Qualitative Research For Two Types Of Schr(?)dinger Equations With Nonlinear Second-order Derivative Terms

This thesis considers the Cauchy problems for two types of nonlinear Schr¨o-dinger equations(NLS).More precisely,we devote to the well-posedness of the Cauchy problem for a nonlinear fourth-order Schr¨odinger equation(NL4S)with a second-order nonlinear derivative term,and the sharp thresholds of global existence and blow-up at finite time for the solution to the Cauchy problem of superfluid equations.The thesis is organised as follows:In Chapter 1,physical backgrounds,current research on NLS and main results are introduced.Chapter 2 considers the following Cauchy problem for a nonlinear fourth-order Schr¨odinger equation(NL4S)with a second-order nonlinear derivative term:We investigate the well-posedness,properties of blow-up solution,scatterings and stability.For the local well-posedness,we apply the established Strichartz esti-mates for NL4S to set up a series of a priori estimates,which coupled with Kato’s method yields the local well-posedness.Moreover,we obtain the blow-up alterna-tive,which is vital for the proofs of following results.Also,based on some a priori estimates,we can get the global well-posedness and scatterings under smallness of the initial value.Furthermore,we argue the stability of this problem in some sense(the exact meaning of“stability”is introduced in Chapter 2).Chapter 3 concerns the Cauchy problem for the following superfluid films equations:The sharp thresholds of global existence and blow-up at finite time of the solu-tion to the above problem will be investigated.We construct a range of con-strained functional minimization problems and then obtain corresponding invari-ant sets(they are also called invariant manifolds).Combined with some inequalities and concavity method,consequently one have the sharp thresholds result for global existence and blow-up at finite time of the solution to the above-mentioned prob-lem.