On the two-power nonlinear Schr\"odinger equation with non-local terms in Sobolev-Lorentz spaces

TL;DR

This paper studies a class of two-power nonlinear Schrödinger equations with non-local terms in Sobolev-Lorentz spaces, establishing global existence, scattering, decay, and asymptotic stability results for solutions with singular initial data.

Contribution

It extends existing results by analyzing solutions in Sobolev-Lorentz spaces with non-local nonlinearities, including asymptotic self-similarity and stability, in a broader functional framework.

Findings

Proved global existence and scattering in Sobolev-Lorentz spaces.

Established decay properties and asymptotic stability of solutions.

Extended analysis to equations with non-local terms and multiple nonlinearities.

Abstract

We are concerned with the two-power nonlinear Schr\"odinger-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 [F. Weissler, ADE 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.