On a sharp lower bound on the blow-up rate for the $L^2$ critical nonlinear Schrödinger equation

Abstract:

We consider the $L^2$ critical nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^{\frac {4}{N}}u$ with initial condition in the energy space $u(0,x)=u_0\in H^1$ and study the dynamics of finite time blow-up solutions. In an earlier sequence of papers, the authors established for a certain class of initial data on the basis of dispersive properties in $L^2_{loc}$ a sharp and stable upper bound on the blow-up rate: $|\nabla u(t)|_{L^2}\leq C\left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}$. In an earlier paper, the authors then addressed the question of a lower bound on the blow-up rate and proved for this class of initial data the nonexistence of self-similar solutions, that is, $\lim _{t\to T}\sqrt {T-t}|\nabla u(t)|_{L^2}=+\infty .$ In this paper, we prove the sharp lower bound \[ |\nabla u(t)|_{L^2}\geq C \left (\frac {\log |\log (T-t)|}{T-t}\right )^{\frac {1}{2}}\] by exhibiting the dispersive structure in the scaling invariant space $L^2$ for this log-log regime. In addition, we will extend to the pure energy space $H^1$ a dynamical characterization of the solitons among the zero energy solutions.