Persistence Properties Of Solutions To The Cauchy Problem Of A Kind Of Coupled Nonlinear Systems

In this thesis,the persistence properties and optimal decay index of the Cauchy problem for the interacting system of the Camassa-Holm and Degasperis-Procesi equations and a generalized cross-coupled Camassa-Holm system with higher-order nonlinearities are studied,respectively.The persistence proper-ties imply that the strong solutions,initially decaying together with its spacial derivative at infinity,must retain this properties.And the optimal decay index describes the optimality of the persistence properties.The method of estimate with weight function is an effective one to study these properties,by which properties of many typical equations have been proved.On the basis of the local well-posedness results,the purpose of this thesis is to establish the persistence properties and optimal decay index of the strong solutions for these Cauchy problems above.The main contents of this thesis are as follows:Firstly,we briefly describe the purpose and significance of the persistence properties of the solutions and the results which the researchers have obtain about the persistence properties both at home and abroad,and give the theorems and lemmas which are needed in the proof of these properties.Secondly,we prove the persistence properties of the Cauchy problem for the interacting system of the Camassa-Holm and Degasperis-Procesi equations.At last,we establish the persistence properties of the Cauchy problem for a generalized cross-coupled Camassa-Holm system with higher-order nonlinear-ities.