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 {document}iuₜ+Δ² u-uΔ|u|²+λ|u|ᵖu = 0{document} , where {document}t{document} and {document}x∈ Rⁿ{document} . First of all, for initial data {document}φ(x)∈ H²(Rⁿ){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 {document}φ(x){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.