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18 pages, 351 KiB

Open AccessArticle

The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

by Jin Xie, Han Yang, Dingshi Li and Sen Ming

Fractal Fract. 2024, 8(1), 19; https://doi.org/10.3390/fractalfract8010019 - 26 Dec 2023

Viewed by 1372

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term. Full article

(This article belongs to the Special Issue Stochastic and Fractional Differential Equations: Attractor, Invariant Measure and Their Relationship)

10 pages, 302 KiB

Open AccessArticle

Exponential Scattering for a Damped Hartree Equation

by Talal Alharbi, Salah Boulaaras and Tarek Saanouni

Fractal Fract. 2023, 7(1), 51; https://doi.org/10.3390/fractalfract7010051 - 1 Jan 2023

Cited by 2 | Viewed by 1210

Abstract

This note studies the linearly damped generalized Hartree equation [...] Read more.

This note studies the linearly damped generalized Hartree equation

$i \overset{?}{u} ? {(? ?)}^{s} u + i a u = \pm {| u |}^{p ? 2} (J_{?} {? | u |}^{p}) u, 0 < s < 1, a > 0, p ? 2 .$ Indeed, one proves an exponential scattering of the energy global solutions, with spherically symmetric datum. This means that, for large time, the solution goes exponentially to the solution of the associated free problem

$i \overset{?}{u} ? {(? ?)}^{s} u + i a u = 0$ , in

$H^{s}$ norm. The radial assumption avoids a loss of regularity in Strichartz estimates. The exponential scattering, which means that

$v : = e^{a t} u$ scatters in

$H^{s}$ , is proved in the energy sub-critical defocusing regime and in the mass-sub-critical focusing regime. This result is presented because of the gap due to the lack of scattering in the mass sub-critical regime, which seems not to be well understood. In this manuscript, one needs to overcome three technical difficulties which are mixed together: the first one is a fractional Laplace operator, the second one is a Choquard (non-local) source term, including the Hartree-type term when

$p = 2$ and the last one is a damping term

$i a u$ . In a work in progress, the authors investigate the exponential scattering of global solutions to the above Schrödinger problem, with different kind of damping terms. Full article

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