Higher Order Dispersive Equations:Unique Continuations And Kernel Estimates

This thesis studies unique continuation properties and kernel estimates for higher order dispersive equations.We first discuss two types of unique continuation properties for higher order Schr???dinger equations:global unique continuation across non-characteristic hyper-plane and quantitative unique continuation.At last,we consider point-wise kernel estimates and global smoothing effects for general dispersive equations.The thesis comprises 5 chapters.Chapter 1 first introduces the physical and mathematical backgrounds of dispersive equations as well as their generalizations,the relation between Schr???dinger operator and unique continuation property along with its development in elliptic equations,and the prob-lem sources of kernel estimates for dispersive equations relating oscillatory integrals.Then we place our main subjects studied in this thesis.Chapter 2 is to study global unique continuation across non-characteristic hyperplane{?t,x??R¹⁺ⁿ;|t|<A,x_n=0}for the time-dependent higher order Schr???dinger operator i^-1?_t+?-?_x?^m.Questions of this type were usually answered locally in second order cases,and in higher order cases in one dimension,while no enough attention has been paid to the higher order cases in high dimension.Our approach is to establish Carleman estimates with two parameters to gain global results.We also give similar results for higher order parabolic equations,as well as local and weak unique continuation results.Chapter 3 studies a type of quantitative unique continuation property for the higher order Schr???dinger equation i^-1?_tu=D_x^2mu+V?x?u in one dimension,while the required vanishing of the solution is only u?0,x?,u?1,x??L²(e^{?|x|2m/?2m-1?}dx),where?>0 is sufficiently large.This type of results is closely related to the Hardy's uncertainty principle in second order case,which has been a major topic in the recent two decades.In higher order case,we first use higher order heat kernel estimates and approximation by higher order heat equations to establish a uniform energy estimate in weighted space L²(e^{?|x|2m/?2m-1?}dx)for the Schr???dinger equation of order 2m,the part of which actually works in high dimension;and we finally establish a one dimensional quantitative Carleman estimate which matches the uniform weighted energy estimate to prove our uniqueness result.Chapter 4 considers general dispersive equations?_tu=ia?D_x?u with function type symbols.Under the philosophy of stationary phase method,we prove point-wise estimates by time and space parameters for two types of oscillatory integrals,which,for a large class of a,leads to estimates for dispersive kernel F^-1(e^ita?·?)?x?and its spatial derivatives of certain orders,and our results match many existing works in special cases.Accordingly we obtain global smoothing effects for dispersive equations,including the L^p-L^qtype and Strichartz type.We also consider fractional Schr???dinger equations with complex valued potentials and establish the L^pestimates.Chapter 5 gives further discussion about the previous three problems.