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Blow-up criteria for linearly damped nonlinear Schrödinger equations

  • * Corresponding author: Van Duong Dinh

    * Corresponding author: Van Duong Dinh
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  • We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations

    $ i\partial_t u + \Delta u + i a u = \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb R^N, $

    where $ a>0 $ and $ \alpha>0 $. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up $ H^1 $ solutions to the focusing problem in the mass-critical and mass-supercritical cases.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35Q44.


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