯定的答案。为此，我们介绍了\ textIt {semivirial}功能的概念，并考虑最小化问题$ m_c $，上面有规定的质量$ c $。我们证明，对于任何$ c \ in（0，\ infty）$，变量问题$ m_c $具有基态优化器$ u_c $，它也解决了常驻波方程$$ - δ__{x，y} u_c+β_c+β_c+β_cu_c u_c = | U | U | U | U | U |^αu$ $ $β__C_c_c>> 0 $ 0 $ 0 $ 0。此外，我们证明存在一个关键数字$ c _*\在（0，\ infty）$中，以便\ in（initizize} \ item $ c \ in（0，c _*）$，任何优化的$ _c $ u_c $ of $ m_c $ of $ _c $ of $ \ u_c $ of $ m_c $必须满足$ \ pt_y u_c \ neq \ neq 0 $。 \ for $ c \ in（c _*，\ infty）$，任何优化器$ u_c $ of $ m_c $必须满足$ \ pt_y u_c = 0 $。 \ end {inatize}最后，我们证明了先前构造的接地状态表征了散射和有限的时间爆破溶液的分叉的尖锐阈值，以依赖于半病毒的符号。
We study the focusing NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^αu\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}^d\times\mathbb{T}$ in the intercritical regime $α\in(\frac{4}{d},\frac{4}{d-1})$. By assuming that the \eqref{nls_abstract} is independent of $y$, it reduces to the focusing intercritical NLS on $\mathbb{R}^d$, which is known to have standing wave and finite time blow-up solutions. Naturally, we ask whether these special solutions with non-trivial $y$-dependence exist. In this paper we give an affirmative answer to this question. To that end, we introduce the concept of \textit{semivirial} functional and consider a minimization problem $m_c$ on the semivirial-vanishing manifold with prescribed mass $c$. We prove that for any $c\in(0,\infty)$ the variational problem $m_c$ has a ground state optimizer $u_c$ which also solves the standing wave equation $$-Δ_{x,y}u_c+β_c u_c=|u|^αu $$ with some $β_c>0$. Moreover, we prove the existence of a critical number $c_*\in(0,\infty)$ such that \begin{itemize} \item For $c\in(0,c_*)$, any optimizer $u_c$ of $m_c$ must satisfy $\pt_y u_c\neq 0$. \item For $c\in(c_*,\infty)$, any optimizer $u_c$ of $m_c$ must satisfy $\pt_y u_c=0$. \end{itemize} Finally, we prove that the previously constructed ground states characterize a sharp threshold for the bifurcation of scattering and finite time blow-up solutions in dependence of the sign of the semivirial.
