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On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation

  • Van Duong DinhEmail author
Article
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Abstract

We consider the focusing \(L^2\)-supercritical fractional nonlinear Schrödinger equation
$$\begin{aligned} i\partial _t u - (-\varDelta )^s u = -|u|^\alpha u, \quad (t,x) \in \mathbb {R}^+ \times \mathbb {R}^d, \end{aligned}$$
where \(d\ge 2, \frac{d}{2d-1} \le s <1\) and \(\frac{4s}{d}<\alpha <\frac{4s}{d-2s}\). By means of the localized virial estimate, we prove that the ground-state standing wave is strongly unstable by blowup. This result is a complement to a recent result of Peng–Shi (J Math Phys 59:011508, 2018) where the stability and instability of standing waves were studied in the \(L^2\)-subcritical and \(L^2\)-critical cases.

Keywords

Fractional nonlinear Schrödinger equation Standing wave Instability Localized virial estimate Blow-up 

Mathematics Subject Classification

35B44 35Q55 

Notes

Acknowledgements

The author would like to express his deep gratitude to his wife—Uyen Cong for her encouragement and support. He would like to thank his supervisor Prof. Jean-Marc Bouclet for the kind guidance and constant encouragement. He also would like to thank the reviewer for his/her helpful comments and suggestions.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Toulouse UMR5219Université Toulouse CNRSToulouse Cedex 9France
  2. 2.Department of MathematicsHCMC University of PedagogyHo Chi MinhVietnam

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