Properties Of Solutions To Shallow Water Wave Equations With Peakons

In this paper,we study qualitative properties of solutions to Cauchy problems for a class of shallow water wave equations,including blow-up criterion,blow-up rate,persistence properties and so on.These equations arise from modern mechanics and physics,and they are important subjects in the study of nonlinear science,especially in the field of soliton and integrable system.This thesis is organized as follows:In Chapter one,we introduce the history and development of shollow water wave equations and present the purpose and significance of our study.In Chapter two,some useful definitions,theorems and important inequalities are collected.In Chapter three,we study a k-abc equation with(k + 1)-degree nonlinearities,which is a 4-parameter family of nonlocal evolution equations.The equation possesses peakon traveling wave solutions for all values of the four parameters k,a,b and c.In the first section of this chapter,we discuss the decay estimates at infinity of the solution to the equation.In the second section,we show the exact spatial asymptotic profile of the solutions.By taking full advantage of features of the admissible weight function ?,we show some persistence results for solutions of the equation in weighted L~p spaces.In Chapter four,we consider a two-component shallow water system.In the first section of this chapter,we investigate the blow-up of the system.A new criterion guaranteeing the blow-up phenomenon of the system is presented via togethering the standard particle trajectory,with the system's structure and the conservation law.Especially,for the case ? = 1,we show the exact blow-up rate of the breaking-wave solutions.In the second section,some persistence results for solutions of the system in weighted Lpspaces are given.In Chapter five,we summarize the results obtained in this thesis and prospect for future research.