Sharp scattering for focusing intercritical NLS on high-dimensional waveguide manifolds

Abstract

We study the focusing intercritical NLS NLS $$\begin{aligned} i\partial _t u+\Delta _{x,y}u=-|u|^\alpha u \end{aligned}$$ on the semiperiodic waveguide manifold $$\mathbb {R}^d_x\times \mathbb {T}_y$$ with $$d\ge 5$$ and $$\alpha \in (\frac{4}{d},\frac{4}{d-1})$$ . In the case $$d\le 4$$ , with the aid of the semivirial vanishing theory (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), 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 $$d\ge 5$$ and the underlying domain possesses an anisotropic nature, the proof in Luo (Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), 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 Luo (Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022) continues to hold for all $$d\ge 5$$ . Together with Tzvetkov–Visciglia (Rev. Mat. Iberoam. 32:1163–1188, 2016) and the author (Luo in Normalized ground states and threshold scattering for focusing NLS on Rd $$\times $$ T via semivirial-free geometry, 2022), we thus give a complete characterization of the large data scattering for (NLS) in both defocusing and focusing case and in arbitrary dimension.