Blow-up of 𝐻¹ solutions for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity

Abstract

We investigate the blow-up of solutions in H 1 ( R ) {H^1}({\mathbf {R}}) with negative energy for the one-dimensional nonlinear Schrödinger equation with critical power nonlinearity: \[ i u t = − u x x − | u | 4 u , t > 0 , x ∈ R , u ( 0 , x ) = u 0 ( x ) , x ∈ R . . \begin {gathered} i{u_t} = - {u_{xx}} - |u{|^4}u,\quad t > 0,x \in {\mathbf {R}}, \hfill \\ u(0,x) = {u_0}(x),\quad x \in {\mathbf {R}}. \hfill \\ \end {gathered} . \] In our result we remove the weight condition x u 0 ∈ L 2 ( R ) x{u_0} \in {L^2}({\mathbf {R}}) , which was always assumed to show the blow-up of solutions in the previous papers.