Qualitative Theory Of Nonlinear Evolution Equations (group) Initial Value Problem

Nonlinear evolution equations, in general, describe the states or processes in physics and other scientific fields that evolve as the time goes, and are a collective name of many important nonlinear partial differential equations that depend on the time variable. They arise not only from many fields of mathematics, but also from other branches of science such as Physics, biological science and mechanics. The study of qualitative properties is one of the main concern in the field of nonlinear evolution equations. In this dissertation, we main discuss the qualitative properties for the initial value problem of some nonlinear dispersive equations. We divide the dissertation into the following parts:Chapter1is a preliminary chapter in which we not only recall the back-ground and history about the related models but also state the main results obtained in this dissertation. Some frequently used inequalities are also pre-sented.In Chapter2, the initial value problem and traveling-wave solutions to the two-component μ-Camassa-Holm system in the periodic setting are inves-tigated. Applying Kato’s theory for the abstract quasi-linear evolution equa-tions and a prior estimates of solution, we get the existence of global solutions and the blow-up of solution in finite time under some conditions. Finally, the existence of smooth periodic traveling-wave solutions is determined.Chapter3deals with the global existence of weak solutions for a general-ized two-component μ-Camassa-Hohn system in the periodic setting. Global existence for strong solutions to the system with smooth approximate initial value is derived. Then we show that the limit of approximate solutions is a global-in-time weak solution of the generalized two-component μ-Camassa- Holm system.In Chapter4, we consider the dynamical stability of the peaked soliton for the modified μ-Camassa-Holm equation with cubic nonlinearity. It is shown that the single peakons are orbitally stable under small perturbations in the energy space by constructing certain Lyapunov functionals.In Chapter5, we investigate a generalized two-component Camassa-Holm system which can be derived from the theory of shallow water waves moving over a linear shear flow. We establish a sufficient condition on the initial data to guarantee wave-breaking for the system. More importantly, the persistence properties and the unique continuation property of the strong solutions are also analyzed.In Chapter6, we give a simply introduction of three-component Camassa-Holm equation. Some problems are also proposed for further research and explore.