The Cauchy Problem Of The Camassa-Holm Tape Equation

In this thesis,we focus on the qualitative research for a family of Camassa-Holm type equations,which includes the large time behavior,ill-posedness properties of the equation(this is the contribution of [13])and persistence properties of the solutions.Four chapters comprise this thesis.In Chapter 1,we introduce the background of the Camassa-Holm type equations and some related achievements.Moreover,we present some problems which will be discussed in this thesis.In Chapter 2,we investigate the qualitative problems of the CamassaHolm type equation which has(k + 1)-degree nonlinearities.we focus on the large time behavior of momentum support for a Camassa-Holm type equation with(k + 1)-degree nonlinearities.In Chapter 3,we discuss the ill-posedness of the equation.Then,we discuss the persistence property of the solution in weighted space by means of moderate weight functions.In chapter 4,we summarize the paper.