Remark on the global existence and large time asymptotics of solutions for the quadratic NLS

Nakao Hayashi, Pavel I. Naumkin

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)

Abstract

We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions i ∂tu+12Δu=|u|2,t>0,x R4,u(0,x)= u0(x),x R4. We prove that if the initial data u0 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.

Original languageEnglish
Pages (from-to)6950-6964
Number of pages15
JournalNonlinear Analysis, Theory, Methods and Applications
Volume74
Issue number18
DOIs
Publication statusPublished - 2011 Dec
Externally publishedYes

Keywords

  • Global existence
  • Nonlinear Schrödinger equations
  • Quadratic nonlinearities

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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