The Unique Continuation Property Of Solutions For Cauchy Problem Of Five Order(3+1)Dimensional Kadomtsev-Petviashvili ? Equation

In this thesis,we consider the solutions of the Cauchy problem for a nonlin-ear partial differential equation:the fifth-order(3+1)-dimensional Kadomtsev-Petviashvili ? equation.The unique continuation property is one of important properties of the solutions to the integrable systems and the methods of proving the unique continuation property for the nonlinear partial differential equation-s(PDE)have been explored and improved constantly.It is well known that the classical methods are:Carleman estimates,Fourier transformation,Bessel po-tential operator and inverse scattering transform methods.In this thesis,we emphatically recall the method of using the Fourier transformation and Car-leman estimates to prove the unique continuation property of the initial value problem associated with a fifth-order(3+1)-dimensional Kadomtsev-Petviashvili II equation.The results are expressed respectively:if a sufficiently smooth solu-tion u =u(x,y,z,t)to the initial value problem associated with the fifth-order(3 + 1)-dimensional Kadomtsev-Petviashvili ? equation is supported compact-ly in a nontrivial time interval,then it vanishes identically;if a sufficiently s-mooth solution u = u(x,y,z,t)to the initial value problem associated with the fifth-order(3 + 1)-dimensional Kadomtsev-Petviashvili ? equation has compact support for two different instants of time,then it is identically zero.The thesis is arranged as follows:At first,we briefly present the research background and the significance of the unique continuation property.In addition,the progress of the methods for this property are also mentioned both at home and abroad.Secondly,we provide the preliminary definitions and theories needed in this thesis.Thirdly,we describe the lemmas,corollaries and the proof of the fifth-order(3 + 1)-dimensional Kadomtsev-Petviashvili II equation by using Fourier transformation.At last,we describe the lemmas,corollaries and the proof of the fifth-order(3 + 1)-dimensional Kadomtsev-Petviashvili II equation by using Carleman esti-mates.