Contents Online

# Cambridge Journal of Mathematics

## Volume 3 (2015)

### Number 4

### Type II blow up for the energy supercritical NLS

Pages: 439 – 617

DOI: https://dx.doi.org/10.4310/CJM.2015.v3.n4.a1

#### Authors

#### Abstract

We consider the energy super critical nonlinear Schrödinger equation\[i\partial_tu+\Delta u+u|u|^{p-1}=0\]in large dimensions $d\geq11$ with spherically symmetric data. For all $p>p(d)$ large enough, in particular in the super critical regime\[s_c=\frac d2-\frac{2}{p-1}>1,\]we construct a family of $\mathcal C^{\infty}$ finite time blow up solutions which become singular via concentration of a universal profile\[u(t,x)\sim\frac{1}{\lambda(t)^{\frac2{p-1}}}Q\left(\frac{r}{\lambda(t)}\right)e^{i\gamma(t)}\]with the so called type II quantized blow up rates:\[\lambda(t)\sim c_u(T-t)^{\frac\ell\alpha}, \ \ \ell\in\Bbb N^*, \ \ 2\ell>\alpha=\alpha(d,p).\]The essential feature of these solutions is that all norms below scaling remain bounded\[\limsup_{t\uparrow T}\|\nabla^su(t)\|_{L^2}<+\infty\ \ \mbox{for}\ \ 0\leq s<s_c.\]Our analysis fully revisits the construction of type II blow up solutions for the corresponding heat equation in [15], [34], which was done using maximum principle techniques following [26]. Instead we develop a robust energy method, in continuation of the works in the energy critical case [38], [31], [39], [40] and the $L^2$ critical case [22]. 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.

Published 9 December 2015