Persistence Properties And Analyticity Of Solutions To The Cauchy Problem Of A Kind Of Integrable Two-Component Systems

Persistence properties describe the long-time behavior of the solution of an equation or system.Specifically,a strong solution of an equation or system,initially decaying in some form together with its spacial derivative at infinity,must decay in this form at any later time.In this thesis,we are concerned with the persistence properties of solutions of the Cauchy problem associated with an integrable two-component Camassa-Holm system with cubic nonlinearity.It is shown that a strong solution of the two-component Camassa-Holm system,initially decaying exponentially together with its spacial derivative,also decays exponentially at infinity by using weight functions.Furthermore,we give an optimal decaying estimate of the momentum.Analyticity is an important property of the solution of an equation or system.The generalized Ovsyannikov theorem is an effective method for studying the analyticity of the solution of an equation or system in the Sobolev-Gevrey spaces.Thanks to this theorem and the basic properties of Sobolev-Gevrey spaces,we prove the Gevrey regularity and analyticity of solutions of the Cauchy problem associated with an integrable twocomponent Camassa-Holm system with cubic nonlinearity and a two-component Degasperis-Procesi system.The main content of this thesis is organized as follows:In chapter 1,we describe the research background and current research status of the persistence properties and analyticity of the solution of nonlinear development equations;In chapter 2,some basic definitions,theorems and notations are introduced;In chapter 3,we study the persistence properties of solutions of the Cauchy problem associated with an integrable two-component Camassa-Holm system with cubic nonlinearity and give an optimal decaying estimate of the momentum.We also obtain the Gevrey regularity and analyticity of this system;In chapter 4,we investigate the Gevrey regularity and analyticity of solutions of the Cauchy problem to a two-component Degasperis-Procesi system.