The Cauchy Problem For A Class Of Nonlinear Boussinesq Equation

The Boussinesq equation is built as a kind of mathematical model which is used for fitting near shore wave propagation deformation, crushing, wave climbing and wave flow interaction. In this paper, we study the Cauchy problem of the class of Boussinesq equation. The thesis is organized as follows:In chapter 2, we deal with the well-posedness and asymptotic behavior of the solution for a six order Boussinesq equation. At first, making use of the Fourier transformation, we obtain the well-posedness of the equation in the Sobolev space Furthermore, the asymptotic expression of solution is eatablished by using the method of the undermined coefficient.In chapter 3, we study the Cauchy problem of a six order Boussinesq equation in one-dimensional space. Firstly, some linear estimates are given. Then the existence and uniqueness the local solution are proved by using the contraction mapping theory.Finally, under some assumptions about energy, we obtain the blow- up result by using concavity principle.In chapter 4, we consider the cauchy problem of a six order Boussinesq equation in R~n. The key feature of the analysis is to transform the nonlinear equation into a linear equation by constructing an operator. We get the existence and uniqueness of the local solution by contraction mapping theory. Moreover, we obtain some sufficient conditions of the blow-up of the solution for this problem.