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

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Abstract

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.

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Acknowledgements

V.B. was partially supported by FCT Project PTDC/MAT-PUR/28177/2017, with national funds, and by CMUP (UID/MAT/00144/2019), which is funded by FCT with national (MCTES) and European structural funds through the programs FEDER, under the partnership agreement PT2020. L.C.F.F. was partially supported by CNPq and FAPESP, Brazil. A.P. was partially supported by CNPq Grants 402849/2016-7 and 303098/2016-3.

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Authors and Affiliations

  1. Instituto de Matemática, Universidade Federal da Bahia, Av. Adhemar de Barros, Ondina, Salvador, Bahia, 40170-110, Brazil

    Vanessa Barros

  2. IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária, Campinas, São Paulo, 13083-859, Brazil

    Lucas C. F. Ferreira & Ademir Pastor

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  1. Vanessa Barros
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Barros, V., Ferreira, L.C.F. & Pastor, A. On the two-power nonlinear Schrödinger equation with non-local terms in Sobolev–Lorentz spaces. Nonlinear Differ. Equ. Appl. 26, 39 (2019). https://doi.org/10.1007/s00030-019-0584-4

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