Mode stability for self-similar blowup of slightly supercritical NLS: I. low-energy spectrum

Abstract

We consider self-similar blowup for (NLS) $i{\partial }_{t}u+\Delta u+u|u{|}^{p-1}=0$ in $d\ge 1$ and slightly mass-supercritical range $0<s_c:=\frac{d}{2}-\frac{2}{p-1}\ll 1$. The existence and stability of such dynamics [Merle-Raphaël-Szeftel, 2010] and construction of suitable profiles [Bahri-Martel-Raphaël, 2021] lead to the question of asymptotic stability. Based on our previous work [Li, 2023], this nonlinear problem is reduced to linear mode stability of the matrix linearized operator. In this work, we prove mode stability for the low-energy spectrum in $d\ge 1$ as a perturbation of the linearized operator around ground state for mass-critical NLS. The main difficulty of this spectral bifurcation problem arises from the non-self-adjoint, relatively unbounded and high-dimensional nature, for which we exploit the Jost function argument from [Perelman, 2001], qualitative WKB analysis generalized from [Bahri-Martel-Raphaël, 2021], matched asymptotics method and uniform estimates for high spherical classes based on special functions.

155 pages. Comments are welcome