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On fractional nonlinear Schrödinger equation with combined power-type nonlinearities

  • * Corresponding author: Binhua Feng

    * Corresponding author: Binhua Feng
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  • We undertake a comprehensive study for the fractional nonlinear Schrödinger equation

    $ i\partial_t u - (-\Delta)^s u = \mu_1 |u|^{\alpha_1} u + \mu_2 |u|^{\alpha_2} u, \quad u(0) = u_0, $

    where $ \frac{d}{2d-1} \leq s <1 $, $ 0 < \alpha_1 <\alpha_2 < \frac{4s}{d-2s} $. Firstly, we establish the local and global well-posedness results for non-radial and radial $ H^s $ initial data, radial $ \dot{H}^{s_c}\cap \dot{H}^s $ initial data, where $ s_c = \frac{d}{2}-\frac{2s}{\alpha_2} $. Secondly, we study the asymptotic behavior of global radial $ H^s $ solutions. Of particular interest is the $ L^2 $-critical case and the results in this case are conditional on a conjectured global existence and spacetime estimate for the $ L^2 $-critical fractional nonlinear Schrödinger equation. Thirdly, we obtain sufficient conditions about existence of blow-up radial $ \dot{H}^{s_c} \cap \dot{H}^s $ solutions, and derive the sharp threshold mass of blow-up and global existence for this equation with $ L^2 $-critical and $ L^2 $-subcritical nonlinearities. Finally, we obtain the dynamical behaviour of blow-up solutions in both $ L^2 $-critical and $ L^2 $-supercritical cases, including mass-concentration and limiting profile.

    Mathematics Subject Classification: Primary: 35B44; Secondary: 35Q55.


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