TY - JOUR

T1 - Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case

AU - Hayashi, Nakao

AU - Naumkin, Pavel I.

N1 - Funding Information:
The work of N.H. is partially supported by JSPS KAKENHI Grant Number 25220702 . The work of P.I.N. is partially supported by CONACYT project I0017-0166579 and PAPIIT project IN100113.
Publisher Copyright:
© 2015 Elsevier Ltd.

PY - 2015/4

Y1 - 2015/4

N2 - 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.

AB - 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.

KW - Fourth-order nonlinear

KW - Large time asymptotics

KW - Schrödinger equation

KW - Self similar solution

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U2 - 10.1016/j.na.2014.12.024

DO - 10.1016/j.na.2014.12.024

M3 - Article

AN - SCOPUS:84922724486

VL - 116

SP - 112

EP - 131

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

ER -