$ H^2 $ blowup result for a Schrödinger equation with nonlinear source term

  • Received: 01 February 2020 Revised: 01 April 2020
  • Primary: 35L71; Secondary: 35B30, 35B44

  • In this paper, we consider the nonlinear Schrödinger equation on $ \mathbb{R}^N, N\ge1 $,

    $ \partial_tu = i\Delta u+\lambda|u|^\alpha u , $

    with $ H^2 $-subcritical nonlinearities: $ \alpha>0, (N-4)\alpha<4 $ and Re$ \lambda>0 $. For any given compact set $ K\subset\mathbb{R}^N $, we construct $ H^2 $ solutions that are defined on $ (-T, 0) $ for some $ T>0 $, and blow up exactly on $ K $ at $ t = 0 $. We generalize the range of the power $ \alpha $ in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.

    Citation: Xuan Liu, Ting Zhang. $ H^2 $ blowup result for a Schrödinger equation with nonlinear source term[J]. Electronic Research Archive, 2020, 28(2): 777-794. doi: 10.3934/era.2020039

    Related Papers:

  • In this paper, we consider the nonlinear Schrödinger equation on $ \mathbb{R}^N, N\ge1 $,

    $ \partial_tu = i\Delta u+\lambda|u|^\alpha u , $

    with $ H^2 $-subcritical nonlinearities: $ \alpha>0, (N-4)\alpha<4 $ and Re$ \lambda>0 $. For any given compact set $ K\subset\mathbb{R}^N $, we construct $ H^2 $ solutions that are defined on $ (-T, 0) $ for some $ T>0 $, and blow up exactly on $ K $ at $ t = 0 $. We generalize the range of the power $ \alpha $ in the result of Cazenave, Han and Martel [5]. The proof is based on the energy estimates and compactness arguments.



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