Random data Cauchy problem for the fourth order Schrödinger equation with the second order derivative nonlinearities

2020 ◽  
Vol 190 ◽  
pp. 111608
Author(s):  
Mingjuan Chen ◽  
Shuai Zhang
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Second Order ◽  
Order Derivative ◽  
Random Data

scholarly journals GLOBAL WELL-POSEDNESS FOR THE GENERALISED FOURTH-ORDER SCHRÖDINGER EQUATION

2012 ◽  
Vol 85 (3) ◽  
pp. 371-379 ◽  
Author(s):  
YUZHAO WANG
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Fourth Order ◽  
Schrodinger Equation ◽  
Vries Equation ◽  
Well Posedness ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Image Position ◽  
Appl Math

AbstractWe study the Cauchy problem for the generalised fourth-order Schrödinger equation for data u0 in critical Sobolev spaces $\dot {H}^{1/2-3/2k}$. With small initial data we obtain global well-posedness results. Our proof relies heavily on the method developed by Kenig et al. [‘Well-posedness and scattering results for the generalised Korteweg–de Vries equation via the contraction principle’, Commun. Pure Appl. Math.46 (1993), 527–620].

scholarly journals The Property of the Solution about Cauchy Problem for Fourth-Order Schrödinger Equation with Critical Time-Oscillating Nonlinearity

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Cuihua Guo ◽  
Hongping Ren ◽  
Shulin Sun
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Periodic Function ◽  
Asymptotic Property ◽  
Fourth Order ◽  
Sobolev Spaces ◽  
Schrodinger Equation ◽  
Critical Time ◽  
The Cauchy Problem

We study the property of the solution in Sobolev spaces for the Cauchy problem of the following fourth-order Schrödinger equation with critical time-oscillating nonlinearityiut+Δ2u+θ(ωt)|u|8/(n-4)u=0, whereω,t∈R,x∈Rn, andθis a periodic function. We obtain the asymptotic property of the solution for the above equation asω→∞under some conditions.

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