The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation

Abstract

We consider the critical nonlinear Schrödinger equation $iu_t=-\Delta u-|u|^{\frac{4}{N}}u$ with initial condition $u(0,x)=u_0$ in dimension $N=1$. For $u_0\in H^1$, local existence in the time of solutions on an interval $[0,T)$ is known, and there exist finite time blow-up solutions, that is, $u_0$ such that $\lim_{t\uparrow T\lt +\infty}|u_x(t)|_{L^2}=+\infty$. This is the smallest power in the nonlinearity for which blow-up occurs, and is critical in this sense. The question we address is to understand the blow-up dynamic. Even though there exists an explicit example of blow-up solution and a class of initial data known to lead to blow-up, no general understanding of the blow-up dynamic is known. At first, we propose in this paper a general setting to study and understand small, in a certain sense, blow-up solutions. Blow-up in finite time follows for the whole class of initial data in $H^1$ with strictly negative energy, and one is able to prove a control from above of the blow-up rate belowthe one of the known explicit explosive solution which has strictly positive energy. Under some positivity condition on an explicit quadratic form, the proof of these results adapts in dimension $N>1$.