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The Dynamics of the 3D Radial NLS with the Combined Terms

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Abstract

In this paper, we show the scattering and blow-up result of the radial solution with the energy below the threshold for the nonlinear Schrödinger equation (NLS) with the combined terms

$$iu_{t} + \Delta{u} = -|u|^{4}u + |u|^{2}u \qquad \qquad \qquad \qquad {\rm (CNLS)}$$

in the energy space \({{H^{1}(\mathbb{R}^{3})}}\) . The threshold is given by the ground state W for the energy-critical NLS: iu t +  Δu =  −|u|4 u. This problem was proposed by Tao, Visan and Zhang in (Comm PDEs 32:1281–1343, 2007). The main difficulty is lack of the scaling invariance. Illuminated by (Ibrahim et al., in Analysis & PDE 4(3):405–460, 2011), we need to give the new radial profile decomposition with the scaling parameter, then apply it to the scattering theory. Our result shows that the defocusing, \({{\dot{H}^{1}}}\) -subcritical perturbation |u|2 u does not affect the determination of the threshold of the scattering solution of (CNLS) in the energy space.

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Authors and Affiliations

  1. Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing, 100088, China

    Changxing Miao & Guixiang Xu

  2. University of Science and Technology of China, Hefei, China

    Lifeng Zhao

Authors
  1. Changxing Miao
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  2. Guixiang Xu
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  3. Lifeng Zhao
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Correspondence to Changxing Miao.

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Communicated by P. Constantin

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Miao, C., Xu, G. & Zhao, L. The Dynamics of the 3D Radial NLS with the Combined Terms. Commun. Math. Phys. 318, 767–808 (2013). https://doi.org/10.1007/s00220-013-1677-2

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  • DOI: https://doi.org/10.1007/s00220-013-1677-2

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