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Energy scattering for a class of inhomogeneous nonlinear Schrödinger equation in two dimensions

Van Duong Dinh

Van Duong Dinh

Laboratoire Paul Painlevé, UMR 8524, Université de Lille CNRS, 59655 Villeneuve d’Ascq Cedex, France

Department of Mathematics, HCMC University of Education, 280 An Duong Vuong, Ho Chi Minh, Vietnam

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219891621500016Cited by:10 (Source: Crossref)

Abstract

We consider a class of $L^{2}$ -supercritical inhomogeneous nonlinear Schrödinger equations in two dimensions

i \partial_{t} u + Δ u = \pm | x |^{- b} | u |^{α} u, (t, x) \in ℝ \times ℝ^{2},

where

0 < b < 1

and

α > 2 - b

. Using a new approach of Arora et al. [Scattering below the ground state for the 2D radial nonlinear Schrödinger equation, Proc. Amer. Math. Soc. 148 (2020) 1653–1663], we show the energy scattering for the equation with radially symmetric initial data. In the focusing case, our result extends the one of Farah and Guzmán [Scattering for the radial focusing INLS equation in higher dimensions, Bull. Braz. Math. Soc. (N.S.) 51 (2020) 449–512] to the whole range of

b

where the local well-posedness is available. In the defocusing case, our result extends the one in [V. D. Dinh, Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation, J. Evol. Equ. 19(2) (2019) 411–434], where the energy scattering for non-radial initial data was established in dimensions

N \geq 3

Communicated by F. Merle

Keywords:

AMSC: 35Q55, 35P25

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