The Cauchy Problems For Some Classes Of Boussinesq Equations

In this paper, we study the existence and uniqueness of local and global solutionsfor the Cauchy problem to some classes of Boussinesq equations, and give the sufficientconditions of blow-up of solutions for the above problems. We also discuss the globalexistence and nonlinear scattering for small amplitude solutions of the problems. Fur-thermore, we study the instability of the ground state solution for the nonlinear Boussi-nesq equations. The main results include the following five aspects:In Chapter 2, we study the Cauchy problem for a class of generalized Boussinesqequation. At first, by using the contraction mapping theorem, we obtain the existenceof the local solution. Under some assumptions about energy, we obtain the blow-upresult by establishing a differential inequality for a functional of the solution. At last, weconsider global small-amplitude solution to this problem and their nonlinear scatteringby using the contraction mapping theorem and utilizing an estimate for the uniformdecay of solutions of the linearized version.In chapter 3, we study a class of generalized Boussinesq equation in Rn. For thespecial case , the existence and the uniqueness of the global solution for the problem areproved. Moreover, we obtain some sufficient conditions of the blow-up of the solutionfor this problem.In chapter 4, we study the global existence of small amplitude solutions and non-linear scattering of a class of generalized Boussinesq equation in Rn. The strategy isto write Boussinesq equation as an integral equation, treat in the nonlinearity as a smallperturbation of the linear part of the equation, then use the contraction mapping theoremand utilized an estimate for the uniform decay of solutions of the linearized version toobtain a priori estimates of solutions. Using these estimates, we establish the results. In chapter 5, we consider the Cauchy problem for a class of nonlinear Boussinesqequation. We obtain the blow-up of solution and the instability of the ground state solu-tion of the nonlinear Boussinesq equation. Furthermore, we obtain the global boundedsolutions with initial data in different regions that are related to traveling wave solutionsof the equation.In chapter 6, we consider the Cauchy problem of a class of generalized Boussinesqequation. The strong instability of solitary-wave is proved by constructing a functionalof the solution and establishing the invariant regions. Furthermore, an improved blowup result related to the solitary-wave solution is also obtained.