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On the well-posedness and stability for the fourth-order Schrödinger equation with nonlinear derivative term

Huafei Di is supported by the NSF of China (11801108, 11801495), the Scientific Program of Guangdong Province (2021A1515010314), and the College Scientific Research Project (YG2020005) of Guangzhou University
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  • Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term $ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $, where $ t\in\mathbb{R} $ and $ x\in \mathbb{R}^n $. First of all, for initial data $ \varphi(x)\in H^2(\mathbb{R}^{n}) $, we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value $ \varphi(x) $, we demonstrate the global well-posedness of the solutions by applying two different methods, and at the same time give the scattering behavior of the solutions. Finally, based on founded a priori estimates, we investigate the stability of solutions by the short-time and long-time perturbation theories, respectively.

    Mathematics Subject Classification: Primary: 35K30; Secondary: 35A01, 35B44, 35B40.

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