Lower bounds for the \(L^2\) minimal periodic blow-up solutions of critical nonlinear Schrödinger equation. (English) Zbl 1016.35018

Summary: We consider the nonlinear Schrödinger equation with critical power \[ iu_t=-\Delta u-|u|^{4/N}u,\quad t\geq 0,\quad x\in \mathbb{T}^N \] (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 \(\mathbb{R}^N\).


35J10 Schrödinger operator, Schrödinger equation
35B40 Asymptotic behavior of solutions to PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)