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Small data global well-posedness for the inhomogeneous biharmonic NLS in Sobolev spaces

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  • In this paper, we study the Cauchy problem for the inhomogeneous biharmonic nonlinear Schrödinger equation (IBNLS)

    $ iu_{t} +\Delta^{2} u = \lambda |x|^{-b}|u|^{\sigma}u, u(0) = u_{0} \in H^{s} (\mathbb R^{d}), $

    where $ \lambda \in \mathbb R $, $ d\in \mathbb N $, $ 0<s<\min \{2+\frac{d}{2}, \frac{3}{2}d\} $ and $ 0<b<\min\{4, d, \frac{3}{2}d-s, \frac{d}{2}+2-s\} $. Under some regularity assumption for the nonlinear term, we prove that the IBNLS equation is globally well-posed in $ H^{s}(\mathbb R^{d}) $ if $ \frac{8-2b}{d}<\sigma< \sigma_{c}(s) $ and the initial data is sufficiently small, where $ \sigma_{c}(s) = \frac{8-2b}{d-2s} $ if $ s<\frac{d}{2} $, and $ \sigma_{c}(s) = \infty $ if $ s\ge \frac{d}{2} $.

    Mathematics Subject Classification: Primary: 35Q55; Secondary: 35A01.

    Citation:

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