Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case
We consider the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation {i∂tu+14∂x4u=iλ∂x(|u|3u),t>0,x∈R,u(0,x)=u0(x),x∈R with critical nonlinearity, where λ∈R. We assume that the initial data are such that u0∈H1,1, with sufficiently small norm |u0|H1,1. We prove that there e...
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Published in | Nonlinear analysis Vol. 116; pp. 112 - 131 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Ltd
01.04.2015
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Subjects | |
Online Access | Get full text |
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Summary: | We consider the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation {i∂tu+14∂x4u=iλ∂x(|u|3u),t>0,x∈R,u(0,x)=u0(x),x∈R with critical nonlinearity, where λ∈R. We assume that the initial data are such that u0∈H1,1, with sufficiently small norm |u0|H1,1. We prove that there exists a unique global solution e−it4∂x4u∈C([0,∞);H1,1) of the Cauchy problem for the nonlinear fourth-order nonlinear Schrödinger equation such that |u(t)|L∞≤C(1+t)−14. Moreover we show that if the total mass 12π∫Ru0(x)dx≠0, then the large time asymptotics is determined by the self-similar solution. |
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Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
ISSN: | 0362-546X 1873-5215 |
DOI: | 10.1016/j.na.2014.12.024 |