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Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

  • Van Duong DinhEmail author
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Abstract

In this paper, we consider a class of the defocusing inhomogeneous nonlinear Schrödinger equation
$$\begin{aligned} i\partial _t u + \varDelta u - |x|^{-b} |u|^\alpha u = 0, \quad u(0)=u_0 \in H^1, \end{aligned}$$
with \(b, \alpha >0\). We first study the decaying property of global solutions for the equation when \(0<\alpha <\alpha ^\star \) where \(\alpha ^\star = \frac{4-2b}{d-2}\) for \(d\ge 3\). The proof makes use of an argument of Visciglia (Math Res Lett 16(5):919–926, 2009). We next use this decay to show the energy scattering for the equation in the case \(\alpha _\star<\alpha <\alpha ^\star \), where \(\alpha _\star = \frac{4-2b}{d}\).

Keywords

Inhomogeneous nonlinear Schrödinger equation Scattering theory Virial inequality Decaying solution 

Mathematics Subject Classification

35G20 35G25 35Q55 

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Toulouse UMR5219, CNRSUniversité ToulouseToulouse Cedex 9France
  2. 2.Department of MathematicsHCMC University of PedagogyHo Chi MinhVietnam

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