Studies On Camassa-Holm System And Its Relevant Problems

This dissertation amasses my main research results obtained in pursuit of my mas-ter’s degree, mainly involving blow-up problem for the two component Camassa-Holm system(CH2), blow-up problem for the modified two component Camassa-Holm sys-tem(MCH2), blow-up problem for the weakly dissipative Novikov equation and infi-nite propagation speed for the two component b-family system.The whole dissertation is divided into six chapters. The first chapter introduces the background and significance of our problems and their corresponding study progress, then finally illustrates several important lemmas.The second and third chapters focus on establishing the blow-up conditions for CH2and MCH2and separately give our blow-up theorems for them with its conditions independent of the initial energy E1(0) or E(0) but only dependent on the sign of the initial values po(x) and yo(x), not the size of them. They are purely morphological conclusions. Additionally another blow-up theorem for MCH2will be presented in the third chapter.The fourth chapter establishes the blow-up theorem, that has nothing to do with the initial energy either, for the weakly dissipative Novikov equation.The fifth chapter discusses the infinite propagation speed for the two component b-family system and give the specific profile of u(x, t).Finally, problems unsolved but expected to have a breakthrough are collected in the sixth chapter.