Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

We consider the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation {i∂tu+14∂x4u=iλ∂x(|u|3u),t>0,xεR,u(0,x)=u0(x),xεR with critical nonlinearity, where λεR. We assume that the initial data are such that u0εH1,1, with sufficiently small norm |u0|H1,1. We prove that there exists a unique global solution e-it4∂x4uεC([0,∞);H1,1) of the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation such that |u(t)|L∞ ≤ C (1+t)-1/4. Moreover we show that if the total mass 12π∫R u0(x)dx ≈ 0, then the large time asymptotics is determined by the self-similar solution.

Original languageEnglish
Pages (from-to)112-131
Number of pages20
JournalNonlinear Analysis, Theory, Methods and Applications
Volume116
DOIs
Publication statusPublished - 2015 Apr
Externally publishedYes

Keywords

  • Fourth-order nonlinear
  • Large time asymptotics
  • Schrödinger equation
  • Self similar solution

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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