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

Antonini, Christophe

Differ. Integral Equ. 15, No. 6, 749-768 (2002).

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 \(\mathbb{R}^N\).

MSC:

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

Keywords:

nonlinear Schrödinger equation with critical power; blow-up solution; blow-up rate