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

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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.

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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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  1. Institut de Mathématiques de Toulouse UMR5219, Université Toulouse CNRS, 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. On instability of standing waves for the mass-supercritical fractional nonlinear Schrödinger equation. Z. Angew. Math. Phys. 70, 58 (2019). https://doi.org/10.1007/s00033-019-1104-4

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  • DOI: https://doi.org/10.1007/s00033-019-1104-4

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