Lower bounds for the $L^2$ minimal periodic blow-up solutions of critical nonlinear Schrödinger equation

Abstract

We consider the nonlinear Schrödinger equation with critical power $iu_t=-\Delta u-|u|^{4/N}u$, $t\geq 0$ and $x\in\mathbb T^N$ (the space-periodic case) in $H^1$. We consider a blow-up solution with minimal mass. We obtain in this context an optimal lower bound for the blow-up rate (that is, for $ | {\nabla u(t)}| _{L^2}$), and we observe that this lower bound equals the blow-up rate (which is explicitly known) of the minimal blow-up solutions in $\mathbb R^N$.