The global well‐posedness for nonlinear Schrödinger equation with the delta potential in the energy space

摘要

In this paper, we concern with the global well‐posedness for nonlinear Schrödinger equation with a Dirac delta potential ( δ$$ delta $$‐NLS): i∂tu+∂x2u+δu+up−1u=0,(t,x)∈ℝ+×ℝ,p≥5.$$ i{partial}_tuamp;amp;#x0002B;{partial}_xamp;amp;#x0005E;2uamp;amp;#x0002B;updelta uamp;amp;#x0002B;{leftamp;amp;#x0007C;urightamp;amp;#x0007C;}amp;amp;#x0005E;{p-1}uamp;amp;#x0003D;0,kern3.0235pt left(t,xright)in {mathbb{R}}amp;amp;#x0005E;{amp;amp;#x0002B;}times mathbb{R},kern3.0235pt pge 5. $$ Firstly, we give a new variational characterization of the solitary waves of ( δ$$ delta $$‐NLS) by making use of the symmetric decreasing rearrangement and constrained minimization argument. Secondly, we construct an invariant set Hα,β,ω+$$ {mathcal{H}}_{alpha, beta, omega}amp;amp;#x0005E;{amp;amp;#x0002B;} $$, via the variational characterization of the solitary waves. Finally, we show the solutions, with initial data in Hα,β,ω+$$ {mathcal{H}}_{alpha, beta, omega}amp;amp;#x0005E;{amp;amp;#x0002B;} $$, exist globally.