Global existence for the defocusing Sobolev critical Schrödinger equation under the finite variance condition of initial data

In unbounded domain RN,N≥1, we consider the defocusing nonlinear Schrödinger equation with power-type nonlinearity containing −|u|4N−2su. For 2s∗=2N−2s, with s<N2 and s≤1, the corresponding scaling invariant space is homogeneous Sobolev Ḣxs(RN) and in this case implies that the critical regulari...

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Bibliographic Details
Published inApplied mathematics letters Vol. 134; p. 108332
Main Author Au, Vo Van
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.12.2022
Subjects
Conservation laws
Critical problems
Global existence
Nonlinear Schrödinger equation
Conservation laws
Critical problems
Global existence
Nonlinear Schrödinger equation
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Summary:In unbounded domain RN,N≥1, we consider the defocusing nonlinear Schrödinger equation with power-type nonlinearity containing −|u|4N−2su. For 2s∗=2N−2s, with s<N2 and s≤1, the corresponding scaling invariant space is homogeneous Sobolev Ḣxs(RN) and in this case implies that the critical regularity is sc=N2−12s∗≡s. Under the finite variance condition of initial data and the solutions of the problem satisfy the pseudo-conformal conservation law, we investigate the global existence of the solutions in Ltq−Lxp spaces for some constants p,q≥2 depending on N,s. The new results of this study encompass the existing results on mass critical (sc=0) in Dodson (2012) and energy critical (sc=1) in Killip and Visan (2010).
ISSN:0893-9659
1873-5452
DOI:10.1016/j.aml.2022.108332