On stability of self-similar blowup for mass supercritical NLS

Abstract

We consider the mass supercritical (NLS) in dimension $d\ge 1$ in the mass-supercritical range. The existence of self-similar blow up dyamics is known [Merle-Rapha\"el-Szeftel, 2010], and suitable self-similar blow up profiles were constructed [Bahri-Martel-Rapha\"el, 2021]. In this work, we prove the finite codimensional nonlinear asymptotic stability of a large class of self-similar profiles. The heart of the proof is, following the approach of Beceanu [Beceanu, 2011], the derivation of Strichartz dispersive estimates for matrix operators with a deformed Laplacian $\Delta_b = \Delta + ib\left(\frac d2 + x\cdot\nabla\right)$ in homogeneous Sobolev spaces and energy space $H^1$. The deformed Laplacian $\Delta_b$ arises from the renormalization and its operator group $e^{it\Delta_b}$ exhibits self-similar dispersion, which not only recovers the free Strichartz but also enables an extension of resolvent families. Compared with Strichartz estimates based on $\Delta$, this one has a larger admissible region, works for arbitrarily small polynomial decaying potential, and requires no spectral assumption.