论文标题
关于$ \ mathbb {r}^d \ times \ mathbb {t} $的重点关键能源NLS的长时间行为
On long time behavior of the focusing energy-critical NLS on $\mathbb{R}^d\times\mathbb{T}$ via semivirial-vanishing geometry
论文作者
论文摘要
我们研究焦点临界nls \ begin {align} \ label {nls_abstract} i \ partial_t u++++Δ_{x，y} u = - | | | | | | | | |^{\ frac {4} {d-1} {d-1}} {d-1}} {d-1}} $ \ mathbb {r} _x^d \ times \ times \ mathbb {t} _y $带有$ d \ geq 2 $。我们揭示了一种有些违反直觉的现象，即尽管非线性潜力具有能量临界，但\ eqref {nls_abstract}的长时间动力学纯粹是由具有能量产生智力特征的半生殖器变化几何学决定的。作为起点，我们考虑了一个最小化问题$ m_c $，在半疗法歧管上定义了规定的质量$ c $。我们证明，对于所有足够大的质量，变量问题$ m_c $具有唯一的优化器$ u_c $满足$ \ partial_y u_c = 0 $，而对于所有足够小的质量，$ m_c $的任何优化器都必须具有非谋取$ y $ y $ $ e $依赖性。之后，我们证明$ m_c $是有限时间爆破（$ d = 2,3 $）和全球散射（$ d = 3 $）的\ eqref {nls_abstract}的解决方案的尖锐阈值。据作者所知，本文还给出了第一个大数据散射结果，以将NLS集中在能量环境中的产品空间上。
We study the focusing energy-critical NLS \begin{align}\label{nls_abstract} i\partial_t u+Δ_{x,y} u=-|u|^{\frac{4}{d-1}} u\tag{NLS} \end{align} on the waveguide manifold $\mathbb{R}_x^d\times\mathbb{T}_y$ with $d\geq 2$. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of \eqref{nls_abstract} are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem $m_c$ defined on the semivirial-vanishing manifold with prescribed mass $c$. We prove that for all sufficiently large mass the variational problem $m_c$ has a unique optimizer $u_c$ satisfying $\partial_y u_c=0$, while for all sufficiently small mass, any optimizer of $m_c$ must have non-trivial $y$-dependence. Afterwards, we prove that $m_c$ characterizes a sharp threshold for the bifurcation of finite time blow-up ($d=2,3$) and globally scattering ($d=3$) solutions of \eqref{nls_abstract} in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting.
