Type II blow up for the energy supercritical NLS

TL;DR

This paper constructs smooth finite time blow-up solutions for the energy supercritical nonlinear Schrödinger equation in high dimensions, revealing universal singularity formation mechanisms through a new energy method.

Contribution

It introduces a robust energy method to construct type II blow-up solutions in the supercritical regime, extending previous heat equation techniques and highlighting the role of solitary waves.

Findings

Existence of finite time blow-up solutions with bounded sub-scaling norms.

Universal blow-up profile characterized by concentration phenomena.

New energy-based approach applicable to supercritical regimes.

Abstract

We consider the energy super critical nonlinear Schr\"odinger equation $$i\pa_tu+\Delta u+u|u|^{p-1}=0$$ in large dimensions $d\geq 11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime, we construct a family of smooth finite time blow up solutions which become singular via concentration of a universal profile with the so called type II quantized blow up rates. The essential feature of these solutions is that all norms below scaling remain bounded. Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation, which was done using maximum principle techniques following. Instead we develop a robust energy method, in continuation of the works in the energy and mass critical cases. This shades a new light on the essential role played by the solitary wave and its tail in the type II blow up mechanism, and the universality of the corresponding singularity formation in both energy critical and super critical regimes.