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

• 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
• ISSN: 1021-9722; 1420-9004
• DOI: 10.1007/s00030-019-0584-4

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.