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

Abstract

<p style='text-indent:20px;'>Considered herein is the well-posedness and stability for the Cauchy problem of the fourth-order Schrödinger equation with nonlinear derivative term <inline-formula><tex-math id="M1">\begin{document}$ iu_{t}+\Delta^2 u-u\Delta|u|^2+\lambda|u|^pu = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ t\in\mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ x\in \mathbb{R}^n $\end{document}</tex-math></inline-formula>. First of all, for initial data <inline-formula><tex-math id="M4">\begin{document}$ \varphi(x)\in H^2(\mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, we establish the local well-poseness and finite time blow-up criterion of the solutions, and give a rough estimate of blow-up time and blow-up rate. Secondly, under a smallness assumption on the initial value <inline-formula><tex-math id="M5">\begin{document}$ \varphi(x) $\end{document}</tex-math></inline-formula>, 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.</p>