Unique Continuation Property Of Solutions To Cauchy Problem For A Class Of Fifth-order Korteweg-de-Vries Equations

In this thesis, we consider the unique continuation property of the Cauchy problem for a kind of nonlinear partial differential equations.We know that the unique continuation property is one of important proper-ties of the solutions to the integrable systems. The methods of proving the unique continuation property for the nonlinear partial differential equations (PDE) have been explored and applied consistently. It is well known that applying Car-leman estimates, Fourier transformation, Besscl potential operator and inverse scattering transformation are four typical methods. In this thesis, we recalled the method of using Fourier transform to discuss the unique continuation property of the initial value problem associated with a class of fifth-order Korteweg-de-Vries (KdV) equation. It is proved that, if a sufficiently smooth solution u= u(x, t) to the initial value problem associated with the fifth-order KdV equations is supported compactly in a nontrivial time interval, then it vanishes identically.The thesis is arranged as follows:At first, we briefly describe the significance of the unique continuation prop-erty and the progress of the methods for this property are also mentioned both at home and abroad. In addition, we classify the corresponding results of the unique continuation property.Secondly, we illustrate related definitions and notations of the unique con-tinuation property of establishing the unique continuation property and list the lemmas and corollaries needed.At last, we establish the unique continuation property of the solutions to the Cauchy problem associated with a class of fifth-order KdV equations.