On long time behavior of the focusing energy-critical NLS on $\mathbb{R}^d\times\mathbb{T}$ via semivirial-vanishing geometry

Abstract: We study the focusing energy-critical NLS \begin{align}\label{nls_abstract} i\partial_t u+\Delta_{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.