Initial Boundary Value Problem For The General Shallow Water Wave Equations Without Dispersion Term

In this paper we study the initial boundary value problems of the general shallow water wave equation without dispersion term and two-component Camassa-Holm equation. By applying some knowledge of function analysis, Sobolev inequalities, some PDE knowledge and so on, we obtain the local well-posedness, blow-up and global existence of solutions to equations on the half-line and on a finite interval subject. There are four sections in this paper:In Chapter One, we introduce the background and actualities and summarize the main results. In Chapter Two, we introduce a number of important definitions and theorems. In Chapter Three, we consider the initial boundary value problem of the general shallow water wave equation without dispersion term on half-line and a finite interval subject, with extension results which come from some lemmas, we get the local well-posedness and the Blow-up of solution in finite time under some conditions. In Chapter Four, we consider the initial boundary value problem of the two-component Camassa-Holm equation on half-line and a finite interval subject, by extension results, some sufficient condition that guarantee the local well-posedness, Blow-up of solution in finite time under some conditions and global existence of solutions are obtained.