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

Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term \begin{document}$ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $\end{document} , where \begin{document}$ t\in\mathbb{R} $\end{document} and \begin{document}$ x\in \mathbb{R}^n $\end{document} . First of all, for initial data \begin{document}$ \varphi(x)\in H^2(\mathbb{R}^{n}) $\end{document} , we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value \begin{document}$ \varphi(x) $\end{document} , 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.