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

We are concerned with the two-power nonlinear Schrodinger-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 L2(-)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 systems26CNPQ - Conselho Nacional de Desenvolvimento Científico e TecnológicoFAPESP – Fundação de Amparo à Pesquisa Do Estado De São Paulo402849/2016-7, 303098/2016-3não te