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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 4 (September 2021)   > Article

Scattering theory in weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

Théorie de diffusion dans les espaces L2 pondérés pour une classe de l’équation de Schrödinger non-linéaire inhomogène défocalisée


VAN DUONG DINH
Université de Lille CNRS



Published on 6 September 2021   DOI : 10.21494/ISTE.OP.2021.0724

Abstract

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In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS)
itu+Δu+μ |x|b|u|αu=0,(t,x)×d
with b,α > 0. First, we revisit the local well-posedness in H1(d) for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. μ=1 when 0 < α < α where α=42bd2 for d3, and α= for d=1,2
by assuming that the initial data belongs to the weighted L2 space Σ={uH1(d): |x|uL2(d)}. Finally, we combine the local theory and the decaying property to show the scattering in Σ for the defocusing (INLS) in the case α < α < α, where α=42bd.

In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS)
itu+Δu+μ |x|b|u|αu=0,(t,x)×d
with b,α > 0. First, we revisit the local well-posedness in H1(d) for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. μ=1 when 0 < α < α where α=42bd2 for d3, and α= for d=1,2
by assuming that the initial data belongs to the weighted L2 space Σ={uH1(d): |x|uL2(d)}. Finally, we combine the local theory and the decaying property to show the scattering in Σ for the defocusing (INLS) in the case α < α < α, where α=42bd.

Inhomogeneous nonlinear Schrödinger equation Local well-posedness Decay solutions Virial identity Scattering Weighted L2 space

Decay solutions Local well-posedness Inhomogeneous nonlinear Schrödinger equation Virial identity Scattering Weighted L2 space

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