Some Studies On Shallow Water Equations

This thesis obtains the main results came from my studying for master degree. The major research object are one and two variable shallow water equation.First, we investigate the modified Novikov equation and DGH equation. For the modified Novikov equation, we through the analysis of the blow-up results that ob-tained by others before, we established a new blow-up condition, which makes results more general. At the same time, for the DGH equation, we present a new blow-up criteria which improve previous results.Secondly, we study the two-component Camassa-Holm equation and modified two-component Camassa-Holm equation. We give a local condition to these equations. However, we can obtain the same result without the condition of y0(x0)0, As a contrast, the local condition can be achieved easier.Finally, in this section, we use the classical energy method to get a conserved quan-tity. Meanwhile, the precise blow-up scenarios for sufficiently regular solutions was described. According to this, we give two types of sufficient condition for blow-up, one only depend on the initial energy, and the other depend on the sign of the initial values po(x) and y0(x). At the same time, we study the infinite propagation speed for this equation.