On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces

We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev–Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting wit...

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Bibliographic Details
Published in:Nonlinear differential equations and applications 2019-10, Vol.26 (5), p.1-29, Article 39
Main Authors: Barros, Vanessa, Ferreira, Lucas C. F., Pastor, Ademir
Format: Article
Language:English
Subjects:
Analysis
Asymptotic properties
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear equations
Scaling
Schrodinger equation
Self-similarity
Stability analysis
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Summary:We are concerned with the two-power nonlinear Schrödinger-type equations with non-local terms. We consider the framework of Sobolev–Lorentz spaces which contain singular functions with infinite-energy. Our results include global existence, scattering and decay properties in this singular setting with fractional regularity index. Solutions can be physically realized because they have finite local L 2 -mass. Moreover, we analyze the asymptotic stability of solutions and, although the equation has no scaling, show the existence of a class of solutions asymptotically self-similar w.r.t. the scaling of the single-power NLS-equation. Our results extend and complement those of Weissler (Adv Differ Equ 6(4):419–440, 2001), particularly because we are working in the larger setting of Sobolev-weak- L p spaces and considering non-local terms. The two nonlinearities of power-type and the generality of the non-local terms allow us to cover in a unified way a large number of dispersive equations and systems.
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-019-0584-4