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 $ 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.