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

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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 \(\alpha _\star = \frac{4-2b}{d}\).

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse UMR5219, CNRS, Université Toulouse, 31062, Toulouse Cedex 9, France

    Van Duong Dinh

  2. Department of Mathematics, HCMC University of Pedagogy, 280 An Duong Vuong, Ho Chi Minh, Vietnam

    Van Duong Dinh

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Dinh, V.D. Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation. J. Evol. Equ. 19, 411–434 (2019). https://doi.org/10.1007/s00028-019-00481-0

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