Energy critical Schrödinger equation with weighted exponential nonlinearity: Local and global well-posedness

Bensouilah, Abdelwahab; Draouil, Dhouha; Majdoub, Mohamed

doi:10.1142/s0219891618500194

Cited by 4 publications

(3 citation statements)

References 26 publications

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“…The situation in the case b > 0 is less understood. Recently, in [5] the authors established the global well-posedness in the energy space for 0 < b < 1. A natural question to ask then is the long time behavior of global solutions, that is the scattering.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Scattering in the weighted <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

Bensouilah¹,

Dinh²,

Majdoub³

2019

Communications on Pure &Amp; Applied Analysis

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Scattering in the weighted <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

Bensouilah¹,

Dinh²,

Majdoub³

2019

Communications on Pure &Amp; Applied Analysis

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“…This fact was confirmed and extensively studied in the last decade for α = 0, that is the 2D-NLS with exponential non-linearity. See [6,7,10,13,23] and the references therein. For the inverse-square potential and for both NLS and Klein-Gordon equations, see [11].…”

Section: ) Suppose Thatmentioning

confidence: 99%

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

Majdoub¹,

Saanouni²

2023

Preprint

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We consider the magnetic nonlinear inhomogeneous Schrödinger equationwhere α ∈ R \ Z, ̺ > 0, p > 1. We prove a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground state threshold in the inter-critical regime. The scattering is obtained by using the new approach of Dodson-Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, Proc. Am. Math. Soc. ( 2017)). This method is based on Tao's scattering criteria and Morawetz estimates. The novelty here is twice: we investigate the case ̺α = 0 and we consider general energy initial data (not necessarily radially symmetric). The particular case α = 0, known as INLS, was widely investigated in the few recent years. Moreover, the particular case ̺ = 0, which gives the homogeneous regime, was considered recently by X. Gao and C. Xu (Scattering theory for NLS with inverse-square potential in 2D, J. Math. Anal. Appl. ( 2020)), where the scattering is proved for spherically symmetric datum. In the radial framework, the above problem translate to the INLS with inverse square potential, which was widely investigated in space dimensions higher than three. The Hardy inequality |x|fails in two space dimensions. Thus, it is not clear how to treat the NLS with inverse square potential in H 1 for two space dimensions. This article seems to be the first one dealing with the NLS with Aharonov-Bohm magnetic potential in the inhomogeneous regime, namely ̺ = 0.

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“…We stress that the 2D energy‐critical counterpart has received more attention in the past decade. See, among others, previous studies [31–34].…”

Section: Introductionmentioning

confidence: 99%

Long‐time dynamics for the radial focusing fractional INLS

Majdoub

Saanouni

2023

Math Methods in App Sciences

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We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: where , , , and . We prove the ground state threshold of global existence and scattering versus finite time blowup of energy solutions in the inter‐critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson–Murphy. This method is based on Tao's scattering criteria and Morawetz estimates. We describe the threshold using some non‐conserved quantities in the spirit of the recent paper by Dinh. The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of a non‐local fractional Laplacian operator. The second one is the presence of a singular weight in the nonlinearity. The greater part of this paper is devoted to the scattering of global solutions in . The Lorentz spaces and the Strichartz estimates play crucial roles in our approach.

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Energy critical Schrödinger equation with weighted exponential nonlinearity: Local and global well-posedness

Abstract: We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearitywhere 0 < b < 1 and α = 2π(2 − b). We establish local and global well-posedness in the subcritical and critical regimes.

Cited by 4 publications

References 26 publications

Scattering in the weighted <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

Scattering in the weighted <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-space for a 2D nonlinear Schrödinger equation with inhomogeneous exponential nonlinearity

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

Long‐time dynamics for the radial focusing fractional INLS

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