Type II blow up for the energy supercritical NLS

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.

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Published 9 December 2015