Existence and stability of a solution blowing up on a sphere for an L2-supercritical nonlinear Schrödinger equation

Abstract

We consider the quintic two-dimensional focusing nonlinear Schrödinger equation ${\mathrm{iu}}_t=-\Delta u-\left|u\right|{}^{4}u$ which is $L^2$-supercritical. Even though the existence of finite-time blow-up solutions in the energy space $H^1$ is known, very little is understood concerning the singularity formation. Numerics suggest the existence of a stable blow-up dynamic corresponding to a self-similar blowup at one point in space. We prove the existence of a different type of dynamic and exhibit an open set among the $H^1$-radial distributions of initial data for which the corresponding solution blows up in finite time on a sphere. This is the first result of an explicit description of a blow-up dynamic in the $L^2$-supercritical setting