On The Well-posedness Problem And Limit Behavior For The General Shallow Water Wave Equations

In this paper we study the well-posedness of the Cauchy problem and the limit behavior of the solutions to a class of nonlinear dispersive wave equations, named the general shallow water wave equation. We obtain the local well-posedness, blow-up and global existence of solutions to this equation in Sobolev space by applying Sobolev inequalities, some PDE knowledge and Kato's theory. Using some knowledge of function analysis, we study the relationship between the solution to the general shallow water wave equation and the solutions to the KdV equation, b -family of equations, the Camassa-Holm equation and the Degasperis-Procesi equation, respectively.There are six 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 obtain the local well-posedness, blow-up and global existence of solutions to this equation in Sobolev space by applying Sobolev inequalities, some PDE knowledge and Kato's theory.In Chapter Four, we demonstrate that the solutions to this equation converge to the solution to the corresponding KdV equation and the b -family of equations as the parametersεandαtend to different values, respectively.In Chapter Five, we give the condition of existing compacton and peakon in this equation.In Chapter Six, we present some results on the weak solutions to this equation with a class of initial values.